CN113705819B - Quantum bit interaction error perception CNOT line nearest neighbor comprehensive method - Google Patents

Abstract

The application discloses a CNOT line nearest neighbor comprehensive method for quantum bit interaction error perception, which comprises the following steps: step S0, generating a corresponding Boolean matrix according to the CNOT line; step S1, constructing a minimum noise Steiner tree for each column of a Boolean matrix, wherein the minimum noise Steiner tree of each column is obtained according to a minimum noise path under an actual quantum system structure; step S2, gaussian elimination of elements below a main diagonal and Gao Sixiao elements above the main diagonal are sequentially carried out on the Boolean matrix, 1 is set on a stepiner point with a zero column value in a minimum noise stepiner tree of each column according to a minimum stepiner noise path in the Gaussian elimination process, the minimum noise path and the minimum stepiner noise path both consider adjacent quantum bit interaction error rate in an actual quantum system structure, nearest neighbor synthesis of a CNOT quantum line is realized on the premise of ensuring line reliability, and meanwhile, nearest neighbor synthesis cost of the quantum line is reduced.

Description

Quantum bit interaction error perception CNOT line nearest neighbor comprehensive method

Technical Field

The application belongs to the field of quantum computation, and particularly relates to a CNOT line nearest neighbor comprehensive method for quantum bit interaction error perception.

Background

With the continuous development of quantum computing, potential or direct subversion effects that may be brought about in many fields are becoming increasingly accepted by students. Although early quantum computing has many problems in the practical process, in recent years, with a series of technical difficulties such as a two-level system quantum processor (ion trap, superconductor, etc.), a quantum algorithm (Shor algorithm, etc.), quantum coding (QECC quantum error correction code, etc.), etc., quantum computing has been paid attention to by more and more students, especially with the disclosure of real quantum systems such as IBM-Q.

Because ion traps and superconducting techniques impose connection limitations on qubit interactions, executable quantum circuits must be required to satisfy nearest neighbor constraints, i.e., the operation of two qubits can only be performed on two physical qubits that are physically adjacent. The conventional method is to map a logic quantum circuit onto an actual physical architecture by using initial mapping, and fig. 1 shows a general process of mapping a quantum logic circuit composed of 4 CNOT quantum gates onto a two-dimensional quantum architecture. The mapping process may be as close as possible to achieve a CNOT gate and then by inserting a SWAP gate, achieve the full nearest neighbor of the quantum wire.

In the A Hardware-Aware Heuristic for the Qubit Mapping Problem in the NISQ document by Niu S et al, nearest neighbor of the quantum wires is achieved by SWAP or bridge gates. Since one SWAP gate consists of 3 CNOT gates and one bridge gate is equivalent to 4 CNOT gates, the use of these methods not only results in an increase in the number of CNOT gates by a factor of 3 in the integration process, but also increases the error rate generated based on the CNOT-error in the operation process, and although nearest neighbor of the line is achieved, there is no better reduction in the error rate to achieve the reliability of the quantum line running on a real quantum computer. With the gradual realization and commercialization of quantum computers, we need to further study a method for reducing the number of CNOT gates in the comprehensive process as much as possible on the premise of ensuring the reliability.

Quantum wire synthesis is another way to achieve quantum wire nearest neighbor. By representing the quantum wires as a corresponding boolean matrix. The process of converting the corresponding Boolean matrix into an identity matrix by using Gao Sixiao elements constrained by a physical architecture is nearest neighbor synthesis of CNOT quantum circuits. Each row operation in Gaussian elimination corresponds to one CNOT gate, the total CNOT gate cost is the number of row operations, and the reverse order operation of all CNOT gates is the execution order of the quantum circuits on an actual quantum computer. The CNOT quantum circuit is a basic circuit for realizing quantum algorithm by a quantum computer. Therefore, the research on the synthesis of the CNOT quantum circuit is very important for the reliable operation of the quantum circuit on an actual quantum chip. The current real quantum system is called as a medium-scale noisy quantum computer (NISQ) because the number of integrated quantum bits is not large and noise affecting the calculation result such as quantum decoherence exists. The quantum operation process in an NISQ system is "error-prone" due in part to the presence of an interaction error rate CNOT-error between adjacent qubits.

