TWI783297B - 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 Computational System for Quantum Amplitude Estimation

The present invention relates to a method and computing 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. Key application areas include chemistry and materials, biosciences and bioinformatics, logistics and finance. Quantum computing has received a surge of attention recently, in part due to a wave of improvements in the performance of off-the-shelf quantum computers. However, near-term quantum devices are still extremely limited in resources, thus hindering the deployment of quantum computers on practical problems of concern.

A recent wave of approaches to meeting the constraints of near-term quantum devices has come under intense scrutiny. Such methods include variational quantum eigensolvers (VQE), quantum approximate optimization algorithms (QAOA) and variants, variational quantum linear system solvers, other quantum algorithms using variational principles, and Quantum Machine Learning Algorithms. Despite innovations in such algorithms, many of these approaches have proven impractical for commercially relevant problems due to their high cost in the number of measurements and runtime. However, for moderately sized problem instances, methods that provide quadratic speedups in run time, such as phase estimation, require quantum resources well beyond the capabilities of recent devices.

The number of measurements required by hybrid quantum classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many practical value problems. Quantum algorithms (eg, quantum amplitude and phase estimation) that reduce this cost require error rates that are too low for near-term implementations. Embodiments of the invention include hybrid quantum-classical (HQC) computers, and methods performed by HQC computers, which exploit available quantum coherence to maximize the ability to sample noisy quantum devices, thereby Reduces the number of measurements and shortens run time compared to VQE. Such embodiments take inspiration from quantum methods, phase estimation, and more recently the "α-VQE" proposal, leading to general formulations that are robust to errors and do not require satellite qubits. The central object of this method is the so-called "Engineered Likelihood Function" (ELF), which is used to perform Bayesian inference. Embodiments of the present invention use the ELF formalism to enhance the quantum advantage in sampling as physical hardware transforms from the state of a noisy mesoscale quantum computer to that of a quantum error corrected computer. The technology accelerates a central component of many quantum algorithms, with applications in chemistry, materials, finance and other fields.

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

關於量子狀態

之預期值

。Embodiments of the present invention relate to a hybrid classical quantum computer (HQC) performing quantum amplitude estimation. Referring to FIG. 4 , there is shown a flowchart of an HQC 430 including both a quantum computer 432 and a classical computer 434 that implements a quantum amplitude estimation method according to one embodiment of the present invention. In block 404 performed by the classical computer 434, a plurality of quantum circuit parameter values are selected to optimize the rate of accuracy improvement of the statistic estimating the observable

About quantum states

expected value of

.

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 an embodiment, the statistic is a sample mean calculated from a plurality of values sampled from a random variable. In this discussion, the sampled values are obtained by quantifying the qubits of the subcomputer 432 . However, the statistic may alternatively be skewness, kurtosis, quantile, or another type of statistic without departing from the scope of the present invention. The statistic is the expected value

and can be biased or unbiased. Complex values can be modeled according to a probability distribution, in which case the statistic represents a parameter of the probability distribution. For example, the statistic may represent the mean of a Gaussian distribution, as described in more detail in Section 3.2 below.

Quantum circuit parameter values are real numbers that control how the quantum gate operates on the qubits. 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 an angle by which the state of one or more qubits is rotated in the corresponding Hibert space.

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

In some embodiments, the plurality of quantum circuit parameter values are selected using one of coordinate ascent and gradient descent. 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 block 406 performed by the quantum computer 432, a 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 state

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

and

are instances of the first and second generalized reflection operators, respectively. The operator introduced in Equation 19

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

Represents a complex number of quantum circuit parameter values. Sequences and Observables of the First and Second Generalized Reflection Operators

The bias of the engineered likelihood 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, also performed by quantum computer 432, the observable

A plurality of qubits in reflective quantum states are measured to obtain a set of measurement results. In block 410 performed by classical computer 434, statistics are updated with the set of measurements to obtain

estimates with higher accuracy.

The method may further include outputting statistics after the updating. Alternatively, the method may perform blocks 404 , 406 , 408 and 410 iteratively. In some embodiments, the method further includes updating the estimate of the accuracy of the statistic on the classical computer 434 and using the set of measurements. Accuracy estimates are calculated values of the accuracies (eg, variance) described above. In these embodiments, the method iteratively performs blocks 404, 406, 408, and 410 until the accuracy estimate falls below a threshold.

In some embodiments, in block 410, the statistics are updated by updating the ex-ante distribution with the plurality of measurements to obtain the ex-post distribution and computing the updated statistics from the ex-post distribution.

In some embodiments, the plurality of quantum circuit parameter values are selected based on the statistic and an estimate of the accuracy of the statistic. A plurality of quantum circuit parameter values may be selected further based on fidelity representative of errors occurring 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 Bézierian perspective results in Bézier phase estimation techniques that are more suitable than earlier proposals for noisy quantum devices capable of realizing finite depth quantum circuits. Using the above notation, the circuit parameters

and the goal is to estimate the operator

eigenvalue of

phase in

. It should be noted that the approximate 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。Common to many contexts other than Bayesian phase estimation, where

and

are Chebyshev polynomials of the first and second kind, respectively. This commonality yields Bayesian inference machines for tasks related to phase estimation, such as Hamiltonian characterization. Among the exponential advantages of Bayesian inference with Gaussian priors over other non-adaptive sampling methods is that by showing the expected post hoc variance

Established by exponential decay with the number of inference steps. This exponential convergence takes the amount required for each inference step to be

at the expense of quantum coherence. This scaling 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。Armed with the Bayesian phase estimation technique and the perspective of overlap estimation as an amplitude estimation problem, a Bayesian inference method for operator measurements that smoothly interpolates between standard sampling and phase estimation can be designed. This is proposed as "

-VQE", where the asymptotic scaling used to perform operator measurements is

, where the extremum

corresponds to the standard sampling pattern (usually implemented in VQE), and

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

and

between

value.

The lower the value, the deeper the quantum circuit required for Bayesian phase estimation. This achieves a compromise between quantum coherence and an asymptotic speed-up 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 where the Heisenberg limit for amplitude estimation can be achieved. In a previous study, the authors considered estimating the quantum state

parameters

task. Propose a parallel strategy in which using for generating

of the parametric circuit of

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

zoom to

status. Such amplification may also produce an likelihood function similar to that in Equation 1. It has been shown in previous studies that by randomizing quantum operations and Bayesian inference, information can be extracted with fewer iterations than classical sampling, even in the presence of noise. In quantum amplitude estimation, consider the number of iterations with varying

and the number of measurements

the circuit. A set of specially selected pairs

An approximate function is generated that can be used to extrapolate the amplitude to be estimated. The Heisenberg limit is shown for one particular likelihood function construction given by the authors. Both studies emphasize the ability to parameterize the likelihood function, making it attractive to study its performance on imperfect hardware.

1.2. main results

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 expected

The system and method, wherein the state

quantum circuit

ready to make

. Embodiments of the invention may use a series of quantum circuits such that when the circuits follow

When deepening with more repetitions of , it allows as

Approximate functions for polynomials of even higher degree. As described with specific examples in the next section, a direct consequence of this increase in polynomial degree is an increase in inference power, 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 quantum circuits and make the resulting approximate functions tunable. Embodiments of the present invention may optimize parameters during each inference step for maximum information gain. The following lines of insight stem from our efforts:

1. The role of noise and error in amplitude estimation: Previous studies have revealed the impact of noise on the output estimation of the likelihood function and the Hamiltonian spectrum. The disclosure herein investigates the above effects for the amplitude estimation scheme used by the embodiments of the present invention. The description herein demonstrates that noise and errors do not necessarily introduce systematic bias in the output of an estimation algorithm when they do not increase the runtime required to produce an output within a certain statistical error tolerance. Systematic bias in estimation can be suppressed by using active noise clipping techniques and calibrating for noise effects.

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 enhanced sampling schemes can outperform VQE in terms of sampling efficiency. Experimental results have also shed light on the perception that tolerating errors in quantum algorithm implementations with higher fidelity does not necessarily lead to better computational performance. In a particular embodiment of the invention, there is approximately

The optimal circuit fidelity of , at which the augmentation scheme yields the greatest amount of quantum speedup.

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 role of the tunability of the approximate function: The parameterized approximate function can be used in phase estimation or amplitude estimation routines. It is known that all current methods focus on the Chebyshev form of the approximate function (Eq. 1). For these Chebyshev-like functions (CLFs), in the presence of noise, there are parameters

Specific values ("dead points") of , for which CLF is used for inference efficiency and

significantly lower than the other values. Embodiments of the present invention can remove these dead spots by engineering an approximate function with an angle parameter via a generalized reflection operator that enables tunability.

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. Run-time model for estimation as error rate decreases: Previous work has shown that asymptotic cost scaling from VQE

to the phase estimated

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

The means come in target accuracy

Estimated run time on

to advance this line of thinking (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

function in

Calibration and

Interpolate between scales. Such bounds also allow analysis of embodiments of the present invention for quantum speedups as a function of hardware specifications such as number of qubits and two-qubit fidelity, and thus use real parameters of current and future hardware to Estimated runtime.

