CN114529003A - Dividing method for multi-quantum bit quantum Fourier transform line - Google Patents

Dividing method for multi-quantum bit quantum Fourier transform line Download PDF

Info

Publication number
CN114529003A
CN114529003A CN202210112495.1A CN202210112495A CN114529003A CN 114529003 A CN114529003 A CN 114529003A CN 202210112495 A CN202210112495 A CN 202210112495A CN 114529003 A CN114529003 A CN 114529003A
Authority
CN
China
Prior art keywords
quantum
sub
line
bit
fourier transform
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN202210112495.1A
Other languages
Chinese (zh)
Inventor
朱畅华
邵磊
马树泉
权东晓
何先灯
易运晖
赵楠
陈南
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Xidian University
Original Assignee
Xidian University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Xidian University filed Critical Xidian University
Priority to CN202210112495.1A priority Critical patent/CN114529003A/en
Publication of CN114529003A publication Critical patent/CN114529003A/en
Pending legal-status Critical Current

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena

Landscapes

  • Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Data Mining & Analysis (AREA)
  • Evolutionary Computation (AREA)
  • Condensed Matter Physics & Semiconductors (AREA)
  • Computational Mathematics (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Computing Systems (AREA)
  • General Engineering & Computer Science (AREA)
  • Mathematical Physics (AREA)
  • Software Systems (AREA)
  • Artificial Intelligence (AREA)
  • Complex Calculations (AREA)

Abstract

A partitioning method for a multi-quantum bit quantum Fourier transform line is provided. The invention provides a multi-bit quantum Fourier transform line segmentation method, which mainly solves the problem that the large-scale multi-bit quantum Fourier transform line cannot be operated due to insufficient quantum bit quantity of the existing small-scale quantum computer. The scheme is as follows: performing line adjustment on the quantum Fourier transform line, and dividing the adjusted line into a plurality of sub-lines; respectively carrying out initial quantum state preparation on the input and the output of each sub-circuit and setting a measuring module to form a plurality of sub-quantum circuits, and operating the sub-quantum circuits on a small-scale quantum computer; and performing classical calculation on the operation results of all the sub-quantum lines to restore the operation results into the operation results of the original large-scale quantum Fourier transform line. The invention optimizes the projection measurement of partial sub-lines, reduces the number of quantum lines running on a quantum computer, effectively reduces the width of the quantum lines through quantum-classical mixed calculation, and can be used for phase estimation of multiple quantum bits in quantum communication and quantum calculation.

Description

Dividing method for multi-quantum bit quantum Fourier transform line
Technical Field
The invention belongs to the technical field of quantum information, and particularly relates to a method for segmenting a multi-quantum-bit quantum Fourier transform line, which can be used for phase estimation of multi-quantum bits in quantum communication and quantum computation.
Background
The quantum fourier transform, which is a key component of quantum factorization and various quantum algorithms, is not only a key to the general process of phase estimation, but also to many quantum algorithms. Limited qubit memory is an important limiting factor affecting the recent development of large-scale quantum circuits. Large-scale quantum algorithms under the existing conditions are difficult to realize in a single quantum computer. Under the current situation, the line segmentation method can solve the difficulty: the large-scale quantum circuit is decomposed into a plurality of small-scale mold quantum circuits, and the operation result of the sub-circuit is subjected to classical post-processing operation and then is reduced into the operation result of the original large-scale circuit. The quantum Fourier transform is used as a key module of most quantum algorithms requiring phase estimation, and the operation of quantum gate adjustment, line segmentation and classical post-processing is carried out on the key module, so that the requirement of the key module on quantum bits can be effectively reduced, and a small-scale quantum computer can operate a large-scale quantum Fourier transform line.
In order to make large scale quantum wires operable on small scale quantum computers, developers have proposed many solutions: in 2020, Tianyi Peng firstly proposes to construct the quantum wires into a corresponding tensor network, and decompose the quantum wires through the decomposability of the tensor network. In 2021, Thomas Ayral proposed a study of the effect of different noise sources on quantum wires after their decomposition. However, the quantum wire decomposition methods proposed by the above studies are not directed to any particular quantum wire, and quantum wire decomposition frameworks and noise studies have been proposed. Because the quantum Fourier transform circuit is not optimized and preprocessed, the method is difficult to be directly applied to the multi-bit quantum Fourier transform circuit. The multi-bit quantum Fourier transform line is subjected to more overhead of quantum computation and classical computation after being divided.
The invention plans to decompose a large-scale quantum Fourier transform circuit into a small-scale quantum Fourier transform circuit and a plurality of sub-quantum circuits, optimize and perform parallel quantum computation on a sub-circuit group, then perform classical post-processing on the result, and combine a quantum computer and a classical computer to achieve the purpose of reducing the width and the depth of the quantum circuits.
Disclosure of Invention
The invention aims to provide a method for dividing a multi-bit quantum Fourier transform line into parallel sub-lines and restoring an original multi-bit quantum line through quantum-classical mixed calculation. The method aims to reduce the number of quantum bits required by a multi-quantum bit quantum Fourier transform circuit and realize the operation of a large-scale quantum Fourier transform circuit on a small-scale quantum computer.
The invention provides a multi-bit quantum Fourier transform circuit, and aims to solve the problem that a small-scale quantum computer cannot operate the multi-bit quantum Fourier transform circuit due to insufficient quantum bit quantity. The invention adjusts the quantum gate of the quantum Fourier transform circuit, and divides the quantum Fourier transform circuit into a plurality of small-scale sub-circuits, so that the quantum Fourier transform circuit can be operated on a small-scale quantum computer, and the operation result of the sub-circuits is reduced into the operation result of the original circuit before division by a classical calculation method, thereby realizing the operation of the multi-bit quantum Fourier transform circuit in the small-scale quantum computer.
