CN108170646A - The reconstruction of quantum states method with sparse disturbance of Fast Convergent - Google Patents

The reconstruction of quantum states method with sparse disturbance of Fast Convergent Download PDF

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CN108170646A
CN108170646A CN201810106897.4A CN201810106897A CN108170646A CN 108170646 A CN108170646 A CN 108170646A CN 201810106897 A CN201810106897 A CN 201810106897A CN 108170646 A CN108170646 A CN 108170646A
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丛爽
胡志林
张娇娇
李克之
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University of Science and Technology of China USTC
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Abstract

The invention discloses a kind of reconstruction of quantum states methods with sparse disturbance of Fast Convergent, consider under quantum state constraints, utilize neighbour's gradient algorithm, the subproblem in relation to density matrix and sparse interference is solved hand to hand to obtain closing solution, tunable steps update Lagrange multiplier is used to accelerate convergence rate, the algorithm greatly reduces the restructing operation time under the premise of reconstruction accuracy is ensured, achievees the purpose that optimize quantum state restructing algorithm.

Description

Fast-convergence quantum state reconstruction method with sparse disturbance
Technical Field
The invention relates to the technical field of quantum state estimation, in particular to a fast-convergence quantum state reconstruction method with sparse disturbance.
Background
The state density matrix p of a quantum system with n qubits is d × d (d ═ 2) in hilbert spacen) Matrix with d × d ═ 2n×2n＝4nThe number of parameters, and therefore the quantum state parameters to be estimated, increases exponentially with the increase in n, in other words, a standard quantum state estimation requires O (d)2) The secondary measurement configuration. The quantum states of interest in practical experiments are often pure or nearly pure, where ρ is a hermitian with rank r low. With this a priori information, one applies the compressive sensing theory proposed in 2006 by Candes, Donaho et al to quantum state estimation: firstly, projecting an original signal to a low-dimensional space through a measurement matrix A; and then, an optimization problem is solved, and an original signal is accurately reconstructed from a small number of measured values. The compressive sensing theory reduces the number of measurements to O (rdlogd). In quantum state estimation based on compressive sensing, there are two important problems to be solved: 1) at least the number of measurements can be guaranteedThe low-rank RIP condition required by compressed sensing is met, so that enough information can be contained in a small amount of selected measurement data, and a density matrix rho is reconstructed; 2) an efficient and robust reconstruction algorithm needs to be designed so that a solution of a high-precision optimization problem can be achieved with the minimum measurement rate given by a compressive sensing theory.
For problem 1), one has concluded from the theory related to compressive sensing: when the number of measurements M satisfies the theoryWhen the lower bound condition is researched, the observation matrix A can meet the rank RIP theory with high probability. At this time, one can measure the number of times from the completely measured O (d)2) Reduced to O (rdlogd). For the problem 2), the convex optimization algorithm based on the quantum state estimation of the compressive sensing is difficult to effectively solve due to the large dimension of the related parameters. For example, Smith et al generalize the quantum state estimation problem as a least-squares problem (LS) or a compressive sensing problem and solve with a convex optimization toolkit, but for high-dimensional quantum systems, reconstruction is difficult to achieve as computation time and memory space increase. Li uses ADMM for solving the quantum state reconstruction problem based on compressive sensing for the first time, and obtains an algorithm which is fast and has robustness. However, in the Li algorithm, the computation of a large number of high-order matrix inversions leads to a long time consumption of the algorithm, and for example, in the case of a qubit n of 7, it takes approximately 3 hours to reconstruct a density matrix with 92.43% accuracy in an Intel Xeon E5-2407CPU, 2 cores, a main frequency of 2.4GHz, and a memory of 16G. In addition, in the optimization problem defined by Li, the special condition that the density matrix and the measured value simultaneously contain noise is considered, and under the condition of high-dimensional quantum state estimation, the reconstruction difficulty of the optimization algorithm based on the problem is increased, and the effect is poor.
