CN118070914A

CN118070914A - Quantum reading calibration method and system based on finite element

Info

Publication number: CN118070914A
Application number: CN202410283895.8A
Authority: CN
Inventors: 谭思危; 尹建伟; 卢丽强; 余加; 张涵禹; 郎聪亮
Original assignee: Zhejiang University ZJU
Current assignee: Zhejiang University ZJU
Priority date: 2024-03-13
Filing date: 2024-03-13
Publication date: 2024-05-24

Abstract

The invention discloses a quantum reading calibration method and a system based on finite elements, comprising a plurality of rounds of iterative calibration, wherein each round of iterative calibration comprises a characterization flow and a calibration flow, and the characterization flow comprises the following steps: determining the weight between any two quantum bits based on the reference probability distribution of a reference circuit in the current round, constructing a weighted quantum bit diagram, and grouping the quantum bit partitions in the weighted quantum bit diagram based on the finite element thought to obtain a grouping scheme; the calibration process comprises the following steps: and calculating a sub-noise matrix corresponding to each quantum bit group based on the quantum bit set, the reference probability distribution and the grouping scheme which are acquired by measurement, so as to obtain a sub-noise matrix set, and calibrating the calibration output of the previous round based on the sub-noise matrix set so as to obtain the calibration output of the current round. Therefore, the finite element method is expanded to the quantum calibration process, and quantum bit interaction characteristics are comprehensively considered to improve the accuracy and efficiency of quantum reading calibration.

Description

Quantum reading calibration method and system based on finite element

Technical Field

The invention belongs to the technical field of quantum computing, and particularly relates to a quantum reading calibration method and system based on finite elements.

Background

Quantum computing paradigms demonstrate great potential beyond classical computing in solving complex problems, such as physical simulation, combinatorial optimization, and many examples in the field of artificial intelligence. In solving these problems on practical quantum hardware, computation inevitably encounters various noise sources, with reading errors (noise generated by reading the output from the quantum device and converting it into classical data) proving to be the most significant source of error. For most physical quantum implementations, this reading error reaches 1% -10% in each qubit operation, thus severely hampering the application of quantum computing.

The fidelity of the readings may be improved by hardware-level optimization techniques, including frequency scaling and sapphire-based processor fabrication. These techniques make the generated quantum states easier to distinguish by isolating the qubit from external noise. In contrast, a more cost-effective approach is to employ software-level optimizations such as matrix-based calibrations, machine learning, domain-specific patterns, and bayesian estimation. Among these methods, matrix-based alignment methods have been experimentally proven effective in superconducting, ion trap and optical quantum hardware, and thus are widely used in commercial quantum cloud platforms such as IBMQ and Rigetti.

Although matrix-based calibration can be easily deployed on a software stack, it faces a tradeoff between scalability and calibration accuracy. To maximize accuracy, i.e. to accurately characterize the matrix, a large number of reference circuits need to be implemented in detail to cover all possible measurement outputs, the number of which grows exponentially with the number of qubits. For example, obtaining such a matrix of 16 qubits requires the execution of 6.5X10-4 circuits, which takes about 18.2 hours on IBMQ Guadalupe quantum devices. In addition to matrix characterization, the MVM (matrix vector multiplication) step is equally computationally overwhelming, as the size of the matrix grows exponentially with the number of qubits. Taking the recently developed M3 of IBMQ as an example, it utilizes a pruning strategy based on hamming distance threshold, however, 87.5PB of memory and more than one year are still required to perform the MVM step.

Quantum reading errors proved to be the most significant source of error, greatly affecting measurement fidelity. Matrix-based calibration has proven effective in a variety of quantum platforms. However, existing approaches have fundamental limitations in terms of scalability or accuracy.

Existing methods fall into two categories: quantum independent methods and sparse sensing methods. The first category includes IBU and Yang et al, which calculate the noise matrix by calculating the tensor product of a series of element matrices (single bit 2 x2 matrix). Meanwhile, M3 and Nation are two representations of sparse perceptions methods that approximate the noise matrix using the underlying sparse mode.

Currently, matrix-based techniques are mainly focused on improving scalability and accuracy. To improve scalability, methods like M3 and Yang, etc. exploit the sparsity of the allocation matrix, although they show the limitation that only a few matrix elements can be pruned. Other approaches only simulate qubit independent errors, resulting in a large loss of accuracy.

In terms of scalability, both the characterization time and the MVM time are exponentially related to the number of qubits measured. The quantum independent method reveals the time to linearly characterize the noise matrix, requiring only 2N _m reference circuits to be implemented. Thus, they support the calibration of more number of quantization bits than sparse perceptual methods. However, they have to introduce a linear equation solver to calculate the inverse matrix (calibration matrix). The complexity of these solvers limits the upper bound to 127 qubits. On the other hand, sparse perceptions prune the matrix at the threshold of hamming distance, however, this does not reduce complexity and results in a slight increase in scalability. Furthermore, they need to be characterized for each calibration run, which further increases complexity.