In order to achieve the purpose, the technical scheme adopted by the invention comprises the following steps:
(1) initializing a multi-quantum bit quantum Fourier transform circuit and a small-scale quantum computer:
the N-bit quantum Fourier transform line consists of N H gates and (N-1) xN/2 controlled phase gates CP;
initializing N-bit quantum Fourier transform line CNThe qubits of (a) are: q ═ q1,q2,...qa,...,qNThe quantum gate is: g ═ g1,g2,...,gb,...,gMAnd the measurement module is as follows: t ═ t1,t2,...,ta,...tNWherein q isaA measurement module denoted as a-th qubit, gbDenoted as the b-th quantum gate, in an N-bit quantum Fourier transform circuit, taDenoted as qubit qaM is the number of quantum gates, M ═ 1+ N) N/2; in the N-bit quantum Fourier transform circuit, if the number of the measuring gates and the number of the quantum bits are both N, then C is addedNQFT (quad flat no-lead) module for N-bit quantum Fourier transformNAnd measurement gate t ═ t1,t2,...,ta,...tNA combination of { overshrinking };
setting a small-scale quantum computer to have the maximum quantum bit number n, wherein the small-scale quantum computer at least needs to have 2 quantum bits;
(2) for N bit quantum Fourier transform circuit CNAnd (3) adjusting:
2a) setting the initial cycle number i to be 0;
2b) QFT (quad flat no-lead) module in N-i bit quantum Fourier transformN-iIn (1), will affect qubit qN-iThe quantum gate is arranged on the N-i bit quantum Fourier transform module QFTN-iAnd keeping the relative position of the moved quantum gate unchanged; after adjustment, QFTN-iThe module can be divided into (N-1-i) quantum Fourier transform module QFTN-1-iAnd quantum line module CPN-1-iIn combination, wherein, CPN-1-iComposed of (N-1-i) CP gates and one at qN-iH gate of bit;
2c) judging whether i is larger than or equal to N-N-1, and optimally, judging whether i is equal to N-N-1:
if yes, the final original N bit quantitysub-Fourier transform line CNIs adjusted to be composed of QFTnModule and CPn,CPn+1,...,CPN-1The module and the measuring module t ═ t1,t2,...,ta,...tNA new quantum wire F composed ofNAnd performing (3),
if not, making i equal to i +1, and returning to 2 b);
(3) for new quantum wire FNAnd (3) carrying out segmentation:
3a) performing circuit division twice for each cycle, and enabling the initial cycle number i to be 0;
3b) for new quantum wire F according to the current cycle timesN-iThe segmentation of (1):
when i is 0, the new quantum wire F is applied to the adjusted N bitsNPerforming a first line division to divide FNSub-quantum wire F segmented into N-1 bitsN-1And a slave CPN-1Module and measurement gate t ═ t1,t2,...,ta,...tNA sub-quantum line S composed ofN-1；
For N-i bits of sub-quantum wires F when i ≠ 0N-iPerforming the first line division in the current round to divide the sub-quantum line FN-iSub-line F divided into (N-1-i) bitsN-1-iAnd from CPN-1-iModule formed sub-line SN-1-i；
3c) Sub-circuit SN-1-iPerforming the second line division of the round, and selecting the cutting point and the cutting times x of the ith round according to the maximum quantum bit number n of the small-scale quantum computeriAnd N is not less than (N-i)/(x)i+1), forming S after divisionN-1-iSub-line group of
Figure BDA0003495227910000031
Wherein the content of the first and second substances,
Figure BDA0003495227910000032
is a sub-line SN-1-iThe e-th sub-line from top to bottom according to the quantum bit serial number;
3d) judging whether i is larger than or equal to N-N-1, and optimally, if i is larger than N-N-1:
if yes, all the finally formed sub-lines are 1 n-bit quantum Fourier transform modules QFTnFormed sub-circuit FnThe sub-line cluster is S ═ SN-1,SN-2,...,Sf,...SnIn which S isfRepresenting the result of the first division through the N-f-th round, formed by f CP gates and a qubit qf+1Sub-line of the H-gate combination of (1), and sub-line SfAfter the second division of the N-f th wheel, the formed sub-line group
Figure BDA0003495227910000033
Figure BDA0003495227910000034
Is SfThe sub-circuit executes (4) according to the sub-circuit from the top to the bottom according to the quantum bit serial number,
if not, making i equal to i +1, and returning to 3 b);
(4) treatment, operation and measurement of the sublines:
4a) defining a segmentation point, and forming a pair of corresponding output end and input end after segmentation; in each sub-line, three different projection measurements of X.Y.Z are performed on each output terminal, and four quantum state |0 are performed on each input terminal>，|1>，|+>，|i>So that each sub-line can form a plurality of actual quantum lines that can be run on a quantum computer; calculating the actual quantum circuit number T required by each sub-circuit cs according to the input end and output end number of each sub-circuitcs；
4b) Fourier transform sub-circuit F from n-bit quantanHaving only output, FnThe actual number of quantum wires required
Figure BDA0003495227910000041
To pair
Figure BDA0003495227910000042
The output of each actual quantum circuit is measuredThen, the measurement result is recorded after the operation on a quantum computer;
4c) for all sub-line groups
Figure BDA0003495227910000043
Sub-line in
Figure BDA0003495227910000044
Is/are as follows
Figure BDA0003495227910000045
Processing the output end and the input end of each actual quantum circuit, namely performing X.Y.Z three different projection measurements on each output end, fully arranging 4 preparations of the input ends, operating the actual quantum circuits in a quantum computer, and recording the final result;
(5) the classical post-processing operation is performed on the measurement results of all the actual quantum wires:
5a) setting the initial cycle renx i to 0;
5b) according to the current cycle timesN-1-iThe measurement results of the sub-line groups are subjected to classical post-processing operations:
5b1) for the case that i is 0, the probability density matrix of the division point is set as A, and an integration formula of the sub-line can be deduced according to a decomposition formula of A; by integrating the formula of the sub-lines, S can be expressedN-1Each sub-line in the sub-line group
Figure BDA0003495227910000046
The measurement results are integrated into S through classical calculationN-1The measurement result of the sub-line and the final calculation result is named SN-1The result of the classical calculation of (2);
5b2) for i ≠ 0, the probability density matrix of the segmentation points is set as A, and the integration formula of the sub-line can be deduced according to the decomposition formula of A; by the integration formula of the sub-lines, S can be expressedN-1-iEach sub-line in the sub-line group
Figure BDA0003495227910000047
The measurement results are integrated into CS through classical calculationN-1-iMeasurement of the sub-line group, and subsequent combination of the measurement with S obtained from the upper wheel cycleN-iThe classical calculation result is again processed by classical calculation, and the obtained calculation result is SN-1-iThe result of the classical calculation of (2);
5c) judging whether i is more than or equal to N-N-1, optimally, i is equal to N-N-1, and if so, executing 5 d); otherwise, let i equal i +1, return to 5 b);
5d) when the loop is jumped out under the condition that i is more than or equal to N-N-1, the relation that i is equal to N-N-1 is obtained under the optimal condition, and S is addedN-1-iIs shown as SnIn obtaining SnAfter the classical calculation of (2), for FnMeasured result of (2) and SnThe classical calculation result is subjected to the classical calculation again, and the N-bit quantum Fourier transform C can be reducedNThe result of the routing.
Compared with the prior art, the invention has the following advantages:
1. according to the invention, the circuit adjustment is carried out on the quantum Fourier transform circuit, the large-scale quantum Fourier transform circuit is divided into the combination of the small-scale quantum Fourier transform module and the quantum CP gate module, and the quantum gate which influences the quantum bit beyond the width of the quantum computer is arranged at the rear end of the circuit on the premise of not influencing the operation result of the original circuit, so that the quantum circuit can be conveniently divided for many times.
2. The invention divides the original large-scale quantum Fourier transform circuit into a plurality of sub-quantum circuits, and all the sub-circuits can run in the small-scale quantum computer, thereby not only reducing the requirement of the original circuit on the quantum bit quantity of the quantum computer, but also enabling the large-scale quantum Fourier transform circuit which needs a large amount of quantum bits to run on the small-scale quantum computer under the condition of insufficient quantum bit resources.
3. The invention greatly reduces the number of sub-lines of actual operation quantity required by the invention because the division points are selected and the sub-lines are processed and the sub-lines only with output ends are optimized.
4. According to the invention, the measurement results of the sub-lines are integrated into the operation results of the original quantum Fourier transform line through the integrated classical calculation, and the quantum lines needing large-scale quantum computer operation are realized through the mixed calculation of a small-scale quantum computer and a classical computer.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
fig. 2 is a circuit diagram of an original quantum wire tuning process according to embodiment 1 of the present invention.
FIG. 3 is a sub-flowchart of the quantum wire splitting of the present invention;
FIG. 4 is a diagram of all the processed sub-quantum wires in example 1 of the present invention.
Detailed description of the preferred embodiments
Embodiments of the present invention are described in further detail below with reference to the accompanying drawings.
Referring to fig. 1, the implementation steps of this example are as follows:
step 1, initializing a multi-quantum bit quantum Fourier transform circuit and a small-scale quantum computer.
