CN116451797A - Quantum circuit segmentation method based on quantum process chromatography - Google Patents
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Abstract
A quantum circuit segmentation method based on quantum process chromatography comprises the following steps: determining a sub-line group SC; estimating a quantum process matrix of the SC by utilizing quantum process chromatography; optimizing the input state and the output end measurement basis of the SC by utilizing the operator property and spectrum decomposition in the Hilbert space; quantum using SC a process matrix is provided which is a function of the process matrix, calculating the probability of each sub-line measurement result; and carrying out classical post-processing on the probability of each sub-line measurement result in the SC to reconstruct the probability of the original quantum line measurement result. The invention overcomes the defect that the number of sub-lines after segmentation increases exponentially along with the segmentation times in the prior art, reduces the expenditure of calculation resources, estimates the quantum process matrix of the sub-lines by utilizing the quantum process chromatography, and obtains the effective probability distribution of the sub-line measurement result by using the quantum process matrix, thereby ensuring the reliability of the original line reconstruction result.
Description
Technical Field
The invention belongs to the technical field of quantum information, and further relates to a quantum circuit segmentation method based on quantum process chromatography in the technical field of quantum circuit segmentation. The invention can be used for dividing the multi-quantum bit line into a plurality of small-scale sub-lines to independently operate in a virtual environment, and realizes the simulation operation of the large-scale quantum line through the classical post-processing of the sub-line measurement result.
Background
Currently, quantum computing has entered the critical era of Quantum technology development, the noise-containing mesoquantum NISQ (Noisy Intermediate-Scale Quantum) era. In the NISQ era, quantum computers face two challenges: one is that the quantum bit of the quantum computer operation is unstable, contain the great noise; the other is that the total number of the quantum bits provided by the medium quantum computer is 50 to hundreds, and only a specific algorithm can be realized, so that the practical requirement cannot be met. To solve the above problems, the idea of quantum wire division is proposed, which is generally as follows: firstly, dividing a large-scale quantum circuit into a plurality of small quantum bit circuits; then, the measurement results and the corresponding probabilities of each sub-line are obtained by respectively distributing the measurement results to different quantum computers for operation; and finally, carrying out classical post-processing on the measurement result of each sub-line, and restoring the measurement result and probability distribution of the original quantum line.
The paper "Simulating Large Quantum Circuits on a Small Quantum Computer" by Tianyi Peng et al (Physical Review Letters 2020:150504) discloses a quantum wire segmentation method based on tensor networks. The method comprises the following implementation steps: firstly, converting a large-scale quantum circuit into a tensor network diagram by utilizing basic knowledge of a tensor network; secondly, providing a cluster simulation method, wherein the method utilizes the decomposability of a tensor network and cluster parameters among quantum circuits to divide the quantum circuits into a plurality of weakly-correlated small-scale sub-circuits; then, using a quantum computer to respectively operate the sub-circuits to obtain a measurement result; and finally, restoring the measurement result of the original quantum circuit by applying the relation between the original circuit and the sub-circuit after segmentation derived by the tensor network. The method has the following defects: the number of sub-lines after segmentation will increase exponentially with the number of segmentations, consuming a lot of quantum and classical computing resources.
Wei Tang et al in its published paper "CutQC: using Small Quantum Computers for Large Quantum Circuit Evaluations" (Proceedings of the 26th ACM International conference on architectural support for programming languages and operating systems 2021:473-486) propose a scalable quantum-classical hybrid computing method. The method comprises the following implementation steps: firstly, selecting proper dividing positions to complete the division of quantum circuits, and obtaining a plurality of sub-circuits; secondly, respectively operating the sub-circuits by using a quantum computer, and observing measurement results under different measurement bases; and finally, carrying out classical post-processing on the measurement result to reconstruct probability distribution of the original quantum circuit measurement result. The method has the following defects: the negative number of the sub-line measurement result corresponding probability after the segmentation can appear under the influence of equipment noise and the sub-line operation times, so that the difference between the reconstructed original quantum line result and the original quantum line measurement result is larger.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a quantum circuit segmentation method based on quantum process chromatography. The method is used for solving the problem that the number of sub-lines after segmentation can be exponentially increased along with the increase of segmentation times, so that a large amount of quanta and classical computing resources are consumed; and the negative number of the sub-line measurement result after segmentation can appear under the influence of equipment noise and the running times of the sub-line, so that the difference between the reconstructed original quantum line result and the original quantum line measurement result is larger.
