Compressed quantum state chromatography method based on priori knowledge

Abstract

The invention discloses a compressed quantum state chromatography method based on priori knowledge. The compressed quantum state chromatography method comprises the following specific steps: (1) setting proper parameters; (2) randomly sampling a Source operator; (3) traversing all sampled Fouli operators, constructing corresponding quantum measurement, and estimating an expected value of a target quantum state under the operator; and (4) processing the obtained expected value by using an optimization function containing priori knowledge, finally obtaining the estimation of the quantum state, and outputting the estimation. According to the compressed quantum state chromatography technology provided by the invention, based on the low-rank property of the target quantum state, the accurate estimation of the target quantum state is obtained by using the convex optimization function containing the prior knowledge of the target quantum state, and the quantum resources required by the quantum chromatography are further reduced.

Description

Technical Field

[0001] The present invention relates to quantum information technology, and in particular to a method for efficiently acquiring quantum state information. Background Art

[0002] Quantum computing, as the core of the next generation of computing technology, has become the key driving force for a new round of industrial transformation. Related technologies are developing at a rapid pace. More and more emerging quantum technologies are emerging, quantum hardware technology is also improving year by year, and at the same time, the fields of quantum communication and quantum Internet are also continuing to grow and develop. In this context, quantum state tomography is crucial to verify the performance of existing noisy intermediate-scale quantum computing (NISQ) devices. By measuring the state of the actual system and comparing it with the expected state, researchers can identify and correct errors, thereby improving the reliability of quantum computing. However, for large quantum systems, traditional quantum state tomography technology consumes exponential quantum resources, which limits its feasibility in practical applications. Therefore, how to efficiently perform large-scale quantum state tomography is an important problem that needs to be solved urgently.

[0003] The state of an n-bit quantum system can be described using a density matrix ρ. The density matrix is ​​a semidefinite matrix that contains all the information about the quantum system. The goal of quantum state tomography is to reconstruct the quantum state ρ from a series of measurement results. Specifically, a set of measurement operations is performed, using a set of measurement operators {M i} i To represent these operations. When the quantum state ρ is completely unknown, it is necessary to ensure that this set of measurement operators satisfies the information completeness condition, so that a series of expected values ​​{Tr[M i ρ]} i (calculated from the measurement results) to reconstruct the target quantum state. However, to meet the information completeness condition, at least 4 n different measurement operators, which means that an exponential amount of quantum resources are required to achieve this, making the technique only applicable to a small number of qubits.

[0004] The existing quantum state tomography method via compressed sensing can significantly reduce the required measurement resources by effectively utilizing the low-rank nature of quantum states. Specifically, it assumes that the target n-bit quantum state ρ is low-rank, that is, the rank of the quantum state ρ is less than a given value r and satisfies r<<d, where d=2 n is the dimension of the system where the quantum state is located. The low-rank assumption allows accurate quantum state reconstruction results to be obtained with fewer measurements. However, this method has the following disadvantages: it still consumes a lot of resources, requiring a total of O(rd log 2 d) The measurement operator of the order of magnitude, where O(·) is the big-O notation in complexity theory. This method is not applicable to large-scale quantum devices. The present invention introduces prior knowledge into the compressed quantum state tomography technology, where the prior knowledge here refers to the density matrix of the output quantum state corresponding to the quantum device that realizes a certain function under ideal conditions. The compressed quantum state tomography technology based on prior knowledge proposed in the present invention can effectively reduce the measurement operator and obtain an estimate of the target quantum state, which is suitable for large-scale quantum devices. Summary of the invention

[0005] Technical problem: In order to further reduce the number of measurement operators required for quantum state tomography, the present invention provides a compressed quantum state tomography method based on prior knowledge, which can further reduce the number of measurement operators required for quantum state tomography tasks on the basis of existing compressed quantum state tomography. The prior knowledge here refers to: the matrix information of the ideal quantum state Φ is fully known, and the noisy quantum state ρ is very close to the ideal quantum state Φ.

