CN110011671B - Quantum error correction code marking bit symptom measuring method based on dynamic time slot allocation - Google Patents
Quantum error correction code marking bit symptom measuring method based on dynamic time slot allocation Download PDFInfo
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Abstract
The invention discloses a quantum error correction code marking bit symptom measuring method based on dynamic time slot allocation, which mainly solves the problem of time slot resource allocation in quantum stable sub-symptom measurement. The invention uses the sequence of the measured quanta to be sequenced by constructing the quanta bit sequence number matrix, uses the dynamic time slot allocation method to allocate the time slot, constructs the error symptom matrix to analyze the error symptoms which may appear during the measurement, and adjusts the sequence of the time slot arrangement according to the error symptoms, thereby deploying the quantum error correction code mark bit measurement circuit diagram to measure the same type of stable quanta CSS code at the same time. The invention ensures the fault tolerance of the quantum error correction code marking bit symptom measuring line, improves the parallelism of the measuring line, reduces the time required by symptom measurement and optimizes the resource expenditure.
Description
Technical Field
The invention belongs to the technical field of quantum information processing, and further relates to a quantum error correction code flag bit symptom measuring method based on dynamic time slot allocation in the technical field of quantum error correction codes. The method can be applied to measuring the error symptoms of the quantum CSS stabilizer subcode in the quantum error correcting code, and the CSS is called bank-Shor-Steane for short.
Background
Due to the coherence of quantum states, quantum computers have potential great advantages over classical computers when solving certain problems with super-polynomial complexity, and in order to perform reliable quantum computation, quantum error correction codes are used to overcome the decoherence effect caused by the mutual influence of the quantum states and the environment. However, in the process of transferring quantum information by using a quantum error correction code, a quantum error caused by noise inevitably occurs, and therefore, it is necessary to constantly perform quantum measurement to obtain an error symptom corresponding to the quantum error, and then correct the error in time based on the error symptom to obtain correct quantum information. The qubits collapse once measured, so that the information qubits cannot be directly measured in a quantum computer, and the auxiliary qubits must be added to copy the quantum information in the information bits, and then the error symptoms are obtained by measuring the auxiliary qubits, thereby correctly correcting the errors.
Among the existing quantum error correcting codes, quantum stabilizer codes become the most basic class due to the complete research architecture, and are different from other quantum error correcting codes in the mode of directly researching quantum states, and the quantum stabilizer codes convert the influence of noise and other dynamic processes on the quantum states into the influence on quantum state stabilizers. Therefore, when symptom measurement is performed on a quantum error occurring in the quantum CSS stabilizer code, only the respective stabilizers thereof need to be measured in sequence, wherein the stabilizers of the quantum CSS stabilizer code are only two types, i.e., the X-type stabilizer and the Z-type stabilizer.
Shor in its published paper "Fault-tall quantum computing [ J ]. fundamentals of computer science,1996: 56-65" discloses a quantum stabilizer code symptom measurement method based on cat state. The method solves the problem that a single auxiliary quantum bit cannot tolerate faults when symptom measurement is carried out, before the stabilizator is measured, the same number of auxiliary quantum bits are prepared according to the weight of the stabilizator, the auxiliary quantum bits are prepared into a cat state, a verification bit is additionally applied to verify whether the cat state is prepared correctly, if the verification fails, the cat state is discarded for re-preparation, and until the verification succeeds, each information quantum bit only interacts with one auxiliary quantum bit, error symptoms are obtained through measurement of the auxiliary quantum bits, and then error correction is carried out according to the error symptoms. However, the method still has the disadvantages that a large number of auxiliary qubits are needed for preparing the cat state when symptom measurement is carried out, and additional auxiliary bits are also needed for verifying the correctness of cat state preparation, and the resource consumption required by measurement is very large.
Reichardt in its published paper "Quantum Error Correction with Only Two Extra bits [ J ]. Physical Review Letters,2018,121 (5)" discloses a Quantum Error Correction code marker bit symptom measurement method. Aiming at a quantum stabilizer code with the code distance of 3, the method can distinguish all error symptoms only by using two auxiliary qubits when measuring each stabilizer, wherein one auxiliary qubit is called a measuring bit and is used for acquiring symptoms of X-base measurement and Z-base measurement; another ancillary qubit, called the marker bit, is used to obtain the marker bit syndrome. The position of quantum error in the measuring line and the error operator caused by the quantum error are deduced by combining the two error symptoms, and then the corresponding recovery operator is applied to correct the error. The method has the disadvantages that each stabilizer needs to be measured in turn when symptom measurement is carried out, the required measurement time is long, and the parallelism degree of a measurement line is low.
Disclosure of Invention
The invention aims to provide a quantum error correction code flag bit symptom measuring method based on dynamic time slot allocation, aiming at the defects of the prior art. The invention uses the sequence number matrix of the quantum bit to sort the quantum bit to be measured, uses the dynamic time slot allocation method to allocate the time slot, constructs the error symptom matrix to analyze the error symptom which may appear during the measurement, and improves the parallelism of the measurement line, reduces the time required by the symptom measurement and optimizes the resource expense while ensuring the fault tolerance of the quantum error correction code marking bit symptom measurement line.