In order to achieve the above purpose, the invention optimizes the number of input states and the number of times of measurement of output ends of the sub-circuit based on the linear property and the spectral decomposition of operators in the Hilbert space, firstly uses the identity of the operators to equivalently express the input density matrix I < - > -and I < - > -i| to reduce the quantum state of each quantum input end of the sub-circuit to 4, then uses the measuring basis I and Z basis to have the same eigenvectors after the spectral decomposition, and reduces the measuring basis of each quantum output end of the sub-circuit to 3 in total, thereby solving the problem that the number of sub-circuits after the segmentation in the prior art can be exponentially increased along with the increase of the segmentation times, and consuming a large number of quanta and classical computing resources. According to the invention, the quantum process of each sub-line after being divided is estimated in sequence by utilizing a quantum process chromatography technology of linear reconstruction, a corresponding quantum process matrix is obtained, and then the measurement result and probability distribution condition of each sub-line after being divided are calculated by using the quantum process matrix, so that the operation condition of an original quantum line is reconstructed, and the problems that the prior art is influenced by equipment noise and the operation times of the sub-line, the corresponding probability of the measurement result of the sub-line after being divided can generate negative numbers, and the difference between the measurement result of the reconstructed original quantum line and the measurement result of the original quantum line is larger are solved.
The technical scheme of the invention is as follows:
step 1, determining a sub-line group SC;
step 2, estimating a quantum process matrix of the sub-line group SC by utilizing quantum process chromatography;
step 3, optimizing the quantum bit input state of the sub-line group SC and the measurement basis of the output end by utilizing the linear property and spectrum decomposition of an operator in the Hilbert space;
step 4, calculating probability distribution corresponding to each sub-line measurement result by using a quantum process matrix of the sub-line group SC estimated by using the quantum process chromatography;
and 5, carrying out classical post-processing on the probability of each sub-line measurement result in the sub-line group SC, and reconstructing the probability distribution condition of the original sub-line measurement result.
Compared with the prior art, the invention has the following advantages:
firstly, the invention optimizes the input states and output end measurement bases of the sub-circuits after division by utilizing the linear property and spectrum decomposition of operators in the Hilbert space, reduces the quantum states of each quantum input end in each sub-circuit to 4 and reduces the measurement bases of the quantum output ends to 3, thereby overcoming the defect that the number of sub-circuits after division in the prior art can be exponentially increased along with the increase of the division times, reducing the cost of calculation resources and improving the efficiency of quantum circuit division.
Secondly, the quantum process matrix of each sub-line is obtained by utilizing the quantum process chromatography to estimate the quantum process of each sub-line after division, and then the quantum process matrix is used for calculating the occurrence probability of the measurement result of each sub-line after division, so that the defect that the negative number can appear in the corresponding probability of the measurement result of the sub-line after division due to the influence of equipment noise and the operation times of the sub-line in the prior art is overcome, the effective probability distribution of the measurement result of the sub-line after division is obtained, and the reliability of the reconstruction result of the original quantum line is ensured.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a quantum circuit diagram of an embodiment of the present invention;
FIG. 3 is a diagram of a quantum wire segmentation process according to an embodiment of the present invention;
fig. 4 is a simulation diagram of an embodiment of the present invention.
Detailed Description
The steps for implementing the present invention will be further described with reference to fig. 1 and the embodiment.
The implementation steps of the embodiment of the present invention will be further described with reference to fig. 1.
The embodiment of the invention takes a maximum entangled GHZ (Greenberger-Horne-Zeilinger) state preparation line of 3 qubits as an example for quantum line segmentation, and the quantum line diagram is shown in figure 2.
As can be seen from fig. 2, the single qubit H gate acts on the qubit q 0 Two-quantum bit CNOT 1 Gate action on qubit q 0 And q 1 ，CNOT 2 Gate action on qubit q 0 And q 2 . The sub-circuit obtained after the segmentation in the embodiment of the invention operates on a quantum computer with 2 quantum bits.
Step 1, determining a sub-line group SC.
Step 1.1, finding the maximum quantum bit gate from the quantum circuit to be segmented, and comparing the quantum bit total number a of the quantum bit gate with the quantum bit total number k of a quantum computer running the segmented sub circuit: if a is less than or equal to k, executing the step 1.2; otherwise, step 1.3 is performed. Where a represents the total number of qubits of the largest quantum bit gate of the quantum circuit to be split and k represents the total number of qubits of the quantum computer.
The maximum qubit gate of the embodiment of the invention is CNOT 1 Gate and CNOT 2 Gate with total number of qubits a=2. And (2) executing the step (1.2) by executing the quantum bit total number k=2 of the quantum computer of the sub-line after the division, wherein the quantum bit total number is equal to the quantum bit total number of the maximum quantum bit gate.
Step 1.2, all the qubit bits of the maximum qubit gate are assembled into an initial set of split points.
