Bayesian process tensor chromatography method based on non-Markov noise

Abstract

The invention discloses a Bayesian process tensor chromatography method based on non-Markov noise, and belongs to the technical field of quantum information. The method is realized through the following steps: setting initial parameters of a quantum circuit, including a time step length, a sample number, an initial mean value and a covariance matrix; performing multi-dimensional normal distribution sampling on the process tensor; traversing the sample and calculating a likelihood function after complete positive and causal projection; and screening samples by using rejection filtering, and updating the mean value and the covariance matrix until convergence. According to the method, the rejection filtering technology is introduced to replace traditional Bayesian integral operation, the calculation complexity is remarkably reduced, meanwhile, the overfitting problem of maximum likelihood estimation is avoided, the accuracy and efficiency of process tensor estimation are effectively improved, and the method is suitable for quantum noise modeling and error correction optimization.

Description

Technical Field

[0001] The present invention relates to the field of quantum information technology, and specifically to a Bayesian process tensor tomography method based on non-Markovian noise, which is used to efficiently model non-Markovian noise in quantum circuits and optimize quantum error correction and device performance. Background Art

[0002] With the development of quantum technology, compared with classical computers, quantum computers can perform more complex and groundbreaking work in the future. However, the noise existing in quantum circuits will seriously affect the scale of executable quantum circuits, restricting the size, depth, and number of qubits of the circuits. According to whether there is a memory effect in the environment where the quantum circuit is located, the noise can be divided into Markovian noise and non-Markovian noise. Markovian noise has no memory effect, and the state of the environment is not correlated before and after any moment; non-Markovian noise is caused by the coupling between the quantum circuit and the environment, and the state of the environment is correlated before and after any moment. An idealized quantum circuit is closed and only has Markovian noise. However, in a real physical environment, all quantum circuits are open, and there is an interaction between the circuit and the environment, indicating that both Markovian noise and non-Markovian noise exist in the system. Characterizing non-Markovian noise helps to understand the noise characteristics of quantum circuits, design effective error-correcting codes, and further optimize the performance of quantum devices. Therefore, how to efficiently characterize non-Markovian noise is an important issue.

[0003] Process tensor tomography can be used to determine the unknown initial state and coupling process of the circuit-environment, mapping all possible time correlations to spatial correlations, and is a method for describing non-Markovian noise. Among them, the process tensor is a 2 2k ×2 2kThe complex matrix needs to satisfy complete positivity and causality. Here, k is the time step at which the quantum circuit evolution process is divided. The existing maximum likelihood estimation - process tensor tomography and Bayesian process tensor tomography are the two main methods for solving the process tensor. Maximum likelihood estimation is a method based on existing measurement data. By optimizing the likelihood function, the prediction results are fitted to the limited measurement data to obtain the unknown parameters. Although the execution steps of this method are relatively simple, it is highly dependent on training data and has the problem of overfitting, which is not applicable to quantum systems. Because when measuring the output quantum state, there is a certain probability of obtaining results that did not appear in the previous training set each time, and maximum likelihood estimation will assign zero probability to these results that did not appear in the training set during training, which easily leads to an increase in the measured probability error. Bayesian process tensor tomography calculates and updates the probability distribution of the process tensor through Bayes' formula, which can take into account more situations and reduce the impact brought by the uncertainty of measurement results. However, there are also certain limitations in the flexibility of Bayesian estimation. In classical Bayesian estimation, the posterior probability of the process tensor needs to be obtained through matrix integral operations, which means that achieving this goal requires extremely high computational complexity and exponential - level computational resources, making it difficult to apply this method under existing conditions.