Disclosure of Invention

In order to solve the problem that the existing CNOT quantum circuit nearest neighbor algorithm cannot simultaneously have the advantages that the error rate of a quantum circuit is executed in an actual quantum system structure and the cost of generating the nearest neighbor circuit is low, the application provides a CNOT circuit nearest neighbor comprehensive method for realizing the interaction error rate of adjacent quantum circuits in the actual quantum system structure, firstly, a minimum noise Steiner tree of each column is generated by utilizing a minimum noise path, then, 1 is set for a Steiner point with zero column value in the minimum noise Steiner tree of each column according to the minimum Steiner noise path in a Gaussian elimination process, and the minimum noise path and the minimum Steiner noise path both consider the nearest neighbor comprehensive cost of the CNOT quantum circuits under the premise of ensuring the circuit reliability.

In order to achieve the above object, the present application adopts a technical scheme as follows:

a CNOT line nearest neighbor synthesis method for quantum bit interaction error perception comprises the following steps:

step S0, generating a corresponding Boolean matrix according to the CNOT line;

step S1, constructing a minimum noise Steiner tree for each column of a Boolean matrix, wherein the minimum noise Steiner tree of each column is obtained according to a minimum noise path under an actual quantum system structure;

step S2, gaussian elimination of elements below a main diagonal and Gao Sixiao elements above the main diagonal are sequentially carried out on the Boolean matrix, and 1 is set on a steper point with zero column value in a minimum noise steper tree of each column according to a minimum steper noise path in the Gaussian elimination process.

Further, the set of steper points of the minimum noise steper tree of a certain column is recorded as s= { S ₀ ,s ₁ ,…,s _i The vertex set with the current column value of 1 is v= { V } ₀ ,v,…,v _j The element Gaussian elimination process under the main diagonal in the step S2 is specifically as follows:

step S20, taking one element S from S at a time _t ；

Step S21, find the AND S through Floyd algorithm _t Adjacent and distance s _t The point p with the smallest noise weight;

step S22, if p ε V, i.e. p has a value of 1, then S will be S with p _t Put 1 and put s _t Deleted from set S, otherwise if the point value is 0, i.e., the point is in set S, S will be _t Updating to p, turning to step S21 until the set S is empty, and turning to step S23;

in step S23, all the stepiner points are set to 1, and Gaussian elimination is performed from the leaf node.

Further, in step S2, the gaussian elimination process of the element above the main diagonal is specifically:

step S24, taking one element S from S each time _t ；

Step S25, finding out index larger than S through Floyd algorithm _t And distance s _t A neighbor point p with the smallest noise weight;

step S26, if p ε V, i.e. p has a value of 1, then S will be S with p _t Put 1 and put s _t Deleted from set S, otherwise if the point value is 0, i.e., the point is in set S, S will be _t Updating to p, turning to step S25 until the set S is empty, and turning to step S27;

in step S27, all the stepiner points are set to 1, and Gaussian elimination is performed from the leaf node.

Compared with the prior art, the technical scheme of the application has the following advantages:

(1) Considering adjacent qubit interaction error rate, reliability is high towards quantum circuit

The application provides a CNOT line nearest neighbor comprehensive method for quantum bit interaction error perception, which comprises the steps of firstly generating a minimum noise Steiner tree of each column by utilizing a minimum noise path, then setting 1 for a Steiner point with zero column value in the minimum noise Steiner tree of each column according to the minimum Steiner noise path in the Gaussian elimination process, wherein the minimum noise path and the minimum Steiner noise path both consider adjacent quantum bit interaction error rate in an actual quantum system structure, and realizing nearest neighbor comprehensive of a CNOT quantum line on the premise of ensuring line reliability;

(2) The cost of the nearest neighbor integrated circuit is low

The Gaussian elimination element is adopted to perform line synthesis, namely the Boolean matrix corresponding to the whole line is directly operated, and a similar linear relation that SWAP gates are inserted to cause the number of CNOT gates to be increased by a multiple of 3 does not exist between the number of non-adjacent CNOT gates, so that the number of CNOT gates in the synthesis cannot be increased due to the increase of the non-adjacent CNOT gates. Therefore, the matrix synthesis mode can reduce the nearest neighbor synthesis cost of the quantum circuit.