The maximum quantum bit gate CNOT of the embodiment of the invention 1 All qubits of the gate make up the initial set of split points 1{q 0 ,q 1 Maximum qubit gate CNOT 2 All qubits of the gate make up the initial set of split points 2{q 0 ,q 2 }。
And step 1.3, performing equivalent conversion on the maximum qubit gate by using a single qubit gate and a controlled NOT gate of two qubits, and executing step 1.1 after obtaining a conversion circuit of the quantum circuit.
And 1.4, after finding all the quantum bit gates with the total number of quantum bits equal to a from all the quantum bit gates connected with the maximum quantum bit gate, updating the initial segmentation point set.
In the embodiment of the invention, the maximum quantum bit gate CNOT 1 The gates are connected, and the qubit gate with the total number of qubits equal to 2 is CNOT 2 Gate, updating initial set of partitioning points 1 to { q 0 }. And maximum qubit gate CNOT 2 The gates are connected, and the total number of qubits is equal to 2Is CNOT 1 Gate, updating initial set of split points 2 to { q 0 }. In summary, the set of segmentation points of the embodiment of the invention is { q 0 }。
And step 1.5, dividing all the quantum bits determined by the updated dividing point set to obtain a sub-line group SC of the quantum line to be divided. And SC represents a sub-line set obtained after the quantum lines to be segmented are segmented.
The embodiment of the invention divides the point set { q } 0 Quantum bit q determined 0 Dividing to obtain a sub-line group SC= { SC 1 ,sc 2 }。
And 2, estimating a quantum process matrix of the sub-line group SC by using quantum process chromatography.
Step 2.1, generating each sub-line SC of the sub-line group SC in the k-qubit quantum computer j . Dimension is as followsDensity matrix ρ of (1) j Input to each sub-line sc j Through each sub-line sc j Projection measurement operator M of (2) j Measuring each sub-line sc j To obtain each sub-line sc j Is measured by the measurement result R of (2) j And R is j Probability of occurrence P j . Wherein sc j Represents the jth sub-line, Q, in the sub-line group SC j Representing the total number of qubits, ρ, of the jth sub-line j A density matrix representing the j-th sub-line input, < >>·>Representing a quantum state symbol, < >>Representing tensor product operations, M j Representing the projection measurement operator of the jth sub-line,R j representing the measurement result of the jth sub-line, +.>P j Representing the measurement result R j Probability of occurrence.
The embodiment of the invention uses a quantum bit gate provided by a quantum computer with k=2 quantum bits to act on the quantum bit to build a sub-line SC of a sub-line group SC 1 And sc 2 . The sub-line sc of the embodiment of the invention 1 Total number of qubits Q of (2) 1 =2, sub-line sc 2 Total number of qubits Q of (2) 2 =2, so the sub-line sc 1 And sc 2 Input density matrixSub-line sc 1 And sc 2 Projection measurement operator of (2)Projection measurement operator M 1 Measurement sub-line sc 1 Is output to obtain the measurement result R 1 ＝{|00>,|01>,|10>,|11>Sum R 1 Probability of occurrenceProjection measurement operator M 2 Measurement sub-line sc 2 Is output to obtain the measurement result R 2 ＝{|00>,|01>,|10>,|11>Sum R 2 Probability of occurrence->
Step 2.2, utilizeThe expected value of each set of measurements in each sub-line measurement is calculated. Wherein (1)>Representing the jth sub-line measurement result R j Expected value of the s-th set of measurement results in (b), t represents R j Sequence number of each group of measurement results, Σ represents summing operation, +.>R represents j Probability of occurrence of the (t) th measurement of the(s) th group, is->R represents j Eigenvalues of the projection measurement operator used for the(s) th set of (t) th measurement results, are->
The sub-line sc of the embodiment of the invention 1 Input density matrix ρ 1 ＝|10><10|, sub-line sc 1 Projection measurement operator of (2)Using M 1 Measurement sub-line sc 1 To obtain the output of the sub-line sc 1 Is measured by the measurement result R of (2) 1 ＝{|00>,|01>，|10>，|11>Sum R 1 Probability of occurrence P 1 ＝{0.256,0.253,0.24,0.251}，M 1 Eigenvalue V of (2) 1 = {1, -1,1}, then the expected value E of the set of measurements 1 =1×0.256+ (-1) x 0.253+ (-1) x 0.24+1×0.251=0.014. Obtaining a sub-line sc by imitating the calculation process 1 And sc 2 Expected values for all measurements.