[0004] In summary, in the prior art, maximum likelihood estimation - process tensor tomography depends on limited measurement data and is prone to overfitting; although the Bayesian method can reduce the impact of uncertainty, it requires high - complexity matrix integral operations and is difficult to be practical. The complexity of non - Markovian noise further exacerbates the limitations of traditional methods. The present invention combines rejection filtering and the Bayesian framework to propose an efficient and low - complexity process tensor estimation method, which solves the above problems. Summary of the Invention

[0005] Object of the Invention: To solve the overfitting problem existing in the existing maximum likelihood estimation - process tensor tomography and the matrix integral operation problem in Bayesian process tensor tomography, the present invention provides a Bayesian process tensor tomography method based on rejection filtering, which can reduce the computational complexity on the basis of accurately estimating the probability distribution of the process tensor.

[0006] Technical Solution: To achieve the above object, the present invention proposes a Bayesian process tensor tomography method based on non - Markovian noise. The technical solution adopted is as follows:

[0007] A Bayesian process tensor tomography method based on non - Markovian noise includes the following steps:

[0008] (1) Add an auxiliary qubit to simulate the environment where the quantum circuit is located, and set the initial parameters and convergence conditions for the n - qubit quantum circuit;

[0009] (2) Perform multi - dimensional normal distribution sampling on the process tensor to generate a sample set

[0010] (3) Traverse the sample set, perform full positivity and causality projections on each sample, and calculate the corresponding likelihood function;

[0011] (4) Use rejection filtering to process the likelihood function, screen the samples that meet the conditions, and update the mean and covariance matrix;

[0012] (5) If the estimation error does not reach the convergence condition, return to step (2) to continue the iteration, otherwise output the optimized process tensor estimation.

[0013] Preferably, the initial parameters and convergence conditions to be set in step (1) include:

[0014] The time step k for quantum circuit partitioning, which is used to determine the structure of the quantum circuit and the dimension of the process tensor;

[0015] The number of samples m, the initial mean μ, and the covariance matrix ∑, which are used to collect process tensor samples;

[0016] Use Pauli operators As an informationally complete basis for quantum gate operations in Choi form n is the number of qubits, Is the tensor product, and the Pauli operator is expressed as

[0017]

[0018] Combined with the projection operator Of the Choi form Where

[0019]

[0020] Calculate the output quantum state probability distribution corresponding to each sample

[0021]

[0022] Measure the quantum circuit to obtain the measurement result set Used to calculate the likelihood function corresponding to each sample;

[0023] Set the estimation error threshold ε, which is used to determine the convergence condition.

[0024] Preferably, the likelihood function in step (3) is in the following form:

[0025]

[0026] Where Is the probability corresponding to the projection operator E j Of, n jIt is the projection to obtain E j The frequency of

[0027] Preferably, the rejection filter in step (4) is in the following form:

[0028] For each likelihood function, randomly sample a constant u from the uniform distribution on (0, 1):

[0029] u ∼ Uniform(0, 1)

[0030] If

[0031] P(E|Υ i ; μ, ∑) / k c ≥ u

[0032]

[0033] Then accept the sample Υ i .

[0034] Preferably, the complete positivity and causality projection includes: ensuring its physical realizability by constraining the process tensor to be a completely positive definite matrix and satisfying the causal time sequence relationship.

[0035] Preferably, the determination condition of the estimation error threshold is: the Euclidean distance of the process tensor means in adjacent iterative steps is less than or equal to the estimation error threshold ε.

[0036] The present invention also provides a quantum noise modeling and error correction optimization system, which uses the method described above to estimate the process tensor of non-Markovian noise, and optimizes the design of quantum error correction codes and the device performance regulation based on this process tensor.

[0037] Beneficial effects: The Bayesian process tensor tomography technology based on non-Markovian noise provided by the present invention directly updates the probability distribution of the process tensor by using Bayesian rules and rejection filtering. Compared with the classical Bayesian estimation, it does not need to perform integral operations, greatly reducing the computational complexity. In addition, compared with the maximum likelihood estimation - process tensor tomography, the present invention can reduce the influence brought by the measurement uncertainty and obtain a more accurate process tensor estimation. Description of the Drawings

[0038] Figure 1 It is the workflow diagram of the present invention;

[0039] Figure 2 It is the specific implementation manner of the present invention. Specific Implementation Manner

[0040] The following further describes the present invention in conjunction with the drawings and embodiments.