The idea for realizing the purpose of the invention is as follows: when quantum stabilizer subcode symptom measurement is carried out, the quantum bits to be measured are sequenced according to the stabilizer weight and the quantum bit weight by constructing the quantum bit sequence number matrix, so that the quantum bits with more measurement times can obtain the right of preferentially allocating time slots. By using the dynamic time slot allocation method, quantum bits are arranged in each time slot as many as possible, so that the parallelism of the measuring line is improved. Error symptoms which can occur when a single-bit quantum error occurs at any position of a measuring line are analyzed by constructing an error symptom matrix, so that corresponding errors can be corrected according to the error symptoms, and the fault tolerance of the measuring line is ensured.
The method comprises the following specific steps:
(1) constructing a quantum bit sequence number matrix:
(1a) inputting a binary generator matrix of quantum stabilizer codes, wherein each row in the matrix represents a stabilizer, and each column represents a quantum bit;
(1b) all rows of the binary generator matrix are arranged in a descending order according to the size of each stable sub-weight by utilizing a matrix elementary row transformation mode;
(1c) constructing a quantum bit serial number matrix, wherein the number of rows is equal to the number of the stable subunits, and the number of columns is equal to the maximum stable subunit weight value;
(1d) traversing the sorted binary generator matrix, sequentially extracting the column serial numbers of the elements with the row value of 1 from left to right according to the rows, sequentially filling the extracted column serial numbers into the rows corresponding to the qubit serial number matrix, and complementing by 0 elements if a blank part exists;
(1e) arranging each row of elements in the qubit sequence number matrix in a descending order according to the weight of each qubit to obtain an ordered qubit sequence number matrix;
(2) allocating time slots by using a dynamic time slot allocation method:
(2a) constructing a time slot distribution matrix, wherein the number of rows is equal to the number of stable subunits, and the number of columns is equal to the maximum stable subunit weight value;
(2b) directly filling the first row element values in the quantum bit sequence number matrix into the first row of the time slot distribution matrix;
(2c) judging whether the element value on the ith row of the jth column in the qubit sequence number matrix is equal to the element value in the 1 st to i-1 th rows of the jth column, if so, executing the step (2d), otherwise, executing the step (2 e); wherein j represents a column serial number, the value range of j is [1, omega ], i represents a row serial number, the value range of i is [2, m ], the value sequence of j and i is from small to large, m represents the number of the stable subunits, and omega represents the maximum weight value of the stable subunits;
(2d) selecting an element which is different from all element values in the 1 st to i-1 th rows of the jth column of the qubit sequence number matrix according to the sequence of the rows from left to right from the element value of the ith row of the qubit sequence number matrix, filling the element into the v th row of the c row of the qubit sequence number matrix, and executing the step (2c) after exchanging the same element value with the selected different element value in the qubit sequence number matrix, wherein the values of c and i are correspondingly equal, and the values of v and j are correspondingly equal;
(2e) directly filling the element value on the ith row of the jth column in the quantum bit serial number matrix into the c row of the vth column of the time slot allocation matrix;
(2f) if a blank part exists in the time slot allocation matrix, complementing the blank part by using 0 elements, so that the values of all other elements except the 0 element in each column are different, wherein each column represents a time slot, the element value in each time slot represents a qubit to be measured in the time slot, and the 0 element represents that any qubit is not measured;
(3) analyzing error symptoms:
(3a) construct t pieces with the size of omegaxX m error symptom matrix, where t is equal to m, ωxThe value of (a) is equal to the corresponding stable sub-weight of the xth row element in the time slot distribution matrix, and the value range of x is [1, q ]]The value of q is equal to m;
(3b) if the odd number of elements in the r-th row to the p-th row of the time slot allocation matrix appears in the h-th row of the time slot allocation matrix, filling 1 in the f-th row of the s-th error symptom matrix, and filling 0 in the f-th row of the s-th error symptom matrix if the even number of elements exists, wherein the value ranges of k, s and h are all [1, q]And r has a value range of [1, p ]]The value of p and ωxThe values of s and k are equal correspondingly, the values of u and r are equal correspondingly, and the values of f and h are equal correspondingly;
(3c) judging whether more than two uplink element values in each error symptom matrix are completely the same, if so, executing the step (4), otherwise, executing the step (5) after obtaining a time slot distribution matrix;
(4) adjusting the time slot arrangement sequence:
exchanging positions of any two columns of element values in the time slot allocation matrix with each other, obtaining a time slot allocation matrix after updating the column sequence, and then executing the step (3);
(5) deploying quantum error correction code marking bit symptom measurement circuit diagram:
(5a) if the stable subtype of the quantum error correcting code to be measured is X type, setting the initial state of the measurement bit in the quantum error correcting code mark bit symptom measurement circuit diagram corresponding to the X type stabilizer as | + >, and if the stable subtype to be measured is Z type, setting the initial state of the measurement bit in the quantum error correcting code mark bit symptom measurement circuit diagram corresponding to the Z type stabilizer as |0 >;
(5b) in a quantum error correction code mark bit symptom measurement circuit diagram, each column of a time slot distribution matrix corresponds to a time slot, each row corresponds to a stable sub, a quantum bit to be measured in each time slot is connected with a measurement bit corresponding to each stable sub by using a quantum controlled non-CNOT gate, when connection is carried out, if the measured stable sub type is an X type, the control quantum bits of each quantum controlled non-CNOT gate are measurement bits, the control quantum bits are the quantum bits to be measured in each time slot, if the measured stable sub type is a Z type, the control quantum bits of each quantum controlled non-CNOT gate are the quantum bits to be measured in each time slot, and the control quantum bits are the measurement bits;
(5c) adding a marker bit measurement slot after the first column of slots and before the last column of slots;
(5d) and finishing the deployment of the quantum error correction code marking bit symptom measurement circuit diagram.