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 Subsequent chapters of this disclosure are organized as follows. Section 2 presents a concrete example of a scheme implemented according to one embodiment of the invention. Subsequent sections then detail the general formulation of this scheme according to various embodiments of the invention. Section 3 describes in detail the construction of general quantum circuits for realizing ELF, and analyzes the structure of ELF in both noisy and non-noisy environments. In addition to quantum circuit schemes, embodiments of the present invention also involve: 1) tuning circuit parameters to maximize information gain, and 2) updating related

Bayesian inference of the current confidence in the distribution of the true value of . Section 4 presents the heuristic algorithms used for both. Numerical results are presented in Section 5, comparing 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 implications of the revealed results from the broad perspective of quantum computing. plan Bayesian inference noise considerations Fully Tunable LF request attachment 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 scheme with related proposals appearing in the literature. Here, the list of features includes whether the quantum circuits used in the scheme require satellite qubits in addition to qubits that hold states used for overlap estimation, whether the scheme uses Bayesian inference, whether any noise resilience is considered, whether The initial state is the eigenstate, and the likelihood function is either fully tunable like the proposed ELF or restricted to the Chebyshev approximation 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 for estimating an operator

expected value of

. Quantum amplitude estimation methods provide demonstrable quantum speedups relative to specific computational models. However, in order to achieve the accuracy of the estimated

, the required circuit depth scaling in this method is

, making it impractical for near-term quantum computers. The variational quantum eigensolver uses standard sampling for amplitude estimation. Standard sampling allows for low-depth quantum circuits, making them more suitable for implementation on near-term quantum computers. In practice, however, the inefficiency of this approach makes VQE impractical for many problems of interest. This section introduces an upsampling method for amplitude estimation that may be used by embodiments of the present invention. This technique attempts to maximize the statistical power of noisy quantum devices. This method is described as starting from a simple analysis of a standard sample 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 subroutine of VQE estimates the amplitude with respect to the Baoli series. For decomposition into packet strings

Linear combinations of and "supposed states"

Hamiltonian, the expected value of the energy is estimated as a linear combination of estimates of the expected value of Pauli

(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：in

for

The (amplitude) estimate of . VQE uses a standard sampling approach to construct an estimate of the expected value for a proposed state, which can be summarized as follows:

Standard sampling:

並且量測算子

，接收成果

。1. Prepare

and measure the operator

, receiving the result

.

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

result.

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) available as time

To quantify the performance of this estimation strategy, the mean square error of the estimator of the function of

Time cost for each measurement. Since the estimator is unbiased, the mean square error is simply 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

, ensuring that the mean square error

The required running time of the 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 the energy estimate in VQE is the sum of the running times of the individual Baoli expected value estimation running times. For the problem of interest, this runtime may be prohibitively expensive, even when certain parallelization techniques are used. The source of this cost is the standard sampling estimation process for

Insensitivity to small deviations in the

The expected information gain is low.

次重複的估計過程的資訊增益

(6) Usually, the standard sampling can be measured by the Fisher information

The information gain of the 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) in

for the standard sampling

A set of repeated results. Fisher Information will approximate the function

identified as responsible for information gain. The lower bound of the mean square error of the (unbiased) 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 the Fisher information is additive with sample size, we get

,in

is a self-similar function

Extracted Fisher information for a single sample. Using the Kramer-Laou bound, a lower bound on the running time of the estimated process can be found

, (8)

It shows that embodiments of the present invention can augment Fisher information in order to reduce the runtime of the estimation algorithm.

，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 upsampling is to reduce the runtime of overlap estimation by engineering the likelihood function to increase the information gain rate. Consider the simplest case of enhanced sampling, which is illustrated in Figures 5A-5C. Embodiments of the invention may prepare hypothetical states for data generation

, apply the operation

, apply a phase flip about the proposed state, and then measure

. The phase inversion of the proposed state can be obtained by applying the reciprocal of the proposed circuit

, applying a phase flip about the initial state

, and then apply the proposed circuit

to achieve. 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

sub-Chebyshev polynomials. The disclosure herein refers to such approximate functions as Chebyshev approximate functions (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) To compare the Chebyshev approximation function for augmented sampling with the Chebyshev approximation function for standard sampling, consider

Case. here,

, so the 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 approximate function is in

The slope at is steeper than the slope of the standard sampling likelihood function. In each case the single-sample Fisher-Information evaluation is

(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 using enhanced sampling can reduce the number of measurements needed to achieve the target error by at least a factor of nine. As will be discussed later, embodiments of the present invention can be achieved by measuring

previously applied

Floor

To further increase Fisher Information. In fact, Fisher Information

with

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。Estimation schemes for converting augmented sampling measurements into estimates have not been described. One complication introduced by enhanced sampling is that when embodiments of the present invention collect measurement data, changes

s Choice. In this case, given the change from

A set of measurement results of the circuit,

and

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

For the information, the embodiment of the present invention can use Bayesian inference. Section 2 describes a specific example 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, one could try to point out that the comparison between standard sampling and augmented sampling is unfair, since in the case of standard sampling only

A query of , while in the enhanced sampling scheme using the pair

of the three queries. It appears that if one considers the likelihood function resulting from three standard sampling steps, a cubic polynomial form can also be obtained in the likelihood function. In fact, assuming three separate standard sampling steps are performed, the result

, and by the self-distribution

Sampling to classically produce binary results

. The likelihood function then takes the 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

can be changed by changing the distribution

Parameters tuned classically. More specifically,

,in

as a bit string

The Hamming weights. Assuming hope

is equal to the

. This hint

,

,

and

, which clearly exceeds the classical tunability of the approximate function in Equation 12. This is evidence that the approximate function produced by the quantum scheme in Equation 9 exceeds 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

with

grow linearly. This linear increase in the number of circuit layers and the quadratic increase in Fisher-Info results in a lower bound on the expected runtime,

, (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 is based on the assumption that there is a fixed and unbiased estimator

In the case of formula estimation strategy. In practice, operations implemented on quantum computers are subject to 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 simply to slow down the rate of information gain. When the number of circuit layers

When increasing, errors in quantum circuits accumulate. Thus, beyond a certain number of circuit layers, the gain (or reduction in runtime) with respect to Fisher information receives a diminishing return. Estimation algorithms can then attempt to balance these competing factors in order to optimize 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 error poses another problem for estimation. In error-free conditions, for all

,exist

The per-sample Fisher Information Gain is greater than or equal to the upsampled case of

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

The value of caused a sharp drop in Fisher Information. Such regions are referred to herein as estimated dead spots. This observation inspired the concept of engineering likelihood functions (ELFs) to enhance their statistical power. by putting

and

Operation generalized to generalized reflection

and

, embodiments of the present invention may use the rotation angle such that the information gain increases near such dead centers. Even for deeper upsampling circuits, engineering the likelihood function allows embodiments of the present invention to mitigate the effects of estimated dead points.

3. Engineered Likelihood Function

This section describes methods for engineering approximate functions for amplitude estimation that may be used by embodiments of the present invention. A quantum circuit for extracting samples corresponding to an engineered likelihood function is first described, and then a technique for tuning circuit parameters and performing Bayesian inference with the resulting likelihood function is described.

3.1. Quantum Circuits for Engineering Likelihood Functions

(18) Techniques for designing, implementing, and executing a program on a computer (e.g., a quantum computer or a hybrid quantum classical computer) for estimating 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。in

,in

The qubit operator,

The eigenvalues are

of

Qubit Hermitian operator, and introduce

To facilitate later Bayesian inferences. In constructing the estimation algorithms disclosed herein, it may be assumed that embodiments of the present invention are capable of performing the following primitive operations. First, embodiments of the present invention may prepare to compute the base state

, and applying a quantum circuit to it

, thus obtaining

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

Realize the positive operator

. Finally, embodiments of the present invention implement

measurement, the

Modeled to have individual achievement markers

projected value measure

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

,in

and

. follow the convention,

and

will be referred to in this paper as the

of

Eigenspace and state

The generalized reflection of , where

and

are the angles of these generalized reflections, respectively.

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) Embodiments of the present invention may use the free-attached (AF) scheme in Figure 7 (we refer to this scheme as "attachment-free" (AF) because this scheme does not involve any appended qubits. In Appendix A, we consider named A variant of the "attachment-based" (AB) scheme that involves an appendage qubit) quantum circuit to generate an engineered likelihood function (ELF) given the unknown to be estimated

case outcome

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

Thereafter, embodiments of the present invention may be applied to

generalized reflection

,

,

,

,

, thus changing the rotation angle in each operation

. For your convenience,

In this paper will be referred to as the circuit's first

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

。in

is a vector of tunable parameters. Finally, embodiments of the present invention may perform projection measurements on this state

, so as to receive the result

.

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, the generalized reflection

and

Ensuring the quantum state for any

are kept in the two-dimensional subspace

in (to ensure

is two-dimensional, assuming

,that is,

or

). Assume

for

Orthogonal to

state (unique, phase-dependent), that is,

(20)

及

分別寫為

及

。To aid in the analysis, consider this two-dimensional subspace as qubits, whereby

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

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

, can be

rewrite as

(twenty one)

及

重寫為

(22) and will generalize reflect

and

rewrite 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) in

is a tunable parameter. Subsequently, by

The 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, while

,

and

depending on the unknown

depends. As a result, it is easier 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 Likelihood Functions (i.e., Measuring Outcomes

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

and observable

depends.