[0041] Figure 1This is the workflow diagram corresponding to the present invention. The specific steps include:

[0042] (1) Add an auxiliary qubit to simulate the environment where the quantum circuit is located, and set appropriate initial parameters and convergence conditions for the n-qubit quantum circuit;

[0043] (2) Perform multi-dimensional normal distribution sampling on the process tensor to obtain a set

[0044] (3) Traverse the process tensors sampled in step (2), perform complete positivity and causality projections on each sample, and calculate the corresponding likelihood function;

[0045] (4) Use rejection filtering to process the likelihood function obtained in step (3), recalculate the mean and covariance matrix for the samples that pass the screening. If the convergence condition is reached, output the estimated process tensor distribution; otherwise, return to step (2) and continue the next iteration.

[0046] The initial parameters and convergence conditions to be set in step (1) include:

[0047] The time step k for quantum circuit partitioning, which is used to determine the structure of the quantum circuit and the dimension of the process tensor;

[0048] The number of samples m, the initial mean μ, and the covariance matrix ∑, which are used to collect process tensor samples;

[0049] Use the Pauli operator as the informationally complete basis for the quantum gate operation in Choi form n is the number of qubits, is the tensor product, and the Pauli operator is expressed as

[0050]

[0051] Combined with the projection operator in Choi form where

[0052]

[0053] Calculate the output quantum state probability distribution corresponding to each sample

[0054]

[0055] Measure the quantum circuit to obtain a set of measurement results which is used to calculate the likelihood function corresponding to each sample;

[0056] The estimated error value ε, which is used to represent the maximum error that can be tolerated between the estimated probability distribution of the output quantum state and the actual probability distribution.

[0057] The likelihood function in step (3) is in the following form:

[0058]

[0059] where is the probability corresponding to the projection operator E j and n j is the frequency of projecting to obtain E j .

[0060] The rejection filtering in step (4) is in the following form:

[0061] For each likelihood function, a constant u is randomly sampled from the uniform distribution on (0, 1):

[0062] u ∼ Uniform(0, 1)

[0063] If

[0064] P(E|Υ i ; μ, ∑) / k c ≥ u

[0065]

[0066] then accept the sample γ i .

[0067] Embodiment 1

[0068] Figure 2 The following shows the specific implementation process of applying the present invention to characterize non - Markovian noise on a 1 - qubit circuit, including the following steps:

[0069] (1) Add an auxiliary qubit to simulate the environment where the quantum circuit is located, and set appropriate initial parameters and convergence conditions for the n - qubit quantum circuit. In this embodiment, the number of qubits of the quantum circuit is 1, and the specific parameter settings are as follows:

[0070] Set the time step k of the quantum circuit to 2, then the process tensor is a 16×16 matrix;

[0071] The number of samples m = 10000, the initial mean μ = I, and the initial covariance matrix

[0072] ∑ = diag(0.1, 0.1,..., 0.1);

[0073] When the number of qubits of the quantum circuit is 1, 16 Pauli operators are required, which can be expressed as

[0074] {II, IX, IY, IZ, XI, XX, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ};

[0075] Perform 25 measurements on the quantum circuit to obtain the measurement result set E;

[0076] Estimate the error value ε = 0.0001.

[0077] (2) Perform multi-dimensional normal distribution sampling on the process tensor to obtain a set with a sample number m = 10000

[0078] (3) Traverse the process tensors sampled in step (2), perform complete positivity and causality projection on each sample, and calculate the corresponding likelihood function:

[0079]

[0080] (4) Use rejection filtering to process the likelihood function obtained in step (3):

[0081] For each likelihood function, randomly sample a constant u from the uniform distribution (0, 1):

[0082] u ∼ Uniform(0, 1)

[0083] If

[0084] P(E|γ i ; μ, ∑) / k c ≥ u

[0085]

[0086] Then accept the sample γ i .