Disclosure of Invention
The invention aims to provide a fast-convergence quantum state reconstruction method with sparse disturbance, which solves the subproblems related to density matrix and sparse disturbance by using a neighbor gradient algorithm in an approximation manner under the quantum state constraint condition to obtain a closed solution, and updates a Lagrangian multiplier by adopting an adjustable step length to accelerate the convergence speed.
The purpose of the invention is realized by the following technical scheme:
a fast-convergence quantum state reconstruction method with sparse perturbation comprises the following steps:
step 1, obtaining a measurement matrix A and a measurement vector b corresponding to the measurement matrix A;
step 2, initializing a density matrix rhokSparse matrix SkLagrange multiplier ykAnd the iteration number k is 1;
step 3, updating the density matrix rhokSparse matrix SkLagrange multiplier yk: adding quantum state constraint conditions to the subproblems of the density matrix to obtain the density matrix rho in the k +1 iterationk+1To obtain the density matrix p at the k +1 th iterationk+1(ii) a For the sparse interference correlation sub-problem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1(ii) a Combined pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1；
Step 4, judging whether the current state meets a preset stop condition; if yes, turning to step 5; if not, continuing the iterative operation in step 3;
step 5, performing density matrix rho in the k +1 th iterationk+1As an estimate of the density matrixAnd calculates the normalized density matrix estimation error.
A fast converging quantum state reconstruction system with sparse perturbations, comprising:
a measurement matrix and measurement vector obtaining module, configured to perform step 1: acquiring a measurement matrix A and a measurement vector b corresponding to the measurement matrix A;
an initialization module for performing step 2: initializing the density matrix ρkSparse matrix SkLagrange multiplier ykAnd the iteration number k is 1;
updating mouldA block for performing step 3: updating the density matrix ρkSparse matrix SkLagrange multiplier yk: adding quantum state constraint conditions to the subproblems of the density matrix to obtain the density matrix rho in the k +1 iterationk+1To obtain the density matrix p at the k +1 th iterationk+1(ii) a For the sparse interference correlation sub-problem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1(ii) a Combined pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1；
A judging module, configured to perform step 4: judging whether a preset stopping condition is met or not at present; if yes, turning to step 5; if not, continuing the iterative operation in step 3;
an estimated value determination and estimation error calculation module, configured to perform step 5: the density matrix rho at the k +1 th iterationk+1As an estimate of the density matrixAnd calculates the normalized density matrix estimation error.
The technical scheme provided by the invention can show that the method mainly has the following beneficial effects:
firstly, aiming at the quantum state estimation problem with sparse interference, the ADMM algorithm is applied to reconstruction of a high quantum bit state based on compressive sensing under the condition that the reconstructed density matrix meets the quantum state constraint condition, and the algorithm can find a more accurate solution more quickly.
Secondly, the ADMM subproblems are approximately solved by utilizing a neighbor gradient algorithm to obtain a closed solution and large-scale matrix operation, so that the operation complexity is greatly reduced.
Thirdly, the Lagrange multiplier is updated by adopting the adjustable step length, so that the convergence speed is increased, and meanwhile, the condition meeting the convergence of the algorithm is given, so that a basis is provided for parameter selection.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the description of the embodiments are briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on the drawings without creative efforts.
Fig. 1 is a flowchart of a fast converging quantum state reconstruction method with sparse perturbation according to an embodiment of the present invention;
FIG. 2 is a comparison result of the estimated errors of the reconstructed quantum state density matrix of the approximation ADMM algorithm (I-ADMM) of the present invention and the conventional ADMM and IST-ADMM algorithms at different sampling rates when the number of iterations provided by the embodiment of the present invention is 20;
FIG. 3 is a comparison of convergence rate of the approximation ADMM (I-ADMM) of the present invention with conventional ADMM and IST-ADMM algorithms at a sampling rate of 0.3 according to an embodiment of the present invention;
fig. 4 is a schematic diagram of a fast-converging quantum state reconstruction system with sparse perturbation according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention are clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.