(25) Precisely, the engineered likelihood function is

(25)

(26) in

(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 bias of the likelihood function (from now on, we will use

and

to represent

and

about

derivative of ). In particular, if

, then get

. That is, this

The error of the approximate function of is

of (the first category)

sub-Chebyshev polynomials. For this reason, the

The likelihood function of will be referred to as the Chebyshev likelihood function (CLF) in this paper. Section 5 explores the performance gap between CLF and general-purpose 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 models into the approximate function.

In practice, the establishment of the noise model may utilize a procedure for calibrating the likelihood function for the particular device used. Regarding Bayesian inference, the parameters of this model are called redundant parameters; the target parameters do not directly depend on the redundant parameters, but the redundant parameters determine the degree of correlation between the data and the target parameters, so 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 effects of model errors are 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) Assume that each circuit layer

The noisy version achieves the target operation and a fully depolarized channel acting on the same input state (the depolarization model assumes that the gates containing each layer are random enough to prevent the systematic accumulation of coherence errors. There is a more exact 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) in

This is the fidelity of this layer. Under the composition of such imperfect operations,

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 for , and then for

imperfect measurement. In the context of randomized benchmarks, such errors are known as state preparation and measurement (SPAM) errors. The embodiment of the present invention can also model the SPAM error by the depolarization model, so that

noisy preparation system

, and make

POVM

. Combining SPAM error parameters into

, the model of the approximate function with noise is obtained

(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。in

is the fidelity of the entire process used to generate the ELF, and

is the bias of the ideal likelihood function, as defined in equation (26) (henceforth, this disclosure will use

To represent

derivative of ). It should be noted that the overall effect of noise on ELF is that noise rescales the bias

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

，對於

的一些(複數值)函數

及

，

可寫為

， (30) Before proceeding to discuss Bayesian inference by ELF, it is worth mentioning the following property of the engineered likelihood function, as it will come into play 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) then multivariate function

is triangular-multilinear, and will

and

known as

about

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。then multivariate function

is trigonometrically-multiquadratic, and will

,

and

known as

about

The cosine-sine-bias-decomposition (CSBD) coefficient function of . The concepts of trigonometric-multilinear and trigonometric-multiquadratic also extend naturally to linear operators. That is, if each term of a linear operator (arbitrarily written) is trigonometric-multilinear (or polyquadratic) in one set of variables, then the operator is trigonometric-multilinear in the same variables (or trigonometric - multi-quadratic). Now, equations (22), (23) and (24) imply

for

The triangular-multilinear operator of . Then, from equation (26), we get,

for

The trigonometric-multiquadratic function of . Furthermore, it is revealed that the

time assessment

about any

function of the CSBD coefficients, and this significantly facilitates the tuning circuit angle used in Section 4.1

The structure of the algorithm.

3.2 Bayesian Inference via Engineered Likelihood Functions

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 a model of the (noisy) engineered likelihood function is in place, the parameters used to tune the circuit will be described

And an embodiment of the present invention that performs Bayesian inference with the resulting approximate 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 pair to estimate

Begin with a high-level overview of an embodiment of the algorithm. For convenience, such embodiments can

valid instead of

efficient. Embodiments of the present invention may use a Gaussian distribution to represent

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

the true value of . Embodiments of the present invention can be obtained from

Start with an initial distribution of (which can be generated by standard sampling or domain knowledge), and transform it into

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

current knowledge) maximizing results from measurements

The circuit parameters of the information gain

. Then, by optimizing the parameters

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

. Finally, embodiments of the present invention can be implemented by using Bayesian law, with

to update the condition

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

The final distribution of is transformed into

The final distribution of , and the mean of this distribution is set as

final estimate of . See Figure 8 for a conceptual diagram of this algorithm.

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 inference process, embodiments of the present invention use a Gaussian distribution to track

The credibility of the value of . That is, at each round, for some prior mean

and ex ante variance

,

with prior 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

Afterwards, the embodiment of the present invention can be calculated by using Bayesian rule

The post hoc distribution of :

, (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 regularization factor or model evidence is defined as

(recall

Fidelity of the process used to generate the ELF). Although the true posterior distribution will not be Gaussian, embodiments of the present invention may approximate it to be so. Following the previous approach, embodiments of the present invention can use the same mean and variance (although embodiments of the present invention can directly compute the post hoc distribution by definition

, but this method is time-consuming because it involves numerical integration. Instead, embodiments of the present invention can speed up this process by exploiting specific properties of engineered likelihood functions. See Section 4.2 for more details) to replace the true posterior with a Gaussian distribution, and set it to be

beforehand. Embodiments of the present invention may repeat this measurement and Bayesian update procedure until

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) Since the algorithm mainly

works, and we finally agree

interested, embodiments of the invention are available at

and

to convert between estimated values. This process proceeds as follows. suppose in bout

,

The prior distribution of

,and

The prior distribution of

(Notice,

,

,

and

are random variables because they depend on time

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

and

The estimates for

and

. given

Distribution

, the embodiment of the present invention can calculate

average value

and variance

, and will

set as

Distribution. This step can be done analytically, as if

, then

, (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 calculate

average value

and variance

(Will

clamped 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 embodiments of the present invention can approximate it as such, and this has been found to have negligible impact 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

。For tuning circuit angle

The method can be realized by the embodiments of the present invention as follows. Ideally, the angles can be chosen carefully so that as

increase,

estimator of

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

The squared bias of can be smaller than its variance, that is,

,in

for

the true value of .

Variation of

often close

The variance of , that is,

with high probability. Combining these facts, one learns that

with high probability. Therefore, embodiments of the present invention may instead find the minimum

Variation of

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, suppose

with prior distribution

. After receiving the measurement results

after,

The expected post hoc variance of is

， (36) =

, (36)

(37) in

(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) in

is the bias of the ideal likelihood function, as defined in equation (26), and

is the fidelity of the process used to generate the likelihood function. Now, introducing the quantity used to engineer the likelihood function, and referred to in this paper as the variance reduction factor,

. (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. Furthermore, in order to quantify

The growth rate (per time step) of the inverse variance of , the following quantity 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。in

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

for

monotonic function of , where

. Therefore, when the number of circuit layers

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

to maximize

(about

). Additionally, when

very young,

Approximate with

proportional to, that is,

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

is proportional to the number of times the proposed set is called in the circuit, thus setting

, where time is in units of the proposed 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, will reveal the method to find for a given

,

and

maximize the variance reduction factor

parameters

technology. Often, this optimization problem becomes intractable. Fortunately, in practice, embodiments of the present invention can assume

The ex ante variance of

very small (e.g., at most

), and in this case,

Likelihood function

exist

is approximated by the Fisher information below, that is,

,

(42)

(43)

(44) in

(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) is a double-outcome approximate function

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

, an embodiment of the present invention optimizes Fisher Information

, which can be efficiently accomplished by embodiments of the present invention using the algorithm in Section 4.1.1. Furthermore, the fidelity of the process used to generate the ELF

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 such cases, embodiments of the present invention may optimize

, which is related to the approximate function

exist

is proportional to the slope below and this task can be efficiently accomplished by embodiments 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, embodiments of the present invention predict with

when growing

estimator of

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

for fixed. when

, this gives

.

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

when, get

,

,

and

with high probability, where

and

respectively

and

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

,

. (46)

，

， (47) Therefore, by 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) in

. because

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

Approximate, predictive 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 of should be approximately

, (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。in

about

get optimized. We will compare this rate with the

The empirical growth rate of the inverse MSE is compared.

4. An Efficient Heuristic Algorithm and Bayesian Inference for Tuning Circuit Parameters

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

, and describe how embodiments of the present invention can efficiently perform Bayesian inference with the resulting likelihood function.

4.1. Efficient Maximization of Proxies for Variation Reduction Factors

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) According to the embodiment of the present invention, it is used to tune the circuit angle

The algorithm is based on maximizing the variance reduction factor

The two agents of (likelihood function

Fisher information and slope). All such algorithms require the evaluation of bias

and its derivative

about

An efficient procedure for the CSBD coefficient function of , where

. Recall that it was shown in 3.1 that the bias

with

for triangular-multiquadratic. That is, for any

,exist

The function

,

and

, so that

(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。with

is also trigonometric-polyquadratic, where

,

,

respectively

,

,

about

bias. The result is that given

and

,

, available at

time calculation

,

,

,

and

each of the

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 the

prove. See Appendix C.

4.1.1. Fisher information for maximizing the likelihood 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) Embodiments of the present invention may be implemented to maximize the likelihood function

at a given point

(that is,

One or more of the two algorithms for the Fisher information of the prior average of . Suppose the goal is to find the maximization of

. (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 same

Steps proportional to the gradient of , until the convergence criterion is met. Specifically, let

for iterative

The parameter vector at . Embodiments of the present invention can update it 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) in

schedule for the step size (in the simplest case,

is a constant. However, for better performance, it may be desirable to

). This requirement calculates

about each

The partial derivative of , this calculation can be done as follows. Embodiments of the present invention first use the procedure in Lemma 1 for each

calculate

,

,

,

,

and

. this get

, (54)

, (55)

, (56)

; (57)

關於

的部分導數：

(58) After knowing these quantities, the embodiments of the present invention can be calculated as follows

about

Partial derivatives of :

(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。Embodiments of the present invention may be directed to

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

. Each iteration of the algorithm costs

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

depends. See Algorithm 65 for more details.

