Fault-tolerant measurement method for quantum Hamming codes

Abstract

The present invention discloses a fault-tolerant measurement method for quantum Hamming codes, which specifically comprises the following steps: step 1, encoding information bits to be transmitted and generating codewords of the quantum Hamming code; step 2, constructing a sequence for implementing fault-tolerant measurement; step 3, using the fault-tolerant measurement sequence constructed in step 2 to perform auxiliary measurement on each codeword in the quantum bit information of the quantum Hamming code in step 1; if an error occurs, forming an error pattern table according to the measurement results; step 4, correcting the measured errors according to the error pattern table in step 3, and handing the corrected quantum information bits to the next-level quantum device for information transmission and quantum computing. The method constructs a measurement sequence in an error correction process to generate a fault-tolerant measurement sequence, thereby overcoming the problem that internal errors and device errors cannot be processed during the error correction process, resulting in error accumulation and error increase, and improving the fault tolerance and measurement anti-interference capability of the quantum Hamming code.

Description

Technical Field

[0001] The present invention belongs to the technical field of quantum computing and quantum error correction codes, and relates to a fault-tolerant measurement method for quantum Hamming codes. Background Art

[0002] Existing quantum computing error correction methods primarily rely on measurement error correction, but this method is affected by the quantum computing environment and the performance of controlled gates, leading to errors during the measurement error correction process. By encoding the quantum Hamming code measurement error correction process using a single-shot decoding measurement protocol, it is possible to achieve fault tolerance for potential errors in measurement error correction. Currently, fault-tolerant measurement coding implemented in single-shot decoding measurement primarily relies on large-scale repeated measurements.

[0003] In their jointly published patent (application number: 202011488325.0, application date: 2020.12.16), Zhijiang Laboratory and Zhejiang Gongshang University proposed a method for fast measurement and estimation of positive real-valued probability amplitude. The steps of this method are as follows: S1 gives an arbitrary angle and establishes a rotation operator gate; S2, performs a tensor product operation on the quantum state to be measured; S3, performs a cross-unitary operation on the above result to obtain the difference probability amplitude between the probability amplitude angle of the quantum state to be measured and the given angle; S4, performs a projection measurement on the difference probability amplitude and judges the sign of the measurement result. If the measurement result is zero, the result to be estimated is obtained, and the operation is stopped. The given angle is the angle of the probability amplitude corresponding to the quantum state to be measured; otherwise, the given angle is updated according to the sign. If the result is positive, the estimated value is updated and reduced by binary search. If the result is negative, the estimated value is updated and increased by binary search. Repeat steps 1 to 4 for a preset number of times or until the measurement result is zero, and then stop the operation. The disadvantage of this method is that when performing error correction measurement on quantum Hamming code, the above method cannot determine the errors that occurred during the measurement process, and therefore cannot achieve fault tolerance for the measurement results.

[0004] Nicolas Delfosse et al. proposed a method for correcting errors that occur in quantum Hamming code measurements in single-shot decoding measurements in their published paper (Quantum Physics, Submitted on 12Aug2020, Short Shor-style syndrome sequences arxiv.org). The steps of this method are: 1. Fault-tolerant measurement encoding is performed on single-type errors of quantum Hamming code by repeatedly adding a stabilizer measurement sequence; 2. For three types of errors, X, Y, and Z, measurement encoding is also performed by repeatedly adding a mixed stabilizer sequence. On this basis, it is ensured that the obtained error pattern can fully represent all types of errors, and fault-tolerant measurement can be completed. Although this method can effectively correct the measurement errors generated in the quantum Hamming code measurement error correction process, the method still has its shortcomings. Since fault tolerance is achieved by simply repeatedly adding a stabilizer measurement sequence, the length of the fault-tolerant measurement coding sequence is relatively long, which increases the physical resource overhead of quantum measurement, causes some redundant and redundant measurement operations, and leads to low efficiency. Summary of the Invention

[0005] The purpose of the present invention is to provide a fault-tolerant measurement method for quantum Hamming codes, which has the characteristics of effectively reducing the physical resource overhead in quantum measurement and improving the fault-tolerant efficiency and fault-tolerant capability of quantum Hamming codes.