Step 2.3, utilizeA density matrix is calculated for each sub-line output. Wherein ρ is j ' represents the density matrix of the j-th sub-line output, s represents the sequence number of the j-th sub-line measurement result expected value,/or%>Measurement basis representing the 1 st qubit output of the jth sub-line, +.>Measurement basis representing the 2 nd qubit output of the j-th sub-line,/and>q representing the jth sub-line j The basis for the measurement of the output of the individual qubits,
the sub-line sc of the embodiment of the invention 1 Total number of qubits Q of (2) 1 =2, then the density matrix of the sub-line outputs is as follows:
wherein ρ is 1 ' represent sub-line sc 1 The density matrix of outputs, s denotes the sub-line sc 1 The sequence number of the expected value of the measurement result,representing sub-line sc 1 Is the expected value of the s-th set of measurements, is>Representing sub-line sc 1 Measurement basis of 1 st qubit output of (a), a +.>Representing sub-line sc 1 Measurement basis of the 2 nd qubit output of (a), a +.>
Substituting the expected value of the measurement result obtained in the step 2.2 into the above formula to calculate a sub-line sc 1 Output density matrix ρ 1 '. Sub-line sc 2 Total number of qubits Q of (2) 2 =2, still using the above formula, getTo sub-line sc 2 Output density matrix ρ 2 ′。
Step 2.4, utilizeA quantum process matrix is calculated for each sub-line. Wherein m and n respectively represent the serial numbers of the Kraus operator group, K m Representing the mth Kraus operator, < +.>Conjugate transpose representing the nth Kraus operator,>representing conjugate transpose operation,/->Representing the nth row and column elements of the quantum process matrix of the jth sub-line.
The embodiment of the invention selects Kraus operator with easy propertySub-line sc 1 All the input density matrixes and the corresponding output density matrixes are substituted into a formula to calculate a sub-line sc 1 Elements in a quantum process matrix of (2) to obtain a sub-line sc 1 Is a quantum process matrix χ 1 . Sub-line sc 2 All the input density matrixes and the corresponding output density matrixes are substituted into a formula to calculate a sub-line sc 2 Elements in a quantum process matrix of (2) to obtain a sub-line sc 2 Is a quantum process matrix χ 2 。
And 3, optimizing the quantum bit input state of the sub-line group SC and the measurement basis of the output end by utilizing the linear property and the spectrum decomposition of an operator in the Hilbert space.
And 3.1, finding out the quantum bit determined by the dividing point set from all the quantum bit of each sub-line, and judging the type of the quantum bit of the sub-line at the dividing point. If the quantum bit does not execute the quantum bit gate operation before the division point, the division point is judged as a quantum input end; otherwise, the division point is determined as a quantum output end.
The segmentation point set { q }, of the embodiment of the invention 0 Determination of qubit q 0 In sub-line sc 1 Before the segmentation point of (a) H gate and CNOT are performed 1 Gate operation, sub-line sc 1 In the qubit q 0 The division point at is the quantum output. Sub-line sc 2 In the qubit q 0 The split point of (1) is a sub-line sc if no qubit gate operation is performed before the split point 2 Is provided.
Step 3.2, obtaining the total number of the quantum bits of classical input of each sub-line according to the type of the quantum bits in the sub-lineQuantum bit total number of Quantum inputs->Quantum bit total number of classical outputs->And the total number of qubits of the quantum output +.>Wherein (1)>Representing the total number of qubits, +.>Total number of qubits representing quantum input of jth sub-line,/-> Representing the total number of qubits of the classical output of the jth sub-line,total number of qubits representing quantum output of jth sub-line,/->
The sub-line sc in the embodiment of the invention 1 Quantum bit population for classical inputsQuantum bit total number of Quantum inputs->Quantum bit total number of classical outputs->Quantum bit total number of quantum output->Sub-line sc 2 Quantum bit total of classical input +.>Quantum bit total number of Quantum inputs->Quantum bit total number of classical outputs->Quantum bit total number of quantum output->
Step 3.3, by each sub-lineAnd->Obtaining a density matrix A of the sub-line input j And measuring base B at output j . Wherein A is j Representing the density matrix of the j-th sub-line input,B j representing the measurement basis at the output of the jth sub-line,
from the linear properties of the operators in Hilbert space, the density matrix |><-|＝|0><0|+|1><1|-|+><+|，|-i><-i|＝|0><0|+|1><1|-|+i><The operator in +i, i.e. single-qubit Hilbert space can be used with |0><0|,|1><1|,|+><+|,+i><+i is linearly expressed. If the qubit of the sub-line is a quantum input, the qubit has 4 input density matrices { |0><0|,1><1,|+><+|,|+i><+i|. If the qubit of the sub-line is a classical input, the density matrix of the qubit input is |0><0|. The sub-line sc of the embodiment of the invention 1 Input density matrix A 1 ＝|00><00, sub-line sc 2 Input density matrixPerforming spectrum decomposition on the measurement base I and the measurement base Z to obtain the same projection measurement operator|0><0|and |1><1, so the process of measuring the quantum output end of the sub-line by the I base can be omitted. If the output of the qubit of the sub-line is a quantum output, the measurement basis of the qubit output is { X, Y, Z }. If the output of the qubit of the sub-line is a classical output, the measurement basis of the output of the qubit is the Z basis. The sub-line sc of the embodiment of the invention 1 Measuring base for output endSub-line sc 2 Measuring base of output>The quantum circuit segmentation process of the embodiment of the invention is shown in FIG. 3, and is applied to the quantum bit q 0 CNOT of (C) 1 Gate and CNOT 2 Dividing the gates to obtain sub-line sc 1 And sc 2 . Sub-line sc 1 Is of the quantum bit q 0 The output end of the (C) is quantum output, and the measuring base of the output end is { X, Y, Z }; sub-line sc 2 Is of the quantum bit q 0 ' is a quantum input whose density matrix is { |0><0|,|1><1|,|+><+|,|+i><+i|}。
And 4, calculating probability distribution corresponding to each sub-line measurement result by using a quantum process matrix of the sub-line group SC estimated by using the quantum process chromatography.