In terms of accuracy, quantum independent methods achieve better scalability by sacrificing accuracy, because they cannot capture noise, such as crosstalk, between qubits. More importantly, inaccuracy will expand as the tensor product length increases. For example, when calibrating the output of more than 81 qubits, such inaccuracy becomes intolerable. Furthermore, they are mainly applicable to the GHZ and BV algorithms, since they consist of fewer bit strings of non-zero probability, which limits their versatility. The sparse sensing method shows a small hilbert-schmitt distance, however, it is still insufficient for calibration.

Disclosure of Invention

In view of the foregoing, it is an object of the present invention to provide a method and system for quantum reading calibration based on finite elements, which extends the Finite Element Method (FEM) to the quantum calibration process and fully considers the quantum bit interaction characteristics to improve the accuracy and efficiency of quantum reading calibration.

In order to achieve the above object, the embodiments of the present invention provide a quantum reading calibration method based on finite elements, which includes a plurality of iterative calibrations, each iterative calibration including a characterization procedure and a calibration procedure,

The characterization flow includes: determining the weight between any two quantum bits based on the reference probability distribution of a reference circuit in the current round and constructing a weighted quantum bit diagram, and grouping the quantum bit partitions in the weighted quantum bit diagram based on a finite element idea to obtain a grouping scheme, wherein the reference probability distribution of the first round is obtained by sampling the output of the reference circuit, and the reference probability distribution of other rounds which are not the first round is obtained by calibrating output in the iterative process;

The calibration process comprises the following steps: and calculating a sub-noise matrix corresponding to each quantum bit group based on the quantum bit set, the reference probability distribution and the grouping scheme which are acquired by measurement, so as to obtain a sub-noise matrix set, and calibrating the calibration output of the previous round based on the sub-noise matrix set so as to obtain the calibration output of the current round.

Preferably, the reference circuit is obtained by setting a threshold value, and includes: and executing the candidate circuit, obtaining a measurement state of each quantum bit of the candidate circuit under a given ideal ground state, calculating an interaction of two quantum bits and an index for distinguishing key interaction based on the ideal ground state and the measurement state, and obtaining the reference circuit by screening the candidate circuit of which the index exceeds a set threshold value.

Preferably, each time a candidate circuit is implemented, the operation and read output of each qubit is recorded as a triplet (ideal, measured, ef), wherein,Ideal ground state for recording preparation in a circuit, ideal=0 representing an ideal ground state bit value of 0, ideal=1 representing an ideal ground state bit value of 1,/>Indicating that the ideal ground state bit value is not measured,/> For recording the measurement state output after the execution of the sample of the circuit, measured=0 representing a measurement state bit value of 0, measured=1 representing a measurement state bit value of 1, Indicating that the measured state bit value is not measured, ef e {0,1} indicating an error flag, ef=0 indicating that the measured state matches the ideal ground state, ef=1 indicating that an error occurred;

The interaction of two qubits is denoted as interaction (q _i.ideal＝x→q_j. Ideal=y), calculated as:

interact(q_i.ideal＝x→q_j.ideal＝y)＝p(q_j.ef＝1∣C1,C2)-p(q_j.ef

＝1∣C2)

C1:q_i.ideal＝x；

C2:

Where q _i. Ideal=x denotes the ideal ground state of the ith quantum bit q _i as x, q _j. Ideal=y denotes the ideal ground state of the jth quantum bit q _j as y, interaction (→·) denotes the former to latter interaction, q _j. Ef=1 denotes the error flag of the jth quantum bit q _j as 1, C1 and C2 denote intermediate variables for referring to the ideal state of the bit, and p (·|·) denotes the conditional probability distribution of the actual value measured at the ideal value as the reference probability distribution;

The index θ that distinguishes the key interactions calculated based on the interactions of the two qubits is expressed as:

Wherein num (. Fwdarw. Cndot.) represents the number of interactions.

Preferably, the first round of reference probability distribution is obtained for output samples of the reference circuit, comprising: acquiring a measurement state of each qubit in the reference circuit in an ideal ground state, recording a triplet, and calculating the conditional probability distribution of each qubit based on the triplet to serve as the reference probability distribution of the first round;

the reference probability distribution for other rounds than the first round is derived from the calibration output in the iterative process, including: and taking the calibration output of the iterative process of the current round as an element value to form the reference probability distribution of the next round.

Preferably, determining weights between any two qubits based on a reference probability distribution of a reference circuit at a current round and constructing a weighted qubit map includes:

the weight (q _i,q_j) between any two qubits is expressed as:

in the constructed weighted quantum bit diagram, quantum bits are nodes, the continuous edges between the nodes represent the interaction between the quantum bits, and the weight value of the continuous edges is marked by weight (q _i,q_j);

preferably, grouping the qubits in the weighted qubit map based on the finite element idea results in a grouping scheme comprising:

And carrying out quantum bit group division on the weighted quantum bit diagram with the aim of maximizing the locality in the group, namely maximizing the weight sum in the group, so as to obtain a grouping scheme.

