TW202121267A - Method and calculation system for quantum amplitude estimation - Google Patents

Abstract

A hybrid quantum-classical (HQC) computer takes advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing measurement number and runtime compared to VQE. The HQC computer derives inspiration from quantum metrology, phase estimation, and the more recent “alpha-VQE” proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits. The HQC computer uses the “engineered likelihood function” (ELF)to carry out Bayesian inference. The ELF formalism enhances the quantum advantage in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers into that of quantum error corrected ones. This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond.

Description

Method and calculation system for quantum amplitude estimation

The present invention relates to a method and calculation system for quantum amplitude estimation.

Quantum computers promise to solve key industry problems that cannot be solved or can only be solved extremely inefficiently with classical computers. The main application areas include chemistry and materials, biological sciences and bioinformatics, logistics and finance. Recently, the attention to quantum computing has soared, partly due to a wave of advances in the performance of off-the-shelf quantum computers. However, the resources of quantum devices are still extremely limited recently, which hinders the deployment of quantum computers on practical concerns.

A recent batch of methods that cater to the limitations of recent quantum devices has received close attention. These methods include variable quantum intrinsic solver (VQE), quantum approximate optimization algorithm (QAOA) and variants, variable quantum linear system solver, other quantum algorithms using the principle of variation, and Quantum machine learning algorithm. Despite the innovation of such algorithms, many of these methods have become impractical for business-related problems due to their high cost in the number of measurements and running time. However, for medium-sized problem examples, the quantum resources required for methods that provide secondary acceleration in runtime (such as phase estimation) are far beyond the capabilities of recent devices.

The number of measurements required by hybrid quantum classical algorithms such as the VQE is too high due to many practical value issues. The error rate required by quantum algorithms that reduce this cost (for example, quantum amplitude and phase estimation) is too low for recent implementations. Embodiments of the present invention include hybrid quantum classical (HQC) computers and methods executed by HQC computers. These computers and methods utilize available quantum coherence to maximize the ability to sample noisy quantum devices, thereby Compared with VQE, it reduces the number of measurements and shortens the running time. Such embodiments are inspired by quantum methods, phase estimation, and the more recent "α-VQE" proposal, resulting in a general formula that is robust to errors and does not require accessory qubits. The central object of this method is the so-called "Engineered Probability Function" (ELF), which is used to perform Bayesian inference. The embodiment of the present invention uses the ELF formal theory to enhance the quantum advantage in sampling, because the form of the intermediate-scale quantum computer with its own noise in the physical hardware is transformed into the form of a quantum error-corrected computer. This technology accelerates the central components of many quantum algorithms, and its applications include chemistry, materials, finance, and other fields.

Other features and advantages of the various aspects and embodiments of the present invention will become apparent from the following description and the scope of the patent application.

關於量子狀態

之預期值

。The embodiment of the present invention relates to a hybrid classical quantum computer (HQC) that performs quantum amplitude estimation. 4, there is shown a flow chart of an HQC 430 including both a quantum computer 432 and a classical computer 434, which executes the quantum amplitude estimation method according to an embodiment of the invention. In the block 404 executed by the classical computer 434, a plurality of quantum circuit parameter values are selected to optimize the accuracy improvement rate of the statistic, which is estimated to be observable

About quantum states

Expected value

.

的估計量，並且可為有偏或不偏的。可根據機率分佈對複數個值進行建模，在此種情況下統計量可表示機率分佈的參數。例如，統計量可表示高斯分佈之平均值，如下文在第3.2節中更詳細描述。In the embodiment, the statistic is a sample average calculated from a plurality of values sampled from a random variable. In this discussion, these sampling values are obtained by measuring the qubits of the subcomputer 432. However, without departing from the scope of the present invention, the statistic may alternatively be skewness, kurtosis, quantile, or another type of statistic. Statistics are expected

The estimate of, and can be biased or unbiased. A plurality of values can be modeled according to the probability distribution, in which case the statistics can represent the parameters of the probability distribution. For example, the statistic can represent the average of the Gaussian distribution, as described in more detail in Section 3.2 below.

The quantum circuit parameter value is a real number that controls how the quantum gate operates on the qubit. In this discussion, each quantum circuit can be represented as a sequence of quantum gates, where each quantum gate of the sequence is controlled by one of the quantum circuit parameter values. For example, each of the quantum circuit parameter values may represent the angle at which the state of one or more qubits rotates in the corresponding Hibbert space.

The accuracy improvement rate is a function that expresses the degree of improvement of the corresponding accuracy of the statistics for each iteration of the method embodiment of the present invention. The accuracy improvement rate is a function of quantum circuit parameters, and may additionally be a function of statistics (e.g., average). Accuracy Any quantitative measurement of the error of the system measurement. For example, accuracy can be the mean square error, standard deviation, variance, mean absolute error, or another moment of error. Alternatively, the accuracy may be an information measure, such as Fisher information or information entropy. Examples of using variance for accuracy are described in more detail below (see, for example, Equation 36). In these examples, the accuracy information rate can be the variance reduction factor introduced in Equation 38. Alternatively, the accuracy information rate may be Fisher information (for example, see Equation 42). However, without departing from the scope of the present invention, the accuracy improvement rate may be another function of the improvement of quantification accuracy.

In some embodiments, one of coordinate rising and gradient descent is used to select a plurality of quantum circuit parameter values. These two techniques are described in more detail in Section 4.1.1.

變換成反射量子狀態。第一及第二廣義反射算子中之每一者係根據複數個量子電路參數值中之對應一者加以控制。第3.1節中描述的算子

及

分別為第一及第二廣義反射算子的實例。方程式19中引入的算子

係交替的第一及第二廣義反射算子之序列之一個實例，其中向量

表示複數個量子電路參數值。第一及第二廣義反射算子之序列以及可觀測

可定義工程化概似函數之偏誤，如下文關於方程式26所描述。In the block 406 executed by the quantum computer 432, the sequence of alternating first and second generalized reflection operators is applied to one or more qubits of the quantum computer 432 to transfer the one or more qubits from the quantum computer 432. status

Transform into a reflective quantum state. Each of the first and second generalized reflection operators is controlled according to the corresponding one of the plurality of quantum circuit parameter values. Operators described in Section 3.1

and

These are examples of the first and second generalized reflection operators, respectively. Operators introduced in Equation 19

Is an example of a sequence of alternating first and second generalized reflection operators, where the vector

Represents multiple quantum circuit parameter values. Sequences and observables of the first and second generalized reflection operators

The deviation of the engineered probability function can be defined, as described below with respect to Equation 26.

量測處於反射量子狀態之複數個量子位元，以獲得一組量測成果。在由古典電腦434執行的方塊410中，用該組量測成果更新統計量，以獲得

之具有更高準確性的估計值。In block 408, which is also executed by quantum computer 432, regarding the observable

Measure a plurality of qubits in the reflective quantum state to obtain a set of measurement results. In block 410 executed by the classical computer 434, the set of measurement results are used to update the statistics to obtain

It has a more accurate estimate.

The method may further include outputting statistics after the update. Alternatively, the method may perform blocks 404, 406, 408, and 410 iteratively. In some embodiments, the method further includes updating the accuracy estimate of the statistics on the classical computer 434 and using the set of measurement results. The accuracy estimate is the calculated value of the accuracy (for example, the variance) described above. In these embodiments, the method performs blocks 404, 406, 408, and 410 iteratively until the accuracy estimate falls below the threshold.

In some embodiments, in block 410, the statistics are updated by updating the pre-distribution with a plurality of measurement values to obtain the post-distribution and calculating the updated statistics from the post-distribution.

In some embodiments, a plurality of quantum circuit parameter values are selected based on the statistics and the accuracy estimates of the statistics. A plurality of quantum circuit parameter values can be further selected based on the fidelity representing the error occurred during the application and measurement.

1.1. introduction

並且目標在於估計算子

的本征值

中的相位

。應重點注意的是此處的概似函數，

(1) The combination of phase estimation and Bayesian perspective produces Bayesian phase estimation technology, which is more suitable for noisy quantum devices capable of realizing finite depth quantum circuits than earlier proposals. Use the above markings, circuit parameters

And the goal is to estimate the operator

Eigenvalue

Phase in

. The important thing to note is the likelihood function here,

(1)

及

分別為第一及第二種類之切比雪夫多項式。此共通性產生用於與相位估計相關的任務(諸如哈密爾頓特徵化)之貝氏推論機器。在藉由高斯先驗進行貝氏推論相比其他非適應性取樣方法的指數優勢中，係藉由展示預期事後變異數

隨推論步驟之數目呈指數衰減來建立的。此種指數收斂以每一推論步驟所要求之數量為

的量子相干性為代價。此種定標在貝氏相位估計的情境下亦得以確認。It is common in many environments except Bayesian phase estimation, among which

and

They are the first and second types of Chebyshev polynomials. This commonality produces Bayesian inference machines for tasks related to phase estimation, such as Hamiltonian characterization. Among the exponential advantages of Bayesian inference by Gaussian priors compared to other non-adaptive sampling methods, is by showing the expected post-event variance

It is established as the number of inference steps decays exponentially. This kind of exponential convergence is based on the quantity required for each inference step

At the expense of quantum coherence. This calibration is also confirmed in the context of Bayesian phase estimation.

-VQE」，其中用於執行算子量測之漸近定標為

，其中極值

對應於標準取樣型態(通常在VQE中實現)，並且

對應於量子增強型態，其中定標達到海森堡限值(通常藉由相位估計實現)。藉由改變貝氏推論的參數，亦可達成在

與

之間的

值。

值愈低，貝氏相位估計所需要的量子電路愈深。此實現了量子相干性與量測過程之漸近加速之間的折衷。Equipped with Bayesian phase estimation technology and the perspective of overlap estimation as an amplitude estimation problem, it is possible to design a Bayesian inference method for operator measurement that smoothly interpolates between the standard sampling pattern and the phase estimation pattern. This is proposed as "

-VQE", where the asymptotic calibration used to perform the operator measurement is

, Where the extreme value

Corresponds to the standard sampling mode (usually implemented in VQE), and

Corresponds to the quantum enhanced state, where the calibration reaches the Heisenberg limit (usually achieved by phase estimation). By changing the parameters of Bayesian inference, it can also be achieved in

versus

between

value.

The lower the value, the deeper the quantum circuit required for Bayesian phase estimation. This achieves a compromise between quantum coherence and the asymptotic acceleration of the measurement process.

的參數

的任務。提議一種並行策略，其中使用用於產生

的參數化電路之

個複製品，以及初始纏結狀態及基於纏結的量測來創建其中參數

放大至

的狀態。此種放大亦可產生類似於方程式1中之概似函數的概似函數。在先前的研究中已展示，藉由隨機化量子運算及貝氏推論，即使在存在雜訊的情況下亦可與古典取樣相比以更少的疊代來提取資訊。在量子振幅估計中，考慮具有變化的疊代數目

及量測數目

之電路。一組特別選擇的對

產生可用於推斷待估計振幅之概似函數。針對作者給出之一個特定概似函數構造展現了海森堡限值。兩項研究均強調參數化概似函數之能力，從而使研究其在不完善硬體狀況下的效能係有吸引力的。It is also worth noting that phase estimation is not the only example that can reach the Heisenberg limit of amplitude estimation. In previous research, the author considered estimating the quantum state

Parameters

Task. Propose a parallel strategy in which the use of

Of the parameterized circuit

Replicas, and the initial entanglement state and entanglement-based measurements to create the parameters

Zoom in to

status. Such amplification can also generate a probability function similar to the probability function in Equation 1. Previous studies have shown that by randomized quantum operations and Bayesian inference, even in the presence of noise, it can extract information with fewer iterations than classical sampling. In the quantum amplitude estimation, consider the number of iterations with variation

And the number of measurements

The circuit. A set of specially selected pairs

Generate a likelihood function that can be used to infer the amplitude to be estimated. The Heisenberg limit is shown for a specific probability function structure given by the author. Both studies emphasize the ability to parameterize the likelihood function, which makes it attractive to study its performance under imperfect hardware conditions.

1.2. Main result

的系統及方法，其中狀態

可由量子電路

準備，以使得

。本發明之實施例可使用一系列量子電路，以使得當電路隨

的更多次重複加深時，其允許作為

的甚至更高次的多項式的概似函數。如下一節中藉由具體實例所描述，多項式次數的此增加之直接後果為推論能力的增加，該增加可由每一推論步驟處的費雪資訊增益來量化。在建立此「增強取樣」技術之後，本發明之實施例可將參數引入至量子電路中，並且使所得的概似函數可調諧。本發明之實施例可在每一推論步驟期間最佳化參數以獲得最大資訊增益。以下多行見解源於吾等的努力：Embodiments of the invention include methods for estimating expectations

System and method, where the state

Quantum circuit

Prepare to make

. The embodiments of the present invention can use a series of quantum circuits, so that when the circuit follows

When more repetitions of deepening, it is allowed as

Likelihood function of even higher degree polynomials. As described by specific examples in the next section, the direct consequence of this increase in polynomial order is an increase in inference ability, which can be quantified by the Fisher information gain at each inference step. After establishing this "enhanced sampling" technique, embodiments of the present invention can introduce parameters into the quantum circuit and make the resulting likelihood function tunable. The embodiment of the present invention can optimize the parameters during each inference step to obtain the maximum information gain. The following lines of insights stem from our efforts:

1. The role of noise and error in amplitude estimation: Previous studies have revealed the influence of noise on the output estimation of the likelihood function and Hamiltonian spectrum. The content disclosed herein is directed to the amplitude estimation scheme used in the embodiments of the present invention to study the above influence. The description in this article shows that when the noise and error do not increase the running time required to produce an output within a certain statistical error tolerance, the noise and error may not introduce system bias in the output of the estimation algorithm. The system error in the estimation can be suppressed by using active noise clipping technology and calibrating the noise effect.

的最佳電路保真度，在此最佳保真度下，增強方案產生最大的量子加速量。Simulations using real error parameters for recent devices have revealed that the enhanced sampling scheme can be superior to VQE in terms of sampling efficiency. The experimental results have also revealed the perception of errors in the implementation of quantum algorithms that tolerate higher fidelity does not necessarily lead to better calculation performance. In certain embodiments of the present invention, there is roughly

The best circuit fidelity of, under this best fidelity, the enhancement scheme produces the largest quantum acceleration.

之特定值(「死點」)，對於此等值，CLF用於推論之效率與

之其他值相比顯著更低。本發明之實施例可藉由利用角度參數經使得可調諧的廣義反射算子工程化概似函數的形式來移除彼等死點。1. The effect of the tunability of the likelihood function: The parameterized likelihood function can be used in phase estimation or amplitude estimation routines. It is reported that all current methods focus on the Chebyshev form of the likelihood function (Equation 1). For this Chebyshev probability function (CLF), in the presence of noise, there are parameters

Specific value ("dead point"). For these values, CLF is used to infer the efficiency and

The other values are significantly lower than those. The embodiments of the present invention can remove these dead points by using the form of an engineered likelihood function of a generalized reflection operator that makes the angle parameter tunable.

至相位估計的

之平滑變換。本發明之實施例藉由研發用於使用具有雜訊度

的裝置來在目標準確性

上估計運行時間

的模型來推進此思路(參見第6節)：

。 (2) 2. Running time model for estimation when the error rate decreases: previous work has shown that the asymptotic cost is calibrated from VQE

To phase estimation

The smooth transformation. The embodiments of the present invention are developed for use with noise

The device comes in target accuracy

Estimated running time

Model to advance this idea (see section 6):

. (2)

的函數在

定標與

定標之間內插。此類界限亦允許分析本發明之實施例以得到作為硬體規格(諸如量子位元數目及雙量子位元保真度)的函數之量子加速，並且因此使用當前及未來硬體之真實參數來估計運行時間。3. Model as

The function is in

Calibration and

Interpolate between calibrations. Such boundaries also allow analysis of embodiments of the present invention to obtain quantum acceleration as a function of hardware specifications (such as the number of qubits and dual-qubit fidelity), and therefore use real parameters of current and future hardware Estimated running time.