The inventionThe embodiment provides a fast-convergence quantum state reconstruction method with sparse disturbance, and the method can accurately solve the quantum state estimation problem with sparse disturbance. Based on the compressive sensing theory, people can reconstruct a low-rank density matrix by solving an optimization problem and utilizing a small amount of measurement. The Alternating Direction Multiplier Method (ADMM) is an efficient algorithm to solve an optimization problem with a separable structure that splits the quantum state estimation problem into two sub-problems, one of which is an optimization problem with respect to the density matrix kernel norm and with quantum state constraints; the other is for l of the sparse interference matrix1Norm optimization problem, neither of which has a closed solution. The scheme of the invention is an approximate ADMM algorithm, the core idea is to approximately solve the subproblems about density matrix and sparse interference by adopting a first-order neighbor gradient method, and the gradient operation is carried out on a smooth least square term; neighbor operation in nuclear norm, l1Norm and an indicative function, to obtain a closed solution of the two subproblems. The scheme of the invention adopts the adjustable step length to update the Lagrange multiplier, thereby accelerating the convergence speed of the algorithm. The scheme of the invention provides conditions for ensuring the convergence of the algorithm and provides a basis for parameter selection. Scheme of the invention calculates the complexity to be O (Md)2) Compared with the existing ADMM algorithm and the existing IST-ADMM algorithm, the quantum state can be accurately reconstructed through fewer measurement times and iteration times.
As shown in fig. 1, the fast converging quantum state reconstruction method with sparse perturbation provided by the present invention mainly includes the following steps:
step 1, obtaining a measurement matrix A and a measurement vector b corresponding to the measurement matrix A.
In the embodiment of the invention, the measurement matrixM is the measurement times, d is the dimension of the quantum state to be estimated: d is 2nN is the number of quantum state bits;the representation of the complex field is represented by a complex field,representing a real number domain.
Step 2, initializing a density matrix rhokSparse matrix SkLagrange multiplier ykAnd the number of iterations k is 1.
In an embodiment of the present invention, a density matrix is initializedSparse matrixLagrange multiplier
Step 3, updating the density matrix rhokSparse matrix SkLagrange multiplier yk: adding quantum state constraint conditions to the subproblems of the density matrix to obtain the density matrix rho in the k +1 iterationk+1To obtain the density matrix p at the k +1 th iterationk+1(ii) a For the sparse interference correlation sub-problem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1(ii) a Combined pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1。
In the embodiment of the invention, the sub-problems about the density matrix and the sparse disturbance are approximately solved by adopting a neighbor gradient algorithm to obtain a closed solution of the sub-problems, which mainly comprises the following steps:
1) adding quantum state constraint conditions to the subproblems of the density matrix to obtain the density matrix rho in the k +1 iterationk+1To obtain the density matrix p at the k +1 th iterationk+1The process of (2) is as follows:
firstly, adding a constraint condition which must be satisfied by any quantum state rho:rho is more than or equal to 0 and tr (rho) is 1; wherein,to solve the conjugate transpose sign of the matrix; tr (-) is a trace-solving operation;
then, a proximity density matrix is calculatedWherein, tau1for the neighboring gradient step associated with the density matrix, vec (X) represents expanding matrix X by column into a column vector, α is a penalty parameter greater than 0;
then toDecomposing the characteristic value to obtain the characteristic value { aiAre further atUnder the condition of (1) solvingThereby obtaining the density matrix rho of the k +1 iterationk+1Characteristic value of { xi}; wherein d is the dimension of the quantum state to be estimated; for how to solve the problemX in (2)ian intermediate variable β may be set, in turn let β be aiI is 1,2, … d. Find out to satisfyAndthe value of (a) intot solves for the optimal value of β asfinally substituting β optimal value of beta to obtain xi＝max{ai-β,0}。
Finally, the density matrix ρ at the k +1 th iteration is calculated using the following equationk+1：
Wherein, diag { xiDenotes a characteristic value xiIs a diagonal matrix;is a unitary matrix of a plurality of discrete elements,representing a complex field.