The multi-bit quantum Fourier transform circuit is a key part of quantum factorization and various quantum algorithms, and is not only key to a general process of phase estimation, but also key to a plurality of quantum algorithms; the small-scale quantum computer refers to a quantum computer having a smaller number of quantum registers than a large-scale quantum line, and on which a large-scale quantum line cannot be directly run. The invention aims to divide a multi-quantum bit quantum Fourier transform line which cannot be directly operated on a small-scale quantum computer into a plurality of sub-lines which can be operated in the small-scale quantum computer, and reduce the operation result of the sub-lines into the operation result of an original line before division by a classical calculation method so as to realize the operation of the multi-quantum bit quantum Fourier transform line in the small-scale quantum computer.
Firstly, determining a bit number value N of an N-bit quantum Fourier transform line and a quantum bit number N of a small-scale quantum computer; the value of N is required to satisfy that N is more than or equal to 3, and the value of N is required to satisfy that N is more than or equal to 2 and less than N;
the N-bit quantum Fourier transform line is composed of N H gates and (N-1) xN/2A controlled phase gate (CP gate); initializing N-bit quantum Fourier transform line CNThe qubits of (a) are: q ═ q1,q2,...qa,...,qNThe quantum gate is: g ═ g1,g2,...,gb,...,gMAnd the measurement module is as follows: t ═ t1,t2,...,ta,...tNWherein q isaA measurement module denoted as a-th qubit, gbDenoted as the b-th quantum gate, in an N-bit quantum Fourier transform circuit, taDenoted as qubit qaM is the number of quantum gates, M ═ 1+ N) N/2; in the N-bit quantum Fourier transform circuit, if the number of the measuring gates and the number of the quantum bits are both N, then C is addedNQFT (quad flat no-lead) module for N-bit quantum Fourier transformNAnd measurement gate t ═ t1,t2,...,ta,...tNA combination of { overshrinking };
in this embodiment, N is 6 and N is 4.
Step 2, for N bit quantum Fourier transform circuit CNAnd (6) adjusting.
The specific implementation of this step is as follows:
2.1) setting the initial cycle number i to 0;
2.2) in the N-i bit quantum Fourier transform module QFTN-iIn (1), will affect qubit qN-iThe quantum gate is arranged on the QFT module QFT of the N-i bit quantumN-iAnd maintaining the relative position of the moved quantum gate unchanged, and the QFTN-iThe module is divided into (N-1-i) quantum Fourier transform module QFTN-1-iAnd quantum line module CPN-1-iWherein, CPN-1-iComposed of (N-1-i) CP gates and one at qN-iH gate of bit;
2.3) judging whether i is more than or equal to N-N-1, and optimally, i is equal to N-N-1:
if so, the final original N-bit quantum Fourier transform line CNIs adjusted to be composed of QFTnModule and CPn,CPn+1,...,CPN-1The module and the measuring module t ═ t1,t2,...,ta,...tNA new quantum wire F composed ofNAnd the step 3 is executed,
if not, let i equal i +1, return to 2.2).
In this embodiment, since N is 6 and N is 4, the termination cycle condition is that i is greater than or equal to 1, and the original 6-bit quantum fourier transform line C6Is adjusted to be composed of QFT4Module and CP4,CP5The module and the measuring module t ═ t1,t2,...t6A new quantum wire F composed of6(ii) a The adjustment process refers to fig. 2, wherein fig. 2a is an original 6-bit quantum fourier transform circuit; fig. 2b is the circuit after completion of cycle renth i ═ 0; fig. 2c shows a circuit in which the cycle rens i is completed by 1, and also a circuit in which the adjustment is finally completed.
Step 3, for new quantum wire FNAnd (6) carrying out segmentation.
Referring to fig. 3, the specific implementation of this step is as follows:
3.1) carrying out two line divisions on each cycle, and enabling the initial cycle number i to be 0;
3.2) when i is 0, the adjusted N-bit quantum Fourier transform circuit FNIs composed of an n-bit quantum Fourier transform module QFTnAnd { CPn,CPn+1,...,CPN-1Is divided by selecting the cutting point at { q }1,q2,...,qN-1The routes of these qubits are located in the CPN-1The position in front of the module entrance is divided into two sub-circuits, namely an n-bit quantum Fourier transform module QFTnAnd { CPn,CPn+1,...,CPN-2N-1 bit sub-quantum circuit F composed of modulesN-1The other is composed of CPN-1Module and measurement gate t ═ t1,t2,...,ta,...tNA sub-quantum line S composed ofN-1；
When i ≠ 0, the sub-quantum wire F of N-i bitsN-iQFT (quad flat no-lead) module formed by n-bit quantum Fourier transformnAnd { CPn,CPn+1,...,CPN-1-iThe module is composed, and when in segmentation, the cutting point is selected to be { q }1,q2,...,qN-2-iThese qubit lanes are located in CPN-1-iThe position in front of the module inlet is divided into two sub-lines, one is an n-bit quantum Fourier transform module QFTnAnd { CPn,CPn+1,...,CPN-2-iSub-quantum circuit F of (N-1-i) bits composed of modulesN-1The other is composed of CPN-1-iModule formed sub-line SN-1-i。
3.3) sub-circuit SN-1-iPerforming the second line division of the round, and selecting the cutting point and the cutting times x of the ith round according to the maximum quantum bit number n of the small-scale quantum computeriAnd N is not less than (N-i)/(x)i+1), forming S after divisionN-1-iSub-line group of
Figure BDA0003495227910000071
Wherein the content of the first and second substances,
Figure BDA0003495227910000072
is a sub-line SN-1-iAccording to the quantum bit serial number, the e sub-line from top to bottom;
Figure BDA0003495227910000073
3.4) judging whether i is more than or equal to N-N-1, and optimally, judging whether i is equal to N-N-1:
if yes, all the finally formed sub-lines are 1 n-bit quantum Fourier transform modules QFTnFormed sub-circuit FnThe sub-line cluster is S ═ SN-1,SN-2,...,Sf,...SnIn which S isfRepresenting the result of the first division through the N-f-th round, formed by f CP gates and a qubit qf+1And the sub-line S of the H-gate combinationfAfter the second division of the N-f th wheel, the formed sub-line group
Figure BDA0003495227910000074
Figure BDA0003495227910000075
Is SfThe sub-circuit executes the step 4 according to the sub-circuit from the top to the bottom according to the quantum bit serial number,
if not, making i equal to i +1, and returning to 3.2);
in this embodiment, since N is 6 and N is 4, the end cycle condition is i ≧ 1, and line F6Is divided into: 1 QFT (quad Flat No-lead) 4-bit quantum Fourier transform module4Formed sub-circuit F4The sub-line cluster is S ═ S5,S4}; wherein
Figure BDA0003495227910000081
Figure BDA0003495227910000082
Figure BDA0003495227910000083
Is S5Two sub-wires of the sub-wire group,
Figure BDA0003495227910000084
is S4Two sub-wires in the sub-wire group.
And 4, processing, operating and measuring all the sub-lines.
4.1) the segmentation point is divided to form two end points, the end point at the tail end of the preorder sub-line is defined as an output end, and the end point at the front end of the subsequent sub-line is defined as an input end;
4.2) in the sub-line, three different projection measurements of X.Y.Z are carried out on each output end, and four quantum state |0 are carried out on each input end>，|1>，|+>，|i>So that each sub-line forms a plurality of actual quantum lines capable of running on a quantum computer; calculating the actual quantum circuit number T required by each sub-circuit cs according to the input end and output end number of each sub-circuitcs＝4d×3uWherein u is the number of output terminals contained in the sub-line, d is the number of input terminals, and the sum of the number of output terminals of all sub-lines is always equal to the number of input terminals no matter how many times the sub-lines are dividedThe sum of the amounts is equal.