Drawings

FIG. 1 is a schematic diagram of a quantum logic circuit mapping to a two-dimensional quantum architecture in the background of the application;

FIG. 2 is a diagram of a weighted ibmq-5_yorkown topology in accordance with an embodiment of the present application;

FIG. 3 is a flowchart of a method for synthesizing nearest neighbors of a CNOT line for qubit interaction error perception in an embodiment of the application;

FIG. 4 is a diagram of a topology change of ibmq_5_yorkown for a 5*5 Boolean matrix using Gaussian elimination of elements below a main diagonal in an embodiment of the application;

fig. 4a is an ibmq_5_yorkown topology change diagram corresponding to the first column of the boolean matrix during the elimination;

FIG. 4b is a diagram of a corresponding ibmq_5_yorkown topology change when the second column of the Boolean matrix is eliminated;

FIG. 4c is a diagram of a corresponding ibmq_5_yorkown topology change when the third column of the Boolean matrix is eliminated;

FIG. 4d is a diagram of the corresponding ibmq_5_yorkown topology change when the fourth column of the Boolean matrix is eliminated;

FIG. 5 is a diagram of a topology change of ibmq_5_yorkown for a 5*5 Boolean matrix using Gaussian elimination of elements above a main diagonal in an embodiment;

fig. 5a is an ibmq_5_yorkown topology change diagram corresponding to the fifth column of the boolean matrix during the elimination;

fig. 5b is a diagram of the ibmq_5_yorkown topology change corresponding to the fourth, third and second columns of the boolean matrix when the elements are eliminated.

DETAILED DESCRIPTION OF EMBODIMENT (S) OF INVENTION

The following description of the embodiments of the present application will be made clearly and fully with reference to the accompanying drawings, in which it is evident that the embodiments described are only some, but not all embodiments of the application. All other embodiments, which can be made by those skilled in the art based on the embodiments of the application without making any inventive effort, are intended to be within the scope of the application.

The current real quantum system is called as a medium-scale noisy quantum computer (NISQ) because the number of integrated quantum bits is not large and noise affecting the calculation result such as quantum decoherence exists. As shown in FIG. 2, the weights on the sides of the ibmq_5_yorkton architecture diagram are the interaction error rate CNOT-error between adjacent qubits for a real quantum computer species offered by IBM Q authorities. Quantum operation processes in NISQ systems are "error-prone" due in part to the presence of CNOT-error

As shown in fig. 3, the method for synthesizing the nearest neighbor of the CNOT line for quantum bit interaction error perception comprises the following steps:

step S0, generating a corresponding Boolean matrix according to the CNOT line;

generating a corresponding boolean matrix according to the CNOT line is prior art and the principle is not repeated here; now, by way of example, the CNOT line acquisition corresponding boolean matrix in fig. 1 is shown as follows:

step S1, constructing a minimum noise Steiner tree for each column of a Boolean matrix, wherein the minimum noise Steiner tree of each column is obtained according to a minimum noise path under an actual quantum system structure;

the topological graph taking CNOT error rate as weight is called a noisy topological graph, and the multisource shortest path set of the current topological graph is W [ i ] [ j ] obtained through Floyd algorithm. The state transition equation is:

W[i][j]＝min{W[i][j],W[i][k]+W[k][j]} (2)

in equation (1), W [ i ] [ j ] represents the shortest distance from vertex i to vertex j, and k is all intermediate points that may pass between i and j. The initial state W [0] [0] =0, when the intermediate point is k according to the Floyd algorithm, the path length W [ i ] [ j ] from i to j in the set is updated, and all the intermediate points which can pass through are traversed to obtain the shortest path with the weight which is globally optimal.