[0087] Recalculate the mean and covariance matrix for the samples that pass the screening. If the estimated error value ε' ≤ ε, it is proved that the convergence condition is reached, and the estimated process tensor probability distribution is output. Otherwise, return to step (2) and continue the next iteration. The final process tensor mean estimate obtained in this embodiment is as follows:

[0088]

[0089] Embodiment 2

[0090] This embodiment provides a quantum noise modeling and error correction optimization system, which uses the described method to estimate the process tensor of non-Markov noise and optimizes the design of quantum error correction codes and the device performance regulation based on this process tensor.

[0091] This system mainly consists of a data acquisition module, a process tensor estimation module, a quantum error correction code design optimization module, and a device performance regulation module.

[0092] Data acquisition module: Responsible for collecting various data during the operation of the quantum system, including the state changes of qubits, environmental parameters, etc.

[0093] Process tensor estimation module: Using the method described above, estimate the process tensor of non-Markovian noise using the collected data.

[0094] Quantum error correction code design and optimization module: Based on the estimated process tensor, design and optimize the quantum error correction code.

[0095] Device performance regulation module: According to the optimization results of the quantum error correction code, regulate the performance of the quantum device.

[0096] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention.

Claims

1. A Bayesian process tensor tomography method based on non-Markov noise, characterized in that: The following steps are involved: (1) Add an auxiliary quantum bit to simulate the environment of the quantum circuit and set the initial parameters and convergence conditions for the n-bit quantum circuit; (2) Perform multidimensional normal distribution sampling on the process tensor to generate a sample set (3) Traverse the sample set, perform complete positive and causal projection on each sample, and calculate the corresponding likelihood function; (4) using rejection filtering to process the likelihood function, screening samples that meet the conditions, and updating the mean and covariance matrix; (5) If the estimated error does not meet the convergence condition, return to step (2) to continue iterating, otherwise output the optimized process tensor estimate.

2. The method according to claim 1, characterized in that The initial parameters in step (1) include: The time step k of the quantum circuit partition is used to determine the dimension of the process tensor; Sample size m, initial mean μ and covariance matrix Σ; Using Pauli operators as the information-complete basis for quantum gate operations, we construct the Choi form of the projection operator. Measure the quantum circuit and get a set of measurement results x i represents the i-th measurement result of the quantum circuit, which is used to calculate the likelihood function of the sample; Set the estimation error threshold ε to determine the convergence condition.

3. The method according to claim 1, characterized in that The calculation formula of the likelihood function in step (3) is: in is the corresponding projection operator E j The probability of n j The projection is E j frequency.

4. The method according to claim 3, characterized in that The specific operation of rejection filtering in step (4) is: For each sample's likelihood function value, generate a uniformly distributed random number u: u~Uniform(0,1) If the conditions are met: P(E|γ i ;μ,∑) / k c ≥u Then accept the sample Υ i , otherwise reject; The mean μ and covariance matrix Σ are recalculated through the screened samples for iterative optimization.

5. The method according to claim 1, characterized in that: The completely positive and causal projection includes: ensuring its physical feasibility by constraining the process tensor to be a completely positive definite matrix and satisfying the causal temporal relationship.

6. The method according to claim 2, characterized in that The Choi form of the Pauli operator is defined as: where I, X, Y, and Z are components of the Pauli operator.

7. The method according to claim 2, characterized in that: The determination condition of the estimation error threshold is: the Euclidean distance of the process tensor means in adjacent iteration steps is less than or equal to the estimation error threshold ε.

8. A quantum noise modeling and error correction optimization system, characterized in that: The method described in any one of claims 1 to 7 is used to estimate the process tensor of non-Markov noise, and the design of quantum error correction code and device performance regulation are optimized based on the process tensor.

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