[0006] Technical solution: To achieve the above purpose, the present invention proposes a compressed quantum state tomography method based on prior knowledge, and the technical solution adopted is:

[0007] A compressed quantum state tomography method based on prior knowledge comprises the following steps:

[0008] 1) Device for generating n-bit quantum states Set appropriate parameters;

[0009] 2) Randomly sample Pauli operators to obtain the set Ω;

[0010] 3) Traverse the Pauli operators sampled in step (2), construct the corresponding quantum measurement, and measure the expectation value of the noisy quantum state ρ under the operator;

[0011] 4) Using an optimization function containing prior knowledge to process the expected value obtained in step (3), finally obtaining an estimate of the quantum state, and outputting the estimate.

[0012] Appropriate parameters to be set in step 1) include:

[0013] Making devices based on quantum states Based on the prior knowledge, the density matrix Φ, reliability λ, and rank r of the output quantum state corresponding to the device under ideal conditions are set.

[0014] The estimated error value ε is used to indicate the tolerable range of the estimated error.

[0015] The number of randomly sampled Pauli operators N, which is usually O(r2 n ) order of magnitude.

[0016] The measurement error value δ is used to indicate the tolerable measurement error range.

[0017] The Pauli operator in step 2) is specifically: for an n-bit quantum state ρ, there are 4 n Pauli operator, which can be expressed as in is the tensor product,

[0018]

[0019] Randomly sample N Pauli operators, and record the sampled Pauli operator set as Ω = {P i}.

[0020] In step 3), all Pauli operators in the set Ω are traversed. For the i-th Pauli operator P obtained by sampling i , construct the corresponding quantum measurement, measure the quantum state ρ total M ~ 4n / ε 2 times, and use the measurement results to estimate the expected value Tr[P i ρ], record the estimated result as y i .

[0021] The optimization function in step 4) is as follows:

[0022]

[0023] Where σ is the quantum state to be optimized, ||A|| * is the trace norm of matrix A, λ represents the degree of trust in prior knowledge Φ, δ is the measurement error value, y i is the expected value Tr[P i The estimation of ρ].

[0024] Beneficial effects: The present invention provides a compressed quantum state tomography technology based on prior knowledge, which uses prior knowledge to perform compressed quantum state tomography to obtain target quantum state estimation. Compared with the existing quantum state tomography method based on compressed sensing, it greatly reduces the number of quantum operators and further reduces the required quantum resources, and is suitable for large-scale quantum devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 It is a work flow chart of the present invention;

[0026] Figure 2 It is a specific embodiment of the present invention. DETAILED DESCRIPTION

[0027] The present invention will be further described below in conjunction with the accompanying drawings.

[0028] Figure 1 The following is a workflow diagram corresponding to the present invention, wherein the specific steps include:

[0029] 1) Device for generating n-bit quantum states Set appropriate parameters; the appropriate parameters to be set include:

[0030] Making devices based on quantum states Based on the prior knowledge, the density matrix Φ, reliability λ, and rank r of the output quantum state corresponding to the device under ideal conditions are set.

[0031] The estimated error value ε is used to indicate the tolerable range of the estimated error.

[0032] The number of randomly sampled Pauli operators N, which is usually O(r2 n ) order of magnitude.

[0033] The measurement error value δ is used to indicate the tolerable measurement error range.

[0034] 2) Randomly sample the Pauli operator to obtain the set Ω; for the n-bit quantum state ρ, there are 4 n Pauli operator, which can be expressed as in is the tensor product,

[0035]

[0036] Randomly sample N Pauli operators, and record the sampled Pauli operator set as Ω = {P i}.

[0037] 3) Traverse the Pauli operators sampled in step (2), construct the corresponding quantum measurement, and measure the expected value of the noisy quantum state ρ under the operator; for the i-th Pauli operator P obtained by samplingi , construct the corresponding quantum measurement, measure the quantum state ρ total M ~ 4n / ε 2 times, and use the measurement results to estimate the expected value Tr[P i ρ], record the estimated result as y i .