Record multisource shortest path set W [ i ]][j]Is W _t [i][j]At this time, the CNOT error rate set corresponding to the edge of the vertex i to the vertex j is { e } ₁ ,e ₂ ,…e _i …,e _n Then at this point:

definition 1: from the qubit with a value of 1 in the current column of the boolean matrix, the Steiner tree with the minimum noise path length obtained from the noisy topology is called the minimum noise Steiner tree, and the path from node i to node j in the Steiner tree is called the minimum Steiner noise path from i to j.

The minimum Steiner noise path length may be obtained by looking up the set of multi-source shortest paths W [ i ] [ j ] of the current topology. Non-leaf nodes with a value of 0 in the Steiner tree are called Steiner points and are used to transfer the state "1" of the qubit where the main diagonal is located or other neighboring qubits.

Step S2, gaussian elimination of elements below a main diagonal and Gao Sixiao elements above the main diagonal are sequentially carried out on the Boolean matrix, and 1 is set on a steper point with zero column value in a minimum noise steper tree of each column according to a minimum steper noise path in the Gaussian elimination process.

Let the set of Steiner points of the minimum noise Steiner tree of a certain column be S= { S ₀ ,s ₁ ,…,s _i The vertex set with the current column value of 1 is v= { V } ₀ ,v,…,v _j The element Gaussian elimination process under the main diagonal in the step S2 is specifically as follows:

step S20, taking one element S from S at a time _t ；

Step S21, find the AND S through Floyd algorithm _t Adjacent and distance s _t The point p with the smallest noise weight;

step S22, if p ε V, i.e. p has a value of 1, then S will be S with p _t Put 1 and put s _t Deleted from set S, otherwise if the point value is 0, i.e., the point is in set S, S will be _t Updating to p, turning to step S21 until the set S is empty, and turning to step S23;

in step S23, all the stepiner points are set to 1, and Gaussian elimination is performed from the leaf node.

In the step S2, the Gaussian elimination process of the element above the main diagonal is specifically as follows:

step S24, taking one element S from S each time _t ；

Step S25, finding out index larger than S through Floyd algorithm _t And distance s _t A neighbor point p with the smallest noise weight;

step S26, if p ε V, i.e. p has a value of 1, then S will be S with p _t Put 1 and put s _t Deleted from set S, otherwise if the point value is 0, i.e., the point is in set S, S will be _t Updating to p, turning to step S25 until the set S is empty, and turning to step S27;

in step S27, all the stepiner points are set to 1, and Gaussian elimination is performed from the leaf node.

As shown in fig. 4 and 5, the nearest neighbor integration method of the present application will now be described with reference to ibmq_5_yorkown topology by using 1 boolean matrix 5*5.

Fig. 4 shows an example of gaussian elimination of the element below the main diagonal, and as shown in fig. 4a, when column 1 is operated, qubit 0 is the root node and qubit 3 is the leaf node. From the physical topology, it can be seen that qubits 0 and 3 are not close neighbors. The steper point is first determined, CNOT-error, although 0 and 2 neighbors ₀₂ 4.18e ^-2 Is greater than CNOT-error ₀₁ And CNOT-error ₁₂ The sum, qubits 1 and 2 are the stepiner points, thus generating a minimum noise stepiner tree. Then, element Gaussian elimination under the main diagonal is carried out: firstly, taking out the quantum bit 1, and calculating to obtain a neighbor quantum bit of which the quantum bit 2 is nearest to 1, wherein the quantum bit 2 is in a steper point set, and the point which can be used for putting the quantum bit 2 into 1 needs to be continuously searched, and the quantum bit 3 is obtained through calculation, so the step of putting the quantum bit 1 into the steper point is to firstly use the quantum bit 3 to put the quantum bit 2 into 1 and then use the quantum bit 2 to put the quantum bit 1 into 1. The operation corresponding to the Boolean matrix is that the 4 th row is exclusive-ored to the 3 rd row, and then the 3 rd row is exclusive-ored to the 2 nd row. The gaussian elimination is then continued, i.e. first with qubit 2 eliminating qubit 3, then with qubit 1 eliminating qubit 2, and finally with qubit 0 eliminating qubit 1. The corresponding matrix operation is that the 3 rd row is exclusive-ored to the 4 th row, the 2 nd row is exclusive-ored to the 3 rd row, and the 1 st row is exclusive-ored to the 2 nd row. The line cost is 5.