In the example, the sub-line with d equal to 0 is optimized on the basis, because the output ends are all positioned at the tail ends of the sub-line, and the measurement results of all the quantum bits at the tail ends are mutually independent and are not influenced, only 3 actual quantum lines are required to run on a quantum computer in the sub-line with d equal to 0, and the measurement results are recorded; the 3 actual quantum wires are: carrying out X projection measurement on all upstream points of the subline, carrying out Y projection measurement on all upstream points of the subline, and carrying out Z projection measurement on all upstream points of the subline; after optimization, the actual quantum wires T required by each sub-wirecsComprises the following steps:
Figure BDA0003495227910000085
4.2) sub-circuit F of Fourier transform based on n-bit quantanHaving only output, FnThe actual number of quantum wires required
Figure BDA0003495227910000086
To pair
Figure BDA0003495227910000087
After the output end of each actual quantum circuit is measured, the actual quantum circuit is operated on a quantum computer, and the measurement result is recorded;
4.3) for all sub-line groups
Figure BDA0003495227910000088
Sub-line in
Figure BDA0003495227910000089
Is/are as follows
Figure BDA00034952279100000810
Processing the output and input of each actual quantum circuit, i.e. making X.Y.Z three different projection measurements for each output, fully arranging 4 preparations for input, and placing these actual quantum circuits in quantum computerRunning and recording the final result;
in this embodiment, the quantum wire diagram after all the sub-wires are processed is shown in fig. 4; wherein FIG. 4a shows a sub-line F4FIG. 4b is a sub-line
Figure BDA0003495227910000091
FIG. 4c is a subline
Figure BDA0003495227910000092
FIG. 4d is a subline
Figure BDA0003495227910000093
FIG. 4e is a subline
Figure BDA0003495227910000094
And 5, performing classical post-processing operation on the measurement results of all the actual quantum lines.
5.1) letting the initial cycle rence i equal to 0;
5.2) according to the current cycle timesN-1-iThe measurement results of the sub-line groups are subjected to classical post-processing operations:
5.2.1) for i ═ 0, pair S by the integration formula for sub-linesN-1Each sub-line in the sub-line group
Figure BDA0003495227910000095
The measurement result of (A) is integrated and reduced into S by classical calculationN-1Measurement of sub-line
Figure BDA0003495227910000096
The method comprises the following steps:
5.2.1a) setting the probability density matrix of the segmentation points as A:
Figure BDA0003495227910000097
further decomposing the matrix a into the pauli matrices and their feature basis combination terms yields the following identity:
Figure BDA0003495227910000098
wherein, Tr (AZ) is a track of a probability density matrix obtained by a Z measurement result of a corresponding output end of the sub-line; tr (AX), Tr (AY) are respectively traces of a probability density matrix obtained by X, Y measurement results of corresponding output ends of the sub-lines; the method comprises the steps that |0> <0|, |1> <1|, | + > <' + |, and | I > < I | are respectively used for preparing probability density matrixes in |0>, |1>, | + >, and | I > states for corresponding input ends, Tr (AI) is a trace of the probability density matrix obtained by an I measurement result of the corresponding output end of a sub-line, and the I measurement and the Z measurement are consistent in physical realization of a quantum line and are always Tr (AI) equal to 1;
5.2.1b) according to SN-1X is common to all sub-lines within a sub-line groupiCorresponding output and input terminals and N-1 isolated input terminals are paired, and the total of 4 are obtained for the N-1 isolated input terminals(N-1)A combination of quantum state preparations; and make the collection of all quantum state preparation combinations of N-1 isolated input ends into
Figure BDA0003495227910000099
Comprising:
Figure BDA0003495227910000101
5.2.1c) setting the initial cycle k of the external circulation to 1;
5.2.1d) maintenance of SN-1The sub-line group is prepared and combined in the quantum state of N-1 isolated input ends of the external circulation of the round
Figure BDA0003495227910000102
Making the initial cycle j of the internal circulation equal to 1;
5.2.1e) deducing an integration formula of the sub-line by using the decomposition formula of A, and combining the original line CNPortable measuring gate, pair
Figure BDA0003495227910000103
And with
Figure BDA0003495227910000104
Performing the integrated classical calculation on the sub-line, namely obtaining the following integrated classical calculation formula by using a formula with the probability density matrix of the quantum bit at the division point as A:
Figure BDA0003495227910000105
wherein:
Figure BDA0003495227910000106
to divide a sub-line by classical calculation
Figure BDA0003495227910000107
Integrating the results of the calculations; p is a radical of1,kFor a sub-line
Figure BDA0003495227910000108
Corresponding output terminal carries out AkThe result of the projection measurement tracing in (1), p2,kTo be on-sub-line
Figure BDA0003495227910000109
Corresponding input terminal of AkIn-quantum state post-fabrication on-sub-line
Figure BDA00034952279100001010
Run result in (A)kThe k-th term decomposed by the probability density matrix a has a value range of k ═ 1,2,3,4, p1,kAnd p2,kRespectively, as follows:
Figure BDA00034952279100001011
Figure BDA00034952279100001012
Figure BDA00034952279100001013
Figure BDA00034952279100001014
Figure BDA00034952279100001015
Figure BDA00034952279100001016
Figure BDA0003495227910000111
Figure BDA0003495227910000112
in the formula (I), the compound is shown in the specification,
Figure BDA0003495227910000113
the relationship of (1);
Figure BDA0003495227910000114
is a sub-line
Figure BDA0003495227910000115
The quantum state to be measured by the original measurement gate,
Figure BDA0003495227910000116
is a sub-line
Figure BDA0003495227910000117
Of the original measurement gate to be measured
Figure BDA0003495227910000118
Figure BDA0003495227910000119
Figure BDA00034952279100001110
Are respectively sub-lines
Figure BDA00034952279100001111
The trace of the probability density matrix is obtained according to the I, Z, X and Y measurement results of the output end;
Figure BDA00034952279100001112
is a sub-line
Figure BDA00034952279100001113
Raw measurement door measurement
Figure BDA00034952279100001114
The probability of (d);
Figure BDA00034952279100001115
are respectively sub-lines
Figure BDA00034952279100001116
Preparing |0 at input>、|1>、|+>、|i>Under the condition of state, the original measuring gate measures the quantum state
Figure BDA00034952279100001117
The probability of (d);
Figure BDA00034952279100001118
to divide a sub-line by classical calculation
Figure BDA00034952279100001119
Integrating the results of the calculations;
5.2.1f) order
Figure BDA00034952279100001120
Judging whether j satisfies that j equals xi：
If so, then there are
Figure BDA00034952279100001121
At this time, the process of the present invention,
Figure BDA00034952279100001122
the sub-line group has been integrated as SN-1Sub-circuits being prepared at isolated inputs
Figure BDA00034952279100001123
Time measurement
Figure BDA00034952279100001124
Has a probability of
Figure BDA00034952279100001125
Order to
Figure BDA00034952279100001126
Execute 5.2.1 g);
if not, let j equal j +1, return to 5.2.1 e);
5.2.1g) to determine if k satisfies k 4(N-1)：
If the condition is satisfied, then at this time,
Figure BDA00034952279100001127
the sub-line group has been fully integrated as SN-1A sub-line; execution 5.2.2);
if not, let k equal to k +1, return to 5.2.1 d);
in this embodiment, the sub-line
Figure BDA00034952279100001128
By classical calculation, reducing them to sub-line S5As a result of (A)