[0038] 4) Use the optimization function containing prior knowledge to process the expected value obtained in step (3), finally obtain an estimate of the quantum state, and output the estimate. The optimization function is in the following form:

[0039]

[0040] Where σ is the quantum state to be optimized, ‖A‖ * is the trace norm of matrix A, λ represents the degree of trust in prior knowledge Φ, δ is the measurement error value, y i is the expected value Tr[P i The estimation of ρ].

[0041] Figure 2 The specific implementation process of the present invention applied to the preparation of a quantum device in a 3-qubit GHZ state is shown, including the following steps:

[0042] (1) Setting appropriate parameters for the target quantum device. It is a 3-qubit GHZ state generation device, and its specific parameters are set as follows:

[0043] Based on the prior knowledge of the 3-qubit GHZ state generation device, set the density matrix Φ, reliability λ, and rank r of the output quantum state corresponding to the device under ideal conditions. In this embodiment, the specific setting value is: Φ = |ψ><ψ|, where Reliability λ=0.8, rank r=1.

[0044] The estimated error value ε=0.01 is used to indicate the tolerable range of the estimated error.

[0045] The number of randomly sampled Pauli operators N=20.

[0046] ·Measurement error value δ = 0.05.

[0047] (2) Randomly sampling Pauli operators to obtain a set Ω. Randomly sample N=20 Pauli operators from a 3-bit Pauli operator set to form a set Ω. In this embodiment, the set Ω obtained by random sampling is as follows:

[0048]

[0049] (3) Traverse all Pauli operators in the set Ω. For the i-th Pauli operator P obtained by sampling i , construct the corresponding quantum measurement, measure the quantum state ρ for a total of M = 1024 times, and use the measurement results to estimate the expected value Tr[P i ρ], record the estimated result as y i .

[0050] (4) Using the optimization function

[0051]

[0052] The expected value y obtained in step (3) i , and finally obtain an estimate of the quantum state. The final quantum state estimate obtained in this embodiment is as follows:

[0053]

[0054] And output the estimate.

[0055] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A compressed quantum state tomography method based on prior knowledge, characterized in that: The method comprises the following steps: 1) Device for generating n-bit quantum states Set appropriate parameters; 2) Randomly sample Pauli operators to obtain the set Ω; 3) Traverse the Pauli operators sampled in step 2), construct the corresponding quantum measurement, and measure the expectation value of the noisy quantum state ρ under the operator; 4) Using an optimization function containing prior knowledge to process the expected value obtained in step 3), finally obtaining an estimate of the quantum state, and outputting the estimate.

2. The compressed quantum state tomography method based on prior knowledge according to claim 1, characterized in that: Appropriate parameters to be set in step 1) include: 1) Making devices based on quantum states The prior knowledge of the device is used to set the density matrix Φ, reliability λ, and rank r of the output quantum state corresponding to the device under ideal conditions; 2) Estimation error value ε, which is used to indicate the tolerable estimation error range; 3) Randomly sample the number of Pauli operators N; 4) Measurement error value δ, which is used to indicate the tolerable measurement error range.

3. The compressed quantum state tomography method based on prior knowledge according to claim 1, characterized in that: The Pauli operator in step 2) is specifically: for an n-bit quantum state ρ, there are 4 n Pauli operator, represented by in is the tensor product, 4. The compressed quantum state tomography method based on prior knowledge according to claim 1, characterized in that: In step 2), the set Ω is obtained by the following method: Randomly sample N Pauli operators, and record the sampled Pauli operator set as Ω = {P i }, P i represents the i-th Pauli operator.

5. The compressed quantum state tomography method based on prior knowledge according to claim 2, characterized in that: In step 3), all Pauli operators in the set Ω are traversed, and for the i-th Pauli operator P obtained by sampling i , construct the corresponding quantum measurement, measure the noisy quantum state ρ total M ~ 4n / ε 2 times, and use the measurement results to estimate the expected value Tr[P i ρ], record the estimated result as y i .

6. The compressed quantum state tomography method based on prior knowledge according to claim 5, characterized in that: The optimization function in step 4) is as follows: where σ is the quantum state to be optimized, ||A|| * is the trace norm of matrix A, λ represents the degree of trust in prior knowledge Φ, and δ is the measurement error value.

Images