Compared with the prior art, the invention has the following advantages:
firstly, the invention utilizes the dynamic time slot allocation method to allocate the time slot, constructs the time slot allocation matrix for the quantum bit to be measured of the same type of steady son to adjust the measurement sequence, fully considers all error positions and corresponding error symptoms, and overcomes the problems that each steady son needs to be measured in sequence when the symptom measurement is carried out in the prior art, the required measurement time is very long, and the parallelism of the measurement line is low, so that the invention has the advantages of capability of simultaneously measuring all the steady sons of the same type with the code distance of 3 quantum CSS steady son codes, less measurement time and higher parallelism of the measurement line.
Secondly, at the stage of deploying the quantum error correction code marking bit measurement circuit diagram after the time slot allocation is completed, each stable sub is measured by using the quantum error correction code marking bit symptom measurement method, and only auxiliary qubits with the number twice that of the stable sub are used, so that the problems that a large number of auxiliary qubits are needed during symptom measurement and resource consumption required by measurement is extremely high in the prior art are solved, and the method has the advantages of being few in auxiliary qubits required during stable sub measurement and small in resource consumption.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a diagram of measurement of the syndrome of the flag bits of the quantum error correction code in embodiment 1 of the present invention;
fig. 3 is a diagram of quantum error correction code flag bit syndrome measurement in embodiment 2 of the present invention.
Detailed Description
The technical solution of the present invention is described in detail below with reference to the accompanying drawings and specific embodiments. Obviously, all other embodiments obtained by a person of ordinary skill in the art based on the embodiments of the present invention without any inventive work belong to the protection scope of the present invention.
The specific steps of the present invention will be described in further detail with reference to fig. 1.
Step 1, constructing a quantum bit sequence number matrix.
A binary generator matrix of the input quantum stabilizer code, each row in the matrix representing one stabilizer and each column representing one qubit.
And all the rows of the binary generator matrix are arranged in a descending order according to the size of each stable sub-weight by utilizing a matrix elementary row transformation mode. The stabilizer weight is the number of 1 included in the row corresponding to each stabilizer in the generator matrix.
And constructing a quantum bit serial number matrix, wherein the number of rows is equal to the number of the stable subunits, and the number of columns is equal to the maximum stable subunit weight value.
Traversing the sorted binary generator matrix, sequentially extracting the column sequence number of each element with the row value of 1 from left to right according to the rows, sequentially filling the extracted column sequence number into the corresponding row of the qubit sequence number matrix, and complementing by 0 element if a blank part exists.
And arranging each row of elements in the qubit sequence number matrix in a descending order according to the weight of each qubit to obtain an ordered qubit sequence number matrix. The qubit weight is the number of 1 contained in the corresponding column of each qubit in the generator matrix.
And 2, allocating time slots by using a dynamic time slot allocation method.
And (2.1) constructing a time slot distribution matrix, wherein the number of rows is equal to the number of stable sub-weights, and the number of columns is equal to the maximum stable sub-weight value.
And (2.2) directly filling the first row element values in the qubit sequence number matrix into the first row of the time slot allocation matrix.
(2.3) judging whether the element value on the ith row of the jth column in the qubit sequence number matrix is equal to the element value in the 1 st to i-1 th rows of the jth column, if so, executing (2.4) of the step, otherwise, executing (2.5) of the step; wherein j represents a column serial number, the value range of j is [1, omega ], i represents a row serial number, the value range of i is [2, m ], the value sequence of j and i is from small to large, m represents the number of stable subunits, and omega represents the maximum weight of the stable subunits.
(2.4) selecting an element with different values from all the element values in the 1 st to i-1 th rows of the jth column of the qubit sequence number matrix from the element values of the ith row of the qubit sequence number matrix which are not filled in the time slot distribution matrix, filling the element into the v th row of the c row of the time slot distribution matrix, and executing the step (2.3) after exchanging the same element value with the selected different element value in the qubit sequence number matrix, wherein the values of c and i are correspondingly equal, and the values of v and j are correspondingly equal.