的真實值的分佈的當前可信度的貝氏推論。第4節呈現用於兩者的啟發式演算法。第5節中呈現數值結果，將本發明之實施例與基於CLF之現有方法進行比較。第6節揭示運行時間模型，並且導出(2)中的表達式。第7節揭示從量子計算的廣泛視角來看的所揭示結果之意義。 方案 貝氏推論 雜訊考慮 完全可調諧LF 要求附屬 要求本征狀態 Knill等人 否 否 否 是 否 Svore等人 否 否 否 是 是 Wiebe及Grenade 是 是 否 是 是 Wang等人 是 是 否 是 是 O’Brien等人 是 是 否 是 否 Zintchenko及Wiebe 否 是 否 否 否 Suzuki等人 否 否 否 否 否 本研究(第1節) 是 是 是 否 否 本研究(附錄A) 是 是 是 是 否 The subsequent chapters of this disclosure are organized as follows. Section 2 presents a specific example of a solution implemented according to an embodiment of the present invention. Subsequently, the following chapters detail the general formula of this scheme according to various embodiments of the present invention. Section 3 describes in detail the general quantum circuit structure used to implement ELF, and analyzes the ELF structure in both noise and noise-free environments. In addition to the quantum circuit solution, the embodiments of the present invention also involve: 1) tuning circuit parameters to maximize information gain, and 2) for updating related

Bayesian inference of the current credibility of the distribution of true values. Section 4 presents heuristic algorithms for both. Numerical results are presented in Section 5, comparing the embodiments of the present invention with existing methods based on CLF. Section 6 reveals the runtime model and derives the expression in (2). Section 7 reveals the significance of the revealed results from a broad perspective of quantum computing. Program Bayesian Corollary Noise considerations Fully tunable LF Request attachment Required eigenstate Knill et al no no no Yes no Svore et al no no no Yes Yes Wiebe and Grenade Yes Yes no Yes Yes Wang et al Yes Yes no Yes Yes O'Brien et al Yes Yes no Yes no Zintchenko and Wiebe no Yes no no no Suzuki et al no no no no no This study (section 1) Yes Yes Yes no no This study (Appendix A) Yes Yes Yes Yes no

Table 1. Comparison of our proposal with related proposals in the literature. Here, the feature list includes whether the quantum circuit used in the scheme requires auxiliary qubits in addition to the qubits used for overlap estimation, whether the scheme uses Bayesian inference, whether any noise elasticity is considered, and whether it is required The initial state is the intrinsic state, and the likelihood function is completely tunable like the ELF proposed in this article or is it limited to the Chebyshev likelihood function.

2. First instance

之預期值

。量子振幅估計方法相對於特定計算模型提供可證明的量子加速。然而，爲了達成估計值的精確度

，此方法中需要的電路深度定標為

，從而使其對於近期量子電腦不切實際。變分量子本征解算器使用標準取樣來進行振幅估計。標準取樣允許低深度量子電路，從而使其更適合在近期量子電腦上實現。然而，在實踐中，此方法的低效率使得VQE對於許多感興趣的問題不切實際。本節介紹一種可由本發明之實施例使用的用於振幅估計之增強取樣方法。此技術試圖最大化有雜訊的量子裝置之統計能力。此方法經描述為從對如VQE中所使用之標準取樣的簡單分析開始。There are two main strategies to estimate a certain operator

Expected value

. The quantum amplitude estimation method provides a demonstrable quantum speedup relative to a specific calculation model. However, in order to achieve the accuracy of the estimate

, The circuit depth required in this method is calibrated as

, Making it impractical for recent quantum computers. The variable quantum eigen solver uses standard sampling to estimate the amplitude. Standard sampling allows low-depth quantum circuits, making them more suitable for implementation on recent quantum computers. However, in practice, the low efficiency of this method makes VQE impractical for many interesting problems. This section introduces an enhanced sampling method for amplitude estimation that can be used by embodiments of the present invention. This technique attempts to maximize the statistical power of noisy quantum devices. This method has been described as starting with a simple analysis of standard samples as used in VQE.

的線性組合及「擬設狀態」

的哈密爾頓，能量預期值估計為包立預期值估計值之線性組合

(3) The energy estimation sub-routine of VQE relates to Baoli string estimation amplitude. For decomposing into a string

Linear combination and "proposed state"

Hamilton, the energy expected value is estimated as a linear combination of the estimated value of Baoli

(3)

為

的(振幅)估計值。VQE使用標準取樣方法來關於擬設狀態建置包立預期值估計值，其可概述為如下：among them

for

Estimated value of (amplitude). VQE uses a standard sampling method to establish an estimated value of the expected value of the proposed state, which can be summarized as follows:

Standard sampling:

並且量測算子

，接收成果

。1. Prepare

And measure operator

To receive results

.

次，接收標記為

的

個成果以及標記為

的

個成果。2. Repeat

Times, received marked as

of

Results and marked as

of

Results.

為

。3. Estimate

for

.

的函數之估計量之均方差來量化此估計策略之效能，其中

為每一量測的時間成本。因為估計量為不偏的，所以均方差僅為估計量的變異數，

。 (4) Can be used as time

The mean square error of the estimator of the function to quantify the effectiveness of this estimation strategy, where

The time cost for each measurement. Because the estimator is unbiased, the mean square error is only the variance of the estimator,

. (4)

，確保均方差

所需要的演算法之運行時間為

。 (5) For a specific mean square error

To ensure the mean square error

The running time of the required algorithm is

. (5)

中的小偏差之不靈敏度：標準取樣量測成果資料中所包含的有關

之預期資訊增益為低。The total running time of energy estimation in VQE is the sum of the running time of the estimated running time of the individual including the expected value. For problems of interest, this runtime may be too costly, even when using specific parallelization techniques. The source of this cost is the standard sampling estimation process

The insensitivity of small deviations in the medium: the relevant information contained in the standard sampling measurement result data

The expected information gain is low.

次重複的估計過程的資訊增益

(6) Usually, Fisher information can be used to measure the standard sampling

Information gain of repeated estimation process

(6)

為來自標準取樣的

次重複的一組成果。費雪資訊將概似函數

識別為負責資訊增益。(不偏)估計量之均方差的下限可藉由克拉瑪-拉歐(Cramer-Rao)界限獲得

(7) among them

Is from standard sampling

A set of results repeated times. Probability function

Identified as responsible for information gain. (Unbiased) The lower bound of the mean square error of the estimator can be obtained by the Cramer-Rao bound

(7)

，其中

為自概似函數

提取之單個樣本之費雪資訊。使用克拉瑪-拉歐界限，可找到估計過程之運行時間之下限

， (8) Using the fact that Fisher information is additive with the number of samples, we get

,among them

Self-likelihood function

Fisher information of a single sample extracted. Using the Klamah-Raou boundary, you can find the lower bound of the running time of the estimation process

, (8)

It shows that, in order to shorten the running time of the estimation algorithm, the embodiment of the present invention can increase the Fisher Information.

，應用運算

，應用有關擬設狀態的相位翻轉，然後量測

。有關擬設狀態的相位翻轉可藉由應用擬設電路之倒數

、應用有關初始狀態的相位翻轉

，隨後再應用擬設電路

來達成。在此情況下，概似函數變為

。 (9) One purpose of enhanced sampling is to shorten the running time of overlap estimation by increasing the engineered likelihood function of the information gain rate. Consider the simplest case of enhanced sampling, which is illustrated in Figures 5A to 5C. In order to generate data, the embodiment of the present invention can prepare a proposed state

, Applied computing

, Apply the phase reversal of the proposed state, and then measure

. The phase reversal of the proposed state can be achieved by applying the inverse of the proposed circuit

、Apply the phase flip of the initial state

, And then apply the proposed circuit

Come to reach. In this case, the likelihood function becomes

. (9)

的

次切比雪夫多項式。本文中之揭示內容將此類概似函數稱為切比雪夫概似函數(CLF)。The error is

of

Second Chebyshev polynomial. The disclosure in this article refers to this type of likelihood function as Chebyshev likelihood function (CLF).

的情況。這裡，

，因此費雪資訊與概似函數之斜率之平方成正比

。 (10) In order to compare the Chebyshev probability function of enhanced sampling with the Chebyshev probability function of standard sampling, consider

Case. Here,

, So Fisher information is proportional to the square of the slope of the likelihood function

. (10)

處之斜率與標準取樣概似函數之斜率相比更傾斜。在每一情況下單個樣本費雪資訊評估為

 

(11) As seen in Figure 5B, the Chebyshev probability function is

The slope is more sloping than the slope of the standard sampling probability function. In each case, a single sample Fisher Information is evaluated as

(11)

之前應用

層

來進一步增大費雪資訊。事實上，費雪資訊

隨

二次增長。This demonstrates how simple variants of quantum circuits can enhance information gain. In this example, the simplest case of using enhanced sampling can reduce the number of measurements required to achieve the target error by at least nine times. As will be discussed later, the embodiments of the present invention can be measured by measuring

Previously applied

Floor

To further increase Fisher Information. In fact, Fisher

Follow

Secondary growth.

的選擇。在此情況下，給定來自具有變化的

之電路之一組量測成果，

及

計數的樣本平均值失去其意義。代替使用樣本平均值，為了將量測成果處理成有關

的資訊，本發明之實施例可使用貝氏推論。第2節描述使用貝氏推論進行估計的特定實施例。The estimation scheme for converting the enhanced sampling measurement data into an estimate has not been described. One of the complications introduced by enhanced sampling is that when the embodiment of the present invention collects measurement data, it changes

s Choice. In this case, given the

One of the measurement results of the circuit,

and

The counted sample average loses its meaning. Instead of using the sample average, in order to process the measurement results into relevant

For information, the embodiments of the present invention can use Bayesian inference. Section 2 describes a specific embodiment of estimation using Bayesian inference.

的一個查詢，而在增強取樣方案中使用對

的三個查詢。看起來，若考慮由三個標準取樣步驟產生的概似函數，則亦可在概似函數中得到三次多項式形式。事實上，假設執行三個獨立的標準取樣步驟，得到結果

，並且藉由自分佈

取樣來古典地產生二進製成果

。隨後，概似函數採取如下形式：

， (12) At this point, you can try to point out that the comparison between standard sampling and enhanced sampling is unfair, because in the case of standard sampling only

Is a query, and the enhanced sampling plan is used to

Of three queries. It seems that if one considers the likelihood function generated by the three standard sampling steps, the third degree polynomial form can also be obtained in the likelihood function. In fact, assuming that three independent standard sampling steps are performed, the results are obtained

, And by self-distribution

Sampling to classically produce binary results

. Subsequently, the likelihood function takes the following form:

, (12)

為可經由改變分佈

古典地調諧之參數。更具體地，

，其中

為位元串

之漢明權重。假設希望

等於方程式9中的

。此隱示

、

、

及

，其明顯超出方程式12中的概似函數之古典可調諧性。此係表明由方程式9中之量子方案產生之概似函數超出古典手段的證據。Each of them

Changeable distribution

Classically tuned parameters. More specifically,

,among them

Bit string

The Hamming weight. Suppose hope

Equal to Equation 9

. This implies

,

,

and

, Which clearly exceeds the classical tunability of the likelihood function in Equation 12. This series shows evidence that the likelihood function produced by the quantum scheme in Equation 9 is beyond classical means.

增大時，每樣本的時間

隨

線性地增長。此電路層數目之線性增長以及費雪資訊之二次增長引起預期運行時間之下限，

， (13) When the number of circuit layers

When increasing, the time per sample

Follow

Increase linearly. The linear increase of the number of circuit layers and the second increase of Fisher Information cause the lower limit of the expected operating time

, (13)

式估計策略之情況下。在實踐中，在量子電腦上實現之運算受誤差影響。幸運的是，本發明之實施例可使用貝氏推論，該推論可將此類誤差併入至估計過程中。只要誤差對概似函數之形式的影響得到準確地建模，此類誤差的主要效應就僅僅是減緩資訊增益率。當電路層之數目

增大時，量子電路中的誤差累積。因此，超出電路層之特定數目，就將關於費雪資訊的增益(或運行時間的縮減)接收到遞減的返回。隨後，估計演算法可試圖平衡此等競爭因素，以便最佳化總體效能。This system assumes a fixed unbiased estimator

In the case of formula estimation strategy. In practice, calculations implemented on quantum computers are affected by errors. Fortunately, embodiments of the present invention can use Bayesian inference, which can incorporate such errors into the estimation process. As long as the effect of errors on the form of the likelihood function is accurately modeled, the main effect of such errors is only to slow down the rate of information gain. When the number of circuit layers

When increasing, errors in the quantum circuit accumulate. Therefore, if the specific number of circuit layers is exceeded, the gain (or reduction in running time) on Fisher information is received in a decreasing return. Subsequently, the estimation algorithm can try to balance these competing factors in order to optimize the overall performance.

，在

的增強取樣情況下每樣本費雪資訊增益大於或等於

。如圖6A至圖6B中所示出，在引入即使小幅誤差時，在概似函數平坦之處的

的值引起費雪資訊之大幅下降。此類區域在本文中稱為估計死點。此觀測激發了對概似函數(ELF)工程化以增強其統計能力的概念。藉由將

及

運算推廣至廣義反射

及

，本發明之實施例可使用旋轉角度，以使得資訊增益在此類死點附近升高。即使對於更深的增強取樣電路，對概似函數工程化仍允許本發明之實施例減輕估計死點的效應。The introduction of errors causes another problem for estimation. When there is no error, for all

,in

In the case of enhanced sampling, the Fisher information gain per sample is greater than or equal to

. As shown in Figures 6A to 6B, when even a small error is introduced, where the probability function is flat

The value of caused a sharp drop in Fisher Information. Such regions are referred to herein as estimated dead points. This observation inspired the concept of engineering the likelihood function (ELF) to enhance its statistical power. By

and

Generalized reflection

and

The embodiment of the present invention can use the rotation angle to increase the information gain near such a dead point. Even for deeper enhanced sampling circuits, engineering the likelihood function still allows embodiments of the present invention to reduce the effect of estimating dead points.

3. Engineering Probability Function

This section describes methods that can be used by embodiments of the present invention for engineering the likelihood function for amplitude estimation. First, the quantum circuit for extracting samples corresponding to the engineered likelihood function is described, and then the technique for tuning the circuit parameters and performing Bayesian inference from the obtained likelihood function is described.

3.1. Quantum circuits for engineering probabilistic functions

(18) The techniques for designing, implementing, and executing a program on a computer (for example, a quantum computer or a hybrid quantum classical computer) for estimating the following expected values will now be described

(18)

，其中

量子位元麼正算子，

本征值為

的

量子位元赫米特算子，並且引入

以促進稍後的貝氏推論。在構造本文所揭示的估計演算法時，可假設本發明之實施例能夠執行以下基元運算。首先，本發明之實施例可準備計算基礎狀態

，並且向其應用量子電路

，從而獲得

。其次，本發明之實施例對於任何角度

實現麼正算子

。最後，本發明之實施例執行

的量測，該

經建模為具有各別成果標記

之投射值度量

。本發明之實施例亦可使用麼正算子

，其中

且

。遵循慣例，

及

在本文中將分別被稱為關於

的

本征空間及狀態

的廣義反射，其中

及

分別為此等廣義反射之角度。among them

,among them

Qubit operator,

Eigenvalue

of

Qubit Hemet operator, and introduce

To facilitate Bayesian inference later. When constructing the estimation algorithm disclosed herein, it can be assumed that the embodiment of the present invention can perform the following primitive operations. First, the embodiment of the present invention can be prepared to calculate the basic state

And apply quantum circuits to it

To get

. Secondly, the embodiment of the present invention is suitable for any angle

Realizing the positive operator

. Finally, the embodiment of the present invention executes

Measurement, the

Modeled as having separate achievement markers

Projected value metric

. The embodiment of the present invention can also use a positive operator

,among them

And

. Follow convention,

and

In this article will be referred to as

of

Eigenspace and state

Generalized reflection, where

and

Respectively the angle of this generalized reflection.