[0006] The technical solution adopted by the present invention is a fault-tolerant measurement method for quantum Hamming codes, which is specifically implemented according to the following steps:

[0007] Step 1: Encode the information bits to be transmitted and generate quantum Hamming code codewords;

[0008] Step 2: construct a sequence for implementing fault-tolerant measurement;

[0009] Step 3: Use the fault-tolerant measurement sequence constructed in step 2 to perform auxiliary measurement on each codeword in the quantum bit information of the quantum Hamming code in step 1. If no error occurs after the measurement, the quantum information bit that achieves fault-tolerant measurement is handed over to the next-level quantum device for information transmission and quantum computing. If an error occurs, an error pattern table is compiled based on the measurement results.

[0010] Step 4: Correct the measured errors according to the error pattern table in step 3, and pass the corrected quantum information bits to the next-level quantum device for information transmission and quantum computing.

[0011] The present invention is also characterized in that:

[0012] Step 1 is implemented as follows:

[0013] Step 1.1: Generate stable subgenerators of quantum Hamming code using the Keldbank-Short-Stuyne CSS construction method.

[0014] Step 1.2: Use the quantum Hamming code stabilizer generator encoding method to generate the codeword of the quantum Hamming code.

[0015] Step 1.1 is implemented as follows:

[0016] Step 1.1.1, in the classic Hamming code [2 r -1,2 r -1-r] select the first 2 r -1-r rows form the generator matrix of the quantum Hamming code, where r represents the dimension of the codeword space and the value of r is a positive integer not less than 3;

[0017] Step 1.1.2: Transpose the generator matrix of the quantum Hamming code in step 1.1.1 to find the null space and obtain the parity check matrix of the quantum Hamming code;

[0018] Step 1.1.3: Use each row of the parity check matrix of the quantum Hamming code in step 1.1.2 as a stabilizer generator of the quantum Hamming code to obtain the X stabilizer generator of r rows and the Z stabilizer generator of r rows.

[0019] Step 2 is implemented as follows:

[0020] Step 2.1. Construct a 2r×(2r+1)-order block matrix and divide it into four sub-matrices, denoted as A, B, C, and D. The A and B matrices are both r×r-order square matrices, and the C and D matrices are both (r+1)×r-order matrices.

[0021] Step 2.2: Construct the A matrix in step 2.1 into an r×r unit matrix. Construct the B matrix into an r×r square matrix with the first two columns of the first row vector being 1 and the remaining columns being 0, and the remaining rows being the row vector of the previous row shifted right by one position.

[0022] Step 2.3. Take the first [(r+1) / 2] rows of the C matrix in step 2.1 and the first [(r+1) / 2] rows of the B matrix in step 3.2, and take the first r-[(r+1) / 2] rows of the A matrix in step 3.2.

[0023] Step 2.4. Take the first [(r+1) / 2] rows of the D matrix in step 2.1 and the first [(r+1) / 2] rows of the A matrix in step 3.2, and take the first r-[(r+1) / 2] rows of the B matrix in step 3.2.

[0024] Step 2.5: Combine the four matrices A, B, C, and D constructed in steps 2.1 to 2.4 into a 2r×(2r+1) block matrix, and add a row vector with only the first and last 1s and the rest of the 0s to the last row of the block matrix.

[0025] Step 2.6: Divide the block matrix in step 2.5 into two parts, one of which is the left half (2r+2)×r-order matrix used to represent the Z-stable subsequence correcting the X error, and the other is the right half (2r+2)-order matrix used to represent the X-stable subsequence correcting the Z error;

[0026] Step 2.7: Perform Pauli product on the X stabilizer measurement sequence and the Z stabilizer measurement sequence in step 2.6 at corresponding positions to obtain the final mixed measurement sequence for correcting the Y error.

[0027] Step 2.6 is implemented as follows:

[0028] According to the vector representation of the X-stabilizer generator in step 1, each row of the (2r+2)-order matrix on the right half of the block matrix is ​​converted into an X-stabilizer measurement sequence for correcting the Z error; according to the vector representation of the Z-stabilizer generator in step 1, each row of the (2r+2)×r-order matrix on the left half of the block matrix is ​​converted into a Z-stabilizer measurement sequence for correcting the X error.

[0029] The auxiliary measurement in step 3 is carried out according to the following steps:

[0030] The X-stabilizer measurement sequence, Z-stabilizer measurement sequence, and hybrid measurement sequence obtained in step 2 are used to measure each codeword in the quantum bit information of the quantum Hamming code in step 1. If an error occurs, the measurement results are combined into a corresponding error pattern, and the error type and error position are located according to the error pattern table.