And (2.5) directly filling the element value on the ith row of the jth column in the qubit sequence number matrix into the ith row of the vth column of the time slot allocation matrix.
(2.6) if a blank part exists in the time slot allocation matrix, complementing the blank part by using an element 0 to ensure that all the other element values except the element 0 in each column are different, wherein each column represents a time slot, the element value in each time slot represents the qubit to be measured in the time slot, and the element 0 represents that any qubit is not measured.
And 3, analyzing error symptoms.
Construct t pieces with the size of omegaxX m error symptom matrix, where t is equal to m, ωxThe value of (a) is equal to the corresponding stable sub-weight of the xth row element in the time slot distribution matrix, and the value range of x is [1, q ]]And the value of q is equal to m.
If the odd number of elements in the r-th row to the p-th row of the time slot allocation matrix appears in the h-th row of the time slot allocation matrix, filling 1 in the f-th row of the s-th error symptom matrix, and filling 0 in the f-th row of the s-th error symptom matrix if the even number of elements exists, wherein the value ranges of k, s and h are all [1, q]And r has a value range of [1, p ]]The value of p and ωxThe values of s and k are equal, the values of u and r are equal, and the values of f and h are equal.
And (4) judging whether more than two uplink element values in each error symptom matrix are completely the same, if so, executing the step (4), otherwise, executing the step (5) after obtaining a time slot distribution matrix.
And 4, adjusting the time slot arrangement sequence.
And (3) interchanging the element values of any two columns in the time slot allocation matrix to obtain a time slot allocation matrix after the column sequence is updated, and then executing the step (3).
And 5, deploying a quantum error correction code mark bit symptom measurement circuit diagram.
If the stable subtype of the quantum error correcting code to be measured is X type, setting the initial state of the measurement bit in the quantum error correcting code mark bit symptom measurement circuit diagram corresponding to the X type stable subtype as | + >, and if the stable subtype to be measured is Z type, setting the initial state of the measurement bit in the quantum error correcting code mark bit symptom measurement circuit diagram corresponding to the Z type stable subtype as |0 >. The quantum error correction code marking bit symptom measuring line means that only two auxiliary quantum bits are used when the measuring code distance is 3 quantum stabilizer codes of each stabilizer. One of the auxiliary qubits is called a measurement bit, and is connected with each qubit contained in the measured stabile by using a quantum Controlled Not (CNOT) gate, so as to obtain error symptoms of the X-base measurement and the Z-base measurement; and the other auxiliary quantum bit is called a mark bit, is connected with the corresponding measuring bit by using a quantum Controlled Not (CNOT) gate and is used for acquiring mark bit symptoms, and deducing the position of quantum error in the measuring line and an error operator caused by the quantum error by combining the two error symptoms.
In a quantum error correction code mark bit symptom measurement circuit diagram, each column of a time slot distribution matrix corresponds to a time slot, each row corresponds to a stable sub, a quantum bit to be measured in each time slot is connected with a measurement bit corresponding to each stable sub through a quantum controlled non-CNOT gate, when connection is carried out, if the measured stable sub type is an X type, control quantum bits of each quantum controlled non-CNOT gate are measurement bits, the control quantum bits are all the quantum bits to be measured in each time slot, if the measured stable sub type is a Z type, the control quantum bits of each quantum controlled non-CNOT gate are all the quantum bits to be measured in each time slot, and the control quantum bits are all the measurement bits.
The marker bit measurement slot is added after the first column of slots and before the last column of slots. The marked bit measurement time slot is that only a quantum Controlled Not (CNOT) gate in which each measurement bit and the marked bit interact is placed in one time slot in a quantum error correction code marked bit symptom measurement line.
And finishing the deployment of the quantum error correction code marking bit symptom measurement circuit diagram.
Example 1:
example 1 of the present invention is a syndrome measurement circuit diagram in which a quantum [ [7,1,3] code Z-type stabilizer is subjected to syndrome measurement using the present invention, and a marker bit syndrome measurement circuit diagram of the quantum [ [7,1,3] code Z-type stabilizer is obtained.
And A, constructing a quantum bit sequence number matrix.
A binary generator matrix of an input quantum [ [7,1,3] code is as follows, with each row in the matrix representing a steady state quantum and each column representing a qubit.
And all the rows of the binary generator matrix are arranged in a descending order according to the size of each stable sub-weight by utilizing a matrix elementary row transformation mode. The stabilizer weight is the number of 1 included in the row corresponding to each stabilizer in the generator matrix. The weight of quantum [ [7,1,3] ] code stabilizer is equal to 4, and the arrangement order of the rows is unchanged.
And constructing a sequence number matrix with the size of 3 multiplied by 4 quantum bits.
Traversing the sorted binary generator matrix, sequentially extracting the column sequence number of each row of elements with the value of 1 from left to right according to the rows, and sequentially filling the extracted column sequence numbers into the rows corresponding to the qubit sequence number matrix to obtain the qubit sequence number matrix as follows.