的情況下成果

的機率分佈。電路可例如包括廣義反射之序列。具體地，在準備擬設狀態

之後，本發明之實施例可向其應用

個廣義反射

、

、

、

、

，從而在每一運算中改變旋轉角度

。爲了便利起見，

在本文中將被稱為電路的第

層，其中

。此電路的輸出狀態為

(19) The embodiment of the present invention can use the unattached in Figure 7 (we call this scheme "unattached" (AF) because this scheme does not involve any attached qubits. In Appendix A, we consider naming it as A different scheme of the "attachment-based" (AB) scheme, which involves an attached qubit) quantum circuit to generate an engineered likelihood function (ELF), which is based on a given unknown quantity to be estimated

Results in the case of

Probability distribution. The circuit may, for example, include a sequence of generalized reflections. Specifically, in preparation for the proposed state

After that, the embodiments of the present invention can be applied to it

Generalized reflection

,

,

,

,

To change the rotation angle in each operation

. For convenience,

In this article will be referred to as the circuit’s

Layer, where

. The output state of this circuit is

(19)

為可調諧參數之向量。最後，本發明之實施例可對此狀態執行投射量測

，從而接收成果

。among them

Is a vector of tunable parameters. Finally, the embodiment of the present invention can perform projection measurement on this state

To receive the results

.

及

確保量子狀態對於任何

均保持在二維子空間

中（為了確保

為二維的，假設

，亦即，

或

）。設

為

中正交於

的狀態(唯一的，取決於相位)，亦即，

(20) As in Grover’s algorithm, generalized reflection

and

Ensure that the quantum state is for any

Are kept in two-dimensional subspace

Medium (to ensure

Is two-dimensional, assuming

,that is,

or

). Assume

for

Orthogonal to

State (unique, depending on the phase), that is,

(20)

及

分別寫為

及

。To help the analysis, treat this two-dimensional subspace as a qubit, thus taking

and

Respectively written as

and

.

、

、

及

分別為此虛擬量子位元上的包立算子及恆等算子。隨後，關注子空間

，可將

重寫為

(21) Assume

,

,

and

Respectively, the Baoli operator and the identity operator on this virtual qubit. Then, focus on the subspace

, Can be

Rewritten as

(twenty one)

及

重寫為

(22) Generalized reflection

and

Rewritten as

(twenty two)

(23) and

(twenty three)

為可調諧參數。隨後，由

層電路實現之麼正算子

變為

。 (24) among them

It is a tunable parameter. Subsequently, by

Positive Operator

Becomes

. (twenty four)

為固定的，而

、

及

視未知量

而定。結果是，與在原始「實體」圖片中相比，在此「邏輯」圖片中設計並分析估計演算法更便利。因此，此圖片將用於本揭示案的剩餘部分。It should be noted that in this picture,

Is fixed, and

,

and

Visual unknown

Depends. As a result, it is more convenient to design and analyze estimation algorithms in this "logical" picture than in the original "physical" picture. Therefore, this picture will be used for the remainder of this disclosure.

之機率分佈)視電路的輸出狀態

及可觀測

而定。Engineering probability function (that is, the measurement result

The probability distribution) depends on the output state of the circuit

Observable

Depends.

(25) Precisely, the engineered probability function is

(25)

(26) among them

(26)

及

來分別表示

及

關於

的導數)。特別地，若

，則得到

。亦即，此

的概似函數之偏誤為

的(第一種類之)

次切比雪夫多項式。出於此原因，此

的概似函數在本文中將被稱為切比雪夫概似函數(CLF)。第5節將探索CLF與通用ELF之間的效能間隙。Is the error of the likelihood function (from now on, we will use

and

To express separately

and

on

Derivative). In particular, if

, You get

. That is, this

The error of the likelihood function is

Of (of the first category)

Second Chebyshev polynomial. For this reason, this

The probability function of will be called Chebyshev probability function (CLF) in this article. Section 5 will explore the performance gap between CLF and general ELF.

In fact, quantum devices are affected by noise. In order to make the estimation process robust against errors, embodiments of the present invention may incorporate the following noise model into the likelihood function.

In practice, the establishment of the noise model can use a procedure for calibrating the likelihood function for the specific device used. Regarding Bayesian inference, the parameters of this model are called redundant parameters; the target parameters are not directly determined by the redundant parameters, but the degree of correlation between the redundant parameters and the target parameters. Therefore, the redundant parameters can be incorporated into the inference process. The remainder of this disclosure will assume that the noise model has been calibrated to sufficient accuracy so that the effect of model error is negligible.

之有雜訊的版本實現目標運算及作用於相同輸入狀態的完全去極化通道（去極化模型假設包含每一層的閘足夠隨機以防止相干誤差的系統累積。存在使此去極化模型更準確之技術，諸如隨機化編譯）的混合物，亦即，

， (27) Assuming that each circuit layer

The noisy version achieves the target calculation and a fully depolarized channel acting on the same input state (the depolarization model assumes that the gates of each layer are sufficiently random to prevent the accumulation of coherent errors. The existence of the depolarization model makes this depolarization model more A mixture of accurate techniques, such as randomized compilation, that is,

, (27)

為此層之保真度。在此類不完善運算的組成下，

層電路之輸出狀態變為

(28) among them

The fidelity of this layer. Under the composition of such imperfect calculations,

The output state of the layer circuit becomes

(28)

的不完善準備，並且之後為

的不完善量測。在隨機化基準的情境下，此類誤差被稱為狀態準備及量測(SPAM)誤差。本發明之實施例亦可藉由去極化模型對SPAM誤差建模，從而使

的有雜訊的準備係

，並且使

的有雜訊的量測係POVM

。將SPAM誤差參數組合到

中，得到有雜訊的概似函數之模型

(29) This imperfect circuit was previously

Imperfect preparation, and later for

Of imperfect measurement. In the context of randomized benchmarks, this type of error is called State Preparation and Measurement (SPAM) error. The embodiment of the present invention can also model the SPAM error by the depolarization model, so that

Noisy preparation department

And make

Noisy measurement system POVM

. Combine the SPAM error parameters into

, Get the model of the probability function with noise

(29)

為用於產生ELF之整個過程之保真度，並且

為理想概似函數之偏誤，如方程式(26)中所定義的(自此，本揭示案將使用

來表示

的導數)。應注意，雜訊對ELF的總體效應為，雜訊將偏誤再縮放

倍。此隱示，產生過程中的誤差愈少，所得ELF的斜率愈大(此意謂對於貝氏推論愈有用)，如所預期。among them

Is the fidelity of the whole process used to generate ELF, and

Is the deviation of the ideal probability function, as defined in equation (26) (from now on, this disclosure will use

To represent

Derivative). It should be noted that the overall effect of noise on ELF is that the noise will be biased and then scaled

Times. This implies that the less error in the generation process, the greater the slope of the resulting ELF (which means the more useful for Bayesian inference), as expected.

，對於

的一些(複數值)函數

及

，

可寫為

， (30) Before proceeding with the discussion of Bayesian inference by ELF, it is worth mentioning the following properties of the engineered probability function, because it will play a role in Section 4. The concepts of trigonometric-multilinear and trigonometric-multiquadratic functions are known. Basically, if for any

,for

Some (complex-valued) functions of

and

,

Can be written as

, (30)

為三角-多線性的，並且將

及

稱為

關於

的餘弦-正弦-分解(CSD)係數函數。類似地，若對於任何

，對於

的一些(複數值)函數

、

及

，

可寫為

(31) Multivariate function

Is trigonometric-multilinear, and will

and

Called

on

The cosine-sine-decomposition (CSD) coefficient function of. Similarly, if for any

,for

Some (complex-valued) functions of

,

and

,

Can be written as

(31)

為三角-多二次的，並且將

、

及

稱為

關於

的餘弦-正弦-偏誤-分解(CSBD)係數函數。三角-多線性及三角-多二次性的概念亦可自然地推廣至線性算子。亦即，若線性算子的每一項(任意地編寫)在一組變數中為三角-多線性的(或多二次性的)，則此算子在相同變數中為三角-多線性的(或三角-多二次性的)。現在，方程式(22)、(23)及(24)隱示

為

的三角-多線性算子。隨後，自方程式(26)得到，

為

的三角-多二次性函數。此外，揭示了可在

時間內評估

關於任何

的CSBD係數函數，並且此顯著促進第4.1節中用於調諧電路角度

的演算法的構造。Multivariate function

Is trigonometric-quadratic, and will

,

and

Called

on

The cosine-sine-bias-decomposition (CSBD) coefficient function of. The concepts of trigonometric-multilinear and trigonometric-multiquadratic can also be naturally extended to linear operators. That is, if each term of a linear operator (written arbitrarily) is trigonometric-polylinear (or multiquadratic) in a set of variables, then this operator is trigonometric-multilinear in the same variable (Or Triangular-more quadratic). Now, equations (22), (23) and (24) imply

for

The triangle-multilinear operator. Then, from equation (26),

for

The trigonometric-multi-quadratic function. In addition, it reveals the

Time evaluation

About any

The CSBD coefficient function, and this significantly promotes the angle used to tune the circuit in Section 4.1

The structure of the algorithm.

3.2 Bayesian Inference by Engineering Probability Function

並且藉由用於振幅估計之所得概似函數執行貝氏推論的本發明之實施例。After the (noisy) engineered probability function model is in place, the parameters used to tune the circuit will be described

And an embodiment of the present invention of Bayesian inference is performed by the obtained likelihood function for amplitude estimation.

的演算法的實施例的高階概述開始。爲了便利起見，此類實施例可對

有效，而不是對

有效。本發明之實施例可使用高斯分佈來表示

的知識，並且隨著推論過程繼續進行，使此分佈逐漸收斂至

的真實值。本發明之實施例可從

的初始分佈(其可由標準取樣或域知識產生)開始，並且將其轉換成

的初始分佈。隨後，本發明之實施例可疊代進行以下程序，直至滿足收斂準則為止。在每一回合，本發明之實施例可找到在特定意義上(基於

的當前知識)最大化來自量測成果

的資訊增益的電路參數

。隨後，藉由最佳化參數

來執行圖7中的量子電路，並且接收量測成果

。最後，本發明之實施例可藉由使用貝氏法則、以

為條件來更新

的分佈。一旦此迴圈結束，本發明之實施例就可將

的最終分佈轉換成

的最終分佈，並且將此分佈的平均值設定為

的最終估計值。有關此演算法的概念圖，請參見圖8。From the pair used to estimate

A high-level overview of the implementation of the algorithm begins. For convenience, such embodiments can be

Effective, not right

effective. The embodiment of the present invention can be represented by Gaussian distribution

Knowledge, and as the inference process continues, the distribution gradually converges to

The true value of. The embodiments of the present invention can be obtained from

The initial distribution of (which can be generated by standard sampling or domain knowledge) starts and transforms it into

The initial distribution. Subsequently, the embodiments of the present invention may iteratively perform the following procedures until the convergence criterion is met. In each round, the embodiment of the present invention can be found in a specific sense (based on

Current knowledge) to maximize the results from the measurement

Circuit parameters of information gain

. Then, by optimizing the parameters

To execute the quantum circuit in Figure 7 and receive the measurement results

. Finally, the embodiments of the present invention can use Bayes’ rule to

Condition to update

Distribution. Once this loop is over, the embodiment of the present invention can change

The final distribution is transformed into

The final distribution of and set the average of this distribution to

The final estimate. For a conceptual diagram of this algorithm, see Figure 8.

的值的可信度。亦即，在每一回合，對於某個事前平均值

及事前變異數

，

具有事前分佈

(32) Each component of the above algorithm is described in more detail below. Throughout the whole inference process, the embodiment of the present invention uses Gaussian distribution to track

The credibility of the value. That is, in each round, for a certain ex-ante average

And prior variance

,

Pre-distribution

(32)

後，本發明之實施例可藉由使用貝氏法則來計算

的事後分佈：

， (33) After receiving the measurement results

Later, the embodiment of the present invention can be calculated by using Bayes’ rule

The post-mortem distribution:

, (33)

(回想起

用於產生ELF之過程之保真度)。儘管真實的事後分佈將不會為高斯分佈，但是本發明之實施例可將其近似為如此。遵循先前的方法，本發明之實施例可用相同平均值及變異數（儘管本發明之實施例可直接按定義來計算事後分佈

的平均值及變異數，但是此方法耗時，因為其涉及數值積分。相反，本發明之實施例可藉由利用工程化概似函數之特定性質來加速此過程。有關更多詳情，請參見第4.2節）的高斯分佈來替換真實事後，並且將其設定為下一回合之

的事前。本發明之實施例可重複此量測及貝氏更新程序，直至

的分佈充分集中在單個值附近為止。Where the normalization factor or model evidence is defined as

(recall

The fidelity of the process used to generate ELF). Although the true posterior distribution will not be a Gaussian distribution, the embodiment of the present invention can approximate it to this. Following the previous method, the embodiment of the present invention can use the same average value and variance (although the embodiment of the present invention can directly calculate the post-mortem distribution by definition

The mean and variance of, but this method is time-consuming because it involves numerical integration. On the contrary, embodiments of the present invention can speed up this process by taking advantage of the specific properties of the engineered probability function. For more details, please refer to the Gaussian distribution in Section 4.2) to replace the real hindsight and set it as the next round

Beforehand. The embodiment of the present invention can repeat this measurement and Bayesian update procedure until

The distribution of is sufficiently concentrated around a single value.

有效，並且吾等最終對

感興趣，本發明之實施例可在

與

的估計值之間進行轉換。此過程如下進行。假設在回合

，

的事前分佈為

，並且

的事前分佈為

(注意，

、

、

及

為隨機變數，因為它們視取決於時間

的隨機量測成果之歷史而定)。在此回合，

及

的估計量分別為

及

。給定

的分佈

，本發明之實施例可計算

的平均值

及變異數

，並且將

設定為

的分佈。此步驟可以解析方式完成，就好像

，隨後

， (34)

。 (35) Because the algorithm is mainly for

Effective, and we are ultimately

Interested, the embodiments of the present invention can be found in

versus

Convert between estimated values. This process proceeds as follows. Suppose in round

,

The pre-distribution of is

,and

The pre-distribution of is

(note,

,

,

and

Are random variables because they depend on time

Depends on the history of random measurement results). In this round,

and

The estimates of are

and

. given

Distribution

, The embodiment of the present invention can be calculated

average value

And variance

And will

set as

Distribution. This step can be done analytically, just like

, Followed by

, (34)

. (35)

的分佈

，本發明之實施例可計算

的平均值

及變異數

(將

鉗位至

)，並且將

設定為

的分佈。此步驟可以數值方式完成。儘管高斯變數之

或

函數並非真正的高斯分佈，但是本發明之實施例可將其近似為如此，並且發現此對演算法的效能具有的影響可忽略。Instead, given

Distribution

, The embodiment of the present invention can be calculated

average value

And variance

(will

Clamp to

), and will

set as

Distribution. This step can be done numerically. Although the Gaussian variable

or

The function is not a true Gaussian distribution, but the embodiment of the present invention can approximate it to this, and it is found that this has negligible influence on the performance of the algorithm.

的方法可由本發明之實施例如下實現。理想地，可謹慎選擇角度以使得隨著

增長，

的估計量

之均方差(MSE)儘可能快地減小。然而在實踐中，直接計算此量很難，並且本發明之實施例可尋求其值的代理。估計量之MSE為估計量之變異數與估計量之平方偏誤的總和。

的平方偏誤可小於其變異數，亦即，

，其中

為

的真實值。

的變異數

常常接近

的變異數，亦即，

具有高機率。組合此等事實，得知

具有高機率。因此，本發明之實施例可改為找到最小化

的變異數

的參數

。Used to tune the angle of the circuit

The method can be implemented by the embodiment of the present invention as follows. Ideally, the angle can be chosen carefully so that as

increase,

Estimator of

The mean square error (MSE) is reduced as quickly as possible. However, in practice, it is difficult to directly calculate this quantity, and embodiments of the present invention can seek a proxy for its value. The MSE of the estimator is the sum of the variance of the estimator and the square error of the estimator.

The squared error of can be less than its variance, that is,

,among them

for

The true value of.

Of variance

Often close

The variance of, that is,

Has a high probability. Combine these facts and learn

Has a high probability. Therefore, the embodiment of the present invention can be changed to find the minimize

Of variance

Parameters

.