Figure BDA00034952279100001129
5.2.2) for i ≠ 0, compare SN-1-iEach sub-line in the sub-line group
Figure BDA00034952279100001130
The measurement results are integrated into S through classical calculationN-1-iOverall operational results for sub-line groups
Figure BDA0003495227910000121
5.2.2a) direct product of the quantum states of the individual qubits of the N-1-i inputs to obtain 4(N-1-i)An initial quantum state of a seed input; make the collection of all quantum states as
Figure BDA0003495227910000122
The intra-set elements are represented as follows:
Figure BDA0003495227910000123
5.2.2b) passing N-i outputs through a full permutation of X, Y, Z measurements to obtain 3(N-1-i)Measuring combinations are planted, and the set of all combinations is
Figure BDA0003495227910000124
The intra-set elements are represented as follows:
Figure BDA0003495227910000125
5.2.2c) setting the initial cycle number of the first layer cycle to 1;
5.2.2d) maintenance of SN-1-iThe measurement combination of the N-i isolated output ends is
Figure BDA0003495227910000126
Making k equal to 1 in the initial round of the second layer cycle;
5.2.2e) holding SN-1-iThe quantum state preparation combination of the N-1-i isolated input ends is
Figure BDA0003495227910000127
Making the third layer cycle initial round j equal to 1;
5.2.2f) obtaining the sub-circuit according to the decomposition formula of the probability density matrix A
Figure BDA0003495227910000128
And
Figure BDA0003495227910000129
the following formula for the integrated classical calculation is performed:
Figure BDA00034952279100001210
wherein:
Figure BDA00034952279100001211
to divide a sub-line by classical calculation
Figure BDA00034952279100001212
Integrating the results of the calculations; p is a radical of1,kFor a sub-line
Figure BDA0003495227910000131
Corresponding output terminal carries out AkThe result of the projection measurement tracing in (1), p2,kTo be on-sub-line
Figure BDA0003495227910000132
Corresponding input terminal of AkIn-quantum state post-fabrication on-sub-line
Figure BDA0003495227910000133
Run result in (A)kThe k-th term decomposed by the probability density matrix a has a value k ═ 1,2,3,4, and p1,kAnd p2,kRespectively, as follows:
Figure BDA0003495227910000134
Figure BDA0003495227910000135
Figure BDA0003495227910000136
Figure BDA0003495227910000137
Figure BDA0003495227910000138
Figure BDA0003495227910000139
Figure BDA00034952279100001310
Figure BDA00034952279100001311
wherein the content of the first and second substances,
Figure BDA00034952279100001312
is a sub-line
Figure BDA00034952279100001313
Isolated output terminal of
Figure BDA00034952279100001314
The quantum states to be measured of the combination are measured,
Figure BDA00034952279100001315
is a sub-line
Figure BDA00034952279100001316
Isolated output terminal of
Figure BDA00034952279100001317
Measuring the quantum state of the combination to be measured
Figure BDA00034952279100001318
Figure BDA00034952279100001319
Figure BDA00034952279100001320
Are respectively sub-lines
Figure BDA00034952279100001321
Traces of a probability density matrix obtained corresponding to the output I, Z, X, Y measurements;
Figure BDA00034952279100001322
is a sub-line
Figure BDA00034952279100001323
Isolated output terminal through
Figure BDA00034952279100001324
Measured by a combination of measurements
Figure BDA00034952279100001325
The probability of (d);
Figure BDA00034952279100001326
Figure BDA00034952279100001327
are respectively sub-lines
Figure BDA00034952279100001328
Preparing |0 at corresponding input>、|1>、|+>、|i>Quantum state measurement under the condition of state
Figure BDA00034952279100001329
The probability of (d);
5.2.2g)order to
Figure BDA00034952279100001330
Judging whether j satisfies that j equals xi：
If so, then
Figure BDA00034952279100001331
The subline has been integrated as SN-1-iSub-circuits being prepared at isolated inputs
Figure BDA0003495227910000141
And the selection is measured at the isolated output
Figure BDA0003495227910000142
Time measurement
Figure BDA0003495227910000143
Has a probability of
Figure BDA0003495227910000144
Order to
Figure BDA0003495227910000145
Execution for 5.2.2 h);
otherwise, let j equal j +1, return 5.2.2 f);
5.2.2h) determine if k satisfies k 4(N-i-1)：
If so, execute 5.2.2 i);
otherwise, let k be k +1, return 5.2.2 e);
5.2.2i) determining if l satisfies l-3(N-i)：
If so, then
Figure BDA0003495227910000146
The sub-circuit group has been integrated into CSN-1-iMeasurement of sub-line
Figure BDA0003495227910000147
Execution 5.2.3);
otherwise, let l ═ l +1, return 5.2.2 d);
5.2.3) reaction of CSN-1-iMeasurement of sub-line
Figure BDA0003495227910000148
With S obtained in the previous roundN-iResult of classical calculation of
Figure BDA0003495227910000149
By integrating classical calculation again
Figure BDA00034952279100001410
The following is achieved as a result of the classical calculation:
5.2.3a) integration into CSN-1-iMeasurement of sub-line and resulting SN-iThe result of the classical calculation of (2) is again integrated with the classical calculation:
first, a probability density matrix based on parallel multi-division points
Figure BDA00034952279100001411
By expanding the formula, will
Figure BDA00034952279100001412
The expansion is as follows:
Figure BDA00034952279100001413
wherein A isaCan be decomposed into Aa1,Aa2,Aa3,Aa4：
Figure BDA00034952279100001414
Aa1＝[Tr(AaI)+Tr(AaZ)]|0a><0a|
Aa2＝[Tr(AaI)-Tr(AaZ)]|1a><1a|
Aa3＝Tr(AaX)[2|+a><+a|-|0a><0a|-|1a><1a|]
Aa4＝Tr(AaY)[2|ia><ia|-|0a><0a|-|1a><1a|]
Secondly, A is mixedaSubstitution of a decomposition formula into a probability density matrix
Figure BDA00034952279100001415
Decomposing the formula to obtain:
Figure BDA00034952279100001416
wherein:
Figure BDA0003495227910000151
Figure BDA0003495227910000152
Figure BDA0003495227910000153
Figure BDA0003495227910000154
5.2.3B) mixing of BbTaking the Tr factors measured at multiple corresponding outputs as a function of the sum, and multiplying the sum by the combined probability density matrix at the corresponding input to obtain a calculation SN-1-iClassic calculation result of
Figure BDA0003495227910000155
Figure BDA0003495227910000156
Wherein p is1,kIs a pair of BbBy adding the Tr factors measured at a plurality of corresponding outputs, p2,kIs BbAfter the quantum state of the middle input end is prepared, the quantum state is obtained through classical calculation
Figure BDA0003495227910000157
The result of (1);
5.3) judging whether i is more than or equal to N-N-1, optimally, i is equal to N-N-1, and if so, executing 5.4); otherwise, let i be i +1, return to 5.2);
in this embodiment, S ═ S5,S4All the sub-lines are integrated completely, and the integrated S line result is p (| Q)S4>)；
5.4) jumping out of the cycle according to the condition that when i is more than or equal to N-N-1, S has the relation that i is equal to N-N-1N-1-iIs shown as SnAt the formation of SnAfter the classical calculation of (2), for FnMeasured result of (2) and SnThe classical calculation result is subjected to the classical calculation again, and the N-bit quantum Fourier transform C can be reducedNThe result of the routing. Reduction to N-bit quantum Fourier transform CNThe results of the circuit, the following are achieved:
Figure BDA0003495227910000158
wherein p is1,kIs FnReference B for different measurement results at the outputkA plurality of corresponding output end measurement results of the decomposition formula are obtained by trace-solving operation, p2,kIs BkAfter preparing the quantum state corresponding to the probability density matrix of the corresponding input end, measuring to obtain
Figure BDA0003495227910000159
As a result of (a) to (b),
Figure BDA00034952279100001510
the result of the final N-bit quantum Fourier transform circuit is obtained.
End result of this example
Figure BDA00034952279100001511
Namely the operation result of the original 6-bit quantum Fourier transform line.