2) For the sparse interference correlation sub-problem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1The process of (2) is as follows:
firstly, for the sparse interference correlation sub-problem, the adjacent gradient algorithm is utilized to convert the problem into:
wherein, tau2Is a neighboring gradient step associated with the sparse matrix, | |Frepresenting F norm, α is a punishment parameter larger than 0, gamma is larger than 0, and represents a weighted value;
then, a soft threshold shrink operator is introducedComputing a sparse matrix S at the k +1 th iterationk+1：
Wherein, for the variable s,
3) combined pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1The formula is as follows:
yk+1＝yk-κα(A(vec(ρk+1+Sk+1))-b)；
wherein, kappa is more than 0, and is a parameter for adjusting the update step length of the Lagrange multiplier.
In the embodiment of the invention, a matrix is providedMaximum eigenvalue is λmaxTo ensure convergence, let λ bemaxParameter k for adjusting update step length of Lagrange multiplier and adjacent gradient step length tau related to density matrix1And the adjacent gradient step τ associated with the sparse matrix2The following relationship is satisfied:
λmaxτ1< 1, and τ2λmax+κ＜2。
Step 4, judging whether the current state meets a preset stop condition; if yes, turning to step 5; if not, the step 3 is carried out to continue the iterative operation.
In the embodiment of the present invention, the predetermined stop condition is:
||A(vec(ρk+Sk))-b||2/||b||2＜10-7or k > kmax；
Wherein k ismaxIs the set maximum number of iterations.
Step 5, performing density matrix rho in the k +1 th iterationk+1As an estimate of the density matrixAnd calculates the normalized density matrix estimation error.
In the embodiment of the present invention, the normalized density matrix estimation error may be calculated by using the following formula:
wherein,is the system true density matrix.
To illustrate the superiority of the above-described scheme of the present invention, a comparison is also made with the conventional ADMM and IST-ADMM algorithms.
FIG. 2 shows the result of the variation of the estimation error with different measurement ratios when the density matrix of 5 qubit states is reconstructed by using the approximate ADMM (I-ADMM) of the present invention, the conventional ADMM and the IST-ADMM algorithm, respectively, at a maximum iteration number of 20, wherein the abscissa is the measurement ratio η, and the value range is [0, 0.5%](ii) a The ordinate is density matrix estimationAnd true valueNormalized distance between (system true density matrix)(normalized Density matrix)Estimation error), whereinThe lines with squares, the lines with circles and the lines with diamonds represent the experimental results using the approximation ADMM (I-ADMM), ADMM and IST-ADMM algorithms, respectively. The computer used for the experiment was: intel Xeon E5-2407CPU, 2 cores, main frequency 2.4GHz and memory 16G. From the experimental results shown in fig. 2, it can be seen that the final error of the normalized density matrix estimation using the scheme of the present invention is much smaller than that of the conventional algorithm.
FIG. 3 shows the density matrix estimates for a qubit n of 5 and a system sampling rate of 0.3 using the approximation ADMM, ADMM and IST-ADMM algorithms, respectivelyAnd the optimal solution ρ*Normalized distance betweenExperimental results varying with the number of iterations. Five lines are distinguished from the rightmost side from top to bottom: the first corresponds to the ADMM algorithm (i.e., the line with circles), the second IST-ADMM algorithm (i.e., the line with squares), and the third corresponds to the approximate ADMM (I-ADMM) algorithm provided by the present invention, where the parameter k is 1.4, τ2The fourth corresponds to the approximate ADMM (I-ADMM) algorithm provided by the present invention, where the parameter k is 1.0, τ2The fifth corresponds to the approximate ADMM (I-ADMM) algorithm provided by the present invention, with the parameter k being 0.6, τ21.399. It can also be seen from the results shown in fig. 3 that the final error of the normalized density matrix estimation using the scheme of the present invention is much smaller than that of the conventional algorithm.