具有事前分佈

。在接收到量測成果

之後，

的預期事後變異數為Specifically, assuming

Pre-distribution

. After receiving the measurement results

after that,

The expected post-mortem variance is

， (36) =

, (36)

(37) among them

(37)

為理想概似函數之偏誤，如方程式(26)中所定義，並且

為用於產生概似函數之過程之保真度。現在，引入用於對概似函數工程化的量，並且在本文中將其稱為變異數縮減因數，

。 (38) among them

Is the deviation of the ideal probability function, as defined in equation (26), and

It is the fidelity of the process used to generate the likelihood function. Now, introduce the quantity used to engineer the likelihood function, and call it the variance reduction factor in this article,

. (38)

。 (39) Then get

. (39)

愈大，

的變異數平均減小得愈快。此外，為了量化

的逆變異數的增長率(每時間步驟)，可使用以下量

(40)

， (41)

Bigger,

The variance of the average decreases faster. In addition, in order to quantify

The growth rate of the contravariant anomalous number (per time step), the following quantities can be used

(40)

, (41)

為推論回合的時間成本。應注意，

為

的單調函數，其中

。因此，當電路層之數目

固定時，本發明之實施例可藉由最大化

來最大化

(關於

)。另外，當

很小時，

近似與

成正比，亦即，

。本揭示案之剩餘部分將假設擬設電路最顯著地構成總體電路之持續時間。使

與擬設在電路中被調用的次數成正比，從而設定

，其中時間以擬設持續時間為單位。among them

It is the time cost of the inference round. It should be noted that

for

Monotonic function, where

. Therefore, when the number of circuit layers

When fixed, the embodiments of the present invention can be maximized by

To maximize

(on

). In addition, when

Very small,

Approximately

Is directly proportional, that is,

. The remainder of this disclosure will assume that the proposed circuit most significantly constitutes the duration of the overall circuit. Make

It is proportional to the number of calls to be set in the circuit, thus setting

, Where the time is based on the planned duration.

、

及

最大化變異數縮減因數

的參數

之技術。通常，此最佳化問題變得難以解決。幸運的是，在實踐中，本發明之實施例可假設

的事前變異數

很小(例如，至多

)，並且在此種情況下，

可藉由概似函數

在

下的費雪資訊來近似，亦即，

，

(42) Now, it will be revealed to find for a given

,

and

Maximize the variance reduction factor

Parameters

Of technology. Usually, this optimization problem becomes difficult to solve. Fortunately, in practice, the embodiments of the present invention can assume

Ex ante variance

Very small (e.g. at most

), and in this case,

Likelihood function

in

To approximate the Fisher information below, that is,

,

(42)

(43)

(44) among them

(43)

(44)

的費雪資訊，如在方程式(29)中所定義。因此，代替直接最佳化變異數縮減因數

，本發明之實施例可最佳化費雪資訊

，此可藉由本發明之實施例使用第4.1.1節中的演算法來高效地完成。此外，當用於產生ELF之過程之保真度

為低時，得到

。隨後，

(45) Probability function

The Fisher information of is as defined in equation (29). Therefore, instead of directly optimizing the variance reduction factor

, The embodiment of the present invention can optimize Fisher Information

This can be done efficiently by the embodiment of the present invention using the algorithm in section 4.1.1. In addition, when used to generate the fidelity of the ELF process

When it is low, get

. Subsequently,

(45)

，其與概似函數

在

下的斜率成正比，並且此任務可由本發明之實施例使用第4.1.2節中的演算法來高效地完成。Therefore, in this case, the embodiment of the present invention can be optimized

, Which is the same as the probability function

in

The slope of the lower is proportional, and this task can be efficiently accomplished by the embodiment of the present invention using the algorithm in section 4.1.2.

增長時

的估計量

之MSE有多快。假設在推論過程期間電路層之數目

為固定的。當

時，此給出

。

的逆MSE之增長率可預測如下。當

時，得到

、

、

及

具有高機率，其中

及

分別為

及

的真實值。當此事件發生時，得到對於大的

，

。 (46) Finally, the embodiments of the present invention can be predicted as

When growing

Estimator of

How fast is MSE. Assuming the number of circuit layers during the inference process

Is fixed. when

When, this gives

.

The growth rate of the inverse MSE can be predicted as follows. when

When, get

,

,

and

Has a high probability, where

and

Respectively

and

The true value of. When this event occurs, get the

,

. (46)

，

， (47) Therefore, from equation (35), we know that for large

,

, (47)

。由於

的偏誤通常遠遠小於其標準差，並且後者可由

近似，預測對於大的

，

。 (48) among them

. due to

The bias of is usually much smaller than its standard deviation, and the latter can be determined by

Approximately, forecast for large

,

. (48)

的逆MSE之漸近增長率(每時間步驟)應大致為

， (49) This means

The asymptotic growth rate (per time step) of the inverse MSE should be roughly

, (49)

關於

得到最佳化。將在第5節中將此率與

的逆MSE之經驗增長率進行比較。among them

on

Get optimized. This rate will be compared with

Compare the empirical growth rate of the inverse MSE.

4. Efficient heuristic algorithm and Bayesian inference for circuit parameter tuning

的啟發式演算法的實施例，並且描述本發明之實施例可如何藉由所得概似函數高效地進行貝氏推論。This section describes the parameters used to tune the circuit in Figure 7

An embodiment of the heuristic algorithm, and describes how the embodiment of the present invention can efficiently perform Bayesian inference with the obtained likelihood function.

4.1. The efficiency maximization of the agent of the variance reduction factor

的演算法係基於最大化變異數縮減因數

的兩個代理(概似函數

的費雪資訊及斜率)。所有此等演算法要求用於評估偏誤

及其導數

關於

的CSBD係數函數的高效程序，其中

。回想在第3.1中已示出，偏誤

隨

為三角-多二次性的。亦即，對於任何

，存在

的函數

、

及

，以使得  

(50) Circuit angle for tuning implemented according to embodiments of the present invention

The algorithm is based on maximizing the variance reduction factor

Two proxies (likelihood function

Fisher information and slope). All these algorithms are required to assess bias

And its derivative

on

An efficient program for the CSBD coefficient function, where

. Recall that it was shown in Section 3.1 that the bias

Follow

It is triangular-multiquadratic. That is, for any

,exist

The function

,

and

To make

(50)

(51) Subsequently,

(51)

亦為三角-多二次性的，其中

、

、

分別為

、

、

關於

的偏誤。結果是，給定

及

，

，可在

時間內計算

、

、

、

及

中的每一者。Follow

It is also triangular-multiquadratic, where

,

,

Respectively

,

,

on

The bias. As a result, given

and

,

, Available at

Time calculation

,

,

,

and

Each of them.

及

，可在

時間內計算

、

、

、

、

及

中的每一者。Lemma 1. Given

and

, Available at

Time calculation

,

,

,

,

and

Each of them.

prove. See Appendix C.

4.1.1. Fisher Information for Maximizing Probability Function

在給定點

(亦即，

的事前平均值)的費雪資訊之兩種演算法中的一或多種。假設目標在於找到最大化下式的

。 (52) The embodiment of the present invention can be used to maximize the likelihood function

At a given point

(that is,

One or more of Fisher Information’s two algorithms. Suppose the goal is to find the maximization

. (52)

的梯度成正比的步驟，直至滿足收斂準則為止。具體地，設

為疊代

處的參數向量。本發明之實施例可如下將其更新：

(53) The first algorithm is based on gradient ascent. That is, the first algorithm starts from a random initial point, and keeps taking the

The gradient is proportional to the steps until the convergence criterion is met. Specifically, let

Iterative

The parameter vector at. The embodiment of the present invention can be updated as follows:

(53)

為步長排程（在最簡單情況下，

為常數。但是，為了達成更好的效能，可能希望當

）。此要求計算

關於每一

的部分導數，此計算可如下進行。本發明之實施例首先使用引理1中之程序來針對每一

計算

、

、

、

、

及

。此獲得

， (54)

， (55)

， (56)

； (57) among them

Schedule the steps (in the simplest case,

Is a constant. However, in order to achieve better performance, you may wish to

). This requirement is calculated

About each

Part of the derivative of, this calculation can be performed as follows. The embodiment of the present invention first uses the procedure in Lemma 1 to target each

Calculation

,

,

,

,

and

. This get

, (54)

, (55)

, (56)

； (57)

關於

的部分導數：

(58) After knowing these equal amounts, the embodiment of the present invention can be calculated as follows

on

Part of the derivative:

(58)

重複此程序。隨後，本發明之實施例可獲得

。演算法的每一次疊代耗費

時間。演算法中的疊代次數視初始點、終止準則及步長排程

而定。有關更多詳情，請參見演算法65。The embodiments of the present invention can be aimed at

Repeat this procedure. Subsequently, the embodiment of the present invention can be obtained

. The cost of each iteration of the algorithm

time. The number of iterations in the algorithm depends on the initial point, termination criterion, and step schedule

Depends. For more details, see Algorithm 65.

，直至滿足收斂準則為止。在每一回合的第

個步驟之後，解決以下針對坐標

的單變數最佳化問題：

， (59) The second algorithm is based on coordinate ascent. Unlike gradient ascent, this algorithm does not require a step size and allows each variable to be changed drastically in a single step. Therefore, the second algorithm can converge faster than the previous algorithm. Specifically, the embodiment of the present invention implementing this algorithm can start from a random initial point, and successively maximize the objective function along the coordinate direction

, Until the convergence criterion is met. In the first round of each round

After three steps, solve the following for coordinates

The single variable optimization problem:

, (59)

、

、

、

、

、

可藉由引理1中之程序在

時間中計算。此單變數最佳化問題可藉由基於標準梯度之方法來解決，並且將

設定為其解。針對

重複此程序。此演算法產生序列

、

、

、

，以使得

。亦即，隨

增長，

的值單調地增大。演算法之每一回合耗費

時間。演算法中的回合數視初始點及終止準則而定。among them

,

,

,

,

,

The procedure in Lemma 1 can be used in

Calculated in time. This single-variable optimization problem can be solved by a method based on standard gradients, and the

Set to its solution. against

Repeat this procedure. This algorithm generates a sequence

,

,

,

To make

. That is, with

increase,

The value of increases monotonically. The cost of each round of the algorithm

time. The number of rounds in the algorithm depends on the initial point and termination criteria.

4.1.2. Maximize the slope of the likelihood function

在給定點

(亦即，

的平均值)的斜率之兩種演算法中的一或多種。假設目標在於找到最大化

的

。The embodiment of the present invention can be used to maximize the likelihood function

At a given point

(that is,

One or more of the two algorithms for the slope of the average). Suppose the goal is to find the maximization

of

.

及

評估

、

及

。然而，基於梯度上升的演算法使用上述量來計算

關於

的部分導數，而基於坐標上升的演算法使用上述量來直接更新

的值。分別在演算法1及2中正式描述此等演算法。Similar to the algorithms 65 and 65 for maximizing Fisher Information, the algorithms for maximizing the slope are also based on gradient ascent and coordinate ascent, respectively. Both call the procedure in Lemma 1 for a given

and

Evaluation

,

and

. However, algorithms based on gradient ascent use the above quantities to calculate

on

Part of the derivative of, and the algorithm based on coordinate ascent uses the above quantity to directly update

Value. These algorithms are formally described in Algorithms 1 and 2, respectively.

4.2. Approximate Bayesian Inference by Engineering Probability Function

的演算法就位後，現在描述如何藉由所得概似函數高效地進行貝氏推論。本發明之實施例可在接收量測成果

之後直接計算

的事後平均值及變異數。但此方法耗時，因為其涉及數值積分。藉由利用工程化概似函數之特定性質，本發明之實施例可大幅加速此過程。Used to tune circuit parameters

With the algorithm in place, we now describe how to efficiently perform Bayesian inference with the obtained likelihood function. The embodiment of the present invention can receive measurement results

Calculate directly afterwards

The post hoc mean and variance of the results. But this method is time consuming because it involves numerical integration. By using the specific properties of the engineered probability function, the embodiments of the present invention can greatly speed up this process.

具有事前分佈

，其中

，並且用於產生ELF之過程之保真度為

。本發明之實施例可發現，最大化

(或

)的參數

滿足以下性質：當

接近

時，亦即，

時，得到

(67) Hypothesis

Pre-distribution

,among them

, And the fidelity of the process used to generate ELF is

. The embodiments of the present invention can be found to maximize

(or

) Parameters

Meet the following properties: when

Close to

时, that is,

When, get

(67)

。亦即，本發明之實施例可藉由

在此區域中的正弦函數來近似

。圖13圖示說明一個此種實例。some of them

. That is, the embodiments of the present invention can be obtained by

The sine function in this area to approximate

. Figure 13 illustrates one such example.

及

：

(68) The embodiment of the present invention can find the best fit by solving the following least square problem

and

:

(68)

。此最小平方問題具有以下解析解：

(69) among them

. This least squares problem has the following analytical solution:

(69)

。 (76) among them

. (76)

Figure 13 shows examples of true likelihood functions and fitted likelihood functions.

及

，其就可藉由針對下式的平均值及變異數來近似

的事後平均值及變異數

， (77) Once the embodiment of the present invention is optimal

and

, Which can be approximated by the mean and variance of the following formula

Post hoc mean and variance

, (77)

在回合

處具有事前分佈

。設

為量測成果，並且

為此回合的最佳擬合參數。隨後，本發明之實施例可藉藉由下式來近似

的事後平均值及變異數

(78)

(79) The above formula has an analytical formula. Specifically, assuming

In the round

Pre-distribution

. Assume

For measuring results, and

The best fit parameters for this round. Subsequently, the embodiment of the present invention can be approximated by

Post hoc mean and variance

(78)

(79)

設定為該回合之

的事前分佈。After that, the embodiment of the present invention can continue to the next round, thereby changing

Set as the round

The ex-ante distribution.

遠離

時，亦即，

時，真實概似函數與擬合概似函數之間的差異可能很大。但是，由於事前分佈

隨

呈指數衰減，此類

對計算

的事後平均值及變異數之貢獻很小。因此，方程式(78)及(79)給出

的事後平均值及變異數的高度準確的估計值，並且其誤差對整個演算法之效能具有的影響可忽略。It should be noted that as illustrated in Figure 13, when

keep away

时, that is,

At this time, the difference between the true likelihood function and the fitted likelihood function may be very large. However, due to the prior distribution

Follow

Decay exponentially, such

Pair calculation

The contribution of the post hoc mean and variance is small. Therefore, equations (78) and (79) give

The post-event average and variance are highly accurate estimates, and the error has negligible influence on the performance of the entire algorithm.

5. Simulation result

This section describes the specific results of the simulation of Bayesian inference using the engineered likelihood function for amplitude estimation. These results show the specific advantages of the specific engineered probability function over the unengineered probability function, and the influence of circuit depth and fidelity on the performance of the specific engineered probability function.

5.1 Experiment details

並且執行投射量測

所耗費的時間比實現

少得多。因此，當電路層之數目為

時，推論回合之時間成本大致為

，其中

為

的時間成本(應注意，

層電路使用

及

次。爲了簡單起見，假設在後續論述中

耗費單位時間(亦即，

)。此外，假設在實驗中在量子狀態的準備及量測中沒有誤差，亦即，

。In the experiment, the hypothetical realization

And perform projection measurement

It takes longer than implementation

Much less. Therefore, when the number of circuit layers is

, The time cost of the inference round is roughly

,among them

for

Time cost (note that

Layer circuit use

and

Times. For the sake of simplicity, assume that in the subsequent discussion

It takes a unit of time (that is,

). In addition, it is assumed that there is no error in the preparation and measurement of the quantum state in the experiment, that is,

.

。設

為

在時間

的估計量。應注意，

本身為隨機變數，因為其視取決於時間

的隨機量測成果之歷史而定。藉由

的均方根誤差(RMSE)來量測方案之效能，其由下式給出  

。 (80) Hypothesis is to estimate the expected value

. Assume

for

In time

Estimated amount. It should be noted that

Itself is a random variable, because it depends on time

It depends on the history of random measurement results. By

The root mean square error (RMSE) is used to measure the efficiency of the scheme, which is given by the following formula

. (80)

增長

衰減有多快，該等方案包括基於附屬的切比雪夫概似函數(AB CLF)、基於附屬的工程化概似函數(AB ELF)、無附屬的切比雪夫概似函數(AF CLF)，以及無附屬的工程化概似函數(AF ELF)。The following will describe various scenarios with

increase

How fast is the decay? These schemes include the attached Chebyshev Probability Function (AB CLF), the attached engineered Probability Function (AB ELF), and the Chebyshev Probability Function (AF CLF) without attachment. And the unaffiliated engineering probabilistic function (AF ELF).