[0031] Correction of measurement errors in step 4 is carried out in the following steps:

[0032] Step 4.1: Based on the error pattern obtained from the fault-tolerant measurement of X, use a qubit flip gate on the qubit with a sequence value of 1.

[0033] Step 4.2: Based on the error pattern obtained from the Z fault tolerance measurement, use the quantum phase bit flip gate on the qubit with sequence value 1;

[0034] Step 4.3: cascade the quantum bit flip gate in step 4.1 and the quantum phase bit flip gate in step 4.2 to prepare a universal quantum gate;

[0035] Step 4.4: Based on the error pattern obtained from the fault-tolerant measurement of Y, place the universal quantum gate on the quantum bit circuit corresponding to the measurement information sequence value of 1.

[0036] The beneficial effects of the present invention are:

[0037] The present invention provides a fault-tolerant measurement method for quantum Hamming codes. The method constructs a measurement sequence during an error correction process to generate a fault-tolerant measurement sequence. This overcomes the problem of error accumulation and error increase caused by the inability to process internal errors and device errors during the error correction process, thereby improving the fault tolerance and measurement anti-interference capability of quantum Hamming codes. The method regularly selects appropriate stabilizers for fault-tolerant measurement through an algorithm, and utilizes the concept of hybrid measurement so that each stabilizer measurement contributes to the verification of any error type, reducing the bit resource overhead of the measurement and improving the efficiency of the fault-tolerant measurement. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a flow chart of the fault-tolerant measurement method for quantum Hamming code of the present invention;

[0039] Figure 2 This is a schematic diagram of block proof in the fault-tolerant measurement method for quantum Hamming codes of the present invention. DETAILED DESCRIPTION

[0040] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0041] The present invention is directed to a fault-tolerant measurement method for quantum Hamming codes, such as Figure 1 As shown, the three types of errors X, Y, and Z in quantum computing are analyzed separately to achieve single-type fault-tolerant measurement. By representing the stable subvector, the fault-tolerant measurement sequence is converted into a matrix. The specific implementation is as follows:

[0042] Step 1: Encode the information bits to be transmitted and generate quantum Hamming code codewords;

[0043] Step 1.1, using the Keldbank-Short-Stuyne CSS construction method, generate the stable sub-generator of the quantum Hamming code;

[0044] Step 1.1.1, in the classic Hamming code [2 r -1,2 r -1-r] select the first 2 r -1-r rows form the generator matrix of the quantum Hamming code, where r represents the dimension of the codeword space and the value of r is a positive integer not less than 3;

[0045] Step 1.1.2: Transpose the generator matrix of the quantum Hamming code in step 1.1.1 to find the null space and obtain the parity check matrix of the quantum Hamming code.

[0046] Step 1.1.3: Use each row of the parity check matrix of the quantum Hamming code in step 1.1.2 as a stabilizer generator of the quantum Hamming code to obtain the X stabilizer generator of r rows and the Z stabilizer generator of r rows;

[0047] Step 1.2: Use the quantum Hamming code stabilizer generator encoding method to generate the codeword of the quantum Hamming code.

[0048] Step 2: construct a sequence for implementing fault-tolerant measurement;

[0049] Step 2.1, construct a 2r×(2r+1) order block matrix, such as Figure 2 As shown, the matrix is ​​divided into four sub-matrices, denoted as A, B, C, and D matrices, where A and B matrices are both r×r square matrices, and C and D matrices are both (r+1)×r matrices. A and C matrices will be used for measuring X error tolerance, and C and D matrices will be used for measuring Z error tolerance.

[0050] Step 2.2: Construct the A matrix in step 2.1 into an r×r unit matrix. Construct the B matrix into an r×r square matrix with the first two columns of the first row vector being 1 and the remaining columns being 0, and the remaining rows being the row vector of the previous row shifted right by one position.

[0051] Step 2.3. Take the first [(r+1) / 2] rows of the C matrix in step 2.1 and the first [(r+1) / 2] rows of the B matrix in step 3.2, and take the first r-[(r+1) / 2] rows of the A matrix in step 3.2.

[0052] Step 2.4. Take the first [(r+1) / 2] rows of the D matrix in step 2.1 and the first [(r+1) / 2] rows of the A matrix in step 3.2, and take the first r-[(r+1) / 2] rows of the B matrix in step 3.2.