The foregoing description is only an example of the present invention and is not intended to limit the invention, so that it will be apparent to those skilled in the art that various changes and modifications in form and detail may be made therein without departing from the spirit and scope of the invention.

Claims (6)

1. A segmentation method for a multi-qubit quantum Fourier transform line is characterized by comprising the following steps:
(1) initializing a multi-quantum bit quantum Fourier transform circuit and a small-scale quantum computer:
the N-bit quantum Fourier transform line consists of N H gates and (N-1) xN/2 controlled phase gates CP;
initializing N-bit quantum Fourier transform line CNThe qubits of (a) are: q ═ q1,q2,...qa,...,qNAnd the quantum gate is: g ═ g1,g2,...,gb,...,gMAnd the measurement module is as follows: t ═ t1,t2,...,ta,...tNWherein q isaA measurement module denoted as a-th qubit, gbDenoted as the b-th quantum gate, in an N-bit quantum Fourier transform circuit, taDenoted as qubit qaM is the number of quantum gates, M ═ 1+ N) N/2; in the N-bit quantum Fourier transform circuit, if the number of the measuring gates and the number of the quantum bits are both N, then C is addedNQFT (quad flat no-lead) module for N-bit quantum Fourier transformNAnd measurement gate t ═ t1,t2,...,ta,...tNA combination of (a) };
setting a small-scale quantum computer to have the maximum quantum bit number n, wherein the small-scale quantum computer at least needs to have 2 quantum bits;
(2) for N bit quantum Fourier transform circuit CNAnd (3) adjusting:
2a) setting the initial cycle number i to be 0;
2b) QFT (quad flat no-lead) module in N-i bit quantum Fourier transformN-iIn (1), will affect qubit qN-iThe quantum gate is arranged on the N-i bit quantum Fourier transform module QFTN-iAnd keeping the relative position of the moved quantum gate unchanged; after adjustment, QFTN-iThe module can be divided into (N-1-i) quantum Fourier transform module QFTN-1-iAnd quantum line module CPN-1-iIn combination, wherein, CPN-1-iComposed of (N-1-i) CP gates and one at qN-iH gate of bit;
2c) judging whether i is larger than or equal to N-N-1, and optimally, judging whether i is equal to N-N-1:
if so, the final original N-bit quantum Fourier transform line CNIs adjusted to be composed of QFTnModule and CPn,CPn+1,...,CPN-1The module and the measuring module t ═ t1,t2,...,ta,...tNA new quantum wire F composed ofNAnd performing (3),
if not, making i equal to i +1, and returning to 2 b);
(3) for new quantum wire FNAnd (3) carrying out segmentation:
3a) performing circuit division twice for each cycle, and enabling the initial cycle number i to be 0;
3b) for new quantum wire F according to the current cycle timesN-iThe segmentation of (1):
when i is 0, the new quantum wire F is applied to the adjusted N bitsNPerforming a first line division to divide FNSub-quantum wire F divided into N-1 bitsN-1And a slave CPN-1Module and measurement gate t ═ t1,t2,...,ta,...tNA sub-quantum wire S ofN-1；
For N-i bits of sub-quantum wires F when i ≠ 0N-iPerforming the first line division to divide the sub-quantaLine FN-iSub-line F divided into (N-1-i) bitsN-1-iAnd from CPN-1-iModule formed sub-line SN-1-i；
3c) Sub-circuit SN-1-iPerforming the second line division of the round, and selecting the cutting point and the cutting times x of the ith round according to the maximum quantum bit number n of the small-scale quantum computeriAnd N is not less than (N-i)/(x)i+1), forming S after divisionN-1-iSub-line group of
Figure RE-FDA0003574234160000021
Wherein the content of the first and second substances,
Figure RE-FDA0003574234160000022
as sub-line SN-1-iThe e-th sub-line from top to bottom according to the quantum bit serial number;
3d) judging whether i is larger than or equal to N-N-1, and optimally, if i is larger than N-N-1:
if yes, all the finally formed sub-lines are 1 n-bit quantum Fourier transform modules QFTnFormed sub-circuit FnThe sub-line cluster is S ═ SN-1,SN-2,...,Sf,...SnIn which S isfShowing f CP gates and a q in qubit formed after the first division through the N-f th roundf+1Sub-line of the H-gate combination of (1), and sub-line SfAfter the second division of the N-f th wheel, the formed sub-line group
Figure RE-FDA0003574234160000023
Figure RE-FDA0003574234160000024
Is SfThe sub-circuit executes (4) according to the sub-circuit from the top to the bottom according to the quantum bit serial number,
if not, making i equal to i +1, and returning to 3 b);
(4) treatment, operation and measurement of the sublines:
4a) defining a division pointAfter over-segmentation, a pair of corresponding output end and input end is formed; in each sub-line, three different projection measurements of X.Y.Z are performed on each output terminal, and four quantum state |0 are performed on each input terminal>，|1>，|+>，|i>So that each sub-line can form a plurality of actual quantum lines that can be run on a quantum computer; calculating the actual quantum circuit number T required by each sub-circuit cs according to the input end and output end number of each sub-circuitcs；
4b) Fourier transform sub-circuit F from n-bit quantanHaving only output, FnThe actual number of quantum wires required
Figure RE-FDA0003574234160000025
To pair
Figure RE-FDA0003574234160000026
After the output end of each actual quantum circuit is measured, the actual quantum circuit is operated on a quantum computer, and the measurement result is recorded;
4c) for all sub-line groups
Figure RE-FDA0003574234160000027
Sub-line in
Figure RE-FDA0003574234160000028
Is/are as follows
Figure RE-FDA0003574234160000029
Processing the output end and the input end of each actual quantum circuit, namely performing X.Y.Z three different projection measurements on each output end, fully arranging 4 preparations of the input ends, operating the actual quantum circuits in a quantum computer, and recording the final result;
(5) the classical post-processing operation is performed on the measurement results of all the actual quantum wires:
5a) setting the initial cycle rentimes i to 0;
5b) according to the current cycle timesN-1-iThe measurement results of the sub-line groups are subjected to classical post-processing operations:
5b1) for the case that i is 0, the probability density matrix of the division point is set as A, and an integration formula of the sub-line can be deduced according to a decomposition formula of A; by integrating the formula of the sub-lines, S can be expressedN-1Each sub-line in the sub-line group
Figure RE-FDA0003574234160000031
The measurement results are integrated into S through classical calculationN-1The measurement result of the sub-line and the final calculation result is named SN-1The result of the classical calculation of (2);
5b2) for i ≠ 0, the probability density matrix of the segmentation points is set as A, and the integration formula of the sub-line can be deduced according to the decomposition formula of A; by the integration formula of the sub-lines, S can be expressedN-1-iEach sub-line in the sub-line group
Figure RE-FDA0003574234160000032
The measurement results are integrated into CS through classical calculationN-1-iMeasurement of the sub-line group, and subsequent combination of the measurement with S obtained from the upper wheel cycleN-iThe classical calculation result is again processed by classical calculation, and the obtained calculation result is SN-1-iThe result of the classical calculation of (2);
5c) judging whether i is more than or equal to N-N-1, optimally, i is equal to N-N-1, and if so, executing 5 d); otherwise, let i equal i +1, return to 5 b);
5d) when the condition that i is more than or equal to N-N-1 is met, the loop is jumped out, and the relation that i is equal to N-N-1 is obtained under the optimal condition, and S isN-1-iIs shown as SnIn obtaining SnAfter the classical calculation of (2), for FnMeasured result of (2) and SnThe classical calculation result is subjected to the classical calculation again, and the N-bit quantum Fourier transform C can be reducedNThe result of the routing.