Step 4.1, utilizeA density matrix is calculated for each sub-line output. Wherein A is j ' represents the density matrix of the j-th sub-line output.
The sub-line sc of the embodiment of the invention 1 Density matrix a for all inputs 1 And a quantum process matrix χ 1 Substituting into the above to obtain sub-line sc 1 Density matrix a of all outputs 1 '. Sub-line sc 2 Density matrix a for all inputs 2 And a quantum process matrix χ 2 Substituting into the above to obtain sub-line sc 2 Density matrix a of all outputs 2 ′。
Step 4.2, utilizeThe probability of occurrence of each sub-line measurement is calculated. Wherein p is j (r) represents the probability that the jth sub-line measurement is r, r represents the measurement of the jth sub-line, tr (·) represents the trace operation,/>indicating that the j-th sub-line measurement is r, B is used j Conjugate transpose of projection measurement operator, +.>Indicating that the j-th sub-line measurement is r, B is used j Is provided with a projection measurement operator of (1), indicating the density matrix output when the jth sub-line measurement is r.
The sub-line sc of the embodiment of the invention 1 Measuring base for output endB 1 Projection measurement operator of (2)Measurement result r= { |00>，|01>，|10>，|11>The probability of each measurement of r occurring is as follows:
wherein p1 (|00)>)、p1(|01>)、p1(|10>) And p1 (|11)>) Respectively represent sub-lines sc 1 The measurement result is |00>、|01>(ii) 10 and (11)>Is a function of the probability of (1),and->Respectively represent sub-lines sc 1 The measurement result is |00>、|01>、|10>And the projection measurement operator used at |11, | #>And->Respectively represent sub-lines sc 1 The measurement result is |00>、|01>、|10>Sum |11>A density matrix for output.
Change sub-line sc 1 Projection measurement operator of (2)Obtaining sub-line sc 1 Use of measuring base->And->The probability of the output measurement result. Sub-line sc 2 Measuring base of output>B 2 Projection measurement operator of (2)Measurement result r= { |00>，|01>，|10>，|11>The probability of each measurement of r occurring is as follows:
wherein p is 2 (|00>)、p 2 (|01>)、p 2 (|10>) And p 2 (|11>) Respectively represent sub-lines sc 2 The measurement result is |00>、|01>、|10>Sum |11>Is a function of the probability of (1),and->Respectively represent sub-lines sc 2 The measurement result is |00>、|01>、|10>Sum |11>Projection measurement operator for use in the case of +.>And->Respectively represent sub-lines sc 2 The measurement result is |00>、|01>、|10>Sum |11>A density matrix for output.
Change sub-line sc 2 Input density matrix A 2 Obtaining a sub-line sc 2 The probability of occurrence of the measurement result r when different density matrices are input.
And 5, carrying out classical post-processing on the probability of each sub-line measurement result in the sub-line group SC, and reconstructing the probability distribution condition of the original sub-line measurement result.
By means ofCalculating the occurrence probability of the original quantum circuit measurement result; wherein p is init (r init ) Indicating that the original quantum circuit measurement result is r init Probability of r init Representing the measurement result of the original quantum circuit, II represents the product operation, b represents the sequence number of the occurrence probability of each sub-circuit measurement result, and p j,b (r) represents the b-th probability that the j-th sub-line measurement is r.
Normalizing Pauli matrix { I, X, Y, Z }, to obtainThe set of linear representation dimensions 2 x 2 of the density matrix D is as follows:
where tr (ID), tr (XD), tr (YD) and tr (ZD) represent the expectation of observing the measurement values of the density matrix D under the I-base, X-base, Y-base and Z-base, respectively.