Preferably, calculating the sub-noise matrix corresponding to each qubit group based on the quantum bit set acquired by measurement, the reference probability distribution and the grouping scheme includes:

The collected quantum bit set is Q _M, the grouping scheme is G _l＝{g_l,1,···,g_l,K, wherein l represents the round, K represents the total group number, and G _l,k represents the kth quantum bit group of the first round;

Definition for each g _l,k: g _∩＝Q_M∩g_l,k of the total number of the components, Where g _∩ represents the qubit overlapping g _l,k and Q _M,/>Representing the remaining qubits in g _l,k;

the sub-noise matrix corresponding to each qubit group is calculated as:

Where q denotes a qubit belonging to g _∩, x _q denotes a bit of the qubit q in the bit string x, P denotes a conditional probability, estimated based on the reference probability distribution BP _l, q.measure=x _q denotes a measured state of q as x _q,g_∩, ideal=y as a condition, denotes an ideal ground state of g _∩ as y, As a condition, express/>Is/>M _l,k [ x ] [ y ] represents the values of the x position and the y position in the sub-noise matrix M _l,k corresponding to g _l,k.

Preferably, calibrating the calibration output of the previous round based on the sub-noise matrix set to obtain the calibration output of the current round includes:

Wherein M _l,1,M_1,2,…,M_l,K is K sub-noise matrices in the sub-noise matrix set, P _l+1 represents the calibration output of the first round, P _l represents the calibration output of the first round, and the symbol Representing the matrix tensor product, NZ _l represents the set of qubit strings of non-zero probabilities in P _l, |x _l,k > represents the corresponding sub-bit string of the kth qubit group in the first round.

Preferably, in the calibration procedure, a sparse tensor product engine is introduced to prune tensor products in the calibration calculation process to perform calculation acceleration, including:

first calculate each qubit group Then calculating the tensor product of the matrix vector multiplication values of all the quantum bit groups, then pruning the intermediate values in the tensor product through another threshold value, and finally aggregating P _l (x) with the rest intermediate values in the tensor product to obtain the calibration output.

In order to achieve the above object, the embodiment of the present invention further provides a quantum reading calibration system based on finite elements, which includes a characterization flow module and a calibration flow module, where the characterization flow module and the calibration flow module sequentially iterate to achieve quantum reading calibration;

The characterization flow module is used for determining the weight between any two quantum bits based on the reference probability distribution of the reference circuit in the current round and constructing a weighted quantum bit diagram, and grouping the quantum bit partitions in the weighted quantum bit diagram based on the finite element thought to obtain a grouping scheme, wherein the reference probability distribution of the first round is obtained by sampling the output of the reference circuit, and the reference probability distribution of other rounds which are not the first round is obtained by calibrating output in the iterative process;

The calibration flow module is used for calculating a sub-noise matrix corresponding to each quantum bit group based on the quantum bit set, the reference probability distribution and the grouping scheme, so as to obtain a sub-noise matrix set, and calibrating the calibration output of the previous round based on the sub-noise matrix set so as to obtain the calibration output of the current round.

Compared with the prior art, the invention has the beneficial effects that at least the following steps are included:

the finite element thought is expanded to a quantum calibration process, a weighted quantum bit diagram is constructed by comprehensively considering quantum bit interaction characteristics through a reference circuit, quantum bit divisions in the weighted quantum bit diagram are grouped to obtain a grouping scheme, in the calibration process, a sub-noise matrix is calculated for each group in the grouping scheme, and iterative calibration is carried out based on the sub-noise matrix based on the finite element thought, so that accuracy and efficiency of quantum reading calibration are improved.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.

FIG. 1 is a flow chart of a finite element based quantum reading calibration method provided by an embodiment;

FIG. 2 is a schematic diagram of the extension of the conventional finite element method provided by the embodiment to the quantum finite element method;

FIG. 3 is a quantum reading calibration schematic for 2 iterations provided by the embodiments;

FIG. 4 is a diagram of an example pruning intermediate value for a sparse tensor product engine provided by an embodiment;

fig. 5 is a schematic structural diagram of a quantum reading calibration system based on finite elements provided in an embodiment.

Detailed Description

The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the detailed description is presented by way of example only and is not intended to limit the scope of the invention.

The technical conception of the invention is as follows: the existing quantum reading calibration is derived from the lack of a theoretical basis for simplifying the noise matrix characterization and a theoretical basis for comprehensively making the interaction characteristics of the quantum bits, and the dilemma is avoided only by avoiding the interaction of the quantum bits or directly trimming the matrix, so that the problem of inaccurate calibration can exist. The present invention considers that the reading error is mainly due to local noise between qubits and finds that the Finite Element Method (FEM) can capture and analyze this feature, and FEM has proven to be effective in approximating many complex evolutionary processes, such as aerodynamic and thermodynamic processes, which naturally apply to simulating complex noise during reading. In view of this, the present invention extends the Finite Element Method (FEM) to the calibration process, and proposes a quantum reading calibration method based on finite elements.