As shown in FIG. 4b, when column 2 is operated, qubit 1 is the root node, qubit 4 is the leaf node, and qubit 2 is the intermediate node, which is not only adjacent to qubits 1, 4, but also has a value of 1, so Gaussian elimination can be directly performed. The qubit 2 is used to eliminate the qubit 4, and then the qubit 1 is used to eliminate the qubit 2. The corresponding matrix is operated by exclusive-or the 3 rd row to the 5 th row and exclusive-or the 2 nd row to the 3 rd row. The line cost is 2.

As shown in FIG. 4c, when column 3 is operated, qubit 2 root node, qubit 3 leaf node, qubits 2 and 3 neighbor and there is no shorter path, so Gaussian elimination can be performed directly. Qubit 3 is eliminated with qubit 2. The corresponding matrix operates to exclusive or the 3 rd row to the 4 th row. The line cost is 1.

As shown in fig. 4d, columns 4 and 5 have satisfied that the element below the main diagonal is 0, and no further operation is required. At this time, the lower triangle is integrated. The total cost of the downward triangle is 8.

FIG. 5 is an example of Gaussian elimination of elements above the principal diagonal, as shown in FIG. 5a, with qubit 4 being the root node and qubit 1 bit leaf node operating on column 5. From the physical topology, it can be seen that qubits 4 and 1 are not close neighbors. Firstly, determining a stepiner point, firstly, taking out a quantum bit 2, and calculating to obtain a quantum 1 which is a nearest neighbor quantum bit to the quantum bit 2, but 1<2 it is not possible to place 2 in 1 with 1. It is necessary to continue to find the point that can be used to place qubit 2 at 1, and calculate qubit 3, but CNOT-error ₂₄ Less than CNOT-error ₂₃ And CNOT-error ₃₄ And therefore only qubit 2 is needed as a stepiner point and only qubit 4 is used to place qubit 2 by 1. The process of placing 1 for the stepiner point 2 is to place 1 for the qubit 2 with the qubit 4. The corresponding boolean matrix operates to exclusive-or the 5 th row to the 4 th row. The gaussian elimination is then continued, i.e. first eliminating qubit 1 with qubit 2 and then eliminating qubit 2 with qubit 4. The corresponding matrix is operated by exclusive-or the 3 rd row to the 2 nd row and exclusive-or the 5 th row to the 3 rd row. The line cost is 3.

As shown in fig. 5b, the matrix has now become an identity matrix, thus doing any operation on columns 4, 3, 2, 1.

Experimental results and analysis:

the nearest neighbor comprehensive method is realized by using Python language, the experimental environment is a macOS Big Sur (11.2.3) operating system, intel core i5 7267U@3.1GHZ dual-core processing is performed, and a memory is 16 GB. The quantum circuit of the experiment is a 5-quantum bit random CNOT quantum circuit, the gate number is expanded on the basis of the existing circuit in order to improve the experiment scale, the gate levels corresponding to the quantum circuit of the experiment are 15, 20, 30, 40, 80, 100 and 200, and each quantum gate level is provided with 20 quantum circuits. A HA method for realizing the close neighbor of the CNOT line by taking the insertion of a SWAP gate into consideration of the error rate of the CNOT is given in the A Hardware-Aware Heuristic for the Qubit Mapping Problem in the NISQ document published by Niu S et al. The method for testing HA respectively by using the same line on ibmq_5_yorkown architecture of IBM Q and the method provided by the application have the advantages that the number of CNOT-error and nearest neighbor CNOT gates in the nearest neighbor comprehensive process of CNOT quantum lines under 7 gate levels is counted experimentally, and the average value of 20 lines is taken as the final result of each gate level. The CNOT-error experimental data pair is shown in Table 1.