And arranging each row of elements in the qubit sequence number matrix in a descending order according to the weight of each qubit to obtain an ordered qubit sequence number matrix. The qubit weight is the number of 1 contained in the corresponding column of each qubit in the generator matrix. The qubit weight values corresponding to the qubit numbers 1, 2,3, 4, 5, 6, and 7 are 1, 2, and 3, respectively, the arrangement order of the first row element values 4, 5, 6, and 7 is changed to 7, 5, 6, and 4, and so on for the remaining two rows.
And B, allocating the time slot by using a dynamic time slot allocation method.
Step 1, a time slot allocation matrix with the size of 3 × 4 is constructed.
And step 2, directly filling element values 7, 5, 6 and 4 in a first row in the qubit sequence number matrix into a first row of the time slot allocation matrix.
Step 3, judging whether the element value on the ith row of the jth column in the qubit sequence number matrix is equal to the element value in the 1 st to i-1 th rows of the jth column, if so, executing the step 4 of the step, otherwise, executing the step 5 of the step; wherein j represents a column serial number, the value range of j is [1, 4], i represents a row serial number, the value range of j is [2, 3], and the value sequence of j and i is from small to large.
And 4, selecting an element with different element values from all the element values in the 1 st to i-1 th rows of the jth column of the qubit sequence number matrix from the element values of the ith row of the qubit sequence number matrix which are not filled in the time slot distribution matrix, filling the element into the v th row of the c row of the time slot distribution matrix, and executing the 3 rd step of the step after exchanging the same element value with the selected different element values in the qubit sequence number matrix, wherein the values of c and i are correspondingly equal, and the values of v and j are correspondingly equal.
And 5, directly filling the element value on the ith row of the jth column in the qubit sequence number matrix into the ith row of the v column of the time slot allocation matrix.
And 6, complementing the blank part by using 0 elements if the blank part exists in the time slot allocation matrix, so that the values of all the other elements except the 0 element in each column are different, each column represents a time slot, the element value in each time slot represents the qubit to be measured in the time slot, and the 0 element represents that no qubit is measured.
And C, analyzing error symptoms.
3 error symptom matrices each of size 4 × 3 are constructed.
If odd elements in the r-th to 4-th rows of the kth row of the time slot allocation matrix appear in the h-th row of the time slot allocation matrix, 1 is filled in the f-th row of the s-th error symptom matrix, if even elements exist, 0 is filled in the position, wherein the value ranges of k and h are [1 and 3], the value ranges of r are [1 and 4], the values of s and k are correspondingly equal, the values of u and r are correspondingly equal, and the values of f and h are correspondingly equal.
And (4) judging whether more than two uplink element values in each error symptom matrix are completely the same, if so, executing the step (4), otherwise, executing the step (5) after obtaining a time slot distribution matrix.
And D, adjusting the time slot arrangement sequence.
And (3) interchanging the element values of any two columns in the time slot allocation matrix to obtain a time slot allocation matrix after the column sequence is updated, and then executing the step (3).
And E, deploying a quantum error correction code marking bit symptom measurement circuit diagram.
The type of the steady son to be measured is Z type, and the initial state of the measurement bit in the quantum error correction code marking bit symptom measurement circuit diagram corresponding to the quantum [ [7,1,3] code Z type steady son is set as |0 >. The quantum error correction code marking bit symptom measuring line is characterized in that when the measuring code distance is 3 quantum stabilizer each stabilizer of a quantum stabilizer code, only two auxiliary qubits are used, wherein one auxiliary qubit is called a measuring bit, and a quantum Controlled Not (CNOT) gate is connected with each qubit contained in the measured stabilizer to obtain error symptoms of X-base measurement and Z-base measurement; and the other auxiliary quantum bit is called a mark bit, is connected with the corresponding measuring bit by using a quantum Controlled Not (CNOT) gate and is used for acquiring mark bit symptoms, and deducing the position of quantum error in the measuring line and an error operator caused by the quantum error by combining the two error symptoms.
In a quantum error correction code marking bit symptom measurement circuit diagram of a quantum [ [7,1,3] ] code Z-type stabilizer, each column of a time slot distribution matrix corresponds to a time slot, each row corresponds to a stabilizer, a quantum bit to be measured in each time slot is connected with a measurement bit corresponding to each stabilizer by using a quantum controlled non-CNOT gate, and when connection is carried out, control quantum bits of each quantum controlled non-CNOT gate are all quantum bits to be measured in each time slot, and the controlled quantum bits are all measurement bits.
The marker bit measurement slot is added after the first column of slots and before the last column of slots. The marked bit measurement time slot is that only a quantum Controlled Not (CNOT) gate in which each measurement bit and the marked bit interact is placed in one time slot in a quantum error correction code marked bit symptom measurement line.
The disposition of the quantum error correction code marking bit symptom measurement circuit diagram is completed, and a quantum error correction code marking bit symptom measurement diagram of [ [7,1,3] ] code Z-type stabilizer based on dynamic time slot allocation is obtained and is shown in fig. 2.