In the method of the present invention, the calibration is redefined as a series of tensor products associated with a plurality of sub-noise matrices that iteratively calibrate the readings through a typical FEM workflow, the sub-noise matrices being determined by grouping the qubits into groups, each group attempting to capture interactions between the qubits, similar to FEM. In order to characterize these sub-noise matrices quickly and accurately, a novel generation method is proposed to minimize the number of reference circuits. On the other hand, in order to accelerate the calibration calculation, i.e. a series of tensor products, a sparse tensor product engine has also been developed to exploit the inherent sparsity of the noise matrix. Unlike previous methods that directly prune the noise matrix, the present invention selects pruning intermediate values to achieve higher characterization accuracy. More importantly, the pruning technique of the present invention only reveals polynomial complexity, thereby greatly alleviating significant computational stress. In general, the method of the invention achieves high-speed improvement in end-to-end calibration, and improves the measurement fidelity, which can realize high expandability and high accuracy in calibration, and can be extended to more than 500 qubits.

In the quantum reading calibration task, the quantum reading error can be expressed as a linear transformation from the ideal profile P _ideal to the measured profile P _measured:

P_ideal＝MP_measured (1)

Wherein M is defined as having a size of N _q represents the number of qubits. Theoretically, the noise matrix M [ x ] [ y ] can be characterized by running the benchmark circuit and taking each ideal ground state as an input. After execution, the noise matrix is filled in according to the output probability distribution P (g.measure=x|g.ideal=y):

M[x][y]＝P(g.measure＝x|g.ideal＝y) (2)

Wherein the method comprises the steps of Representing a set of N _q qubits q, P (g.measure=x|g.ideal=y) represents the probability that the qubit q is in the x (i.e. g.measure=x) state under the condition that the ideal ground state is in y (i.e. g.ideal=y). For example, a reference circuit of 4 is implemented for a2 qubit system, the element in (3, 0) is 0.02, which means that when the ideal state is |00>, the probability of observing |11> is 0.02, expressed as M [3] [0] =p (measure=11|preparation=00) =0.02.

The noise matrix may be obtained using its inverse matrix (i.e. a calibration matrix), the calibration being performed by an MVM (matrix-vector multiplication) operation between the measurement distribution and the calibration matrix. In short, by multiplying the measured quantum distribution P _measured by the calibration matrix M ^-1, the probability distribution P _calibrated output after calibration can be obtained.

P_calibrated＝M^-1P_measured (3)

A typical application of the Finite Element Method (FEM) in fluid mechanics is shown in fig. 2 (a), which analyzes the force distribution of a sponge under compression. In this case, FEM adopts a divide-and-conquer strategy to simulate the evolution of the sponge, dividing the sponge into multiple lattices and grouping it into groups. Each lattice is analyzed independently to simplify the modeling process, taking into account the locality of the force distribution. By combining the lattices together, the state of the sponge can be obtained at a certain point in time. Based on the current force distribution, the whole evolution process is iteratively updated using FEM. In view of this, for the quantum reading calibration task described above, in the method proposed by the present invention, reading calibration is performed iteratively according to the workflow of FEM, as shown in fig. 2 (b). Similar to the sponge case, noise between qubits also appears localized. Thus, the qubits are divided into groups, and the sub-noise matrices for each group are formulated independently. Unlike conventional FEM that employ static partitioning, the inventive method employs a diverse grouping scheme during iterative calibration to cover interactions between qubits over the whole.

As shown in fig. 1, the quantum reading calibration method based on finite elements provided by the embodiment of the invention comprises multiple rounds of iterative calibration, wherein each round of iterative calibration comprises a characterization flow and a calibration flow.

Wherein the characterization flow aims at generating parameters required for calibration, namely the grouping scheme G _i and the reference probability distributions BP _l.G_l and BP _l for sub-noise matrix generation in each iteration, to be used in the ith iteration in the calibration flow. In particular, constructing a weighted quantum graph to quantify interactions between qubits and grouping the qubits to generate a grouping scheme in each iteration requires the application of different grouping schemes in the calibration procedure to cover local interactions between the qubits, which grouping schemes aim to maximize population locality. To this end, interactions between qubits are quantized by weighting the qubit map.

Specifically, in each round of iteration, the weight between any two qubits is determined based on the reference probability distribution of the reference circuit in the current round, a weighted qubit diagram is constructed, and the qubits in the weighted qubit diagram are divided into groups based on the finite element idea to obtain a grouping scheme.

The reference probability distribution of the first round is obtained by sampling the output of a reference circuit, wherein the reference circuit is obtained by setting a threshold value and screening, so that the readout noise is accurately represented by adopting a more flexible reference circuit. The reference circuit involves all the qubits of the device, each of which has three possible operations:

1) Setting the qubit to |0> and making measurements;

2) Setting the qubit to |1> and making measurements;

3) Setting the qubit to a random state (|0 > or |1 >) does not make a measurement.