的分佈難以特徵化，並且

解析公式。為了估計此量，本發明之實施例可執行推論過程

次，並且收集

的

個樣本

、

、

、

，其中

為

在第

輪(其中

)中在時間

處的估計值。隨後，本發明之實施例可使用量  

(81) usually,

The distribution of is difficult to characterize, and

Analyze the formula. In order to estimate this amount, the embodiment of the present invention can perform an inference process

Times and collect

of

Samples

,

,

,

,among them

for

In the first

Round (where

) In time

Estimated value at the place. Subsequently, the embodiment of the present invention can be used

(81)

。在實驗中，設定

，並且發現此引起令人滿意的結果。To approximate reality

. In the experiment, set

, And found that this leads to satisfactory results.

。此示出演算法1及2產生相等品質的解，並且演算法5及6亦如此。因此，若改為使用基於梯度上升的演算法1及5來調諧電路角度

，實驗結果將不變。The embodiment of the present invention can use algorithms 2 and 6 based on coordinate ascent to optimize the circuit parameters in the case of no attachment and the case based on attachment, respectively

. This shows that Algorithms 1 and 2 produce solutions of equal quality, and Algorithms 5 and 6 do the same. Therefore, if you change to use gradient-based algorithms 1 and 5 to tune the circuit angle

, The experimental results will remain unchanged.

的事後平均值及變異數。特別地，在ELF之正弦擬合期間，本發明之實施例可設定方程式(68)及(148)中的

(亦即，

含有在

中均勻分佈的

個點)。已發現，此足以獲得ELF的高品質正弦擬合。In order to perform Bayesian update by ELF, the embodiment of the present invention can use the methods in Section 4.2 and Appendix A.2 to calculate the non-attachment case and the attachment-based case respectively

The post hoc mean and variance of the results. In particular, during the sine fitting of ELF, the embodiment of the present invention can set the equations (68) and (148)

(that is,

Contained in

Evenly distributed

Points). It has been found that this is sufficient to obtain a high-quality sine fit of the ELF.

6. Noisy algorithm performance model

的估計值之目標均方根誤差所需要的。此模型可基於兩個主要假設來建置。第一假設為，逆均方差的增長率係由逆變異數率表達式(參見方程式(40))的一半來良好描述。一半係歸因於如下保守估計：變異數及平方偏誤對均方差有同等貢獻(來自先前章節的模擬示出平方偏誤趨於小於變異數)。第二假設為變異數縮減因數之經驗下限，該經驗下限由切比雪夫概似函數之數值研究激發。Embodiments of the present invention can implement a model for runtime that is achieved when scaling to a larger system and running on a device with better gate fidelity

The estimated value of the target root mean square error is required. This model can be built based on two main assumptions. The first assumption is that the growth rate of the inverse mean square error is well described by half of the contravariant rate expression (see equation (40)). Half of the lines are due to conservative estimates: the variance and squared error contribute equally to the mean square error (simulation from the previous section shows that the squared error tends to be smaller than the variance). The second hypothesis is the empirical lower limit of the variance reduction factor, which is inspired by the numerical study of the Chebyshev probability function.

的估計值之MSE進行分析。隨後，將此估計值的MSE轉換成MSE關於

的估計值。策略將為，對方程式(40)中之率表達式

的上限及下限求積分，以得到作為時間的函數之逆MSE的界限。Yes about

The estimated value of MSE is analyzed. Subsequently, the estimated value of MSE is converted into MSE.

Estimated value. The strategy will be the rate expression in the equation (40)

The upper and lower limits of is integrated to obtain the limit of the inverse MSE as a function of time.

，並且藉由引入

及

以使得

來重參數化雜訊的併入方式。To help analysis, make substitutions

And by introducing

and

So that

To re-parameterize the way the noise is incorporated.

。由於切比雪夫概似函數係工程化概似函數之子集，切比雪夫效能的下限給出ELF效能的下限。吾等猜測，在ELF之情況下此率的上限為針對切比雪夫率建立的上限之小的倍數(例如，1.5倍)。The upper and lower limits of this rate expression are based on the discovery of the Chebyshev probability function, where

. Since the Chebyshev probability function is a subset of the engineered probability function, the lower limit of Chebyshev efficiency gives the lower limit of ELF efficiency. We guess that in the case of ELF, the upper limit of this rate is a small multiple (for example, 1.5 times) of the upper limit established for the Chebyshev rate.

、

及

，可示出（對於切比雪夫概似函數，可將變異數縮減因數表達為

(只要

)。隨後，

隱示

）變異數縮減因數達成最大值

，此在

處出現。此表達式小於

，其在

處達成最大值

。因此，因數

無法超過

。將上述全部組合在一起，對於固定的

、

及

，最大率的上限為

。此由如下事實得到：

隨

為單調的，並且

在

處最大化。在實踐中，本發明之實施例可使用最大化逆變異數率之

的值。藉由離散

達成之率無法超過當在

的連續值上最佳化上述上限時獲得之值。此最佳值針對

實現。藉由評估在此最佳值處的

來定義

，  

， (82) The Chebyshev ceiling is established as follows. For fixed

,

and

, It can be shown (for the Chebyshev probability function, the variance reduction factor can be expressed as

(as long as

). Subsequently,

Implied

) The variance reduction factor reaches the maximum

, Dasein

Appear everywhere. This expression is less than

,Its

Maximum value

. Therefore, the factor

Cannot exceed

. Put all of the above together, for a fixed

,

and

, The upper limit of the maximum rate is

. This is derived from the following facts:

Follow

Is monotonous, and

in

Maximization. In practice, the embodiment of the present invention can use the

Value. Discrete

The rate of achievement cannot exceed that of

The value obtained when the above upper limit is optimized on the continuous value of. This optimal value is for

achieve. By evaluating the optimal value

To define

,

, (82)

。 (83) Which gives the upper limit of the Chebyshev rate

. (83)

，逆變異數率在

個點

處為零。由於率對於所有

在此等端點處均為零，

的總體下限為零。然而，並不擔心逆變異數率在此等端點附近的不良效能。當將估計量自

轉換成

時，此等端點附近的資訊增益實際上趨於大的值。爲了建立有用的界限，將

限制在範圍

內。在數值測試（對

之50000個值、自

至

之

值的均勻網格進行搜尋，其中

為用以得到方程式82之最佳化值，並且

及

在

的範圍內。對於每一

對，找到使最大逆變異數率(針對

)為最小值的

。對於查驗的所有

對，此最差情況率始終在

與

之間，其中發現最小值為

）中，發現對於所有

，總是存在使逆變異數率高於上限的

倍的

的選擇。將此等組合在一起，得到  

。 (84) The embodiments of the present invention do not have a lower analytical limit for Chebyshev's probable performance. The lower limit of experience can be established based on numerical verification. For any fixed

, The inverter anomalous rate is at

Points

Where is zero. As the rate for all

Are zero at these endpoints,

The overall lower limit is zero. However, there is no concern about the poor performance of the inverter outlier near these endpoints. When the estimator is measured from

Convert to

At this time, the information gain near these end points actually tends to be a large value. In order to establish useful boundaries, the

Limited to scope

Inside. In the numerical test (to

Of 50000 values, from

to

Of

Search for a uniform grid of values, where

Is used to obtain the optimized value of Equation 82, and

and

in

In the range. For each

Yes, find the maximum contravariant anomalous rate (for

) Is the minimum

. For all inspected

Yes, this worst case rate is always

versus

, Where the minimum value is found to be

), found that for all

, There is always a factor that makes the contravariant anomalous rate higher

Times

s Choice. Combine these to get

. (84)

為連續的，

及

之特定值可引起使

為負的最佳

。因此，此等結果僅在

(其確保

)的情況下適用。預期此模型在大雜訊型態 (亦即，

)下失效。It is important to note that by using

Is continuous,

and

The specific value of can cause

Negative best

. Therefore, these results are only

(It ensures

) Is applicable. This model is expected to be in the form of large noise (i.e.,

) Is invalid.

，記住上限及下限為其中之小的常數因數。Now, let the hypothetical rate track the geometric mean of these two limits, that is,

, Remember that the upper and lower limits are the small constant factors.

所捕獲的差商表達式所給定的率持續增長。使

表示此逆變異數，可將上文的率方程式重算為

的微分方程式，

(85) Assuming that the inverting anomalous number is in the time

The rate given by the captured difference quotient expression continues to increase. Make

Represents this contravariant anomaly, and the above rate equation can be recalculated as

The differential equation of

(85)

，微分方程式變為  

(86) With this expression, both Heisenberg limit behavior and shot noise limit behavior can be identified. for

, The differential equation becomes

(86)

。此係海森堡限值型態之特徵。對於

，率接近常數，  

。 (87) Its integral is the quadratic increase of the inverse square error

. This is the characteristic of Heisenberg limit type. for

, The rate is close to a constant,

. (87)

，此指示散粒雜訊限值型態。This type produces linear growth of inverse square error

, This indicates the type of shot noise limit.

，將此等界限重新表達為，  

(88) To make the integration easier to handle, the upper and lower limit expressions (used in conjunction with the previous limit) that can be integrated can be used to replace the rate expression. Make

, Re-express these boundaries as,

(88)

的函數並且積分，可自上限建立運行時間之下限，  

(89)  

(90) By treating time as

The function of and the integration can establish the lower limit of the running time from the upper limit,

(89)  

(90)

為變異數的兩倍(亦即，變異數等於偏誤)，因此變異數必須達到MSE的一半：

。在最好情況下，假設估計值之偏誤為零，並且設定

。將此等界限與方程式(84)之上限及下限組合，以得到作為目標MSE的函數之估計運行時間之界限，

(91) Similarly, the lower limit can be used to establish the upper limit of the running time. The following assumptions are introduced here. In the worst case, the MSE of the phase estimation

It is twice the variance (that is, the variance equals the bias), so the variance must reach half of the MSE:

. In the best case, assume that the bias of the estimated value is zero, and set

. Combine these bounds with the upper and lower bounds of equation (84) to obtain the bounds of the estimated running time as a function of the target MSE,

(91)

。among them

.

轉換回成振幅估計

。可就相位估計MSE將關於振幅估計之MSE

近似為At this point, the phase can be estimated

Convert back to amplitude estimation

. Can estimate MSE for phase and MSE for amplitude estimate

Approximately

 

， (92)

, (92)

充分達到峰值，以忽略較高階項。此引起

，可將其代入至上述界限表達式中，對於

亦是如此。藉由去掉估計量下標(因為它們僅貢獻常數因數)，可建立低雜訊及高雜訊限值中的運行時間定標，  

(94) The distribution of the assumed estimator is for

The peak value is reached sufficiently to ignore higher-order terms. This caused

, Can be substituted into the above limit expression, for

The same is true. By removing the estimator subscripts (because they only contribute constant factors), it is possible to establish runtime calibrations in the low noise and high noise limits,

(94)

It is observed that the Heisenberg limit calibration and the shot noise limit calibration have been restored respectively.

。Use the properties of Chebyshev's probability function to obtain these limits. As shown in the previous section, by engineering the likelihood function, the estimated running time can be shortened in many cases. Motivated by the numerical discovery of the variance reduction factor of the engineered likelihood function (see, for example, Figure 19), we guessed that using the engineered likelihood function to increase the worst-case contravariant anomaly rate in equation (84) to

.

及雙量子位元閘保真度

表示。考慮估計包立串

關於狀態

的預期值之任務。假設

非常接近零，使得

。設

層中之每一者的雙量子位元閘深度為

。將總層保真度建模為

，其中已經忽略由單量子位元閘引起的誤差。由此，得到

及

。將此等組合起來，得到運行時間表達式，

(95) In order to give this model more meaning, it is subdivided into the number of qubits

And dual qubit gate fidelity

Said. Consider the estimation of Bao Li string

About status

The expected value of the task. Hypothesis

Very close to zero, making

. Assume

The depth of the dual qubit gate for each of the layers is

. Model the total layer fidelity as

, Which has neglected the error caused by the single qubit gate. From this, get

and

. Combine these to get the runtime expression,

(95)

個邏輯量子位元，並且要求雙量子位元閘深度為大約量子位元數目

。此外，預期目標準確性

將需要為大約

至

。運行時間模型依據擬設電路持續時間來量測時間。為了將此時間轉換成秒，假設雙量子位元閘之每一層將耗費時間

s，此係針對當今的超導量子位元硬體之樂觀假設。圖26示出作為雙量子位元閘保真度的函數之此估計運行時間。Finally, put some meaningful numbers into this expression and estimate the required running time (in seconds) as a function of the fidelity of the dual-qubit gate. In order to achieve quantum advantage, it is expected that the problem example will require approximately

Logical qubits, and the dual-qubit gate depth is required to be approximately the number of qubits

. In addition, the expected target accuracy

Will need to be approximately

to

. The running time model measures the time based on the duration of the circuit to be set up. In order to convert this time into seconds, it is assumed that each layer of the dual qubit gate will consume time

s, this is an optimistic hypothesis for today's superconducting qubit hardware. Figure 26 shows this estimated run time as a function of the fidelity of the dual qubit gate.

Reducing the running time to the dual-qubit gate fidelity required by the practical area will most likely require error correction. Performing quantum error correction requires an additional burden, thereby increasing this running time. When designing a quantum error correction protocol, it is important to estimate that the increase in running time does not exceed the improvement of gate fidelity. The proposed model gives a means to quantify this trade-off: when incorporating useful error correction, the product of gate fidelity and gate time (error-corrected) should be reduced. In practice, in order to make a more rigorous statement, there are many subtleties that should be considered. These minorities include considering the fidelity changes of the gates of the circuit, and the time cost of the changes of different types of gates. However, the cost analysis provided by this simple model can be a useful tool in designing quantum gates, quantum chips, error correction schemes, and noise mitigation schemes.

appendix A. Attached-based scheme

為可調諧參數。In this appendix, an alternative solution is presented, which is referred to as an attachment-based solution. In this scheme, the engineered likelihood function (ELF) is generated by the quantum circuit in Fig. 27, where

It is a tunable parameter.

(96) Assuming that there is no noise in the circuit in Figure 27, the engineered likelihood function is given by

(96)

(97) among them

(97)

替換了

。因此，將使用與之前相同的標記(例如，

、

、

、

、

、

)，除非另有說明。特別地，當考慮到圖27中之電路中之誤差時，有雜訊的概似函數由下式給出  

(98) Is the error of the likelihood function. As a result, most of the arguments in Section 3.1 are still valid in the case of attachment, but with

Replaced

. Therefore, the same notation as before will be used (e.g.,

,

,

,

,

,

),Unless otherwise indicated. In particular, when considering the error in the circuit in Fig. 27, the probability function with noise is given by

(98)

為用於產生ELF之過程之保真度。然而，應注意，

與

之間存在差異，因為前者在隨

為三角-多二次性的，而後者隨

為三角-多線性的。among them

It is the fidelity of the process used to generate ELF. However, it should be noted that

versus

There is a difference between because the former is

Is trigonometric-multiquadratic, and the latter varies with

Triangular-multi-linear.

，並且以與第3.2節中類似的方式藉由所得ELF執行貝氏推論。實際上，第3.2節中之論證在基於附屬的情況下仍然成立，只不過需要用

替換

。因此，將使用與之前相同的標記，除非另有說明。具體地，亦如同在方程式(37)及(38)中一樣定義變異數縮減因數

，從而用

替換

。可示出，  

(99) Will tune the circuit angle

, And perform Bayesian inference with the resulting ELF in a similar way as in Section 3.2. In fact, the argument in Section 3.2 is still valid in the case of attachment, but it needs to be used

replace

. Therefore, the same notation as before will be used unless otherwise stated. Specifically, the variance reduction factor is also defined as in equations (37) and (38)

To use

replace

. Can show,

(99)

(100) and

(100)

在

時的費雪資訊及斜率為變異數縮減因數

之兩個代理。由於

的直接最佳化通常很難，將改為藉由最佳化此等代理來調諧參數

。That is, under reasonable assumptions, the likelihood function

in

Fisher information at time and the slope of the variance reduction factor

Of the two agents. due to

The direct optimization of is usually difficult, and the parameters will be tuned by optimizing these agents instead

.

A.1. The efficiency maximization of the agent of the variance reduction factor

的兩個代理(概似函數

的費雪資訊及斜率)之高效啟發式演算法。所有此等演算法使用以下程序來評估偏誤

及其關於

的導數

，其中

。Now, the reduction factor used to maximize the variance

Two proxies (likelihood function

Fisher-Price information and slope) efficient heuristic algorithm. All these algorithms use the following procedure to assess bias

And its about

Derivative of

,among them

.