2. The method according to claim 1, characterized in that the splitting of the sub-quantum wires in 3b) is implemented as follows:
when i is 0, adjustedN-bit quantum Fourier transform circuit FNIs composed of an n-bit quantum Fourier transform module QFTnAnd { CPn,CPn+1,...,CPN-1Is divided by selecting the cutting point at { q }1,q2,...,qN-1The routes of these qubits are located in the CPN-1The position in front of the module entrance is divided into two sub-circuits, namely an n-bit quantum Fourier transform module QFTnAnd { CPn,CPn+1,...,CPN-2N-1 bit sub-quantum circuit F composed of modulesN-1The other is composed of CPN-1Module and measurement gate t ═ t1,t2,...,ta,...tNA sub-quantum line S composed ofN-1；
When i ≠ 0, the sub-quantum wire F of N-i bitsN-iQFT (quad flat no-lead) module formed by n-bit quantum Fourier transformnAnd { CPn,CPn+1,...,CPN-1-iThe module is composed, and when in segmentation, the cutting point is selected to be { q }1,q2,...,qN-2-iThese qubit lanes are located in CPN-1-iThe position in front of the module inlet is divided into two sub-lines, one is an n-bit quantum Fourier transform module QFTnAnd { CPn,CPn+1,...,CPN-2-iSub-quantum circuit F of (N-1-i) bits composed of modulesN-1The other is composed of CPN-1-iModule formed sub-line SN-1-i。
3. The method as claimed in claim 1, wherein in 4a), the actual number of quantum wires T required by each sub-circuit cs is calculated according to the number of the output terminals and the input terminals of each sub-circuitcsThe implementation is as follows:
Figure RE-FDA0003574234160000041
wherein, the number of the output ends contained in the u sub-line, and d is the number of the input ends.
4. The method as claimed in claim 1, wherein in 5b1), for i-0, S is addedN-1-iEach sub-line in the sub-line group
Figure RE-FDA0003574234160000042
The measurement results are integrated into S through classical calculationN-1-iThe results of the sub-line measurements were achieved as follows:
5b11) determining the probability of a pair of corresponding output and input in a classical calculation:
the decomposition formula of A is as follows:
Figure RE-FDA0003574234160000043
further decomposing the matrix A into the Paglie matrix and their feature base combination terms, the following identity is obtained:
Figure RE-FDA0003574234160000044
A1＝[Tr(AI)+Tr(AZ)]|0><0|
A2＝[Tr(AI)-Tr(AZ)]|1><1|
A3＝Tr(AX)[2|+><+|-|0><0|-|1><1|]
A4＝Tr(AY)[2|i><i|-|0><0|-|1><1|]
wherein, Tr (AZ) is a track of a probability density matrix obtained by a Z measurement result of a corresponding output end of the sub-line; tr (AX), Tr (AY) are respectively traces of a probability density matrix obtained by X, Y measurement results of corresponding output ends of the sub-lines; i0><0|、|1><1|、|+><+|、|i><i | preparing |0 for corresponding input ends respectively>、|1>、|+>、|i>The probability density matrix of the state, Tr (AI) is the trace of the probability density matrix obtained by the I measurement result of the corresponding output end of the sub-line, because the I measurement is consistent with the Z measurement in the physical realization of the quantum line, and Tr (AI) is 1 all the time; sN-1All sub-lanes within a sub-lane groupHas a total of xiCorresponding output ends and input ends are paired, and N-1 isolated output ends are provided; n-1 isolated outputs total 4(N-1)A combination of quantum state preparations;
make a quantum state preparation combination of N-1 output terminals
Figure RE-FDA0003574234160000051
Comprising:
Figure RE-FDA0003574234160000052
Figure RE-FDA0003574234160000053
Figure RE-FDA0003574234160000054
Figure RE-FDA0003574234160000055
Figure RE-FDA0003574234160000056
5b12) making k equal to 1 for the initial cycle of the external circulation;
5b13) retention of SN-1The sub-line group is prepared and combined in the quantum state of N-1 input ends circulating in the wheel
Figure RE-FDA0003574234160000057
Making the initial cycle j of the internal circulation equal to 1;
5b14) deducing an integration formula of the sub-line by using a decomposition formula of A, and combining the integration formula with an original line CNPortable measuring gate, pair
Figure RE-FDA0003574234160000058
And
Figure RE-FDA0003574234160000059
performing the integrated classical calculation on the sub-line, namely obtaining the following integrated classical calculation formula by using a formula with the probability density matrix of the quantum bit at the division point as A:
Figure RE-FDA00035742341600000510
wherein:
Figure RE-FDA00035742341600000511
to divide a sub-line by classical calculation
Figure RE-FDA00035742341600000512
Integrating the results of the calculations; p is a radical of1,kFor a sub-line
Figure RE-FDA00035742341600000513
Corresponding output terminal carries out AkThe result of the projection measurement tracing in (1), p2,kTo be on-sub-line
Figure RE-FDA00035742341600000514
Corresponding input terminal of AkIn-quantum state post-fabrication on-sub-line
Figure RE-FDA00035742341600000515
Run result in (A)kThe k-th term decomposed by the probability density matrix a has a value k ═ 1,2,3,4, and p1,kAnd p2,kRespectively, as follows:
Figure RE-FDA00035742341600000516
Figure RE-FDA0003574234160000061
Figure RE-FDA0003574234160000062
Figure RE-FDA0003574234160000063
Figure RE-FDA0003574234160000064
Figure RE-FDA0003574234160000065
Figure RE-FDA0003574234160000066
Figure RE-FDA0003574234160000067
in the formula (I), the compound is shown in the specification,
Figure RE-FDA0003574234160000068
the relationship of (1);
Figure RE-FDA0003574234160000069
is a sub-line
Figure RE-FDA00035742341600000610
The quantum state to be measured by the original measurement gate,
Figure RE-FDA00035742341600000611
is a sub-line
Figure RE-FDA00035742341600000612
Of the original measurement gate to be measured
Figure RE-FDA00035742341600000613
Figure RE-FDA00035742341600000614
Figure RE-FDA00035742341600000615
Are respectively sub-lines
Figure RE-FDA00035742341600000616
The trace of the probability density matrix is obtained according to the I, Z, X and Y measurement results of the output end;
Figure RE-FDA00035742341600000617
is a sub-line
Figure RE-FDA00035742341600000618
Raw measurement door measurement
Figure RE-FDA00035742341600000619
The probability of (d);
Figure RE-FDA00035742341600000620
Figure RE-FDA00035742341600000621
are respectively sub-lines
Figure RE-FDA00035742341600000622
Preparing |0 at input>、|1>、|+>、|i>Under the condition of state, the original measuring gate measures the quantum state
Figure RE-FDA00035742341600000623
The probability of (d);
Figure RE-FDA00035742341600000624
to divide a sub-line by classical calculation
Figure RE-FDA00035742341600000625
Integrating the results of the calculations;
5b15) order to
Figure RE-FDA00035742341600000626
Judging whether j satisfies that j equals xi：
If so, then there are
Figure RE-FDA00035742341600000627
At this time, the process of the present invention,
Figure RE-FDA00035742341600000628
the sub-line group has been integrated as SN-1Sub-circuits being prepared at isolated inputs
Figure RE-FDA00035742341600000629
Time measurement
Figure RE-FDA00035742341600000630
Has a probability of
Figure RE-FDA00035742341600000631
Order to
Figure RE-FDA00035742341600000632
Figure RE-FDA00035742341600000633
Execution 5b 16);
if not, let j ═ j +1, return to 5b 14);
5b16) judging whether k satisfies k being 4(N-1)：
If the condition is satisfied, then at this time,
Figure RE-FDA0003574234160000071
the sub-line group has been fully integrated as SN-1Sub-line
If not, let k be k +1, return to 5b 13).