The scheme of the embodiment of the invention mainly has the following beneficial effects:
firstly, aiming at the quantum state estimation problem with sparse interference, the ADMM algorithm is applied to reconstruction of a high quantum bit state based on compressive sensing under the condition that the reconstructed density matrix meets the quantum state constraint condition, and the algorithm can find a more accurate solution more quickly.
Secondly, the ADMM subproblems are approximately solved by utilizing a neighbor gradient algorithm to obtain a closed solution and large-scale matrix operation, so that the operation complexity is greatly reduced.
Thirdly, the Lagrange multiplier is updated by adopting the adjustable step length, so that the convergence speed is increased, and meanwhile, the condition meeting the convergence of the algorithm is given, so that a basis is provided for parameter selection.
Another embodiment of the present invention further provides a fast converging quantum state reconstruction system with sparse perturbation, as shown in fig. 4, which mainly includes:
a measurement matrix and measurement vector obtaining module, configured to perform step 1: acquiring a measurement matrix A and a measurement vector b corresponding to the measurement matrix A;
an initialization module for performing step 2: initializing the density matrix ρkSparse matrix SkLagrange multiplier ykAnd the iteration number k is 1;
an update module for performing step 3: updating the density matrix ρkSparse matrix SkLagrange multiplier yk: adding quantum state constraint conditions to the subproblems of the density matrix to obtain the density matrix rho in the k +1 iterationk+1To obtain the density matrix p at the k +1 th iterationk+1(ii) a For the sparse interference correlation sub-problem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1(ii) a Combined pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1；
A judging module, configured to perform step 4: judging whether a preset stopping condition is met or not at present; if yes, turning to step 5; if not, continuing the iterative operation in step 3;
an estimated value determination and estimation error calculation module, configured to perform step 5: when the (k + 1) th iteration is performedDensity matrix ρk+1As an estimate of the density matrixAnd calculates the normalized density matrix estimation error.
In the embodiment of the invention, quantum state constraint conditions are added into the subproblems of the density matrix to obtain the density matrix rho at the k +1 th iterationk+1To obtain the density matrix p at the k +1 th iterationk+1The process of (2) is as follows:
firstly, adding a constraint condition which must be satisfied by any quantum state rho:rho is more than or equal to 0 and tr (rho) is 1; wherein,to solve the conjugate transpose sign of the matrix; tr (-) is a trace-solving operation;
then, a proximity density matrix is calculatedWherein, tau1for the neighboring gradient step associated with the density matrix, vec (X) represents expanding matrix X by column into a column vector, α is a penalty parameter greater than 0;
then toDecomposing the characteristic value to obtain the characteristic value { aiAre further atUnder the condition of (1) solvingThereby obtaining the density matrix rho of the k +1 iterationk+1Characteristic value of { xi}; wherein d is the dimension of the quantum state to be estimatedCounting; for how to solve the problemX in (2)ian intermediate variable β may be set, in turn let β be aiI is 1,2, … d. Find out to satisfyAndsubstituting t into the value of t to solve the optimal value of beta asfinally substituting β optimal value of beta to obtain xi＝max{ai-β,0}。
Finally, the density matrix ρ at the k +1 th iteration is calculated using the following equationk+1：
Wherein, diag { xiDenotes a characteristic value xiIs a diagonal matrix;is a unitary matrix of a plurality of discrete elements,representing a complex field.