TABLE 1 CNOT-error comparison

Size represents the number of CNOT gates in the quantum wire. CNOT-error is the sum of error rates obtained through experiments, HA is document A Hardware-Aware Heuristic for the Qubit Mapping Problem in the NISQ to obtain experimental data, and SR is the method provided by the application to obtain experimental data. Reduction1 represents a reduced error rate, optimization rate, of Imp1 for the SR method relative to the HA method.

As can be seen from Table 1, the SR method HAs a larger improvement in reducing the error rate than the HA method, the sum of CNOT-error is optimized, the more the number of gates is, the more obvious the optimizing effect is, and the optimizing rate reaches 93.15% at 200 gate levels. Experimental data shows that the SR method realizes the synthesis of the CNOT quantum circuit with lower error rate, which has great significance for improving the reliability of the quantum circuit.

TABLE 2 comparison of adjacent CNOT gate counts

Table 2 shows the comparison of the number of nearest neighbor CONT gates required for quantum wire synthesis. adjacent CNOT gate counts is the nearest neighbor CNOT gate number required by line synthesis, reduction2 represents the CNOT gate number reduced by the SR method relative to the HA method, and the optimization rate of the CNOT gate number is Imp2.

As can be seen from the experimental data in tables 1 and 2, compared with the HA method in the document A Hardware-Aware Heuristic for the Qubit Mapping Problem in the NISQ, the SR method provided by the application reduces the error rate as much as possible during quantum circuit integration by the proposed algorithm, improves the correctness of the running result of the quantum circuit, and reduces the number of CNOT gates during quantum circuit integration on the premise of ensuring the circuit reliability. The average optimization rate was 61.48%. When the gate number of the CNOT quantum circuit reaches 200, the optimization rate is nearly 92.5 percent.

The foregoing examples illustrate only a few embodiments of the application and are described in detail and are not to be construed as limiting the scope of the application. Various modifications and variations of the present application will be apparent to those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims (2)

1. The nearest neighbor comprehensive method of the CNOT line for quantum bit interaction error perception is characterized by comprising the following steps of:

step S0, generating a corresponding Boolean matrix according to the CNOT line;

step S1, constructing a minimum noise Steiner tree for each column of a Boolean matrix, wherein the minimum noise Steiner tree of each column is obtained according to a minimum noise path under an actual quantum system structure;

step S2, sequentially carrying out Gaussian elimination of an element below a main diagonal and Gao Sixiao elements above the main diagonal on the Boolean matrix, and setting 1 for a Steiner point with zero column value in a minimum noise Steiner tree of each column according to a minimum Steiner noise path in the Gaussian elimination process;

let the set of Steiner points of the minimum noise Steiner tree of a certain column be S= { S ₀ ,s ₁ ,…,s _i The vertex set with the current column value of 1 is v= { V } ₀ ,v,…,v _j The i+j is less than or equal to n, n is the order of a Boolean matrix, and the Gaussian elimination process of the element below the main diagonal in the step S2 is specifically as follows:

step S20, taking one element S from S at a time _t ；

In step S21 of the process,finding the AND s by Floyd algorithm _t Adjacent and distance s _t The point p with the smallest noise weight;

step S22, if p ε V, i.e. p has a value of 1, then S will be S with p _t Put 1 and put s _t Deleted from set S, otherwise if the point value is 0, i.e., the point is in set S, S will be _t Updating to p, turning to step S21 until the set S is empty, and turning to step S23;

in step S23, all the stepiner points are set to 1, and Gaussian elimination is performed from the leaf node.

2. The method for synthesizing the nearest neighbor of the CNOT line by adopting the quantum bit interaction error perception according to claim 1, wherein the Gaussian elimination process of the element above the main diagonal in the step S2 is specifically as follows:

step S24, taking one element S from S each time _t ；

Step S25, finding out index larger than S through Floyd algorithm _t And distance s _t A neighbor point p with the smallest noise weight;

step S26, if p ε V, i.e. p has a value of 1, then S will be S with p _t Put 1 and put s _t Deleted from set S, otherwise if the point value is 0, i.e., the point is in set S, S will be _t Updating to p, turning to step S25 until the set S is empty, and turning to step S27;

in step S27, all the stepiner points are set to 1, and Gaussian elimination is performed from the leaf node.