，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 system is based on coordinate ascent. Unlike gradient ascent, this algorithm does not require a step size and allows each variable to change substantially in a single step. Therefore, the second algorithm can converge faster than the previous algorithm. Specifically, an embodiment of the present invention implementing this algorithm can start from a random initial point and sequentially maximize the objective function along the coordinate direction

, until the convergence criterion is satisfied. in each round

After steps, solve the following for coordinates

The univariate optimization problem for :

, (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。in

,

,

,

,

,

By the procedure in Lemma 1, the

calculated in time. This univariate optimization problem can be solved by standard gradient-based methods, and the

Set as its solution. against

Repeat this procedure. This algorithm produces the sequence

,

,

,

, so that

. That is, with

increase,

The value of is increasing 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

。Embodiments of the present invention may be implemented to maximize the likelihood function

at a given point

(that is,

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

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 Fisher information maximization, the algorithm for slope maximization is also based on gradient ascent and coordinate ascent, respectively. Both invoke the procedure in Lemma 1 for a given

and

Evaluate

,

and

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

about

, while the algorithm based on coordinate ascent uses the above quantities to directly update

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

4.2. Approximate Bayesian Inference via Engineered Likelihood Functions

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。parameters used in tuning the circuit

With the algorithm for , now in place, it is now described how Bayesian inference can be efficiently performed with the resulting approximate function. Embodiments of the present invention can receive measurement results

Calculate directly after

The post-event mean and variance of . But this method is time-consuming because it involves numerical integration. Embodiments of the present invention can greatly speed up this process by exploiting specific properties of engineered likelihood functions.

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) suppose

with prior distribution

,in

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

. Embodiments of the present invention can find, maximize

(or

) parameters

satisfy the following properties: when

near

when, 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, embodiments of the present invention can be implemented by

A sine function in this region to approximate

. Figure 13 illustrates one such example.

及

：

(68) Embodiments of the present invention can find the best fit by solving the following least squares problem

and

:

(68)

。此最小平方問題具有以下解析解：

(69) in

. This least squares problem has the following analytical solution:

(69)

。 (76) in

. (76)

Figure 13 shows examples of true 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 obtains the best

and

, which can then be approximated by the mean and variance for

The post-event mean and variance of

, (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, suppose

in round

ex ante distribution

. Assume

to measure results, and

Best fit parameters for this round. Then, an embodiment of the present invention can be approximated by

The post-event mean and variance of

(78)

(79)

設定為該回合之

的事前分佈。Thereafter, embodiments of the present invention may proceed to the next round, whereby

set to the round

the prior distribution of .

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

when, that is,

, the difference between the true and fitted likelihood functions can be large. However, due to the prior distribution

with

decays exponentially, such

to calculate

The contribution of post-event mean and variance of . Therefore, equations (78) and (79) give

Highly accurate estimates of the post-event mean and variance of , and its error has negligible impact on the performance of the entire algorithm.

5. Simulation results

This section describes specific results of modeling Bayesian inference with engineered likelihood functions for amplitude estimation. These results demonstrate specific advantages of certain engineered approximation functions over non-engineered approximation functions, and the effect of circuit depth and fidelity on the performance of certain engineered approximation functions.

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, it is assumed that the realization

and perform projection measurements

takes longer to implement than

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

, the time cost of the inference round is roughly

,in

for

time cost (it should be noted that

layer circuit using

and

Second-rate. For simplicity, assume that in the subsequent discussion

takes a unit of time (i.e.,

). Furthermore, it is assumed that there are no errors 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) Assumptions aimed at estimating expected values

. Assume

for

at time

estimate of . It should be noted that

itself is a random variable, since it depends on time

Depends on the history of random measurement results. by

The root mean square error (RMSE) to measure the performance of the scheme is given by

. (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 along with

increase

how fast the decay, the schemes include Affiliation Based Chebyshev Likelihood Function (AB CLF), Affiliation Based Engineered Likelihood Function (AB ELF), No Affiliation Chebyshev Likelihood Function (AF CLF), and Unattached Engineered Likelihood Functions (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

Parse the formula. To estimate this quantity, embodiments of the invention may perform an inference process

times, and collect

of

samples

,

,

,

,in

for

on the

wheel (of which

) at time

estimated value at . Subsequently, the embodiment of the present invention can use the amount of

(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 this was found to give 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。Embodiments of the present invention may use coordinate ascent-based algorithms 2 and 6 to optimize circuit parameters for the unattached case and the attached-based case, respectively

. This shows that Algorithms 1 and 2 produce solutions of equal quality, and so do Algorithms 5 and 6. Therefore, if instead using gradient ascent 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。For Bayesian updating by ELF, embodiments of the present invention may use the methods in Section 4.2 and Appendix A.2 to calculate the case without attachment and the case based on attachment, respectively

The post-event mean and variance of . In particular, during the sinusoidal fitting of the ELF, embodiments of the present invention may set the

(that is,

contained in

Evenly distributed in

points). It has been found that this is sufficient to obtain a high quality sinusoidal fit of the ELF.

6. Modeling Algorithmic Performance with Noise

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 models for runtime achieved when scaling to larger systems and running on devices with better gate fidelity

The target root mean square error of the estimated value of is needed. 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 inverse variance rate expression (see equation (40)). Half are due to conservative estimates that the variance and squared bias contribute equally to the mean square error (simulations from previous sections show that the squared bias tends to be smaller than the variance). The second assumption is an empirical lower bound on the variance reduction factor, which is motivated by numerical studies of Chebyshev's approximate functions.

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。right about

The MSE of the estimated value was analyzed. Subsequently, the MSE of this estimate is transformed into an MSE with respect to

estimated value. The strategy will be, for the rate expression in equation (40)

Integrate the upper and lower bounds of , to obtain bounds on 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 aid analysis, substitute

, and by introducing

and

so that

To reparameterize how 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 bounds of this rate expression are based on the discovery of the Chebyshev approximate function, where

. Since the Chebyshev approximate functions are a subset of the engineered approximate functions, the lower bound of the Chebyshev performance gives the lower bound of the ELF performance. We guess that in the case of ELF this rate is capped at a small multiple (eg, 1.5 times) of the cap 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 upper bound is established as follows. for fixed

,

and

, it can be shown that (for the Chebyshev approximate function, the variance reduction factor can be expressed as

(if only

). Subsequently,

implied

) The variation reduction factor reaches the maximum value

, Dasein

appears everywhere. This expression is less than

,Its

reach the maximum

. Therefore, the factor

cannot exceed

. Combining all of the above, for a fixed

,

and

, the upper limit of the maximum rate is

. This follows from the fact that:

with

is monotonic, and

exist

is maximized. In practice, embodiments of the present invention may use the maximization of the inverse variance rate

value. by discrete

The rate of achievement cannot exceed when

Values obtained when optimizing the above upper bound on continuous values of . This optimal value is for

accomplish. By evaluating at this optimum value the

to define

,

, (82)

。 (83) which gives an upper bound on 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) Embodiments of the present invention do not have an analytical lower bound on Chebyshev's approximate power. An empirical lower bound can be established based on numerical inspection. for any fixed

, the inverse variance rate is at

points

is zero. Due to the rate for all

are zero at such endpoints,

The overall lower bound of is zero. However, poor performance of the inverse anomaly ratio near these endpoints is not a concern. When estimating from

converted to

, the information gain near these endpoints actually tends to large values. To establish useful boundaries, set the

limited to

Inside. In numerical tests (for

50000 values, from

to

Of

A uniform grid of values is searched, where

is the optimized value used to obtain Equation 82, and

and

exist

In the range. for each

Yes, find the maximum inverse variance rate (for

) is the minimum

. for all inspections

Yes, this worst case rate is always at

and

between , where the minimum value is found to be

), it is found that for all

, there is always a condition that makes the inverse anomaly rate higher than the upper limit

times

s Choice. Combining these together, we 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

for continuous,

and

A specific value can cause the

negative best

. Therefore, these results are only

(which ensures

) is applicable. The model is expected to be in the large noise regime (i.e.,

) fails.

，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 bounds, that is,

, remembering that the upper and lower bounds are the smaller 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) Assume the inverse variance in time at the inverse variance rate

The rate given by the captured differential quotient expression continues to grow. Make

Representing this contravariance, the rate equation above can be recalculated as

The differential equation of

(85)

，微分方程式變為  

(86) Via this expression, both the Heisenberg limit behavior and the 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 growth of the inverse squared error

. This is a characteristic of the Heisenberg limit type. for

, the rate is close to a constant,

. (87)

，此指示散粒雜訊限值型態。This pattern produces a linear increase in the inverse squared error

, which indicates the shot noise limit type.

，將此等界限重新表達為，  

(88) To make integration tractable, the rate expressions can be replaced by integrable upper and lower bound expressions (used in conjunction with the previous bounds). Make

, reformulating these bounds 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 viewing time as

function of and integrated, a lower bound on the running time can be established from the upper bound,

(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, a lower bound can be used to establish an upper bound on runtime. The following assumptions are introduced here. In the worst case, the MSE of the phase estimation

is twice the variance (that is, the variance is equal to the bias), so the variance must be half of the MSE:

. In the best case, assume that the bias in the estimate is zero, and set

. Combining these bounds with the upper and lower bounds of equation (84) yields bounds on the estimated running time as a function of the target MSE,

(91)

。in

.