In order to reduce the number of reference circuits and maintain high accuracy, the present invention identifies candidate circuits capable of achieving high qubit interactions and analyzes the results of candidate circuit execution, while low-interaction candidate circuits mainly involve qubit independent errors, so their outputs can be easily modeled without executing a program. Obviously, some candidate circuits (4 times the number of qubits) are first randomly generated to obtain an initial value of the interaction between every two qubits. Thereafter, a table is maintained to record updated interaction values and the number of circuits involved in these interactions. Next, an index θ for distinguishing key interactions is defined:

Basically, the interaction is static, and depending on the physical device only, it can be estimated more accurately by executing more reference circuits. In other words, θ naturally decreases with the number of reference circuits. Based on this, the present invention sets a threshold α to specify accuracy, iteratively executes candidate circuits related thereto for interactions of θ > α, and updates the table.

Thus, some candidate circuits were prepared, where q ₁. Ideal=0,And collecting the measurement state of each qubit of the candidate circuit in a given ideal ground state, and calculating the interaction (q _i.ideal＝x→q_j. Ideal=y) of the two qubits and the index θ for distinguishing the key interactions based on the ideal ground state and the measurement state, in this way, the candidate circuit is continuously executed until all the indexes θ of the interactions are smaller than the threshold α, and the reference circuit is obtained by screening the candidate circuits whose indexes exceed the set threshold. In general, ensuring that the index θ of all interactions exceeds a threshold requires more reference circuit execution to produce significant interactions, whereas for interactions near 0, circuit execution is unnecessary because the output profile of each qubit is the same as a single qubit reference circuit.

Each qubit in the circuit is prepared to one state and a subset of them is measured in the circuit. Each time a candidate or reference circuit is implemented, the operation and read output of each qubit is recorded as a triplet (ideal, measured, ef), where,Ideal ground state for recording preparation in a circuit, ideal=0 representing an ideal ground state bit value of 0, ideal=1 representing an ideal ground state bit value of 1,/>Representing an ideal ground state bit value of null,/> For recording the measured state output after a sample of the execution circuit, measured=0 representing a measured state bit value of 0, measured=1 representing a measured state bit value of 1,/> Indicating that the measurement state bit value is null, ef e {0,1} indicates an error flag, ef=0 indicates that the measurement state matches the ideal ground state, ef=1 indicates that an error occurred,/>Indicating that the qubit was not measured and was not output. In this case/>There can be no read-out errors, i.e. >Ef=0. For example, one reference circuit prepares qubit q2 as |1>, respectively. When qubit q1 is not measured and the output of q2 is |0>, the triplet is assigned as follows:

q₂.ideal＝1,q₂.measuerd＝0,q₂.ef＝1。

From the recorded triplets, the read-out errors are observed to be independent of the state of the qubit. Furthermore, the interaction from one qubit to another can change under different operations (i.e., state preparation and readout), resulting in different readout errors. From this observation, the interaction between qubit operations was quantified, expressed as interaction (q _i.ideal＝x→q_j. Ideal=y), calculated as:

interact(q_i.ideal＝x→q_j.ideal＝y)＝p(q_j.ef＝1∣C1,C2)-

p(q_j.ef＝1∣C2) (5)

Wherein c1:q _i ideal = x; c2: q _i. Ideal=x denotes the ideal ground state of the ith quantum bit q _i as x, q _j. Ideal=y denotes the ideal ground state of the jth quantum bit q _j as y, interaction (→·) denotes the former to the latter interaction, q _j. Ef=1 denotes the error flag of the jth quantum bit q _j as 1, C1 and C2 denote intermediate variables for referring to the ideal state of the bit, p (·|·) denotes the conditional probability distribution, as the base probability distribution BP1; equation (5) characterizes the correlation between q _i. Ideal=x and the readout error of q _j.

After the reference circuit is obtained, the measurement state of each qubit in the reference circuit in an ideal ground state is obtained, a triplet is recorded, and the conditional probability distribution of each qubit is calculated based on the triplet and is used as the reference probability distribution of the first round. The interaction between any two qubits (q _i,q_j) is calculated based on the reference probability distribution and the weight (q _i,q_j) is calculated from the interaction, expressed as:

In a weighted qubit map constructed based on weight (q _i,q_j), the qubits are nodes, the edges between the nodes represent interactions between the qubits, the weight values of the edges are marked by weight (q _i,q_j), then the qubits in the weighted qubit map are grouped based on finite element ideas to obtain a grouping scheme, specifically, the weighted qubit map is divided by taking the maximization of locality in the group, namely the maximization of weight sum in the group, to obtain the grouping scheme, and such a grouping scheme ensures that in each iteration, the formulated sub-noise matrix tries to comprehensively capture interactions between the qubits, so that the interactions are closer to an actual noise matrix. The MAXCUT solver may be utilized in particular implementations to divide the qubits in the weighted qubit map into groups.