The spectral decomposition process of the Pauli matrix is as follows:
I＝|0><0|+|1><1|
X＝|+><+|-|-><-|
Y＝|+i><+i|-|-i><-i|
Z＝|0><0|-|1><1|
wherein, |0> <0| and |1> <1| represent eigenvectors of the I-base, |++ > < + > and |- > < - | represent eigenvectors of the X-base, |+i > <+i and |-I > < -i| represent eigenvectors of the Y-base, and |0> <0| and |1> <1| represent eigenvectors of the Z-base.
Bringing the spectral decomposition of the Pauli matrix into a linear expression of the density matrix D, the density matrix D is represented as follows:
D 1 ＝[tr(ID)+tr(ZD)]|0><0|
D 2 ＝[tr(ID)-tr(ZD)]|1><1|
D 3 ＝tr(XD)[2|+><+|-|0><0|-|1><1|]
D 4 ＝tr(YD)[2|+i><+i|-|0><0|-|1><1|]
wherein tr (ID), tr (XD), tr (YD) and tr (ZD) respectively represent a sub-line sc j The density matrix D output by the quantum output end is measured by using an I base, an X base, a Y base and a Z base to obtain an expected value of a measurement result, |0><0|、|1><1|、|+><+|and|+i><+i| represents the sub-line sc, respectively j+1 A density matrix input by the quantum input end.
The quantum state output by the embodiment of the inventionTo reconstruct the output quantum state |111>The probability of occurrence is exemplified by illustrating the sub-line sc 1 And sc 2 The probability of the measurement is subjected to classical post-processing. The output quantum state of the embodiment of the invention is |111>At the time, sub-line sc 1 The relevant measurement result is->Measuring sub-line sc using group I, group X, group Y and group Z 1 The quantum output of (2) to obtain the measurement result +.>The probability of (2) is as follows:
wherein p (|10)>|I)、p(|10>|X)、p(|10>Y) and p (|10)>I Z) represents measuring sub-line sc using I, X, Y and Z groups, respectively 1 Quantum output terminal of (2), sub-line sc 1 The quantum state of the output is |10>P (|11)>|I)、p(|11>|X)、p(11>Y) and p (|11)>I Z) represents measuring sub-line sc using I, X, Y and Z groups, respectively 1 Quantum output terminal of (2), sub-line sc 1 The quantum state of the output is |11>Is a probability of (2).
The output quantum state of the embodiment of the invention is |111>At the time, sub-line sc 2 The relevant measurement results areSub-line sc 2 Is input into the density matrix { |0 by the quantum input end of (C)><0|,|1><1|,|+><+|,|+i><+i| } to obtain measurement resultsThe probability of (2) is as follows:
wherein p (|11)>||0><0|)、p(|11>||1><1|)、p(|11>||+><++ |) and p (|11)>||+i><+i|) respectively represent sub-lines sc 2 Is input into the density matrix |0 by the quantum input terminal of (a)><0|、|1><1|、|+><+|and|+i><At +i|, sub-line sc 2 The probability that the output quantum state is 11.
Using sub-line sc 1 And sc 2 The probability of the result is measured, and the output quantum state |111 is reconstructed according to the embodiment of the invention>The probability of occurrence is as follows:
wherein p is init (|111> init ) Represents the output quantum state |111 of the embodiment of the invention>Is a function of the probability of (1),sub-line sc representing an embodiment of the present invention 1 Measurement using different measurement bases, resulting in measurement results +.>Probability of->Sub-line sc representing an embodiment of the present invention 2 Inputting different density matrixes to obtain a measurement result +.>Is a probability of (2).
The effects of the present invention are further described below in conjunction with simulation experiments:
1. simulation experiment conditions:
the platform of the simulation experiment of the invention is: windows 10 operating system, python 3.8.8 and Qiaskit 0.18.3.
2. Simulation content and result analysis:
the simulation experiment of the invention adopts the invention and two prior arts (a quantum circuit segmentation method based on tensor network and an extensible quantum-classical mixed calculation method) to respectively segment the 3-quantum-bit GHZ state preparation circuit. In order to verify the simulation effect of the invention, the quantum circuit segmentation is carried out by adopting three methods for the simulation experiment of the invention, the fidelity of the reconstruction probability result of the three methods under 9 operation times is calculated respectively, and the fidelity curves corresponding to the three methods are drawn as shown in figure 4.
Two prior art techniques employed in simulation experiments refer to:
the quantum circuit segmentation method based on tensor network in the prior art refers to: the tensor network-based quantum wire segmentation method is proposed by Tianyi Peng et al in its published paper "Simulating Large Quantum Circuits on a Small Quantum Computer" (Physical Review Letters2020: 150504).
The prior art scalable quantum-classical hybrid computing method refers to: scalable quantum-classical hybrid computation methods are presented by Wei Tang et al in its published paper "CutQC: using Small Quantum Computers for Large Quantum Circuit Evaluations" (Proceedings of the 26th ACM International conference on architectural support for programming languages and operating systems 2021:473-486).