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 estimate can be

Convert back into amplitude estimate

. The MSE for the amplitude estimate can be compared to the MSE for the phase 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) where the distribution of the estimator is assumed for

Peaked sufficiently to ignore higher-order terms. This causes

, which can be substituted into the above bound expression, for

The same is true. Run-time scaling in the low-noise and high-noise limits can be established by removing the estimator subscripts (since they only contribute constant factors),

(94)

It is observed that the Heisenberg limit calibration and the shot noise limit calibration are each recovered.

。These bounds are obtained using properties of the Chebyshev approximate function. As has been shown in previous sections, by engineering the likelihood function, the estimation runtime can be reduced in many cases. Motivated by the numerical discovery of the variance reduction factor of the engineered likelihood function (see e.g. Fig. 19), we conjecture that using the engineered likelihood function increases the worst-case inverse variance 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) To give more meaning to this model, it is broken down by the number of qubits

and two-qubit gate fidelity

express. Consider estimating the packet string

about status

task of expected value. suppose

is very close to zero, making

. Assume

The two-qubit gate depth for each of the layers is

. Model the total layer fidelity as

, where the error caused by the single-qubit gate has been neglected. From this, get

and

. Combining these, we 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 runtime (in seconds) as a function of two-qubit gate fidelity. To achieve quantum supremacy, the expected problem example would require approximately

logical qubits, and requires a two-qubit gate depth of approximately the number of qubits

. Furthermore, the expected target accuracy

will require approximately

to

. Run-time models measure time in terms of the duration of a proposed circuit. To convert this time to seconds, assume that each layer of the two-qubit gate will take time

s, which is an optimistic assumption for today's superconducting qubit hardware. Figure 26 shows this estimated runtime as a function of two-qubit gate fidelity.

Reducing run times to the required two-qubit gate fidelity in the practical region will likely require error correction. Performing quantum error correction requires additional overhead, increasing such runtimes. When designing a quantum error correction protocol, it is important to estimate that the increase in runtime does not outweigh the improvement in gate fidelity. The proposed model gives a means to quantify this tradeoff: when incorporating useful error corrections, the product of gate fidelity and (error corrected) gate time should decrease. In practice, there are many subtleties that should be taken into account in order to make a more rigorous statement. Such nuances include accounting for gate fidelity variations among the gates of the circuit, as well as the time cost of varying gates of different types. 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. Affiliate based programs

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

is a tunable parameter.

(96) Assuming that the circuit in Figure 27 has no noise, the engineering approximate function is given by

(96)

(97) in

(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 bias of the approximate function. It turns out that most of the arguments in Section 3.1 still hold in the subordination-based case, just with

replaced

. Therefore, the same markup as before will be used (eg,

,

,

,

,

,

),Unless otherwise indicated. In particular, when considering errors in the circuit in Fig. 27, the noisy approximate function 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。in

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

and

There is a difference between the former because the former

is triangular-multiquadratic, and the latter follows

For triangular-multilinear.

，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 Bayesian inference is performed by the resulting ELF in a similar manner as in Section 3.2. In fact, the argument in Section 3.2 still holds in the case of subordination, but only with

replace

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

, thus using

replace

. can be shown,

(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

exist

The Fisher information and the slope are the variance reduction factors

of two agents. because

Direct optimization of is usually difficult, and parameters will be tuned by optimizing such proxies instead

.

A.1. Efficient Maximization of Proxies for Variation Reduction Factors

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 rendering factor for maximizing the variance reduction factor

The two agents of (likelihood function

An efficient heuristic algorithm for the Fisher information and slope of . All such algorithms evaluate bias using the following procedure

and its about

derivative of

,in

.

A.1.1. Evaluation 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) because

exist

is triangular-multilinear (for any

), there exists

is a trigonometric-multilinear function

and

, so that

. (101)

(102) Subsequently,

(102)

亦為三角-多線性的，其中

及

分別為

及

關於

的導數。with

is also triangular-multilinear, where

and

respectively

and

about

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 an efficient program for a given

and

Evaluate

,

,

as well as

. As a result, such tasks can be found in

completed within 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 the

，

，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 markup is introduced. Assume

,

,in

. In addition, set

,in

. It should be noted that if

is an even number, then

. Then, define if

,otherwise

.

(103) With this notation, it can be shown that

(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。for a given

and

Evaluate

,

,

and

, considered separately

is an even number and

case of odd numbers.

為偶數，其中

。在此情況下，

。使用如下事實  

(105)  

(106)  

(107) • Case 1:

is an even number, where

. In this situation,

. Use the following facts

(105)  

(106)  

(107)

(108) • get

(108)

， (109)

。 (110) • in

, (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 at

time calculation

and

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

and

. This procedure costs only

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

,get

(111)

(111)

(112)

， (113)

(114)

。 (115) Assume

(112)

, (113)

(114)

. (115)

(116) Then, equation (111) yields

(116)

(117)

(117)

， (118)

, (118)

， (119) which caused

, (119)

(120)

。 (121) in

(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 by means of standard dynamic stylization techniques in the total

Compute the following matrix in time:

及

，其中

；

and

,in

;

及

，其中

；

and

,in

;

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, by using equations (113) and (115) to calculate

and

. Thereafter, calculated by equations (120) and (121)

and

. Overall, the program consumes

time.

為奇數，其中

。在此情況下，

。使用如下事實

(122)

(123)

(124) 1. Case 2:

is an odd number, where

. In this situation,

. Use the following facts

(122)

(123)

(124)

， (125) 2. get

, (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 at

time calculation

and

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

and

. This procedure costs only

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 which

,get

。 (128)

. (128)

(129)

(130)

(131)

(132) Assume

(129)

(130)

(131)

(132)

(133) Then, equation (128) gives

(133)

(134)

(134)

(135)

(135)

(136) which caused

(136)

(137)  

(138) in

(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 by means of standard dynamic stylization techniques in the total

Compute the following matrix in time:

及

，其中

；

and

,in

;

及

，其中

；

and

,in

;

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, by using equations (130) and (132) to calculate

and

. Thereafter, calculated by equations (137) and (138)

and

. Overall, the program consumes

time.

W

A.1.2. Fisher information for maximizing the likelihood 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) Proposed for maximizing the likelihood function

at a given point

(that is,

The two algorithms of the Fisher information of the prior average value of . That is, the goal is to find the one that maximizes

(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。These algorithms are similar to Algorithms 1 and 2 for Fisher Information Maximization in the sense that they are also based on gradient ascent and coordinate ascent respectively. The main difference is that the procedure in Lemma 2 is now invoked for the given

and

Evaluate

,

,

and

, and then use them to compute

about

Partial derivatives of (in gradient ascent) or for

Defines a univariate optimization problem (in 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

。Also proposed for maximizing the likelihood function

at a given point

(that is,

The two algorithms of the slope of the prior mean value). 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。These algorithms are similar to Algorithms 3 and 4 for slope maximization in the dependency-based case, in the sense that these algorithms are also based on gradient ascent and coordinate ascent, respectively. The main difference is that the procedure in Lemma 2 is now invoked for the given

and

Evaluate

and

. Then, use these quantities to calculate

about

Partial derivatives of (in gradient ascent) or direct update of

The value of (in coordinate ascent). These algorithms are formally described in Algorithms 7 and 8.

A.2. Approximate Bayesian Inference via Engineered Likelihood Functions

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。parameters used in tuning the circuit

With the algorithm for , now in place, we now briefly describe how Bayesian inference can be efficiently performed with the resulting approximate function. This idea is similar to that 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) suppose

with prior distribution

,in

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

. Discovery maximization

(or

) parameters

satisfy the following properties: when

near

when, 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) in

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

(149)

。 (156) in

. (156)

Figure 34 illustrates examples of true and fitted likelihood functions.

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 the best

and

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

The post-hoc mean and variance of

(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, suppose

in round

ex ante distribution

. Assume

become a measurement result, and set

Best fit parameters for this round. Then, it is approximated by

The post hoc mean and variance of

, (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。Afterwards, proceed to the next round, the

set to the round

the prior distribution of . The approximation errors caused by equations (158) and (159) are small, and for the same reasons as the no-attachment case, they have negligible impact on the performance of the overall algorithm.

c. Lemma Proof

、

、

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 markup is introduced. Assume

,

,

and

,in

,and

. In addition, set

,in

. It should be noted that if

is an odd number, then

. Then, define if

,otherwise

.

   

，    

  By this mark,

,

求導產生  

  In addition, regarding

Derivation produces

 

，  

,

  in

關於

的導數，其中  

  for

about

derivative of , where

關於

的導數。隨後，for

about

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. suppose

,

and

Hibbert space

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

 

，  

,

  and

 

 

 

 

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

Derivation produces

 

 

 

 

 

 

 

 

及

評估

、

、

、

、

及

，分開考慮

為偶數的情況及

為奇數的情況。for a given

and

Evaluate

,

,

,

,

and

, considered separately

is an even number and

case of odd numbers.