The grouping schemes in different iterations aim to capture different noise. To achieve this, after each iteration of the qubit grouping in the characterization process, a calibration is also applied to the reference probability distribution, which eliminates noise captured in the grouping scheme, the reference probability distribution calibration for other rounds than the first round resulting from the calibration output in the iteration process, including: and taking the calibration output of the iterative process of the current round as an element value to form the reference probability distribution of the next round. Since the measurements are made multiple times, taking the current round as the first round as an example, the calibration output of the first round is P ₂, and then P ₂ is taken as the element value of the multiple measurements to form the reference probability distribution BP ₂ of the next round.

In the calibration flow, the calibration of the grouping scheme is focused on removing the residual noise outside the grouping scheme in the output, and the specific flow comprises the following steps: and calculating a sub-noise matrix corresponding to each quantum bit group based on the quantum bit set, the reference probability distribution and the grouping scheme which are acquired by measurement, so as to obtain a sub-noise matrix set, and calibrating the calibration output of the previous round based on the sub-noise matrix set so as to obtain the calibration output of the current round.

The grouping scheme generated by the characterization process is G _l＝{g_l,1,···,g_l,K, where l represents the round, K represents the total number of groups, and G _l,k represents the kth qubit group of the first round. A sub-noise matrix is dynamically generated for each probability distribution due to the different interactions in the different circuits measuring the qubits. The measured set of qubits is defined as Q _M, then for each g _l,k:

Where g _∩ represents the qubit overlapping g _l,k and Q _M, Representing the remaining qubits in g _l,k;

the corresponding sub-noise matrix for each qubit group is expressed as:

The conditional probability P is estimated based on the reference probability distribution BP _i of the current iteration, and the condition is additionally introduced To ensure that the matrix generation takes into account the unmeasured qubits. Furthermore, since the measured output of a qubit depends only on the operation of the qubit, equation (8) is expressed as:

Where q denotes a qubit belonging to g _∩, x _q denotes a bit of the qubit q in the bit string x, q.measure=x _q denotes a measurement state of q of x _q,g_∩. Ideal=y as a condition, denotes an ideal ground state of g _∩ of y, As a condition, express/>Is/>M _l,k [ x ] [ y ] represents the values of the x position and the y position in the sub-noise matrix M _l,k corresponding to g _l,k. Thus, in calculating the sub-noise matrix, the condition g _∩. Ideal=Φ is introduced to improve the accuracy of the characterization, since these qubits are not measured.

Defining the circuit output as P _measured, i.e., P ₁, and the calibration output as P _L+1, the present invention reformulates the calibration as an iterative process that includes a series of sub-noise matrices:

Iter.1:

Iter.2:

···；

Iter.L:

wherein M _l,1,M_l,2,…,M_l,K is K sub-noise matrices in the sub-noise matrix set, M _l,k is the sub-noise matrix of the kth quantum bit group in the first iteration, and the symbol Representing the tensor product of combining these sub-noise matrices, P ₂ to P _L act as intermediate probability distributions for iteratively approximating the ideal probability distribution, L and K representing the number of iterations and the number of groups, respectively, the first iteration calibrating P _l to P _l+1.

Calibration of P _l to P _l+1 for the first iteration, definition of P _l (x) as the probability P _l＝∑P_l (x) |x > -that a bit string x is observed in the P _l distribution, the bit string x can be partitioned into sub-bit strings according to the grouping scheme of the first iteration: i x > = |x _l,1>|x_l,2…|x_l,K >, where x _l,k is the sub-bit string of the qubit group g _l,k, then the calibration for each iteration is rewritten from each segment as follows:

Where NZ _l represents the set of qubit strings of non-zero probabilities in P _l. Equation (11) has two advantages. First, its temporal complexity is linear with the magnitude of the non-zero probability NZ _l, which is typically lower than the number of samples in the readout. Second, matrix vector multiplication for each group can be performed independently, with time determined by the size of the sub-noise matrix, being a constant time.

The iterative characterization process and calibration process of the present invention operate as follows: the calibration procedure follows two iterations to calibrate the target probability distribution. In each iteration, a sub-noise matrix is first calculated according to equations (7) and (9). The grouping scheme g _l,k in equation (7) comes from the characterization flow. The conditional probability in equation (9) is calculated based on the reference probability distribution BP _l in the characterization flow. Next, the calibration procedure applies MVM according to equation (11). If there are more iterations remaining, it uses the MVM result for the next iteration.

The static grouping scheme and the calibration data set are based on the fact that the interaction between the qubits is constant. This consistency is true in current quantum hardware because current quantum operations (including readout) rely on the ability of the hardware to achieve a stable (constant) system hamiltonian. Physically, the interactions are determined after deployment and do not change unless the read-out frequency of the qubits changes. The purpose of the packet is to capture this inherent interaction. In general, a noise matrix describes the perturbation from an ideal probability distribution to a noise distribution, and finite element methods can be used to simulate such perturbation within a group of qubits, through a small sub-noise matrix. Different grouping schemes are then employed to capture interactions across groups.