In the embodiment of the invention, for the sparse interference related subproblem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1The process of (2) is as follows:
firstly, for the sparse interference correlation sub-problem, the adjacent gradient algorithm is utilized to convert the problem into:
wherein, tau2Is a neighboring gradient step associated with the sparse matrix, | |Frepresenting F norm, α is a punishment parameter larger than 0, gamma is larger than 0, and represents a weighted value;vec (X) represents expanding the matrix X by column into a column vector;
then, a soft threshold shrink operator is introducedComputing a sparse matrix S at the k +1 th iterationk+1：
In the embodiment of the invention, rho is combinedk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1The formula of (1) is:
yk+1＝yk-κα(A(vec(ρk+1+Sk+1))-b)；
wherein vec (X) represents that the matrix X is expanded into a column vector according to columns, α is a penalty parameter larger than 0, and kappa is larger than 0 and is a parameter for adjusting the update step length of the Lagrange multiplier;
setting matrixMaximum eigenvalue is λmaxLet λ bemaxParameter k for adjusting update step length of Lagrange multiplier and adjacent gradient step length tau related to density matrix1And the adjacent gradient step τ associated with the sparse matrix2The following relationship is satisfied:
λmaxτ1< 1, and τ2λmax+κ＜2。
In the embodiment of the present invention, the predetermined stop condition is:
||A(vec(ρk+Sk))-b||2/||b||2＜10-7or k > kmax；
Where vec (X) denotes the column-wise expansion of the matrix X into a column vector, kmaxIs the set maximum number of iterations.
It should be noted that, specific implementation manners of functions implemented by the functional modules included in the system are described in detail in the foregoing embodiments, and therefore, detailed descriptions thereof are omitted here.
It will be clear to those skilled in the art that, for convenience and simplicity of description, the foregoing division of the functional modules is merely used as an example, and in practical applications, the above function distribution may be performed by different functional modules according to needs, that is, the internal structure of the system is divided into different functional modules to perform all or part of the above described functions.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (10)

1. A fast converging quantum state reconstruction method with sparse perturbation is characterized by comprising the following steps:
step 1, obtaining a measurement matrix A and a measurement vector b corresponding to the measurement matrix A;
step 2, initializing a density matrix rhokSparse matrix SkLagrange multiplier ykAnd the iteration number k is 1;
step 3, updating the density matrix rhokSparse matrix SkLagrange multiplier yk: adding to the sub-problem of the density matrixObtaining a density matrix rho of the k +1 th iteration under the quantum state constraint conditionk+1To obtain the density matrix p at the k +1 th iterationk+1(ii) a For the sparse interference correlation sub-problem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1(ii) a Combined pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1；
Step 4, judging whether the current state meets a preset stop condition; if yes, turning to step 5; if not, continuing the iterative operation in step 3;
step 5, performing density matrix rho in the k +1 th iterationk+1As an estimate of the density matrixAnd calculates the normalized density matrix estimation error.
2. The fast converging quantum state reconstruction method with sparse perturbation of claim 1, wherein a quantum state constraint condition is added to the subproblem of the density matrix to obtain the density matrix p at the k +1 th iterationk+1To obtain the density matrix p at the k +1 th iterationk+1The process of (2) is as follows:
firstly, adding a constraint condition which must be satisfied by any quantum state rho:rho is more than or equal to 0 and tr (rho) is 1; wherein,to solve the conjugate transpose sign of the matrix; tr (-) is a trace-solving operation;
then, a proximity density matrix is calculatedWherein, tau1for the neighboring gradient step associated with the density matrix, vec (X) represents expanding matrix X by column into a column vector, α is a penalty parameter greater than 0;
then toDecomposing the characteristic value to obtain the characteristic value { aiAre further atUnder the condition of (1) solvingThereby obtaining the density matrix rho of the k +1 iterationk+1Characteristic value of { xi}; wherein d is the dimension of the quantum state to be estimated; in solving forX in (2)iwhen the measured value is larger than the preset value, an intermediate variable beta is set, and the beta is sequentially set as aiI is 1,2, … d; find out to satisfyAndsubstituting t into the value of t to solve the optimal value of beta asfinally substituting β optimal value of beta to obtain xi＝max{ai-β,0}；
Finally, the density matrix ρ at the k +1 th iteration is calculated using the following equationk+1：
Wherein, diag { xi } represents a characteristic value { xiIs a diagonal matrix;is a unitary matrix of a plurality of discrete elements,representing a complex field.