為偶數，其中

。在此情況下，

，並且

。隨後獲得  

   

，   • Case 1:

is an even number, where

. In this situation,

,and

. subsequently obtained

,

   

   

  • in

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 at

time calculation

,

and

. Then, calculate

,

and

. This procedure costs only

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 fact

, any of which

,get

 

 

 

 

   

  Then, get

   

   

   

   

，    

   

，    

。   in

,

,

.

   

，   At the same time, get

,

   

   

。   in

.

  Combining the above facts, we get

in

 

 

 

 

 

 

 

 

 

。  

.

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 by means of standard dynamic stylization techniques in the total

Compute 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

,in

. Thereafter, calculate

,

and

. Overall, the program consumes

time.

為奇數，其中

。在此情況下，

，並且

。隨後，得到  

   

，   5. Case 2:

is an odd number, 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 at

time calculation

,

and

. Then, calculate

,

and

. This procedure costs only

time.

、

及

。使用上式及事實

，其中

，獲得Next, describe how to calculate

,

and

. Use the above formula and fact

,in

,get

 

。  

.

Then, get

，

，

,

,

，    

，    

   

，    

，    

   

，    

，    

   

   

。   in

,

,

,

,

,

,

.

   

，   At the same time, get

,

，

，

。 in

,

,

.

Combining the above facts, we get

in

，

,

。

.

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 by means of standard dynamic stylization techniques in the total

Compute the following matrix in time:

、

、

、

、

；

,

,

,

,

;

及

，其中

；

and

,in

;

及

，其中

。

and

,in

.

、

及

。，其中

此後，計算

、

及

。總體而言，此程序耗費

時間。Then, calculate

,

and

. ,in

Thereafter, calculate

,

and

. Overall, the program consumes

time.

It should be understood that while the invention has been described above in terms of specific embodiments, the foregoing embodiments are provided as illustrative only, and do not limit or define the scope of the invention. Various other embodiments (including but not limited to the following embodiments) are also within the scope of the patent application. For example, elements and components described herein may 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 in accordance with the present disclosure. Generally, the basic data storage unit in quantum computing is a quantum bit (quantum bit or qubit). A qubit is the quantum computing analog of a bit in a classical digital computer system. A classical bit is considered to occupy one of two possible states corresponding to a binary digit (bit) of 0 or 1 at any given point in time. Qubits, by contrast, are implemented in hardware by a physical medium with quantum mechanical properties. Such a medium that physically embodies a qubit may be referred to herein as a "physical realization of a qubit," a "physical embodiment of a qubit," "a medium embodying a qubit," or similar terms, Or simply "qubits" for ease of explanation. 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 medium embodying the qubit.

Each qubit has an infinite number of different potential quantum mechanical states. When physically measuring the state of a qubit, the measurement results from one of two different underlying states resulting from the state resolution of the qubit. Thus, 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 four orthogonal fundamental states; and three qubits Can be in any superposition of 8 orthogonal base 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 outcome of a given measurement. A qubit with a two-dimensional quantum state (ie, with two orthogonal basis 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, measurements of the qubit result from one of d different underlying states of the qubit's state resolution. Any reference herein to qubits should be understood to refer more generally to d-dimensional qubits (d is any value).

Although the specific description of qubits herein may describe such qubits in terms of their mathematical properties, each such qubit may be implemented in a physical medium in any of a variety of different ways. Examples of such solid media include superconducting materials, trapped ions, photons, optical cavities, individual electrons trapped within quantum dots, point defects in solids (e.g., phosphorus donors in silicon or nitrogen vacancy centers in diamond) ), molecules (e.g., alanine, vanadium complexes), or any of the above that exhibit qubit behavior (that is, include quantum states and transitions between them that can be controllably induced or detected) A summary of any of the items.

For any given medium that implements qubits, any of various properties of the medium can be chosen to implement the qubits. For example, if electrons are chosen to implement a qubit, then the x-component of their spin degree of freedom can be chosen as a property of such electrons to represent the state of such qubits. Alternatively, the y-component or the z-component of the spin degree of freedom can be chosen as a property of such electrons to represent the state of such qubits. This is just a specific example of the general feature that for any physical medium that can be chosen to implement a qubit, there can be multiple physical degrees of freedom that can be chosen to represent 0 and 1 (e.g. x in the electron spin example , y and z components). For any particular degree of freedom, the physical medium can be controllably placed in a superposition state, and can then be measured with the chosen degree of freedom to obtain a readout of the qubit value.

A particular implementation of a quantum computer (called a gate-model quantum computer) involves quantum gates. In contrast to classical gates, there are an infinite number of possible single-qubit quantum gates that change the state vector of a qubit. Changing the state of a qubit state vector is often referred to as a single-qubit rotation, and may also be referred to herein as a state change or a single-qubit quantum gate operation. Rotations, state changes, or single-qubit quantum gate operations can be mathematically represented by monoregular 2X2 matrices with complex elements. The rotation corresponds to the rotation of the qubit state in its Hibert space, which can be conceptualized as the rotation of the Bloch sphere. (As is well known to those skilled in the art, a Bloch sphere is a geometric representation of the space of pure states of a qubit.) A multi-qubit gate changes the quantum state of a group of qubits. For example, a two-qubit gate rotates the state of two qubits as a rotation of the two qubits in four-dimensional Hibert space. (As is well known to those skilled in the art, a Hibbert space is an abstract vector space for structures that allow the measurement of lengths and angles for handling inner products. Furthermore, a Hibbert space is complete: there are sufficient constraints on the space to allow the use of calculus techniques.)

A quantum circuit can be specified as a sequence of quantum gates. As described in more detail below, as used herein, the term "quantum gate" refers to the application of a gate control signal (defined below) to one or more qubits so as to cause those qubits to undergo a specific physical transformation and thereby This realizes logic gate operation. To conceptualize a quantum circuit, the matrices 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 matrix of complex numbers representing the same overall state change of n qubits. Therefore, a quantum circuit can be expressed as a single resulting operator. However, designing quantum circuits in terms of constituent gates allows the design to be consistent with a set of standard gates, thus enabling easier deployment. A quantum circuit thus corresponds to the design of actions to be taken on the physical components of a quantum computer.

A given variable quantum circuit can be parameterized in an appropriate device-specific manner. More generally, quantum gates constituting a quantum circuit may have a plurality of tuning parameters associated therewith. For example, in an optical switching based embodiment, tuning parameters may correspond to angles of individual optical elements.

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

Some embodiments described herein generate, measure, or utilize a quantum state that approximates a target quantum state (eg, Hamilton's ground state). As those skilled in the art will appreciate, there are many ways to quantify how "similar" a first quantum state is to a second quantum state. In the following description, any approximate concepts or definitions known in the art may be used without departing from the scope of the present invention. For example, when the first and second quantum states are represented as 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 The first quantum state approximates the second quantum state by a predefined quantity (often denoted ϵ). In this example, fidelity measures how "close" or "similar" the first and second quantum states are to each other. Fidelity represents the probability that a measurement of a first quantum state will give the same result as if the measurement were performed on a second quantum state. Proximity between quantum states can also be quantified by a distance metric such as the Euclidean norm, Hamming distance, or another type of norm known in the art. The approximation between quantum states can also be defined computationally. For example, a first quantum state approximates a second quantum state when polynomial time sampling of the first quantum state gives it some desired information or property that it shares with the second quantum state.

Not all quantum computers are gate model quantum computers. Embodiments of the present invention are not limited to implementation using a gate-model quantum computer. As an alternative, embodiments of the present invention may be implemented in whole or in part using a quantum computer implemented using a quantum annealing architecture that is an alternative to a gate-model quantum computing architecture. More specifically, quantum annealing (QA) is a metaheuristic for finding the overall minimum of a given objective function in a given set of candidate solutions (candidate states) by using the process of quantum fluctuations.

FIG. 2B shows a diagram illustrating operations typically performed by a computer system 250 implementing quantum annealing. System 250 includes both quantum computer 252 and classical computer 254 . Operations shown to the left of vertical dashed line 256 are typically performed by quantum computer 252 , while operations shown to the right of vertical dashed line 256 are typically performed by classical computer 254 .

Quantum annealing begins with classical computer 254 generating initial Hamilton 260 and final Hamilton 262 based on computational problem 258 to be solved and providing initial Hamilton 260 , final Hamilton 262 and annealing schedule 270 as input to quantum computer 252 . The quantum computer 252 prepares a well-known initial state 266 based on the initial Hamiltonian 260 (FIG. 2B, operation 264), such as a quantum mechanical superposition of all possible states (candidate states) with equal weights. The classical computer 254 provides the initial Hamilton 260 , the final Hamilton 262 and the annealing schedule 270 to the quantum computer 252 . The quantum computer 252 starts in an initial state 266 and evolves its state according to an annealing schedule 270 following the time-dependent Schrödinger equation (natural quantum-mechanical evolution of a physical system) (FIG. 2B, operation 268). More specifically, the state of the quantum computer 252 undergoes a time evolution under a time-dependent Hamiltonian starting from an initial Hamilton 260 and terminating at a final Hamilton 262 . If the rate of change of the Hamiltonian of the system is slow enough, the system remains close to the instantaneous Hamiltonian ground state. If the rate of change of the Hamiltonian of the system is accelerated, the system may temporarily leave the ground state, but yields a higher probability of ending up in the ground state of the final problem Hamiltonian (ie, non-adiabatic quantum computation). At the end of the time evolution, the set of qubits on the quantum annealer is in a 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 . The experimental demonstration of successful quantum annealing for random magnets was reported shortly after the initial theoretical proposal.