It should be noted that the present invention has two features: 1) The reference probability distribution BP ₁ is a probability distribution obtained by running the reference circuit, and the other reference probability distributions BP _l in each iteration are updated with equation (11); 2) The sub-noise matrix is dynamically generated from the read measurement qubits in both the characterization and calibration processes to maximize fidelity, since the interactions always change under different combinations of measurement qubits. For the target quantum device, the calibration parameters (G _l,BP_l) are static.

Fig. 3 is a schematic diagram of quantum reading calibration for 2 iterations provided by an embodiment, in which a first iteration (iter=1) obtains grouping schemes G ₁.BP₁ and G ₁ based on a reference probability distribution BP ₁ to be stored for use in the calibration procedure. Each probability distribution of BP ₁ is then updated to construct BP ₂. The second iteration (iter=2) outputs the partition scheme G ₂ of qubits derived based on BP ₂. Finally, the characterization flow outputs cp= [ G ₁,G₂],[BP₁,BP₂ ], which are used as calibration parameters. In the calibration procedure, for any measured probability distribution P ₁ and measured qubit Q _M, the first iteration obtains a sub-noise matrix based on Q _M、G₁ and BP and updates P ₁ to P ₂ using equation (11). The second iteration obtains a sub-noise matrix based on Q _M、G₂ and BP ₃ in the same steps, and calibrations P ₂ to P ₃.P₃ are the resulting probabilities of these two iterative calibrations.

In an embodiment, the iterative calibration based on equation (11) involves tensor products of a number of MVM results with exponential complexity, whereas it is observed that the non-zero elements of the sub-noise matrix are mainly distributed around the diagonal. This sparsity will accumulate to a higher degree during the tensor product. A sparse tensor product engine is therefore proposed to accelerate the computation.

Specifically, each qubit group is first calculatedThen calculating the tensor product of the MVM results of all the quantum bit groups, then pruning the intermediate value in the tensor product through another threshold value beta, and finally polymerizing P _l (x) with the rest intermediate value in the tensor product to obtain the calibration output. FIG. 4 provides an example, ① for a non-zero probability (0.47) at state 000, engine calculation/>And/>MVM results of (a); ② Then calculating tensor products of the two MVM results; ③ The intermediate value of the tensor product is pruned by a threshold β. Similarly, the engine executes ①②③, for a non-zero probability (0.53) in state 011, and aggregates the pruned intermediate values by ④. The sparse tensor product engine chooses to prune over intermediate values rather than directly over the noise matrix so that higher characterization accuracy can be achieved. Furthermore, as the number of tensor product chains grows, the number of intermediate values pruned also increases, which in effect greatly reduces the computational complexity.

As shown in fig. 5, the embodiment further provides a quantum reading calibration system based on finite elements, which comprises a characterization flow module and a calibration flow module, wherein the characterization flow module and the calibration flow module are sequentially iterated to realize quantum reading calibration; the characterization flow module is used for determining the weight between any two quantum bits based on the reference probability distribution of the reference circuit in the current round and constructing a weighted quantum bit diagram, and grouping the quantum bits in the weighted quantum bit diagram based on a finite element thought to obtain a grouping scheme, wherein the reference probability distribution of the first round is obtained by sampling the output of the reference circuit, and the reference probability distribution of other rounds which are not the first round is obtained by calibrating output in the iterative process; the calibration flow module is used for calculating a sub-noise matrix corresponding to each quantum bit group based on the quantum bit set, the reference probability distribution and the grouping scheme, so as to obtain a sub-noise matrix set, and calibrating the calibration output of the previous round based on the sub-noise matrix set so as to obtain the calibration output of the current round.

It should be noted that, when the quantum reading calibration system based on finite elements provided in the foregoing embodiment performs quantum reading calibration, the division of each functional module should be illustrated, and the foregoing functional allocation may be performed by different functional modules according to needs, that is, the internal structure of the terminal or the server is divided into different functional modules, so as to complete all or part of the functions described above. In addition, the quantum reading calibration system based on finite elements provided in the above embodiment and the quantum reading calibration method based on finite elements belong to the same concept, and detailed implementation processes of the quantum reading calibration system based on finite elements are shown in the quantum reading calibration method based on finite elements, which are not described herein.

In the quantum reading calibration method and system provided by the invention, a comprehensive framework of measuring errors is calibrated by using a Finite Element Method (FEM), so that deeper understanding is provided for characterization calibration; meanwhile, the method for generating the reference circuit greatly reduces the characterization time and reduces the polynomial complexity (O (n ²)) unlike the characterization of exponential time complexity (O (2 ⁿ)); furthermore, the proposed sparse computation engine can speed up end-to-end calibration, which is proportional to the cube of the number of qubits only, i.e., O (n ³), as is the memory requirement.