One time slot is indicated between every two dotted lines in fig. 2, the solid lines numbered 1, 2,3, 4, 5, 6, 7 in fig. 2 indicate 7 qubits, the black dots on the solid lines indicate control qubits of a quantum-Controlled Not (CNOT) gate, numbered |0>The solid line represents the measured qubit, labeled | +>The solid line of (a) indicates a marker qubit, the solid line of (b) indicates
Representing a controlled qubit of a quantum Controlled Not (CNOT) gate, the block Z after the solid line represents performing a Z-basis measurement, and the block X after the solid line represents performing an X-basis measurement.
Example 2:
example 2 of the present invention is a syndrome measurement circuit diagram in which a qubit syndrome measurement circuit diagram of a quantum [ [12,2,3] code Z-type stabilizer is obtained by performing syndrome measurement on the quantum [ [12,2,3] code Z-type stabilizer by using the present invention.
And a, constructing a quantum bit sequence number matrix.
A binary generator matrix of an input quantum [ [12,2,3] code is as follows, where each row in the matrix represents a steady state and each column represents a qubit.
All the rows of the binary generator matrix of the quantum [ [12,2,3] ] code are arranged in descending order according to the size of each stable sub-weight by using a matrix elementary row transformation mode. The number of 1 contained in each stable sub-corresponding row in the generator matrix is 1, the stable sub-weight values of each row of the quantum [ [12,2,3] code are respectively 4, 6, 4 and 4, the third row with the stable sub-weight value of 6 is moved to the first row, and all rows of the matrix are arranged in a descending order according to the size of each stable sub-weight.
A qubit number matrix of size 5 × 6 is constructed.
Traversing the sorted binary generator matrix, sequentially extracting the column sequence number of each row of elements with the value of 1 from left to right according to the rows, sequentially filling the extracted column sequence numbers into the rows corresponding to the qubit sequence number matrix, and complementing by 0 elements if a blank part exists to obtain the qubit sequence number matrix as follows.
And arranging each row of elements in the qubit sequence number matrix in a descending order according to the weight of each qubit to obtain an ordered qubit sequence number matrix. The qubit weight is the number of 1 contained in the corresponding column of each qubit in the generator matrix. The qubit weight values corresponding to the qubit numbers 1, 2,3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 are 1, 2,3, 2, 1, 2, and 1, respectively, then the first row element values 4, 5, 6, 7, 8, and 9 are replaced by 5, 8, 4, 6, 7, and 9, and the rest four rows are analogized, so as to obtain the ordered qubit number matrix as follows.
And b, allocating the time slot by using a dynamic time slot allocation method.
In step 1, a slot allocation matrix with the size of 5 × 6 is constructed.
Step 2, the first row element values 5, 8, 4, 6, 7, 9 in the qubit index matrix are filled directly into the first row of the slot assignment matrix, as shown below.
Step 3, judging whether the element value on the ith row of the jth column in the qubit sequence number matrix is equal to the element value in the 1 st to i-1 th rows of the jth column, if so, executing the step 4 of the step, otherwise, executing the step 5 of the step; wherein j represents a column serial number, the value range of j is [1, 6], i represents a row serial number, the value range of j is [2, 5], and the value sequence of j and i is from small to large.
And 4, selecting an element with different element values from all the element values in the 1 st to i-1 th rows of the jth column of the qubit sequence number matrix from the element values of the ith row of the qubit sequence number matrix which are not filled in the time slot distribution matrix, filling the element into the v th row of the c row of the time slot distribution matrix, and executing the 3 rd step of the step after exchanging the same element value with the selected different element values in the qubit sequence number matrix, wherein the values of c and i are correspondingly equal, and the values of v and j are correspondingly equal.
And 5, directly filling the element value on the ith row of the jth column in the qubit sequence number matrix into the ith row of the v column of the time slot allocation matrix.
And 6, complementing the blank part by using 0 elements if the blank part exists in the time slot allocation matrix, so that the values of all the other elements except the 0 element in each column are different, each column of the time slot represents one time slot, the element value in each time slot represents the qubit to be measured in the time slot, the 0 element represents that any qubit is not measured, and the time slot allocation matrix is obtained as follows.
Analyzing error symptoms.
Construct 5 pieces with the size of omegaxX 5 error symptom matrix, ωxThe value of (a) is equal to the corresponding stable sub-weight of the xth row element in the time slot distribution matrix, and the value range of x is [1, 5]]。
If the odd number of the elements of the r-th row to the p-th row of the time slot allocation matrix appears in the h-th row of the time slot allocation matrix, filling 1 in the f-th row of the s-th error symptom matrix, and filling 0 in the f-th row of the s-th error symptom matrix if the even number of the elements exists, wherein the value ranges of k, s and h are all [1, 5%]And r has a value range of [1, p ]]The value of p and ωxThe values of s and k are equal correspondingly, the values of u and r are equal correspondingly, and the values of f and h are equal correspondingly, and 5 error symptom matrixes are obtained as follows.