[0053] Step 2.5: Combine the four matrices A, B, C, and D constructed in steps 2.1 to 2.4 into a 2r×(2r+1) block matrix, and add a row vector with only the first and last 1s and the rest of the 0s to the last row of the block matrix.

[0054] Step 2.6: Divide the block matrix in step 2.5 into two parts, one of which is the left half (2r+2)×r-order matrix used to represent the Z-stable subsequence correcting the X error, and the other is the right half (2r+2)-order matrix used to represent the X-stable subsequence correcting the Z error;

[0055] According to the vector representation of the X-stabilizer generator in step 1, each row of the (2r+2)-order matrix on the right half of the block matrix is ​​converted into an X-stabilizer measurement sequence for correcting the Z error; according to the vector representation of the Z-stabilizer generator in step 1, each row of the (2r+2)×r-order matrix on the left half of the block matrix is ​​converted into a Z-stabilizer measurement sequence for correcting the X error.

[0056] Step 2.7: Perform Pauli product on the X stabilizer measurement sequence and the Z stabilizer measurement sequence in step 2.6 at corresponding positions to obtain the final mixed measurement sequence for correcting the Y error.

[0057] In steps 2.1-2.7 above, the three types of errors, X, Y, and Z, present in quantum computing are analyzed separately to achieve fault-tolerant measurement of a single type. By representing the fault-tolerant measurement sequence as a stabilizer vector, the fault-tolerant measurement sequence is converted into a matrix, and the fault-tolerant measurement conditions for different types of errors are converted into constraints on matrix construction. The matrix is ​​then split using the principle of block matrices. The left matrix is ​​used to achieve fault-tolerant measurement of the X error, and the right matrix is ​​used to achieve fault-tolerant measurement of the Z error. The two matrices are constructed together using modular two multiplication to achieve fault-tolerant measurement of the Y error.

[0058] Step 3: Use the fault-tolerant measurement sequence constructed in step 2 to perform auxiliary measurement on each codeword in the quantum bit information of the quantum Hamming code in step 1. If no error occurs after the measurement, the quantum information bit that achieves fault-tolerant measurement is handed over to the next-level quantum device for information transmission and quantum computing. If an error occurs, an error pattern table is compiled based on the measurement results.

[0059] The X-stabilizer measurement sequence, Z-stabilizer measurement sequence, and hybrid measurement sequence obtained in step 2 are used to measure each codeword in the quantum bit information of the quantum Hamming code in step 1. If an error occurs, the measurement results are combined into a corresponding error pattern, and the error type and error position are located according to the error pattern table.

[0060] Step 4: Correct the measured errors according to the error pattern table in step 3, and pass the corrected quantum information bits to the next-level quantum device for information transmission and quantum computing. Correcting the measured errors is specifically implemented according to the following steps:

[0061] Step 4.1: Based on the error pattern obtained from the fault-tolerant measurement of X, use a qubit flip gate on the qubit with a sequence value of 1.

[0062] Step 4.2: Based on the error pattern obtained from the Z fault tolerance measurement, use the quantum phase bit flip gate on the qubit with sequence value 1;

[0063] Step 4.3: cascade the quantum bit flip gate in step 4.1 and the quantum phase bit flip gate in step 4.2 to prepare a universal quantum gate;

[0064] Step 4.4: Based on the error pattern obtained from the fault-tolerant measurement of Y, place the universal quantum gate on the quantum bit circuit corresponding to the measurement information sequence value of 1.

[0065] In steps 3-4 above, the fault-tolerant implementation of the quantum Hamming code error correction process is achieved by selecting and constructing the stabilizer measurement sequence, and the final error pattern is obtained based on the measurement result sequence. According to the error pattern table, the errors generated in the quantum information transmission and calculation process are effectively corrected. At the same time, internal errors and device measurement errors generated during the error correction process can also be corrected.