The effects of the present invention are further described below in conjunction with the simulation diagram of fig. 4.
Fig. 4 is a graph showing the fidelity of quantum circuit segmentation for 3-qubit GHZ-state fabrication circuits, respectively, by three methods of the simulation experiment of the present invention. The fidelity refers to the approximation degree of the original quantum circuit probability obtained by using the measurement result probability reconstruction of the sub-circuit after segmentation and the theoretical probability of the original quantum circuit measurement result.
The abscissa in fig. 4 represents the number of operations of each sub-line obtained after the 3-qubit GHZ-state preparation line is split, and the ordinate represents the fidelity of the 3-qubit GHZ-state preparation line in quantum line splitting. The solid line in fig. 4 represents the fidelity curve obtained by the method of the present invention under different operation times, the dash-dot line in fig. 4 represents the fidelity curve obtained by the quantum wire division method based on tensor network under different operation times, and the dashed line in fig. 4 represents the fidelity curve obtained by the scalable quantum-classical mixed calculation method under different operation times.
As can be seen from fig. 4, under any fixed number of operation times, the fidelity obtained by the method is higher than that obtained by the other two methods, mainly because the number of sub-lines after being divided by the quantum line dividing method based on the tensor network can be exponentially increased along with the increase of the dividing times, so that a large amount of quantum and classical computing resources are consumed; the extensible quantum-classical mixed calculation method is affected by equipment noise and the running times of the sub-circuits, and the corresponding probability of the sub-circuit measurement results after segmentation can generate negative numbers, so that the difference between the reconstructed original quantum circuit results and the original quantum circuit measurement results is larger.
The simulation experiment shows that: the method can solve the problem of overlarge computing resource expenditure through the linear property and spectrum decomposition of operators in the Hilbert space, improves the efficiency of quantum circuit segmentation, improves the existing quantum circuit segmentation method by utilizing quantum process chromatography, improves the fidelity of the reconstruction probability of the measurement result, and is a very practical quantum circuit segmentation method.
Claims (6)
1. The quantum circuit segmentation method based on quantum process chromatography is characterized in that the linear property and spectral decomposition of operators in Hilbert space are utilized to optimize the measurement basis of the quantum bit input state and output end of a sub-circuit group SC, the quantum process matrix of the sub-circuit group SC estimated by quantum process chromatography is used to calculate the probability distribution corresponding to each sub-circuit measurement result; the method comprises the following specific steps:
step 1, determining a sub-line group SC;
step 2, estimating a quantum process matrix of the sub-line group SC by utilizing quantum process chromatography;
step 3, optimizing the quantum bit input state of the sub-line group SC and the measurement basis of the output end by utilizing the linear property and spectrum decomposition of an operator in the Hilbert space;
step 4, calculating probability distribution corresponding to each sub-line measurement result by using a quantum process matrix of the sub-line group SC estimated by using the quantum process chromatography;
and 5, carrying out classical post-processing on the probability of each sub-line measurement result in the sub-line group SC, and reconstructing the probability distribution condition of the original sub-line measurement result.
2. The quantum wire division method based on quantum process chromatography according to claim 1, wherein the step of determining the sub-wire group SC in step 1 is as follows:
the method comprises the steps of firstly, finding the maximum quantum bit gate from a quantum circuit to be segmented, comparing the total number of quantum bits a of the quantum bit gate with the total number of quantum bits k of a quantum computer running the segmented sub circuit, if a is smaller than or equal to k, executing a second step, otherwise, executing a third step; wherein a represents the total number of quantum bits of the maximum quantum bit gate of the quantum circuit to be segmented, and k represents the total number of quantum bits of the quantum computer;
secondly, forming an initial segmentation point set by all quantum bit positions of the maximum quantum bit gate;
the third step, the equivalent conversion is carried out on the maximum quantum bit gate by using a single quantum bit gate and a controlled NOT gate of two quantum bits, and the first step is executed after the conversion circuit of the quantum circuit is obtained;
fourth, after finding all quantum bit gates with quantum bit total number equal to a from all quantum bit gates connected with the maximum quantum bit gate, updating the initial segmentation point set;
fifthly, dividing all the quantum bits determined by the updated dividing point set to obtain a sub-line group SC of the quantum line to be divided; and SC represents a sub-line set obtained after the quantum lines to be segmented are segmented.