And (4) judging whether more than two uplink element values in each error symptom matrix are completely the same, if so, executing the step (4), otherwise, executing the step (5) after obtaining a time slot distribution matrix.
And d, adjusting the time slot arrangement sequence.
And (3) interchanging the element values of any two columns in the time slot allocation matrix to obtain a time slot allocation matrix after the column sequence is updated, and then executing the step (3).
And e, deploying quantum error correction code marking bit symptom measurement circuit diagram.
The type of the steady sub to be measured is Z type, and the initial state of the measurement bit in the quantum error correction code marking bit symptom measurement circuit diagram corresponding to the quantum [ [12,2,3] code Z type steady sub is set to be |0 >. The quantum error correction code marking bit symptom measuring line is characterized in that when the measuring code distance is 3 quantum stabilizer each stabilizer of a quantum stabilizer code, only two auxiliary qubits are used, wherein one auxiliary qubit is called a measuring bit, and a quantum Controlled Not (CNOT) gate is connected with each qubit contained in the measured stabilizer to obtain error symptoms of X-base measurement and Z-base measurement; and the other auxiliary quantum bit is called a mark bit, is connected with the corresponding measuring bit by using a quantum Controlled Not (CNOT) gate and is used for acquiring mark bit symptoms, and deducing the position of quantum error in the measuring line and an error operator caused by the quantum error by combining the two error symptoms.
In a quantum error correction code marking bit symptom measurement circuit diagram of a quantum [ [12,2,3] ] code Z-type stabilizer, each column of a time slot distribution matrix corresponds to a time slot, each row corresponds to a stabilizer, a quantum bit to be measured in each time slot is connected with a measurement bit corresponding to each stabilizer by using a quantum controlled non-CNOT gate, and when connection is carried out, control quantum bits of each quantum controlled non-CNOT gate are all quantum bits to be measured in each time slot, and the controlled quantum bits are all measurement bits.
The marker bit measurement slot is added after the first column of slots and before the last column of slots. The marked bit measurement time slot is that only a quantum Controlled Not (CNOT) gate in which each measurement bit and the marked bit interact is placed in one time slot in a quantum error correction code marked bit symptom measurement line.
The disposition of the quantum error correction code marking bit symptom measurement circuit diagram is completed, and a quantum error correction code marking bit symptom measurement diagram of [ [12,2,3] ] code Z-type stabilizer based on dynamic time slot allocation is obtained and is shown in fig. 3.
In fig. 3, a time slot is indicated between every two dotted lines, a solid line labeled 1, 2,3, 4, 5, 6, 7, 8, 9, 10, 11, 12 in fig. 3 represents 12 qubits, a black dot on the solid line represents a control qubit of a quantum Controlled Not (CNOT) gate, and is labeled |0>OfLine represents the measured quantum bit, labeled | +>The solid line of (a) indicates a marker qubit, the solid line of (b) indicates
Representing the controlled qubit of a quantum Controlled Not (CNOT) gate, the box Z after the solid line represents the Z-basis measurement, and the box X after the solid line represents the X-basis measurement.
Claims (5)
1. A quantum error correction code marking bit symptom measuring method based on dynamic time slot allocation is characterized in that a quantum bit sequence number matrix is constructed to sort quantum bits to be measured, a time slot is allocated by utilizing a dynamic time slot allocation method, and an error symptom matrix is constructed to analyze error symptoms when quantum error correction code marking bit symptom measurement is carried out; the specific steps of the quantum error correction code mark bit symptom measurement comprise the following steps:
(1) constructing a quantum bit sequence number matrix:
(1a) inputting a binary generator matrix of quantum stabilizer codes, wherein each row in the matrix represents a stabilizer, and each column represents a quantum bit;
(1b) all rows of the binary generator matrix are arranged in a descending order according to the size of each stable sub-weight by utilizing a matrix elementary row transformation mode;
(1c) constructing a quantum bit serial number matrix, wherein the number of rows is equal to the number of the stable subunits, and the number of columns is equal to the maximum stable subunit weight value;
(1d) traversing the sorted binary generator matrix, sequentially extracting the column serial numbers of the elements with the row value of 1 from left to right according to the rows, sequentially filling the extracted column serial numbers into the rows corresponding to the qubit serial number matrix, and complementing by 0 elements if a blank part exists;
(1e) arranging each row of elements in the qubit sequence number matrix in a descending order according to the weight of each qubit to obtain an ordered qubit sequence number matrix;
(2) allocating time slots by using a dynamic time slot allocation method:
(2a) constructing a time slot distribution matrix, wherein the number of rows is equal to the number of stable subunits, and the number of columns is equal to the maximum stable subunit weight value;
(2b) directly filling the first row element values in the quantum bit sequence number matrix into the first row of the time slot distribution matrix;
(2c) judging whether the element value on the ith row of the jth column in the qubit sequence number matrix is equal to the element value in the 1 st to i-1 th rows of the jth column, if so, executing the step (2d), otherwise, executing the step (2 e); wherein j represents a column serial number, the value range of j is [1, omega ], i represents a row serial number, the value range of i is [2, m ], the value sequence of j and i is from small to large, m represents the number of the stable subunits, and omega represents the maximum weight value of the stable subunits;