Claims

1. A fault-tolerant measurement method for quantum Hamming codes, characterized in that: Please follow the steps below to implement it: Step 1: Encode the information bits to be transmitted and generate quantum Hamming code codewords; Step 2: construct a sequence for implementing fault-tolerant measurement; Step 3: Use the fault-tolerant measurement sequence constructed in step 2 to perform auxiliary measurement on each codeword in the quantum bit information of the quantum Hamming code in step 1. If no error occurs after the measurement, the quantum information bit that achieves fault-tolerant measurement is handed over to the next-level quantum device for information transmission and quantum computing. If an error occurs, an error pattern table is compiled based on the measurement results. Step 4: Correct the measured errors according to the error pattern table in step 3, and pass the corrected quantum information bits to the next-level quantum device for information transmission and quantum computing; The step 2 is specifically implemented according to the following steps: Step 2.1, construct a The block matrix of order is divided into four sub-matrices, which are recorded as A, B, C, and D matrices, where A and B matrices are The C and D matrices are both Matrix of order; Step 2.2: Construct the A matrix in step 2.1 into a The B matrix is ​​constructed as the first two columns of the first row vector are 1 and the remaining columns are 0, and the remaining rows are the row vector of the previous row shifted one position to the right. Order square matrix; Step 2.3: Replace the front of the C matrix in step 2.1 with The first row of the B matrix in step 3.2 is taken OK, then Take the front row of the A matrix in step 3.2 OK; Step 2.4: Replace the front of the D matrix in step 2.1 with Take the front row of the A matrix in step 3.2 OK, then The first row of the B matrix in step 3.2 is taken OK; Step 2.5: Combine the four matrices A, B, C, and D constructed in steps 2.1 to 2.4 into A block matrix of order , and add a row vector with only 1 at the beginning and end and 0 at the rest of the positions to the last row of the block matrix; Step 2.6: Divide the block matrix in step 2.5 into two parts, one of which is the left half The matrix of order is used to represent the Z stable subsequence that corrects the X error, and the other part is the right half The matrix of order is used to represent the X stable subsequence that corrects the Z error; Step 2.7: Perform Pauli product of the X stabilizer measurement sequence and the Z stabilizer measurement sequence in step 2.6 to obtain the final mixed measurement sequence for correcting the Y error. The step 2.6 is specifically implemented according to the following steps: According to the vector representation of the X-stable sub-generator in step 1, the right half of the block matrix Each row of the order matrix is ​​converted into an X-stable sub-measurement sequence to correct the Z error; according to the vector representation of the Z-stable sub-generator in step 1, the left half of the block matrix is ​​converted into Each row of the matrix of order is converted into a sequence of Z-stable sub-measurements that corrects the X errors.

2. The fault-tolerant measurement method for quantum Hamming codes according to claim 1, characterized in that: The step 1 is specifically implemented according to the following steps: Step 1.1, using the Keldbank-Short-Stuyne CSS construction method, generate the stable sub-generator of the quantum Hamming code; Step 1.2: Use the quantum Hamming code stabilizer generator encoding method to generate quantum Hamming code codewords.

3. The fault-tolerant measurement method for quantum Hamming codes according to claim 2, wherein: The step 1.1 is specifically implemented according to the following steps: Step 1.1.

1. In the classic Hamming code Before the selection The rows form the generator matrix of the quantum Hamming code, where r represents the dimension of the codeword space and the value of r is a positive integer not less than 3; Step 1.1.2: Transpose the generator matrix of the quantum Hamming code in step 1.1.1 to find the null space and obtain the parity check matrix of the quantum Hamming code. Step 1.1.3: Use each row of the parity check matrix of the quantum Hamming code in step 1.1.2 as a stabilizer generator of the quantum Hamming code to obtain the X stabilizer generator of r rows and the Z stabilizer generator of r rows.

4. The fault-tolerant measurement method for quantum Hamming codes according to claim 1, wherein: The auxiliary measurement in step 3 is specifically implemented according to the following steps: The X-stabilizer measurement sequence, Z-stabilizer measurement sequence, and hybrid measurement sequence obtained in step 2 are used to measure each codeword in the quantum bit information of the quantum Hamming code in step 1. If an error occurs, the measurement results are combined into a corresponding error pattern, and the error type and error position are located according to the error pattern table.

5. The fault-tolerant measurement method for quantum Hamming codes according to claim 1, characterized in that: Correcting the measurement error in step 4 is specifically performed according to the following steps: Step 4.1: Based on the error pattern obtained from the fault-tolerant measurement of X, use a qubit flip gate on the qubit with a sequence value of 1. Step 4.2: Based on the error pattern obtained from the Z fault tolerance measurement, use the quantum phase bit flip gate on the qubit with sequence value 1; Step 4.3: cascade the quantum bit flip gate in step 4.1 and the quantum phase bit flip gate in step 4.2 to prepare a universal quantum gate; Step 4.4: Based on the error pattern obtained from the fault-tolerant measurement of Y, place the universal quantum gate on the quantum bit circuit corresponding to the measurement information sequence value of 1.

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