The final state 272 of the subcomputer 254 is measured, thereby producing a result 276 (ie, a measurement) (FIG. 2B, operation 274). 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 connection with measurement unit 110 in FIG. 1 . Classical computer 254 performs post-processing on measurement results 276 to generate output 280 representing a solution to original computational problem 258 (FIG. 2B, operation 278).

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

The results of each individual measurement are random, but they are related in such a way that the computation always succeeds. Usually, the selection of the basis for later measurements depends on the results of earlier measurements, so the measurements cannot all be performed at the same time.

Any of the functions disclosed herein can be implemented using means for performing those 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 , a diagram of a system 100 implemented according to one embodiment of the present invention is shown. Referring to FIG. 2A , there is shown a flowchart of a method 200 performed by the system 100 of FIG. 1 according to an embodiment of the present invention. System 100 includes quantum computer 102 . Quantum computer 102 includes a plurality of qubits 104, which may be implemented in any of the ways disclosed herein. Any number of qubits 104 may exist in quantum computer 104 . For example, qubits 104 may comprise or consist of 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, up to 64 qubits, up to 128 qubits, up to 256 qubits, up to 512 qubits, up to 1024 qubits, up to 2048 qubits elements, not exceeding 4096 qubits, or not exceeding 8192 qubits. These are examples only, and in practice any number of qubits 104 may exist in a quantum computer 102 .

Any number of gates can exist in a quantum circuit. However, in some embodiments, the number of gates may 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 some linear multiple of the number of qubits 104 in the quantum computer 102 (e.g., 2, 3, 4, 5, 6 or 7).

Qubits 104 may be interconnected in any pattern. For example, they may 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 description below, although element 102 is referred to herein as a "quantum computer," this does not imply that all components of quantum computer 102 utilize quantum phenomena. One or more components of quantum computer 102 may, for example, be classical components that do not utilize quantum phenomena (ie, non-quantum components).

Quantum computer 102 includes a control unit 106, which may include any of various circuits and/or other machinery for performing the functions disclosed herein. The control unit 106 may, for example, consist entirely of classical components. The control unit 106 generates one or more control signals 108 and provides them as outputs to the qubits 104 . The control signal 108 may take any of various forms, such as any kind of electromagnetic signal, such as an electrical signal, a magnetic signal, an optical signal (eg, laser pulses), or any combination thereof.

E.g:

• In embodiments where some or all of the qubits 104 are implemented as photons traveling along a waveguide (also known as a "quantum optics" implementation), the control unit 106 may be a beam splitter (e.g., a heater or mirror ), the control signal 108 can be a signal to control the rotation of a heater or a mirror, the measurement unit 110 can be a photodetector, and the measurement signal 112 can be a photon.

• Some or all of the qubits 104 are implemented as charge qubits (e.g. transmon, X-mon, G-mon) or flux qubits (e.g. flux qubits, capacitive shunt Flux qubit) embodiment (also referred to as a "circuit quantum electrodynamics" (circuit QED) implementation), the control unit 106 may be a busbar resonator activated by a driver, and the control signal 108 may be a resonant cavity mode In this state, the measurement unit 110 may be a second resonator (eg, a low-Q resonator), and the measurement signal 112 may be a voltage measured from the second resonator using a dispersive readout technique.

• In embodiments where some or all of the qubits 104 are implemented as superconducting circuits, the control unit 106 may be a circuit QED assisted control unit or a direct capacitive coupling control unit or an inductive capacitive coupling control unit, and the control signal 108 may be For cavity modes, the measurement unit 110 may be a second resonator (eg, a low-Q resonator), and the measurement signal 112 may be a voltage measured from the second resonator using a dispersive readout technique.

• In embodiments where some or all of qubits 104 are implemented as trapping ions (such as the electronic state of (for example) magnesium ions), control unit 106 may be a laser, control signal 108 may be a laser pulse, measure Unit 110 can be a laser and a CCD or a photodetector (eg, a photomultiplier tube), and measurement signal 112 can be a photon.

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

• In embodiments 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, microwave antenna or coil, and the control signal 108 can be visible light, a microwave signal, or a constant For the electromagnetic field, the measurement unit 110 can be a photodetector, and the measurement signal 112 can be a photon.

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

• In embodiments where some or all of the qubits 104 are implemented as semiconducting materials (eg, nanowires), the control unit 106 can be a micromachined gate, the control signal 108 can be an RF or microwave signal, the measurement unit 110 may be a micromachined gate, and measurement signal 112 may 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, quantum computers known as "one-way quantum computers" or "measurement-based quantum computers" utilize such feedback 114 from the measurement unit 110 to the control unit 106 . Such feedback 114 is also necessary for the operation of error-tolerant quantum computing and error correction.

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

Another example of a control signal 108 that may be output by the control unit 106 and received by the qubit 104 is a gate control signal. Control unit 106 may output such gate control signals, thereby applying one or more gates to qubits 104 . Applying a gate to one or more qubits causes the group of qubits to undergo a physical state change embodying a corresponding logical gate operation (e.g., single-qubit rotation, two-qubit entanglement gate or multi-qubit operations). Thus implied, in response to receiving the gate control signal, qubit 104 undergoes physical transformations that cause qubit 104 to change state in such a way that the state of qubit 104 when measured (see below ) indicates 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 logic gate operations.

It should be understood that the dividing line between the status ready (and corresponding status ready signal) and application of the gate (and corresponding gate control signal) can be chosen arbitrarily. For example, some or all of the components and operations of elements illustrated as "state ready" in FIGS. 1 and 2A-2B may instead be characterized as elements of a gate application. Conversely, for example, some or all of the components and operations of elements illustrated as "gate applications" in FIGS. 1 and 2A-2B may instead be characterized as state-ready elements. As a specific example, the systems and methods of FIGS. 1 and 2A-2B can be characterized as performing only state preparation followed by measurements without any gate application, wherein elements described herein as part of gate application are replaced by Considered part of state readiness. Conversely, for example, the systems and methods of FIGS. 1 and 2A-2B may be characterized as simply performing a gate application followed by a measurement without any state preparation, and wherein the elements described herein as part of the state preparation are modified. is considered part of the gate application.

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

In general, quantum computer 102 can perform the various operations described above any number of times. For example, control unit 106 may generate one or more control signals 108 , thereby causing qubit 104 to perform one or more quantum gate operations. Subsequently, the measurement unit 110 may perform one or more measurement operations on the qubits 104 to read out a set of one or more measurement signals 112 . Measurement unit 110 may repeat such a measurement operation on qubit 104 before control unit 106 generates additional control signal 108 , thereby causing measurement unit 110 to read the same gate that was performed before reading out previous measurement signal 112 . The additional measurement signal 112 obtained by 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. Quantum computer 102 may then aggregate such multiple measurements of the same gate operation in any of a variety of ways.

After measurement unit 110 has performed one or more measurement operations on qubits 104 after a set of gate operations has been performed, control unit 106 may generate one or more additional control signals 108 that may be different from previous control signals 108 , thereby causing 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 may be repeated, with measurement unit 110 performing one or more measurement operations on qubit 104 in its new state (resulting from the most recently performed gate operation).

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

Then, for each of qubit Q 104 (FIG. 2A, operation 216), system 100 measures subbit Q to produce a measurement output representing the current state of qubit Q (FIG. 2A, operation 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 quantum gates in the circuit to the qubit 104, and then measuring the state of the qubit 104; and the system 100 can be directed to One or more circuits perform multiple triggers.

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

Quantum computer component 102 may include a plurality of qubits 104 as described above in connection with FIG. 1 . A single qubit can represent one, zero, or any quantum superposition of the states of those two qubits. The classical computer component 304 can provide the classical state ready signal Y32 to the quantum computer 102, and in response to the classical state ready signal Y32, the quantum computer 102 can be in any of the ways disclosed herein (such as in conjunction with FIGS. The state of qubit 104 is prepared in any of the ways disclosed in FIG. 2B .

Once qubit 104 is ready, classical processor 308 may provide classical control signal Y34 to quantum computer 102, and in response to classical control signal Y34, quantum computer 102 may apply the gate operation specified by control signal Y32 to the qubit 104, whereby the qubit 104 reaches a final state. (It can be realized as described above in conjunction with FIGS. 1 and 2A to 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, the measurement output Y38 Indicates that the state of the qubit 104 collapses to one of its eigenstates. The measurement output Y38 therefore comprises or consists of bits and thus represents the classical state. The quantum computer 102 provides the measured output Y38 to the classical processor 308 . Classical processor 308 may store data representing and/or derived from measured output Y38 in classical memory 310 .

The steps described above may be repeated any number of times, with the state described above as the final state of qubit 104 being used as the initial state for the next iteration. In this manner, classical computer 304 and quantum computer 102 may operate together as co-processors to perform joint computations as a single computer system.