It has also been shown in theory that the complexity of the invention with respect to the number of qubits does not exceed the polynomial complexity. Specifically, 4 quantum cloud platforms are spanned on a quantum device with 7 to 135 qubits, and a data set containing 2500 ten thousand real machine samples is collected. The results show that the present invention achieves a 1.8x10 ⁹ -fold acceleration in 136 qubit calibration and provides 1.2xand 1.4 x fidelity improvements over 18 qubit and 36 qubit real quantum devices, as compared to the most advanced matrix-based calibration techniques.

The foregoing detailed description of the preferred embodiments and advantages of the invention will be appreciated that the foregoing description is merely illustrative of the presently preferred embodiments of the invention, and that no changes, additions, substitutions and equivalents of those embodiments are intended to be included within the scope of the invention.

Claims

1. A quantum reading calibration method based on finite elements is characterized by comprising a plurality of iterative calibration, wherein each iterative calibration comprises a characterization flow and a calibration flow,

2. The finite element based quantum reading calibration method of claim 1, wherein the reference circuit is obtained by setting a threshold value, and comprises: and executing the candidate circuit, obtaining a measurement state of each quantum bit of the candidate circuit under a given ideal ground state, calculating an interaction of two quantum bits and an index for distinguishing key interaction based on the ideal ground state and the measurement state, and obtaining the reference circuit by screening the candidate circuit of which the index exceeds a set threshold value.

3. The finite element based quantum reading calibration method of claim 2, wherein each time a candidate circuit is performed, the operation and reading output of each qubit is recorded as a triplet (ideal, measured, ef) in which,Ideal ground state for recording preparation in a circuit, ideal=0 representing an ideal ground state bit value of 0, ideal=1 representing an ideal ground state bit value of 1,/>Indicating that the ideal ground state bit value is not measured, For recording the measured state output after a sample of the execution circuit, measured=0 representing a measured state bit value of 0, measured=1 representing a measured state bit value of 1,/>Indicating that the measured state bit value is not measured, ef e {0,1} indicating an error flag, ef=0 indicating that the measured state matches the ideal ground state, ef=1 indicating that an error occurred;

interact(q_i.ideal＝x→q_j.ideal＝y)＝p(q_j.ef＝1∣C1,C2)-p(q_j.ef＝1∣C2)

C1:q_i.ideal＝x；

C2:

Wherein num (. Fwdarw. Cndot.) represents the number of interactions.

4. A method of finite element based quantum reading calibration according to claim 3, wherein the first round of reference probability distribution is obtained for output samples of the reference circuit, comprising: acquiring a measurement state of each qubit in the reference circuit in an ideal ground state, recording a triplet, and calculating the conditional probability distribution of each qubit based on the triplet to serve as the reference probability distribution of the first round;

5. A finite element based quantum reading calibration method according to claim 3, wherein determining weights between any two qubits and constructing a weighted qubit map based on a reference probability distribution of a reference circuit at a current round comprises:

the weight (q _i,q_j) between any two qubits is expressed as:

In the constructed weighted qubit graph, the qubits are nodes, the edges between the nodes represent interactions between the qubits, and the weight values of the edges are marked by weight (q _i,q_j).

6. The finite element based quantum reading calibration method of claim 5, wherein grouping the quantum bit partitions in the weighted qubit map based on the finite element concept results in a grouping scheme comprising:

7. The finite element based quantum reading calibration method of claim 1, wherein calculating a sub-noise matrix for each group of qubits based on a set of quantum bits collected by measurement, a reference probability distribution, and a grouping scheme comprises:

Definition for each g _l,k: Where g _∩ represents the qubit overlapping g _l,k and Q _M,/> Representing the remaining qubits in g _l,k;

the sub-noise matrix corresponding to each qubit group is calculated as:

Where q represents a qubit belonging to g _∩, x _q represents a bit of the qubit q in the bit string x, P represents a conditional probability, estimated based on the reference probability distribution BP _l, q.measure=x _q represents that the measured state of q is x _q,g_∩. Ideal=y as a condition, represents that the ideal ground state of g _∩ is As a condition, express/>Is/>M _l,k [ x ] [ y ] represents the values of the x position and the y position in the sub-noise matrix M _l,k corresponding to g _l,k.

8. The finite element based quantum reading calibration method of claim 1, wherein calibrating the calibration output of the previous round based on the set of sub-noise matrices yields the calibration output of the current round, comprising:

9. The finite element based quantum reading calibration method of claim 8, wherein in the calibration flow, a sparse tensor product engine is introduced to prune tensor products in the calibration calculation process to perform calculation acceleration, comprising:

10. The quantum reading calibration system based on the finite element is characterized by comprising a characterization flow module and a calibration flow module, wherein the characterization flow module and the calibration flow module are sequentially iterated to realize quantum reading calibration;