(2d) selecting an element which is different from all element values in the 1 st to i-1 th rows of the jth column of the qubit sequence number matrix according to the sequence of the rows from left to right from the element value of the ith row of the qubit sequence number matrix, filling the element into the v th row of the c row of the qubit sequence number matrix, and executing the step (2c) after exchanging the same element value with the selected different element value in the qubit sequence number matrix, wherein the values of c and i are correspondingly equal, and the values of v and j are correspondingly equal;
(2e) directly filling the element value on the ith row of the jth column in the quantum bit serial number matrix into the c row of the vth column of the time slot allocation matrix;
(2f) if a blank part exists in the time slot allocation matrix, complementing the blank part by using 0 elements, so that the values of all other elements except the 0 element in each column are different, wherein each column represents a time slot, the element value in each time slot represents a qubit to be measured in the time slot, and the 0 element represents that any qubit is not measured;
(3) analyzing error symptoms:
(3a) construct t pieces with the size of omegaxX m error symptom matrix, where t is equal to m, ωxThe value of (a) is equal to the corresponding stable sub-weight of the xth row element in the time slot distribution matrix, and the value range of x is [1, q ]]The value of q is equal to m;
(3b) if odd number of elements in the r-th to p-th row elements of the k-th row of the time slot allocation matrix appear in the h-th row of the time slot allocation matrix, the u-th row and the u-th row of the s-th error symptom matrixColumn f is filled with 1, and if there are even number of elements, 0 is filled in the position, where k, s, and h all have the value range of [1, q ]]And r has a value range of [1, p ]]The value of p and ωxThe values of s and k are equal correspondingly, the values of u and r are equal correspondingly, and the values of f and h are equal correspondingly;
(3c) judging whether more than two uplink element values in each error symptom matrix are completely the same, if so, executing the step (4), otherwise, executing the step (5) after obtaining a time slot distribution matrix;
(4) adjusting the time slot arrangement sequence:
exchanging positions of any two columns of element values in the time slot allocation matrix with each other, obtaining a time slot allocation matrix after updating the column sequence, and then executing the step (3);
(5) deploying quantum error correction code marking bit symptom measurement circuit diagram:
(5a) if the stable subtype of the quantum error correcting code to be measured is X type, setting the initial state of the measurement bit in the quantum error correcting code mark bit symptom measurement circuit diagram corresponding to the X type stabilizer as | + >, and if the stable subtype to be measured is Z type, setting the initial state of the measurement bit in the quantum error correcting code mark bit symptom measurement circuit diagram corresponding to the Z type stabilizer as |0 >;
(5b) in a quantum error correction code mark bit symptom measurement circuit diagram, each column of a time slot distribution matrix corresponds to a time slot, each row corresponds to a stable sub, a quantum bit to be measured in each time slot is connected with a measurement bit corresponding to each stable sub by using a quantum controlled non-CNOT gate, when connection is carried out, if the measured stable sub type is an X type, the control quantum bits of each quantum controlled non-CNOT gate are measurement bits, the control quantum bits are the quantum bits to be measured in each time slot, if the measured stable sub type is a Z type, the control quantum bits of each quantum controlled non-CNOT gate are the quantum bits to be measured in each time slot, and the control quantum bits are the measurement bits;
(5c) adding a marker bit measurement slot after the first column of slots and before the last column of slots;
(5d) and finishing the deployment of the quantum error correction code marking bit symptom measurement circuit diagram.
2. The method according to claim 1, wherein the steady sub-weight in step (1b) is the number of 1's contained in the row corresponding to each steady sub in the generator matrix.
3. The method according to claim 1, wherein the qubit weight in step (1e) is the number of 1's contained in the corresponding column of each qubit in the generator matrix.
4. The method according to claim 1, wherein the quantum error correction code flag bit syndrome measurement circuit in step (5a) is configured to use only two auxiliary qubits for each stabile of a 3-quantum stabile code, where one auxiliary qubit is called a measurement bit and is connected to each qubit included in its measured stabile by a quantum Controlled Not (CNOT) gate for obtaining error syndromes of X-basis measurement and Z-basis measurement; and the other auxiliary quantum bit is called a mark bit, is connected with the corresponding measuring bit by using a quantum Controlled Not (CNOT) gate and is used for acquiring mark bit symptoms, and deducing the position of quantum error in the measuring line and an error operator caused by the quantum error by combining the two error symptoms.
5. The method according to claim 1, wherein the flag bit measurement time slot in step (5c) is a time slot in which only a quantum controlled not gate (CNOT) gate is placed in the quantum error correction code flag bit measurement line, where the measurement bit and the flag bit interact with each other.
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