3. The fast converging quantum state reconstruction method with sparse perturbation according to claim 1, wherein for sparse interference related subproblems, a neighboring gradient algorithm is used to transform the problems, and a soft threshold shrinking operator is introduced, so as to obtain a sparse matrix S at the k +1 th iterationk+1The process of (2) is as follows:
firstly, for the sparse interference correlation sub-problem, the adjacent gradient algorithm is utilized to convert the problem into:
wherein, tau2Is a neighboring gradient step associated with the sparse matrix, | |Frepresenting F norm, α is a punishment parameter larger than 0, gamma is larger than 0, and represents a weighted value;vec (X) represents expanding the matrix X by column into a column vector;
then, a soft threshold shrink operator is introducedComputing a sparse matrix S at the k +1 th iterationk+1：
4. The method of claim 1, wherein the fast convergence method comprises reconstructing the quantum state with sparse perturbationIn combination with pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1The formula of (1) is:
yk+1＝yk-κα(A(vec(ρk+1+Sk+1))-b)；
wherein vec (X) represents that the matrix X is expanded into a column vector according to columns, α is a penalty parameter larger than 0, and kappa is larger than 0 and is a parameter for adjusting the update step length of the Lagrange multiplier;
setting matrixMaximum eigenvalue is λmaxLet λ bemaxParameter k for adjusting update step length of Lagrange multiplier and adjacent gradient step length tau related to density matrix1And the adjacent gradient step τ associated with the sparse matrix2The following relationship is satisfied:
λmaxτ1< 1, and τ2λmax+κ＜2。
5. A fast converging quantum state reconstruction method with sparse perturbations as claimed in any one of claims 1-4, wherein the predetermined stopping conditions are:
||A(vec(ρk+Sk))-b||2/||b||2＜10-7or k > kmax；
Where vec (X) denotes the column-wise expansion of the matrix X into a column vector, kmaxIs the set maximum number of iterations.
6. A fast converging quantum state reconstruction system with sparse perturbation, comprising:
a measurement matrix and measurement vector obtaining module, configured to perform step 1: acquiring a measurement matrix A and a measurement vector b corresponding to the measurement matrix A;
an initialization module for performing step 2: initializing the density matrix ρkSparse matrix SkLagrange multiplier ykAnd number of iterations k＝1；
An update module for performing step 3: updating the density matrix ρkSparse matrix SkLagrange multiplier yk: adding quantum state constraint conditions to the subproblems of the density matrix to obtain the density matrix rho in the k +1 iterationk+1To obtain the density matrix p at the k +1 th iterationk+1(ii) a For the sparse interference correlation sub-problem, the problem is converted by using a neighboring gradient algorithm, and a soft threshold shrinkage operator is introduced, so that a sparse matrix S in the k +1 th iteration is obtainedk+1(ii) a Combined pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1；
A judging module, configured to perform step 4: judging whether a preset stopping condition is met or not at present; if yes, turning to step 5; if not, continuing the iterative operation in step 3;
an estimated value determination and estimation error calculation module, configured to perform step 5: the density matrix rho at the k +1 th iterationk+1As an estimate of the density matrixAnd calculates the normalized density matrix estimation error.