Although some functions may be described herein as being performed by classical computers, and other functions may be described herein as being performed by quantum computers, these are examples only, and should not be considered limitations of the invention. A subset of the functions disclosed herein as being performed by a quantum computer may instead be performed by a classical computer. For example, a classical computer can perform functionality to mimic a quantum computer and provide a subset of the functionality described herein, although the exponential scaling of the simulation would limit the functionality. Functions disclosed herein as being performed by a classical computer can instead be performed by a quantum computer.

The techniques described above may 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 , implemented on classical computers alone or on hybrid classical quantum (HQC) computers. The techniques disclosed herein can be implemented, for example, only on classical computers that mimic the functions of quantum computers disclosed herein.

The techniques described above can be implemented in one or more computer programs that execute on (or can be programmed by) a programmable computer (such as a classical computer, a quantum computer, or HQC). execution), the programmable computer includes any combination of any number of the following: a processor, processor-readable and/or writable storage media (including, for example, volatile and non-volatile memory and and/or storage elements), input devices, and output devices. Code can be applied to input, typed using input devices, to perform the functions described and to generate output using output devices.

Embodiments of the invention include features that are only possible and/or feasible if implemented using one or more computers, computer processors, and/or other elements of a computer system. It may not be possible or practical to implement such features intelligently and/or manually. For example, it is not possible to intelligently or manually generate random samples from complex distributions describing operators P and states |s>.

Any claim herein affirmatively requiring a computer, processor, memory, or similar computer-related element is intended to require such element, and should not be construed to mean that such element is not present in or required by such claim . Such claims are not intended, and should not be construed, to cover methods and/or systems that do not include the stated computer-related components. For example, any method claim herein that states that a claimed method is performed by a computer, processor, memory, and/or similar computer-related element(s) is intended and should be understood only to cover implementation by the stated computer-related element(s) method. Such method claims are not to be read, for example, as covering methods performed intelligently or manually (eg, using a pencil and paper). Similarly, any product claim herein that states that the claimed product includes a computer, processor, memory, and/or similar computer-related component is intended and should be read only as covering a product that includes the stated computer-related component(s). Such product claims should not be read, for example, to cover products that do not include the stated computer-related component(s).

In embodiments where a classical computing component executes a computer program providing any subset of the functionality within the scope of the following claims, the computer program may be implemented in any programming language, such as assembly language, machine language, high-level programming language, or object-oriented programming language. For example, the programming language may be a compiled or interpreted programming language.

Each such computer program may be implemented in a computer program product tangibly embodied in a machine-readable storage device for execution by a computer processor (which may be a classical processor or a quantum processor). Method steps of the invention can be performed by one or more computer processors executing a program tangibly embodied in a computer-readable medium to perform functions by operating on inputs and generating output. Suitable processors include, for example, both general and special purpose microprocessors. Generally, a processor receives (reads) instructions and data from and writes (stores) instructions and data to a memory, such as ROM and/or RAM. 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 drives and memory removable disks; magneto-optical disks; and CD-ROMs. Any of the above can be supplemented or incorporated by specially designed ASICs (Application Specific Integrated Circuits) or FPGAs (Field Programmable Gate Arrays). Classical computers can also typically receive (read) programs and data from, and write (store) programs and data to, a non-transitory computer-readable storage medium such as an internal disk (not shown) or a removable disk. Transient computer readable storage media. These elements will also be found in conventional desktop or workstation computers and other computers suitable for executing computer programs for carrying out the methods described herein, which computers can be used with any digital printing engine or marking engine, Used in conjunction with display monitors, or other raster output devices capable of producing color or grayscale pixels on paper, film, display screen, or other output media.

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

100: system 102: Quantum computer 104: Qubits 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: busbar 316: input data 318: Output data 400: operator 402: Quantum state 404: block 406: block 408: block 410: block 430: Hybrid Classical Quantum Computers (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。1 is a diagram of a quantum computer according to an embodiment of the present invention; FIG. 2A is a flowchart of a method performed by the quantum computer of FIG. 1 according to an embodiment of the present invention; FIG. 2B is a flow chart according to an embodiment of the present invention Figure 3 is a diagram of a hybrid quantum classical computer performing quantum annealing; Figure 3 is a diagram of a hybrid quantum classical computer according to an embodiment of the invention; Figure 4 is a diagram of a hybrid quantum classical computer performing quantum amplitude estimation according to an embodiment of the invention A diagram of a classical computer (HQC); Figures 5A to 5C illustrate standard sampling and enhanced sampling quantum circuits of some embodiments of the present invention and their corresponding approximate functions; Figures 6A to 6B illustrate Fisher Information Graphs of dependence on various likelihood functions; FIG. 7 illustrates an operation for generating samples corresponding to engineered likelihood functions according to one embodiment of the present invention; FIG. Implemented Algorithms; Figures 9A to 9B, Figure 10, Figures 11A to 11B and Figure 12 illustrate various algorithms implemented by embodiments of the present invention; Figure 13 is a schematic representation of various embodiments according to the present invention Graphs of functions and fitted approximate functions; Figures 14A-14B, 15A-15B, 16A-16B, 17A-17B, and 18A-18B illustrate various implementations of the present invention The graph of performance of example; Fig. 19 shows the graph of the embodiment of the present invention

Factor; 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 the performance of various embodiments of the present invention

Factor; 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

Factor; FIG. 26 shows graphs illustrating run time versus target accuracy for various embodiments of the invention; FIGS. 27 to 28 illustrate quantum circuits implemented according to embodiments of the invention; FIGS. 29A to 29B, Figures 30A-30B, 31A-31B and 32 illustrate algorithms implemented according to embodiments of the present invention; and Figure 33 illustrates true and fitted approximate functions according to embodiments of the present invention.

400: operator

402: Quantum state

404: block

406: block

408: block

410: block

430: Hybrid Classical Quantum Computers (HQC)

432:Quantum computer

434: classical computer

Claims (16)

A method for quantum amplitude estimation, comprising: selecting a plurality of quantum circuit parameter values by a classical computer to optimize an accuracy improvement rate of a statistic estimating an observable P with respect to a quantum state | s 〉An expected value 〈 s ｜ P ｜ s 〉, where the statistic comprises a value calculated from a plurality of sampled values sampled from a random variable by measuring a quantum computer One or more qubits are obtained, and the complex number of quantum circuit parameter values is a complex number of real numbers that control how one or more quantum gates operate on the one or more qubits; the alternating first and second A sequence of generalized reflection operators is applied to the one or more qubits of the quantum computer to transform the one or more qubits from the quantum state | s > into a reflection quantum state, the first and Each of the second generalized reflection operators is a function of a corresponding one of the plurality of quantum circuit parameter values; the plurality of qubits in the reflection quantum state are measured with respect to the observable P to obtain a set of measurements; and updating the statistic with the set of measurements on the classical computer. The method for estimating quantum amplitude according to Claim 1, further comprising outputting the statistic after the update. In the method for estimating quantum amplitude according to claim 1, the statistic includes an average value calculated from the plurality of sampled values sampled from the random variable. According to the quantum amplitude estimation method described in Claim 1, the accuracy improvement rate includes a variation reduction factor. According to the quantum amplitude estimation method described in Claim 1, the accuracy improvement rate includes an information improvement rate. In 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 as described in claim 1, wherein the sequence of the first and second generalized reflection operators and the observable P define a bias of an engineered likelihood function. The quantum amplitude estimation method as claimed in claim 1, further comprising iterating the selection, the application, the measurement and the update. The quantum amplitude estimation method as described in Claim 1, which further comprises: updating an accuracy estimate 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, iterate over the selection, the application, the measurement, and the update. The method for estimating quantum amplitude according to claim 1, wherein updating the statistic comprises: updating a prior distribution with the plurality of measured values to obtain a post-hoc distribution; and calculating an updated statistic from the post-hoc distribution. The quantum amplitude estimation method as claimed in claim 1, wherein the selection is based on the statistic and an accuracy estimate of the statistic. The method of claim 11, wherein the selection is further based on a fidelity representative of errors occurring during the application and the measurement. The quantum amplitude estimation method as claimed in claim 1, wherein the selection uses one of coordinate ascent and gradient descent. A computing system for quantum amplitude estimation comprising: a processor; a quantum classical interface communicatively coupling the computing system to a quantum computer; and a memory coupled to the processing 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 statistical The rate of accuracy improvement of a statistic that estimates an expected value of an observable P with respect to a quantum state | s 〉 〈 s | P | s 〉, where the statistic consists of a complex number of samples sampled from The plurality of sampled values are obtained by measuring one or more qubits of the quantum computer, and the plurality of quantum circuit parameter values control how one or more quantum gates respond to one or more a plurality of real numbers operated on by a plurality of qubits; (ii) controlling the quantum computer via the quantum classical interface to use one or more sequences of alternating first and second generalized reflection operators of the quantum computer qubits are transformed from the quantum state | s > into a reflective quantum state, each of the first and second generalized reflective operators being a function of a corresponding one of the plurality of quantum circuit parameter values, (iii) controlling the quantum computer via the quantum classical interface to measure the plurality of qubits in the reflected quantum state with respect to the observable P to obtain a set of measurements, and (iv) using the set of quantities The statistics are updated according to the test results. In the quantum amplitude estimation computing system described in 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 computing system according to claim 14, further comprising the quantum computer.

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