A.1.1. Evaluation of bias and its derivatives CSD Coefficient function

在

中為三角-多線性的(對於任何

)，存在隨

為三角-多線性的函數

及

，以使得  

。 (101) due to

in

Medium is triangular-multilinear (for any

), there is random

Is a trigonometric-multilinear function

and

To make

. (101)

(102) Subsequently,

(102)

亦為三角-多線性的，其中

及

分別為

及

關於

的導數。Follow

Is also triangular-multilinear, where

and

Respectively

and

on

The derivative of.

及

評估

、

、

以及

。結果是，此等任務可在

時間內完成。Our optimization algorithm requires efficient procedures to target a given

and

Evaluation

,

,

as well as

. As a result, these tasks can be

Completed in time.

及

，可在

時間內計算

、

、

及

中的每一者。Lemma 2. Given

and

, Available at

Time calculation

,

,

and

Each of them.

，

，其中

。此外，設

，其中

。應注意，若

為偶數，則

。隨後，定義若

，否則

。prove. For convenience, the following notation is introduced. Assume

,

,among them

. In addition, suppose

,among them

. It should be noted that if

Is an even number, then

. Subsequently, define if

,otherwise

.

(103) With this mark, you can show

(103)

(104) and

(104)

及

評估

、

、

及

，分開考慮

為偶數的情況及

為奇數的情況。In order to target a given

and

Evaluation

,

,

and

, Consider separately

Is an even number and

Is an odd number.

為偶數，其中

。在此情況下，

。使用如下事實  

(105)  

(106)  

(107) • Situation 1:

Is an even number, where

. In this situation,

. Use the following facts

(105)  

(106)  

(107)

(108) • Obtain

(108)

， (109)

。 (110) • among them

, (109)

. (110)

及

，首先在

時間內計算

及

。隨後，藉由方程式(109)及(110)計算

及

。此程序僅耗費

時間。• given

and

, First in

Time calculation

and

. Then, calculate by equations (109) and (110)

and

. This procedure only costs

time.

及

。藉由使用方程式(104)及事實

，對於任何

，獲得Next, describe how to calculate

and

. By using equation (104) and the fact

For any

,obtain

(111)

(111)

(112)

， (113)

(114)

。 (115) Assume

(112)

, (113)

(114)

. (115)

(116) Subsequently, equation (111) produces

(116)

(117)

(117)

， (118)

, (118)

， (119) Which caused

, (119)

(120)

。 (121) among them

(120)

. (121)

及

，首先藉由標準動態程式化技術在總的

時間內計算以下矩陣：given

and

, First of all by standard dynamic programming technology in the total

Calculate the following matrix in time:

及

，其中

；

and

,among them

；

及

，其中

；

and

,among them

；

及

。

and

.

及

。此後，藉由方程式(120)及(121)計算

及

。總體而言，此程序耗費

時間。Then, calculate by equations (113) and (115)

and

. After that, use equations (120) and (121) to calculate

and

. Overall, this procedure costs

time.

為奇數，其中

。在此情況下，

。使用如下事實

(122)

(123)

(124) 1. Situation 2:

Is odd, where

. In this situation,

. Use the following facts

(122)

(123)

(124)

， (125) 2. Obtain

, (125)

， (126)

。 (127) 3. Among them

, (126)

. (127)

及

，首先在

時間內計算

及

。隨後，藉由方程式(126)及(127)計算

及

。此程序僅耗費

時間。4. Given

and

, First in

Time calculation

and

. Then, calculate by equations (126) and (127)

and

. This procedure only costs

time.

及

。藉由使用方程式(104)及事實

，其中任何

，得到Next, describe how to calculate

and

. By using equation (104) and the fact

, Any of them

,get

。 (128)

. (128)

(129)

(130)

(131)

(132) Assume

(129)

(130)

(131)

(132)

(133) Subsequently, equation (128) yields

(133)

(134)

(134)

(135)

(135)

(136) Which caused

(136)

(137)  

(138) among them

(137)  

(138)

及

，首先藉由標準動態程式化技術在總的

時間內計算以下矩陣：given

and

, First of all by standard dynamic programming technology in the total

Calculate the following matrix in time:

及

，其中

；

and

,among them

；

及

，其中

；

and

,among them

；

及

。

and

.

及

。此後，藉由方程式(137)及(138)計算

及

。總體而言，此程序耗費

時間。Then, calculate by equations (130) and (132)

and

. After that, use equations (137) and (138) to calculate

and

. Overall, this procedure costs

time.

W

A.1.2. Fisher Information for Maximizing Probability Function

在給定點

(亦即，

的事前平均值)的費雪資訊之兩種演算法。亦即，目標在於找到最大化下式的

(139) Proposal for maximizing the likelihood function

At a given point

(that is,

The prior average) of Fisher Information’s two algorithms. That is, the goal is to find the maximization

(139)

及

評估

、

、

及

，並且隨後使用它們來計算

關於

的部分導數(在梯度上升中)或針對

定義單變數最佳化問題(在坐標上升中)。在演算法5及6中正式描述此等演算法。In the sense that these algorithms are also based on gradient ascending and coordinate ascending, respectively, these algorithms are similar to the algorithms 1 and 2 used to maximize Fisher information under the condition of attachment. The main difference is that the procedure in Lemma 2 is now called to target a given

and

Evaluation

,

,

and

, And then use them to calculate

on

Partial derivative of (in gradient ascent) or for

Define the single variable optimization problem (in the coordinate ascent). These algorithms are formally described in Algorithms 5 and 6.

A.1.3. Maximize the slope of the likelihood function

在給定點

(亦即，

的事前平均值)的斜率之兩種演算法。亦即，目標在於找到最大化

的

。It is also proposed to maximize the likelihood function

At a given point

(that is,

The two algorithms for the slope of the prior average). That is, the goal is to find the maximum

of

.

及

評估

及

。隨後，使用此等量來計算

關於

的部分導數(在梯度上升中)或直接更新

的值(在坐標上升中)。在演算法7及8中正式描述此等演算法。In the sense that these algorithms are also based on gradient ascent and coordinate ascent respectively, these algorithms are similar to algorithms 3 and 4 for slope maximization in the case of attachment-based. The main difference is that the procedure in Lemma 2 is now called to target a given

and

Evaluation

and

. Then, use this equivalent to calculate

on

Partial derivative of (in gradient ascent) or update directly

The value of (in the ascending coordinates). These algorithms are formally described in Algorithms 7 and 8.

A.2. Approximate Bayesian Inference by Engineering Probability Function

的演算法就位後，現在簡要描述如何藉由所得概似函數高效地執行貝氏推論。此想法類似於第4.2節中用於無附屬方案的想法。Used to tune circuit parameters

With the algorithm in place, now briefly describe how to efficiently perform Bayesian inference with the resulting likelihood function. This idea is similar to the idea used for the unaffiliated scheme in Section 4.2.

具有事前分佈

，其中

，並且用於產生ELF之過程之保真度為

。發現最大化

(或

)的參數

滿足以下性質：當

接近

時，亦即，

時，得到

(147) Hypothesis

Pre-distribution

,among them

, And the fidelity of the process used to generate ELF is

. Discovery maximization

(or

) Parameters

Meet the following properties: when

Close to

时, that is,

When, get

(147)

。藉由解決以下最小平方問題來找到最佳擬合

及

：

， (148) some of them

. Find the best fit by solving the following least squares problem

and

:

, (148)

。此最小方根問題具有以下解析解：

(149) among them

. This least square root problem has the following analytical solutions:

(149)

。 (156) among them

. (156)

FIG. 34 illustrates examples of the true likelihood function and the fitted likelihood function.

及

，就可藉由下式的平均值及變異數來近似

的事後平均值及變異數

(157) Once you get the best

and

, Can be approximated by the mean and variance of the following formula

Post hoc mean and variance

(157)

在回合

處具有事前分佈

。設

成為量測成果，並且設

為此回合的最佳擬合參數。隨後，藉由下式來近似

的事後平均值與變異數

， (158)

(159) The above formula has an analytical formula. Specifically, assuming

In the round

Pre-distribution

. Assume

Become a measurement result, and set

The best fit parameters for this round. Then, approximate by

Post hoc mean and variance

, (158)

(159)

設定為該回合之

的事前分佈。由方程式(158)及(159)引起的近似誤差很小，並且出於與無附屬的情況相同的原因，該等近似誤差對整個演算法之效能具有的影響可忽略。After that, continue to the next round and put

Set as the round

The ex-ante distribution. The approximation errors caused by equations (158) and (159) are very small, and for the same reason as the case without attachment, the effect of these approximation errors on the performance of the entire algorithm is negligible.

C. Proof of Lemma

、

、

及

，其中

，並且

。此外，設

，其中

。應注意，若

為奇數，則

。隨後，定義若

，否則

。For convenience, the following notation is introduced. Assume

,

,

and

,among them

,and

. In addition, suppose

,among them

. It should be noted that if

Is odd, then

. Subsequently, define if

,otherwise

.

   

，    

  With this mark,

,

求導產生  

  In addition, about

Derivative generation

 

，  

,

  among them

關於

的導數，其中  

  for

on

Derivative of where

關於

的導數。隨後，for

on

The derivative of. Subsequently,

 

。  

.

、

及

為希伯特空間

上的任意線性算子。隨後，藉由直接計算，可驗證  

  The following facts will be useful. Hypothesis

,

and

Hibbert Space

Any linear operator on. Then, by direct calculation, it can be verified

 

，  

,

  and

 

 

 

 

求導產生The following facts will also be useful. on

Derivative generation

 

 

 

 

 

 

 

 

及

評估

、

、

、

、

及

，分開考慮

為偶數的情況及

為奇數的情況。In order to target a given

and

Evaluation

,

,

,

,

and

, Consider separately

Is an even number and

Is an odd number.

為偶數，其中

。在此情況下，

，並且

。隨後獲得  

   

，   • Situation 1:

Is an even number, where

. In this situation,

,and

. Then get

,

   

   

  • among them

及

，首先在

時間內計算

、

及

。隨後，計算

、

及

。此程序僅耗費

時間。• given

and

, First in

Time calculation

,

and

. Then, calculate

,

and

. This procedure only costs

time.

、

及

。使用上式及事實

，其中任何

，獲得Then show how to calculate

,

and

. Use the above formula and facts

, Any of them

,obtain

 

 

 

 

   

  Then, get

   

   

   

   

，    

   

，    

。   among them

,

,

.

   

，   At the same time, get

,

   

   

。   among them

.

  Combine the above facts to get

among them

 

 

 

 

 

 

 

 

 

。  

.

及

，首先藉由標準動態程式化技術在總的

時間內計算以下矩陣：given

and

, First of all by standard dynamic programming technology in the total

Calculate the following matrix in time:

、

、

、

、

；

,

,

,

,

；

及

and

及

and

、

及

，其中

。此後，計算

、

及

。總體而言，此程序耗費

時間。Then, calculate

,

and

,among them

. After that, calculate

,

and

. Overall, this procedure costs

time.

為奇數，其中

。在此情況下，

，並且

。隨後，得到  

   

，   5. Situation 2:

Is odd, where

. In this situation,

,and

. Then, get

,

   

，    

  6. Among them

,

及

，首先在

時間內計算

、

及

。隨後，計算

、

及

。此程序僅耗費

時間。7. Given

and

, First in

Time calculation

,

and

. Then, calculate

,

and

. This procedure only costs

time.

、

及

。使用上式及事實

，其中

，獲得Next, describe how to calculate

,

and

. Use the above formula and facts

,among them

,obtain

 

。  

.

Then, get

，

，

,

,

，    

，    

   

，    

，    

   

，    

，    

   

   

。   among them

,

,

,

,

,

,

.

   

，   At the same time, get

,

，

，

。 among them

,

,

.

Combine the above facts to get

among them

，

,

。

.

及

，首先藉由標準動態程式化技術在總的

時間內計算以下矩陣：given

and

, First of all by standard dynamic programming technology in the total

Calculate the following matrix in time:

、

、

、

、

；

,

,

,

,

；

及

，其中

；

and

,among them

；

及

，其中

。

and

,among them

.

、

及

。，其中

此後，計算

、

及

。總體而言，此程序耗費

時間。Then, calculate

,

and

. ,among them

After that, calculate

,

and

. Overall, this procedure costs

time.

It should be understood that although the present invention has been described above in terms of specific embodiments, the foregoing embodiments are only provided for illustration, and do not limit or define the scope of the present invention. Various other embodiments (including but not limited to the following embodiments) are also within the scope of the patent application. For example, the elements and components described herein can be further divided into additional components or combined together to form fewer components to perform the same function.

Various physical embodiments of quantum computers are suitable for use according to the present disclosure. Generally, the basic data storage unit in quantum computing is a quantum bit (qubit). Qubits are the quantum computing analogues of classical digital computer system bits. It is considered that a classical bit occupies one of two possible states corresponding to a binary digit (bit) 0 or 1 at any given point in time. In contrast, qubits are realized in hardware by physical media with quantum mechanical properties. This kind of media that physically manifests qubits can be referred to herein as "the physical realization of qubits", "substantial embodiments of qubits", "media embodying qubits" or similar terms. Or for ease of explanation, it is simply referred to as "qubit". Therefore, it should be understood that the reference to "qubit" in the description of the embodiments of the present invention herein refers to the physical media embodying qubits.

Each qubit has an unlimited number of different potential quantum mechanical states. When the state of the sub-bit is physically measured, the measurement results from one of the two different basic states of the state analysis of the qubit. Therefore, a single qubit can represent one, zero, or any quantum superposition of their two qubit states; a pair of qubits can be in any quantum superposition of 4 orthogonal basic states; and three qubits Can be in any superposition of 8 orthogonal basic states. The function that defines the quantum mechanical state of a qubit is called its wave function. The wave function also specifies the probability distribution of the result of a given quantitative measurement. A qubit with a two-dimensional quantum state (that is, with two orthogonal base states) can be generalized to a d-dimensional "qubit", where d can be any integer value, such as 2, 3, 4 or greater. In the general case of qubits, the measurement of qubits results from one of d different basic states of the qubit's state analysis. Any reference to qubits in this text should be understood as referring more generally to d-dimensional qubits (d is any value).

Although the specific description of qubits in this article can describe such qubits in terms of their mathematical properties, each such qubit can be implemented in physical media in any of a variety of different ways. Examples of such physical media include superconducting materials, trapped ions, photons, optical resonators, individual electrons trapped in quantum dots, point defects in solids (for example, phosphorous donors in silicon or nitrogen vacancy centers in diamonds) ), molecules (for example, alanine, vanadium complexes), or exhibiting qubit behavior (that is, including quantum states and transitions between them, and these transitions can be controllably induced or detected). A summary of any of the items.

For any given medium that realizes qubits, any one of the various properties of the medium can be selected to realize qubits. For example, if an electron is selected to realize a qubit, the x component of its spin degree of freedom can be selected as the property of the electron to express the state of the qubit. Alternatively, the y-component or z-component of the spin degrees of freedom can be selected as the properties of such electrons to represent the state of such qubits. This is just a specific example of the following general characteristics: For any physical media that can be selected to implement qubits, there may be multiple physical degrees of freedom that can be selected to represent 0 and 1 (for example, x in the electron spin example) , Y and z components). For any specific degree of freedom, the physical media can be controllably placed in a superimposed state, and then the selected degree of freedom can be measured to obtain a reading of the qubit value.

The specific implementation of quantum computers (called gate model quantum computers) includes quantum gates. Compared to classical gates, there are an infinite number of possible single-qubit quantum gates that change the state vector of the qubit. The state that changes the state vector of a qubit is usually called a single-qubit rotation, and can also be referred to as a state change or a single-qubit quantum gate operation in this document. Rotation, state change, or single-qubit quantum gate operations can be mathematically represented by a 2X2 matrix with complex elements. The rotation corresponds to the rotation of the qubit state in its Hibbert space, which can be conceptualized as the rotation of the Bloch sphere. (As those familiar with this technology know, Bloch sphere is the geometric representation of the space of the pure state of qubits.) The multi-qubit gate changes the quantum state of a group of qubits. For example, the double-qubit gate rotates the state of two qubits as the rotation of the two qubits in the four-dimensional Hibbert space. (As those familiar with this technology know, Hibbert space is an abstract vector space that allows the measurement of length and angle to process the structure of the inner product. In addition, Hibbert space is complete: there are enough limits in the space To allow the use of calculus techniques.)