7. The fast converging quantum state reconstruction system with sparse perturbation of claim 6, wherein a quantum state constraint condition is added to the subproblem of the density matrix to obtain the density matrix p at the k +1 th iterationk+1To obtain the density matrix p at the k +1 th iterationk+1The process of (2) is as follows:
firstly, adding a constraint condition which must be satisfied by any quantum state rho:rho is more than or equal to 0 and tr (rho) is 1; wherein,to solve the conjugate transpose sign of the matrix;
then, a proximity density matrix is calculatedWherein, tau1for the neighboring gradient step associated with the density matrix, vec (X) represents expanding matrix X by column into a column vector, α is a penalty parameter greater than 0;
then toDecomposing the characteristic value to obtain the characteristic value { aiAre further atUnder the condition of (1) solvingThereby obtaining the density matrix rho of the k +1 iterationk+1Characteristic value of { xi}; wherein d is the dimension of the quantum state to be estimated; in solving forX in (2)iwhen the measured value is larger than the preset value, an intermediate variable beta is set, and the beta is sequentially set as aiI is 1,2, … d; find out to satisfyAndsubstituting t into the value of t to solve the optimal value of beta asfinally substituting β optimal value of beta to obtain xi＝max{ai-β,0}；
Finally, the density matrix ρ at the k +1 th iteration is calculated using the following equationk+1：
Wherein, diag { xiDenotes a characteristic value xiIs a diagonal matrix;is a unitary matrix of a plurality of discrete elements,representing a complex field.
8. The fast converging quantum state reconstruction system with sparse perturbation of claim 6, wherein for sparse interference related subproblems, the problem is transformed by using a neighboring gradient algorithm, and a soft threshold shrinking operator is introduced, thereby obtaining a sparse matrix S at the k +1 th iterationk+1The process of (2) is as follows:
firstly, for the sparse interference correlation sub-problem, the adjacent gradient algorithm is utilized to convert the problem into:
wherein, tau2Is a neighboring gradient step associated with the sparse matrix, | |Frepresenting F norm, α is a punishment parameter larger than 0, gamma is larger than 0, and represents a weighted value;vec (X) represents expanding the matrix X by column into a column vector;
then, a soft threshold shrink operator is introducedComputing a sparse matrix S at the k +1 th iterationk+1：
9. The fast converging sparse perturbation system of claim 6, in combination with pk+1And Sk+1Computing Lagrange multiplier y at the k +1 iterationk+1The formula of (1) is:
yk+1＝yk-κα(A(vec(ρk+1+Sk+1))-b)；
wherein vec (X) represents that the matrix X is expanded into a column vector according to columns, α is a penalty parameter larger than 0, and kappa is larger than 0 and is a parameter for adjusting the update step length of the Lagrange multiplier;
setting matrixMaximum eigenvalue is λmaxLet λ bemaxParameter k for adjusting update step length of Lagrange multiplier and adjacent gradient step length tau related to density matrix1And the adjacent gradient step τ associated with the sparse matrix2The following relationship is satisfied:
λmaxτ1< 1, and τ2λmax+κ＜2。
10. A fast converging sparse perturbed quantum state reconstruction system according to any of claims 6 to 9, wherein the predetermined stopping condition is:
||A(vec(ρk+Sk))-b||2/||b||2＜10-7or k > kmax；
Where vec (X) denotes the column-wise expansion of the matrix X into a column vector, kmaxIs the set maximum number of iterations.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111062482A (en) * 2019-11-28 2020-04-24 南方科技大学 Quantum state reconstruction method, device and system and storage medium
CN113222160A (en) * 2020-01-21 2021-08-06 合肥本源量子计算科技有限责任公司 Quantum state conversion method and device

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111062482A (en) * 2019-11-28 2020-04-24 南方科技大学 Quantum state reconstruction method, device and system and storage medium
CN111062482B (en) * 2019-11-28 2023-07-04 南方科技大学 Quantum state reconstruction method, device, system and storage medium
CN113222160A (en) * 2020-01-21 2021-08-06 合肥本源量子计算科技有限责任公司 Quantum state conversion method and device
CN113222160B (en) * 2020-01-21 2023-08-08 本源量子计算科技（合肥）股份有限公司 Quantum state conversion method and device

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