3. The quantum wire dividing method based on quantum process chromatography according to claim 1, wherein the step of estimating the quantum process matrix of the sub-wire group SC using quantum process chromatography in step 2 is as follows:
first, each sub-line SC of the sub-line group SC is generated in a k-qubit quantum computer j Dimension is as followsDensity matrix ρ of (1) j Input to each sub-line sc j Through each sub-line sc j Projection measurement operator M of (2) j Measuring each sub-line sc j To obtain each sub-line sc j Is measured by the measurement result R of (2) j And R is j Probability of occurrence P j The method comprises the steps of carrying out a first treatment on the surface of the Wherein sc j Represents the jth sub-line, Q, in the sub-line group SC j Representing the total number of qubits, ρ, of the jth sub-line j Representing the density matrix of the j-th sub-line input,|·>representing a quantum state symbol, < >>Representing tensor product operations, M j Projection measurement operator representing the jth sub-line, < +.>R j Representing the measurement result of the jth sub-line, +.>P j Representing the measurement result R j Probability of occurrence;
second step, utilizeCalculating each set of measurements in each sub-line measurementExpected values of the results; wherein (1)>Representing the jth sub-line measurement result R j Expected value of the s-th set of measurement results in (b), t represents R j Sequence number of each group of measurement results, Σ represents summing operation, +.>R represents j Probability of occurrence of the (t) th measurement of the(s) th group, is->R represents j Eigenvalues of the projection measurement operator used for the(s) th set of (t) th measurement results, are->
Third step, utilizeCalculating a density matrix output by each sub-line; wherein ρ' j A density matrix representing the output of the jth sub-line, s representing the sequence number of the expected value of the measurement result of the jth sub-line,/->Measurement basis representing the 1 st qubit output of the jth sub-line, +.>Measurement basis representing the 2 nd qubit output of the j-th sub-line,/and>q representing the jth sub-line j The basis for the measurement of the output of the individual qubits,
fourth step, utilizeCalculating a quantum process matrix of each sub-line; wherein m and n respectively represent the serial numbers of the Kraus operator group, K m Representing the mth Kraus operator, < +.>Conjugate transpose representing the nth Kraus operator,>representing conjugate transpose operation,/->Representing the nth row and column elements of the quantum process matrix of the jth sub-line.
4. The quantum circuit segmentation method based on quantum process chromatography according to claim 1, wherein the step of optimizing the measurement basis of the quantum bit input state and output end of the sub-circuit group SC by using the linear property and spectral decomposition of the operator under the hilbert space in step 3 is as follows:
the method comprises the steps that firstly, a quantum bit determined by a division point set is found from all quantum bits of each sub-line, the type of the quantum bit of the sub-line at the division point is judged, if the quantum bit is not subjected to quantum bit gate operation before the division point, the division point is judged to be a quantum input end, otherwise, the division point is judged to be a quantum output end;
second, the total number of the quantum bits of classical input of each sub-line is obtained according to the type of the quantum bits in the sub-lineQuantum ratio of quantum inputTotal number of special->Quantum bit total number of classical outputs->And the total number of qubits of the quantum output +.>Wherein (1)>Representing the total number of qubits, +.>Total number of qubits representing quantum input of jth sub-line,/-> Representing the total number of qubits, +.>Total number of qubits representing quantum output of jth sub-line,/->
Third step, by each sub-lineAnd->Obtaining the sub-lineInput density matrix A j And measuring base B at output j The method comprises the steps of carrying out a first treatment on the surface of the Wherein A is j Representing the density matrix of the j-th sub-line input,B j representing the measurement basis at the output of the jth sub-line,
5. the quantum wire dividing method based on quantum process chromatography according to claim 1, wherein the step of calculating the probability distribution corresponding to each sub-wire measurement result using the quantum process matrix of the sub-wire group SC estimated by quantum process chromatography in step 4 is as follows:
first step, utilizeCalculating a density matrix output by each sub-line; wherein A 'is' j A density matrix representing the j-th sub-line output;
second step, utilizeCalculating the occurrence probability of each sub-line measurement result; wherein p is j (r) represents the probability that the jth sub-line measurement is r, r represents the measurement of the jth sub-line, tr (·) represents the trace operation,/-)>Indicating that the j-th sub-line measurement is r, B is used j Conjugate transpose of projection measurement operator, +.>Indicating that the j-th sub-line measurement is r, B is used j Is provided with a projection measurement operator of (1), indicating the density matrix output when the jth sub-line measurement is r.
6. The quantum wire dividing method based on quantum process chromatography according to claim 1, wherein the step 5 is characterized in that the probability of each sub-wire measurement result in the sub-wire group SC is subjected to classical post-processing, and the step of reconstructing the probability distribution of the original quantum wire measurement result is as follows:
by means ofCalculating the occurrence probability of the original quantum circuit measurement result; wherein p is init (r init ) Indicating that the original quantum circuit measurement result is r init Probability of r init Indicating the measurement result of the original quantum circuit, pi represents the product operation, b represents the sequence number of the occurrence probability of each sub-circuit measurement result, and p j,b (r) represents the b-th probability that the j-th sub-line measurement is r.
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