The quantum circuit can be designated as a sequence of quantum gates. As described in more detail below, as used herein, the term "quantum gate" refers to the application of gate control signals (defined below) to one or more qubits to cause them to undergo specific physical transformations and This realizes logic gate operation. In order to conceptualize the quantum circuit, the matrix corresponding to the component quantum gates can be multiplied in the order specified by the gate sequence to produce a 2 ⁿ X ^{2 n} complex matrix representing the same overall state change of n qubits. Therefore, the quantum circuit can be expressed as a single resulting operator. However, designing a quantum circuit based on the composition of the gate allows the design to be consistent with a set of standard gates, thus enabling easier deployment. Therefore, the quantum circuit corresponds to the design of actions taken on the physical components of the quantum computer.

A given variable component subcircuit can be parameterized in an appropriate device-specific manner. More generally, the quantum gate that constitutes a quantum circuit may have a plurality of tuning parameters associated with it. For example, in an embodiment based on optical switching, the tuning parameter may correspond to the angle of an individual optical element.

In a particular embodiment of the quantum circuit, the quantum circuit includes both one or more gates and one or more measurement operations. Quantum computers implemented using such quantum circuits are referred to as implementing "measurement feedback" in this article. For example, a quantum computer that implements measurement feedback can execute a gate in a quantum circuit, and then only measure a subset of the qubits in the sub-computer (that is, less than all qubits), and then based on (multiple times) ) The result(s) of the measurement determines which gate(s) will be executed next. In particular, the measurement (multiple times) can indicate the degree of error in the gate operation(s), and the quantum computer can decide which gate(s) to execute next based on the degree of error. Subsequently, the quantum computer can execute the gate(s) indicated by the decision. The process of executing gates, measuring a subset of sub-bits and then determining which gate(s) to execute next can be repeated any number of times. Measurement feedback may be useful for performing quantum error correction, but is not limited to use in performing quantum error correction. For each quantum circuit, with or without measurement feedback, there is an error-corrected implementation of the circuit.

Some embodiments described herein generate, measure, or utilize a quantum state that approximates a target quantum state (eg, Hamilton's ground state). Those who are familiar with this technology will understand that there are many ways to quantify the degree to which the first quantum state is "approximately" to the second quantum state. In the following description, any approximate concepts or definitions known in the art can be used without departing from the scope of the present invention. For example, when the first and second quantum states are represented as the first and second vectors, respectively, when the inner product between the first and second vectors (called the "fidelity" between the two quantum states) is greater than When a predefined quantity (usually marked as ϵ), the first quantum state is similar to the second quantum state. In this example, the fidelity quantifies the degree to which the first and second quantum states are "close" or "similar" to each other. Fidelity represents the probability that the measurement of the first quantum state will give the same result as if the measurement was performed on the second quantum state. The proximity between quantum states can also be quantified by distance metrics (such as Euclidean norm, Hamming distance, or another type of norm known in the art). The approximation between quantum states can also be defined in terms of calculations. For example, when the polynomial time sampling of the first quantum state gives some desired information or properties shared with the second quantum state, the first quantum state is similar to the second quantum state.

Not all quantum computers are gate-model quantum computers. The embodiments of the present invention are not limited to be implemented using a gate model quantum computer. As an alternative example, the embodiments of the present invention can be implemented in whole or in part by using a quantum computer implemented using a quantum annealing architecture, which is an alternative to a gate model quantum computing architecture. More specifically, quantum annealing (QA) is a meta-heuristic used to find the overall minimum of a given objective function in a given set of candidate solutions (candidate states) by using a process of quantum fluctuations.

FIG. 2B shows a diagram illustrating operations generally performed by a computer system 250 that implements quantum annealing. The system 250 includes both a quantum computer 252 and a classical computer 254. The operations shown to the left of the vertical dotted line 256 are usually performed by the quantum computer 252, and the operations shown to the right of the vertical dotted line 256 are usually performed by the classical computer 254.

Quantum annealing starts from the classical computer 254 generating the initial Hamiltonian 260 and the final Hamiltonian 262 based on the calculation problem 258 to be solved, and the initial Hamiltonian 260, the final Hamiltonian 262 and the annealing schedule 270 are provided to the quantum computer 252 as inputs. The quantum computer 252 prepares a well-known initial state 266 (FIG. 2B, operation 264) based on the initial Hamilton 260, such as a quantum mechanical superposition of all possible states (candidate states) with the same weight. The classical computer 254 provides the initial Hamiltonian 260, the final Hamiltonian 262, and the annealing schedule 270 to the quantum computer 252. The quantum computer 252 starts in the initial state 266, and follows the time-dependent Schrodinger equation to evolve its state according to the annealing schedule 270 (natural quantum mechanical evolution of the physical system) (FIG. 2B, operation 268). More specifically, the state of the quantum computer 252 undergoes time evolution under the time-dependent Hamiltonian, which starts from the initial Hamilton 260 and ends at the final Hamilton 262. If the rate of change of the system Hamilton is slow enough, the system remains close to the instantaneous Hamiltonian ground state. If the rate of change of the system Hamiltonian accelerates, the system can temporarily leave the ground state, but there is a higher possibility of ending in the final problem Hamilton's ground state (that is, non-adiabatic quantum calculation). At the end of the time evolution, the group of qubits on the quantum annealing is in the final state 272, which is expected to be close to the ground state of the classical Ising model, which corresponds to the solution of the original optimization problem 258. An experimental demonstration of successful quantum annealing of random magnets was reported shortly after the initial theoretical proposal.

The final state 272 of the sub-computer 254 is measured, thereby generating a result 276 (ie, measured value) (FIG. 2B, operation 274). The measurement operation 274 may be performed, for example, in any of the ways disclosed herein, such as in any of the ways disclosed herein in conjunction with the measurement unit 110 in FIG. 1. The classical computer 254 performs post-processing on the measurement result 276 to generate an output 280 representing the solution of the original calculation problem 258 (FIG. 2B, operation 278).

As yet another alternative example, the embodiments of the present invention can be implemented in whole or in part using a quantum computer implemented using a single vector computing architecture (also known as a measurement-based quantum computing architecture), which is a gate Another alternative to model quantum computing architecture. More specifically, a unidirectional or measurement-based quantum computer (MBQC) is a quantum computing method that first prepares the entangled resource state (usually a cluster state or a graphic state), and then performs a single-qubit measurement on it. This method is "one-way" because the resource state is destroyed by measurement.

The results of each individual measurement are random, but they are related in a way that makes the calculation always successful. Usually, the selection of the basis of the later measurement depends on the results of the early measurement, so the measurement cannot be performed all at the same time.

Any of the functions disclosed herein can be implemented using components for performing their functions. Such components include, but are not limited to, any of the components disclosed herein, such as the computer-related components described below.

Referring to FIG. 1, there is shown a diagram of a system 100 implemented in accordance with an embodiment of the present invention. 2A, there is shown a flowchart of a method 200 executed by the system 100 of FIG. 1 according to an embodiment of the present invention. The system 100 includes a quantum computer 102. The quantum computer 102 includes a plurality of qubits 104, and the plurality of qubits 104 can be implemented in any of the ways disclosed herein. Any number of qubits 104 can exist in the quantum computer 104. For example, the qubit 104 may include or consist of the following: no more than 2 qubits, no more than 4 qubits, no more than 8 qubits, no more than 16 qubits, no more than 32 qubits Qubits, no more than 64 qubits, no more than 128 qubits, no more than 256 qubits, no more than 512 qubits, no more than 1024 qubits, no more than 2048 qubits Yuan, no more than 4096 qubits, or no more than 8192 qubits. These are just examples. In practice, any number of qubits 104 can exist in the quantum computer 102.

There can be any number of gates in a quantum circuit. However, in some embodiments, the number of gates can be at least proportional to the number of qubits 104 in the quantum computer 102. In some embodiments, the gate depth may be no greater than the number of qubits 104 in the quantum computer 102, or no greater than a linear multiple of the number of qubits 104 in the quantum computer 102 (e.g., 2, 3, 4, 5, 6 or 7).

The qubits 104 can be interconnected in any pattern. For example, they can be connected by linear chains, two-dimensional grids, all-to-all connections, any combination thereof, or any sub-graph of any of the foregoing.

As will become apparent from the following description, although the element 102 is referred to herein as a "quantum computer", this does not imply that all components of the quantum computer 102 utilize quantum phenomena. One or more components of the quantum computer 102 may be, for example, classical components (ie, non-quantum components) that do not utilize quantum phenomena.

The quantum computer 102 includes a control unit 106, and the control unit 106 may include any of various circuits and/or other mechanisms for performing the functions disclosed herein. The control unit 106 may be composed entirely of classical components, for example. The control unit 106 generates one or more control signals 108 and provides them as output to the qubit 104. The control signal 108 can take any of various forms, such as any kind of electromagnetic signal, such as an electrical signal, a magnetic signal, an optical signal (for example, a laser pulse), or any combination thereof.

E.g:

• In an embodiment where some or all of the qubits 104 are implemented as photons traveling along a waveguide (also referred to as a "quantum optics" implementation), the control unit 106 may be a beam splitter (for example, a heater or a mirror) ), the control signal 108 can be a signal for controlling the rotation of a heater or a mirror, the measurement unit 110 can be a light detector, and the measurement signal 112 can be a photon.

• Some or all of the qubits 104 are implemented as charge type qubits (for example, transmon, X-mon, G-mon) or flux type qubits (for example, flux qubits, capacitive shunt type In the embodiment of the flux qubit (also known as "circuit quantum electrodynamics" (circuit QED) implementation), the control unit 106 can be a bus resonator activated by a driver, and the control signal 108 can be a cavity mode In this case, the measurement unit 110 may be a second resonator (for example, a low-Q resonator), and the measurement signal 112 may be a voltage measured from the second resonator using a dispersion readout technique.

• In the embodiment where some or all of the qubits 104 are implemented as superconducting circuits, the control unit 106 can be a circuit QED auxiliary control unit or a direct capacitive coupling control unit or an inductive capacitive coupling control unit, and the control signal 108 can be In the cavity mode, the measurement unit 110 may be a second resonator (for example, a low-Q resonator), and the measurement signal 112 may be a voltage measured from the second resonator using a dispersion readout technique.

• In an embodiment where some or all of the qubits 104 are implemented to capture ions (for example, the electronic state of magnesium ions), the control unit 106 may be a laser, and the control signal 108 may be a laser pulse, and the measurement The unit 110 may be a laser and a CCD or a light detector (for example, a photomultiplier tube), and the measurement signal 112 may be a photon.

• In an embodiment where some or all of the qubits 104 are implemented using nuclear magnetic resonance (NMR) (in this case, the qubits may be molecules, for example, in liquid or solid form), the control unit 106 may be radio frequency (RF) antenna, the control signal 108 may be an RF field transmitted by an RF antenna, the measurement unit 110 may be another RF antenna, and the measurement signal 112 may be an RF field measured by a second RF antenna.

• In the embodiment where some or all of the qubits 104 are implemented as nitrogen vacancy centers (NV centers), the control unit 106 can be, for example, a laser, a microwave antenna or a coil, and the control signal 108 can be a visible light, a microwave signal or a constant For the electromagnetic field, the measurement unit 110 may be a photodetector, and the measurement signal 112 may be a photon.

• In an embodiment where some or all of the qubits 104 are implemented as two-dimensional quasi-particles called "anyons" (also known as "topological quantum computer" implementations), the control unit 106 can be a nanowire, The control signal 108 may be a local electric field or a microwave pulse, the measurement unit 110 may be a superconducting circuit, and the measurement signal 112 may be a voltage.

• In an embodiment where some or all of the qubits 104 are implemented as semiconducting materials (for example, nanowires), the control unit 106 can be a micro-machining gate, the control signal 108 can be an RF or microwave signal, and the measurement unit 110 can be a micro-machining gate, and the measurement signal 112 can be an RF or microwave signal.

Although not explicitly shown in FIG. 1 and not required, the measurement unit 110 may provide one or more feedback signals 114 to the control unit 106 based on the measurement signal 112. For example, a quantum computer called a "single quantum computer" or a "quantum computer based on measurement" utilizes such feedback 114 from the measurement unit 110 to the control unit 106. Such feedback 114 is also necessary for the operation of fault-tolerant quantum computation and error correction.

The control signal 108 may, for example, include one or more state preparation signals that, when received by the qubit 104, cause some or all of the qubits 104 to change their states. This type of state preparation signal constitutes a quantum circuit, also known as a "planned circuit". The resulting state of the qubit 104 is referred to herein as the “initial state” or “proposed state”. The process of outputting the state preparation signal(s) so that the qubit 104 is in its initial state is referred to herein as "state preparation" (FIG. 2A, section 206). The special case of state preparation is "initialization" (also called "reset operation"), where the initial state is that some or all of the qubits 104 are in the "zero" state (that is, the default single-qubit state )status. More generally, state preparation may involve any distribution of using state preparation signals to cause some or all of the qubits 104 to be in a desired state. In some embodiments, the control unit 106 may initialize the qubit 104 by first outputting the first set of state preparation signals, and then output the second set of state preparation signals to partially or completely place the qubit 104 in the non- Zero state, to first perform initialization on the qubit 104 and then perform preparation on the qubit 104.

Another example of the control signal 108 that can be output by the control unit 106 and received by the qubit 104 is a gate control signal. The control unit 106 can output such gate control signals, thereby applying one or more gates to the qubit 104. Applying a gate to one or more qubits causes the group of qubits to undergo a physical state change that reflects the corresponding logical gate operation specified by the received gate control signal (e.g., single qubit rotation, Double-qubit entanglement gate or multi-qubit operation). This implies that in response to receiving the gate control signal, the qubit 104 undergoes physical transformations that cause the qubit 104 to change its state in such a way that the state of the qubit 104 is measured when the state of the qubit 104 is measured (see below ) Represents the result of executing the logic gate operation specified by the gate control signal. As used herein, the term “quantum gate” refers to the application of gate control signals to one or more qubits to cause those qubits to undergo the physical transformations described above and thereby implement logical gate operations.

It should be understood that the dividing line between the state preparation (and the corresponding state preparation signal) and the application of the gate (and the corresponding gate control signal) can be arbitrarily selected. For example, some or all of the components and operations of the components illustrated as "state preparation" in FIGS. 1 and 2A to 2B can be changed to be characterized as components for gate applications. Conversely, for example, some or all of the components and operations of the components illustrated as "gate application" in FIGS. 1 and 2A to 2B can be changed to be characterized as state-ready components. As a specific example, the system and method of FIGS. 1 and 2A to 2B can be characterized as only performing state preparation and then performing measurement without any gate application. The components described herein as part of the gate application are changed to It is regarded as part of the state preparation. Conversely, for example, the systems and methods of FIGS. 1 and 2A to 2B can be characterized as only performing gate application and then performing measurement without any state preparation, and the component changes described herein as part of the state preparation It is regarded as part of the gate application.

The quantum computer 102 also includes a measurement unit 110 that performs one or more measurement operations on the qubit 104 to read the measurement signal 112 from the qubit 104 (also referred to herein as "measurement Result”), where the measurement result 112 is a signal representing the state of some or all of the qubits 104. In practice, the control unit 106 and the measurement unit 110 can be completely different from each other, or contain some common components, or be implemented by a single unit (that is, a single unit can realize both the control unit 106 and the measurement unit 110) . For example, the laser unit can be used to generate the control signal 108 and the subbit 104 to provide a stimulus (eg, one or more laser beams) to cause the measurement signal 112 to be generated.

Generally, the quantum computer 102 can perform the various operations described above any number of times. For example, the control unit 106 can generate one or more control signals 108 to cause the qubit 104 to perform one or more quantum gate operations. Subsequently, the measurement unit 110 may perform one or more measurement operations on the qubit 104 to read a set of one or more measurement signals 112. The measurement unit 110 can repeat this type of measurement operation on the qubit 104 before the control unit 106 generates the additional control signal 108, thereby causing the measurement unit 110 to read the same gate that was executed before the previous measurement signal 112 was read. The additional measurement signal 112 obtained by the calculation. The measurement unit 110 can repeat this process any number of times to generate any number of measurement signals 112 corresponding to the same gate operation. Subsequently, the quantum computer 102 can aggregate such multiple measurements of the same gate operation in any of various ways.

After the measurement unit 110 has performed one or more measurement operations on the qubit 104 after a set of gate operations has been performed, the control unit 106 may generate one or more additional control signals 108 that may be different from the previous control signals 108 , Thereby causing the qubit 104 to perform one or more additional quantum gate operations that may be different from a set of previous quantum gate operations. Subsequently, the process described above can be repeated, in which the measurement unit 110 performs one or more measurement operations on the qubit 104 in its new state (obtained from the most recently performed gate operation).

Generally, the system 100 can implement a plurality of quantum circuits as follows. For each quantum circuit C in the plurality of quantum circuits (FIG. 2A, operation 202 ), the system 100 performs a plurality of "triggers" on the qubit 104. The meaning of triggering will become apparent from the following description. For each trigger S in the plurality of triggers (FIG. 2A, operation 204), the system 100 prepares the state of the qubit 104 (FIG. 2A, section 206). More specifically, for each quantum gate G in the quantum circuit C (FIG. 2A, operation 210), the system 100 applies the quantum gate G to the qubit 104 (FIG. 2A, operations 212 and 214).

Subsequently, for each of the qubits Q 104 (FIG. 2A, operation 216), the system 100 measures the sub-bit Q to generate a measurement output representing the current state of the qubit Q (FIG. 2A, operations 218 and 220).

The operations described above are repeated for each trigger S (FIG. 2A, operation 222) and circuit C (FIG. 2A, operation 224). The above description implies that a single “trigger” involves preparing the state of the qubit 104 and applying all the quantum gates in the circuit to the qubit 104, and then measuring the state of the sub-bit 104; and the system 100 can target One or more circuits perform multiple triggers.

Referring to FIG. 3, there is shown a diagram of a hybrid classical quantum computer (HQC) 300 implemented according to an embodiment of the present invention. The HQC 300 includes a quantum computer component 102 (which can be implemented, for example, in the manner shown and described in conjunction with FIG. 1) and a classical computer component 306. A classical computer component can be a machine implemented according to a general computing model established by von Newman, in which the program is written in the form of an ordered list of instructions and stored in the classical (for example, digital) memory 310, and is used by the classical computer of the classical computer. (E.g., digital) processor 308 executes. The memory 310 is classical in the sense that data is stored in the storage medium in the form of bits, and these bits have a single definite binary state at any point in time. The bits stored in the memory 310 may, for example, represent a computer program. The classical computer component 304 usually includes a bus 314. The processor 308 can read bits from the memory 310 and write bits to the memory 310 via the bus 314. For example, the processor 308 can read instructions from a computer program in the memory 310, and can optionally receive instructions from a source external to the computer 302 (such as from a user input device such as a mouse, keyboard, or any other input device) Enter data 316. The processor 308 can use the instructions that have been read from the memory 310 to perform calculations on the data read from the memory 310 and/or the input 316, and generate output from those instructions. The processor 308 may store the output back to the memory 310 and/or provide the output externally as the output data 318 via an output device (such as a monitor, a speaker, or a network device).

The quantum computer component 102 may include a plurality of qubits 104, as described above in conjunction with FIG. 1. A single qubit can represent one, zero, or any quantum superposition of the states of these two qubits. The classical computer component 304 can provide the classical state preparation signal Y32 to the quantum computer 102. In response to the classical state preparation signal Y32, the quantum computer 102 can be any of the methods disclosed herein (such as by combining FIGS. 1 and 2A to Any one of the methods disclosed in FIG. 2B) prepares the state of the qubit 104.

Once the qubit 104 is ready, the classical processor 308 can provide the classical control signal Y34 to the quantum computer 102. In response to the classical control signal Y34, the quantum computer 102 can apply the gate operation specified by the control signal Y32 to the qubit Element 104, whereby the qubit 104 reaches its final state. (It can be implemented as described above in conjunction with FIG. 1 and FIG. 2A to FIG. 2B) The measurement unit 110 in the quantum computer 102 can measure the state of the sub-bit 104 and generate a measurement output Y38, which is a measurement output Y38 It means that the state of the qubit 104 collapses to one of its intrinsic states. Therefore, the measurement output Y38 includes or consists of bits, and therefore represents the classical state. The quantum computer 102 provides the measurement output Y38 to the classical processor 308. The classic processor 308 can store data representing the measurement output Y38 and/or data derived therefrom in the classic memory 310.

The steps described above can be repeated any number of times, where the state described above as the final state of the qubit 104 is used as the initial state of the next iteration. In this way, the classical computer 304 and the quantum computer 102 can operate together as a co-processor to perform joint calculations as a single computer system.

Although some functions may be described herein as being performed by a classical computer, and other functions may be described herein as being performed by a quantum computer, these are only examples and do not constitute a limitation of the present invention. The subset of functions disclosed in this article as being performed by a quantum computer can be changed to being performed by a classical computer. For example, a classical computer can be used to mimic the functionality of a quantum computer and provide a subset of the functionality described in this article, although the simulated exponential scaling will limit the functionality. The functions disclosed in this article as being performed by a classical computer can be changed to being performed by a quantum computer.

The technology described above can be implemented, for example, in hardware, in one or more computer programs, firmware, or any combination thereof tangibly stored on one or more computer-readable media, such as on a quantum computer alone , It can be implemented on a classical computer alone or on a hybrid classical quantum (HQC) computer. The technology disclosed in this article can be implemented, for example, only on a classical computer, where the classical computer mimics the functions of the quantum computer disclosed in this article.

The technology described above can be implemented in one or more computer programs that are executed on a programmable computer (such as a classical computer, a quantum computer, or HQC) (or can be executed by a programmable computer). Execution), the programmable computer includes any combination of any number of the following: processor, processor readable and/or writable storage medium (including, for example, volatile and non-volatile memory and / Or storage components), input devices, and output devices. Program codes can be applied to input typed using input devices to perform the functions described and output devices to generate output.

The embodiments of the present invention include features that are possible and/or feasible only when implemented by using one or more computers, computer processors, and/or other components of a computer system. It is impossible or impractical to implement such features intelligently and/or manually. For example, it is impossible to generate random samples intelligently or manually from the complex distribution of the description operator P and the state |s>.

Any claim that definitely requires a computer, processor, memory, or similar computer-related components in this article intends to require such components, and should not be understood as such components are not present in or required by such claims . Such claims are not intended and should not be understood to cover methods and/or systems that do not contain the stated computer-related components. For example, it is stated in this article that the claimed method is executed by a computer, processor, memory, and/or similar computer-related components. Any method claim is intended and should only be understood as covering the stated computer-related component(s). method. Such method request items should not be understood as covering, for example, methods that are executed intelligently or manually (for example, using pencil and paper). Similarly, any product claims in which the claimed products include computers, processors, memory, and/or similar computer-related components are intended and should only be understood as covering products that include the stated computer-related component(s). Such product claims should not be understood as covering, for example, products that do not include the stated computer-related component(s).

In the embodiment of a classical computing component executing a computer program that provides any subset of the functionality within the scope of the following patent applications, the computer program can be implemented in any programming language, such as assembly language, machine language, and high-level programming Language, or object-oriented programming language. For example, the programming language can be a compilation or interpretation programming language.

Each such computer program can be implemented in a computer program product tangibly embodied in a machine-readable storage device for execution by a computer processor (which can be a classical processor or a quantum processor). The method steps of the present invention can be executed by one or more computer processors executing programs tangibly embodied in a computer-readable medium to perform functions by operating on input and generating output. For example, suitable processors include both general-purpose microprocessors and special-purpose microprocessors. Generally, the processor receives (reads) instructions and data from a memory (such as read-only memory and/or random access memory), and writes (stores) the instructions and data to the memory. Storage devices suitable for tangibly embodying computer program instructions and data include, for example, all forms of non-volatile memory, such as semiconductor memory devices, including EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks and Removable disks; magneto-optical disks; and CD-ROMs. Any of the above can be supplemented by or incorporated into a specially designed ASIC (application-specific integrated circuit) or FPGA (field programmable gate array). Classical computers can usually also receive (read) programs and data from non-transitory computer-readable storage media such as internal disks (not shown) or removable disks, and write (store) the programs and data to the non-transitory computer-readable storage media. Temporary computer-readable storage media. These components will also be found in conventional desktop computers or workstation computers and other computers suitable for executing computer programs that implement the methods described in this article. These computers can be combined with any digital printing engine or markup engine, Display monitors, or other raster output devices that can produce color or grayscale pixels on paper, film, display screens, or other output media are used together.

Any data disclosed herein may, for example, be in one or more data structures tangibly stored on a non-transitory computer-readable medium (such as classical computer-readable media, quantum computer-readable media, or HQC computer-readable media) achieve. The embodiment of the present invention can store such data in this (type) data structure, and read such data from this (type) data structure.

100: System 102: Quantum Computer 104: Qubit 106: control unit 108: Control signal 110: Measuring unit 112: Measurement signal 114: feedback signal 200: method 202: Operation 204: Operation 206: Operation 208: Operation 210: Operation 212: Operation 214: Operation 216: Operation 218: Operation 220: Operation 222: Operation 224: Operation 250: computer system 252: Quantum Computer 254: Classical Computer 256: vertical dotted line 258: question 260: Initial Hamilton 262: Final Hamilton 264: Operation 266: initial state 268: Operation 270: Annealing schedule 272: final state 274: Operation 276: result 278: result 280: output 300: Hybrid Classical Quantum Computer (HQC) 306: Classical Computer Components 308: processor 310: memory 314: Bus 316: Input data 318: output data 400: operator 402: Quantum State 404: Block 406: Block 408: Block 410: Block 430: Hybrid Classical Quantum Computer (HQC) 432: Quantum Computer 434: Classical Computer

因數； 圖20A至圖20B及圖21A至圖21B示出圖示說明本發明之各種實施例之效能的曲線圖； 圖22A至圖22B示出本發明之各種實施例的

因數； 圖23A至圖23B及圖24A至圖24B示出圖示說明本發明之各種實施例之效能的曲線圖； 圖25A至圖25B示出本發明之各種實施例的

因數； 圖26示出圖示說明本發明之各種實施例之運行時間相對目標準確性的曲線圖； 圖27至圖28示出根據本發明之實施例實現之量子電路； 圖29A至圖29B、圖30A至圖30B、圖31A至圖31B及圖32示出根據本發明之實施例實現之演算法；以及 圖33示出根據本發明之實施例之真實概似函數及擬合概似函數。Fig. 1 is a diagram of a quantum computer according to an embodiment of the present invention; Fig. 2A is a flowchart of a method executed by the quantum computer of Fig. 1 according to an embodiment of the present invention; Fig. 2B is an embodiment according to the present invention A diagram of a hybrid quantum classical computer performing quantum annealing; Fig. 3 is a diagram of a hybrid quantum classical computer according to an embodiment of the present invention; Fig. 4 is a diagram of a hybrid quantum computer for performing quantum amplitude estimation according to an embodiment of the present invention A diagram of a classical computer (HQC); Figs. 5A to 5C illustrate standard sampling and enhanced sampling quantum circuits and their corresponding likelihood functions of some embodiments of the present invention; Figs. 6A to 6B illustrate Fisher Information A graph of the dependence of various likelihood functions; FIG. 7 illustrates an operation for generating samples corresponding to an engineered likelihood function according to an embodiment of the present invention; FIG. 8 illustrates an embodiment of the present invention Algorithms implemented; Figs. 9A to 9B, Fig. 10, Figs. 11A to 11B and Fig. 12 illustrate various algorithms executed by embodiments of the present invention; Fig. 13 is a true overview of various embodiments of the present invention Graphs of functions and fitting probability functions; Figures 14A to 14B, 15A to 15B, 16A to 16B, 17A to 17B, and 18A to 18B show diagrams illustrating various implementations of the present invention Example of the performance of the graph; Figure 19 shows the example of the present invention

Factors; Figures 20A to 20B and Figures 21A to 21B show graphs illustrating the performance of various embodiments of the present invention; Figures 22A to 22B show diagrams of various embodiments of the present invention

Factors; Figures 23A to 23B and Figures 24A to 24B show graphs illustrating the performance of various embodiments of the present invention; Figures 25A to 25B show the performance of various embodiments of the present invention

Factors; Figure 26 shows a graph illustrating the relative accuracy of the target of the running time of various embodiments of the present invention; Figures 27 to 28 show quantum circuits implemented according to embodiments of the present invention; Figures 29A to 29B, FIGS. 30A to 30B, FIGS. 31A to 31B, and FIG. 32 show algorithms implemented according to an embodiment of the present invention; and FIG. 33 shows a true likelihood function and a fitting likelihood function according to an embodiment of the present invention.

400: operator

402: Quantum State

404: Block

406: Block

408: Block

410: Block

430: Hybrid Classical Quantum Computer (HQC)

432: Quantum Computer

434: Classical Computer

Claims (16)

A method for quantum amplitude estimation, which includes: selecting a plurality of quantum circuit parameter values by a classical computer to optimize a statistic and an accuracy improvement rate, the statistic estimates an observable

About a quantum state

An expected value of

; Apply a sequence of alternating first and second generalized reflection operators to one or more qubits of a quantum computer, so as to remove the one or more qubits from the quantum state

Transformed into a reflection quantum state, each of the first and second generalized reflection operators is controlled according to the corresponding one of the plurality of quantum circuit parameter values; about the observable

Measure the plurality of qubits in the reflection quantum state to obtain a set of measurement results; and update the statistics with the set of measurement results on the classical computer. The quantum amplitude estimation method according to claim 1, which further includes outputting the statistic after the update. According to the quantum amplitude estimation method described in claim 1, the statistic includes an average value. According to the quantum amplitude estimation method of claim 1, the accuracy improvement rate includes a variance reduction factor. According to the quantum amplitude estimation method of claim 1, the accuracy improvement rate includes an information improvement rate. According to the quantum amplitude estimation method described in claim 5, the information improvement rate includes one of a Fisher information improvement rate and an entropy reduction rate. The quantum amplitude estimation method according to claim 1, wherein the sequence of the first and second generalized reflection operators and the observable

Define an error of an engineered probability function. The quantum amplitude estimation method according to claim 1, further comprising iterating the selection, the application, the measurement, and the update. The quantum amplitude estimation method according to claim 1, which further includes: Update one of the accuracy estimates of the statistic on the classical computer and using the set of measurement results; and When the accuracy estimate is greater than a threshold value, the selection, the application, the measurement, and the update are iterated. The quantum amplitude estimation method according to claim 1, wherein the updating the statistic includes: Use the plurality of measured values to update a pre-distribution to obtain a post-distribution; and An updated statistic is calculated from the post-event distribution. The quantum amplitude estimation method according to claim 1, wherein the selection is based on the statistic and an accuracy estimate of one of the statistic. The quantum amplitude estimation method according to claim 11, wherein the selection is further based on a fidelity representing an error occurring during the application and the measurement. The quantum amplitude estimation method according to claim 1, wherein the selection uses one of coordinate rising and gradient descent. A computing system for quantum amplitude estimation, comprising: a processor; a quantum classical interface that communicably couples the computing system with a quantum computer; and a memory, the memory and the processing The memory is communicatively coupled, and the memory stores machine-readable instructions that, when executed by the processor, control the computing system to: (i) Select a plurality of quantum circuit parameter values to optimize a statistic The accuracy improvement rate of a quantity, the statistic estimates an observable

About a quantum state

An expected value of

, (Ii) controlling the quantum computer through the quantum classical interface to use a sequence of alternating first and second generalized reflection operators to bring one or more qubits of the quantum computer from the quantum state

Transformed into a reflection quantum state, each of the first and second generalized reflection operators is controlled according to the corresponding one of the plurality of quantum circuit parameter values, (iii) the quantum is controlled via the quantum classical interface Computer, with regard to the observable

Measure the plurality of qubits in the reflection quantum state to obtain a set of measurement results, and (iv) update the statistics with the set of measurement results. According to the quantum amplitude estimation computing system of claim 14, the memory stores additional machine-readable instructions, and the additional machine-readable instructions control the computing system to output the statistics when executed by the processor. The quantum amplitude estimation calculation system according to claim 14, which further includes the quantum computer.

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