5. Method according to claim 1, characterized in that for i ≠ 0 in 5b2), S will beN-1-iEach sub-line in the sub-line group
Figure RE-FDA0003574234160000072
The measurement results are integrated into CS through classical calculationN-1-iMeasurement of the sub-line group, and subsequent combination of the measurement with S obtained from the upper wheel cycleN-iThe classical calculation result of (2) is again performed by classical calculation, and is realized as follows:
5b21) directly multiplying the quantum state of a single quantum bit of N-1-i input ends to obtain 4(N-1-i)An initial quantum state of a seed input; make the collection of all quantum states as
Figure RE-FDA0003574234160000073
The intra-set elements are represented as follows:
Figure RE-FDA0003574234160000074
let N-i output ends pass through the full arrangement of X, Y, Z measurement to obtain 3(N-1-i)Measuring combinations are planted, and the set of all combinations is
Figure RE-FDA0003574234160000075
The intra-set elements are represented as follows:
Figure RE-FDA0003574234160000076
5b22) making the initial cycle number l of the first layer cycle equal to 1;
5b23) retention of SN-1-iThe measurement combination of the N-i isolated output ends is
Figure RE-FDA0003574234160000077
Making k equal to 1 in the initial round of the second layer cycle;
5b24) retention of SN-1-iThe quantum state preparation combination of the N-1-i isolated input ends is
Figure RE-FDA0003574234160000078
Making the third layer cycle initial round j equal to 1;
5b25) obtaining a sub-circuit according to a decomposition formula of the probability density matrix A
Figure RE-FDA0003574234160000081
And
Figure RE-FDA0003574234160000082
the following formula for the integrated classical calculation is performed:
Figure RE-FDA0003574234160000083
wherein:
Figure RE-FDA0003574234160000084
to divide a sub-line by classical calculation
Figure RE-FDA0003574234160000085
Integrating the results of the calculations; p is a radical of1,kFor a sub-line
Figure RE-FDA0003574234160000086
Corresponding output terminal carries out AkProjection inMeasuring the result of tracing, p2,kTo be on-sub-line
Figure RE-FDA0003574234160000087
Corresponding input terminal of AkIn-quantum state post-fabrication on-sub-line
Figure RE-FDA0003574234160000088
Run result in (A)kThe k-th term decomposed by the probability density matrix a has a value range of k ═ 1,2,3,4, p1,kAnd p2,kRespectively, as follows:
Figure RE-FDA0003574234160000089
Figure RE-FDA00035742341600000810
Figure RE-FDA00035742341600000811
Figure RE-FDA00035742341600000812
Figure RE-FDA00035742341600000813
Figure RE-FDA00035742341600000814
Figure RE-FDA00035742341600000815
Figure RE-FDA00035742341600000816
wherein the content of the first and second substances,
Figure RE-FDA00035742341600000817
is a sub-line
Figure RE-FDA00035742341600000818
Isolated output terminal of
Figure RE-FDA00035742341600000819
The quantum states to be measured of the combination are measured,
Figure RE-FDA00035742341600000820
as a sub-line
Figure RE-FDA00035742341600000821
Isolated output terminal of
Figure RE-FDA00035742341600000822
Measuring the quantum state of the combination to be measured
Figure RE-FDA00035742341600000823
Figure RE-FDA00035742341600000824
Figure RE-FDA00035742341600000825
Are respectively sub-lines
Figure RE-FDA00035742341600000826
Traces of a probability density matrix obtained corresponding to the output I, Z, X, Y measurements;
Figure RE-FDA00035742341600000827
is a sub-line
Figure RE-FDA00035742341600000828
Isolated output terminal of
Figure RE-FDA00035742341600000829
Measured by a combination of measurements
Figure RE-FDA00035742341600000830
The probability of (d);
Figure RE-FDA0003574234160000091
are respectively sub-lines
Figure RE-FDA0003574234160000092
Preparing |0 at corresponding input>、|1>、|+>、|i>Quantum state measurement under the condition of state
Figure RE-FDA0003574234160000093
The probability of (d);
5b26) order to
Figure RE-FDA0003574234160000094
Judging whether j satisfies that j equals xi：
If so, then
Figure RE-FDA0003574234160000095
The subline has been integrated as SN-1-iSub-circuits being prepared at isolated inputs
Figure RE-FDA0003574234160000096
And the selection is measured at the isolated output
Figure RE-FDA0003574234160000097
Time measurement
Figure RE-FDA0003574234160000098
Has a probability of
Figure RE-FDA0003574234160000099
Order to
Figure RE-FDA00035742341600000910
Execution 5b 27);
otherwise, let j equal j +1, return 5b 25);
5b27) judging whether k satisfies k being 4(N-i-1)：
If so, execute 5b 28);
otherwise, let k be k +1, return 5b 24);
5b28) judging whether l satisfies l-3(N-i)：
If so, then
Figure RE-FDA00035742341600000911
The sub-circuit group has been fully integrated into CSN-1-iMeasurement of sub-line, execution 5b 29);
otherwise, let l ═ l +1, return to 5b 23);
5b29) will integrate into CSN-1-iMeasurement of sub-line and resulting SN-iThe result of the classical calculation of (2) is again integrated with the classical calculation:
first, a probability density matrix based on parallel multi-division points
Figure RE-FDA00035742341600000912
By expanding the formula, will
Figure RE-FDA00035742341600000913
The expansion is as follows:
Figure RE-FDA00035742341600000914
wherein A isaCan be decomposed into Aa1,Aa2,Aa3,Aa4：
Figure RE-FDA00035742341600000915
Aa1＝[Tr(AaI)+Tr(AaZ)]|0a><0a|
Aa2＝[Tr(AaI)-Tr(AaZ)]|1a><1a|
Aa3＝Tr(AaX)[2|+a><+a|-|0a><0a|-|1a><1a|]
Aa4＝Tr(AaY)[2|ia><ia|-|0a><0a|-|1a><1a|]
Secondly, A is mixedaSubstitution of a decomposition formula into a probability density matrix
Figure RE-FDA0003574234160000101
Decomposing the formula to obtain:
Figure RE-FDA0003574234160000102
wherein:
Figure RE-FDA0003574234160000103
Figure RE-FDA0003574234160000104
Figure RE-FDA0003574234160000105
Figure RE-FDA0003574234160000106
5b210) B is to bebTaking the Tr factors measured at multiple corresponding outputs as a function of the sum, and multiplying the sum by the combined probability density matrix at the corresponding input to obtain a calculation SN-1-iResult of classical calculation of
Figure RE-FDA0003574234160000107
Figure RE-FDA0003574234160000108
Wherein p is1,kIs to BbBy addition of the Tr factors measured at a plurality of corresponding outputs, p2,kIs BbAfter the quantum state of the middle input end is prepared, the quantum state is obtained through classical calculation
Figure RE-FDA0003574234160000109
The result of (1).
6. The method of claim 1, wherein C in 5d)nMeasured result of (2) and SnThe classical calculation result is subjected to the classical calculation again and is reduced into N-bit quantum Fourier transform CNThe results of the circuit, the following are achieved:
Figure RE-FDA00035742341600001010
wherein p is1,kIs FnReference B for different measurement results at the outputbA plurality of trace-solving operations measured at corresponding output ends of the decomposition formula are obtained, p2,kIs BbAfter preparing the quantum state corresponding to the probability density matrix of the corresponding input end, measuring to obtain
Figure RE-FDA00035742341600001011
As a result of (a) to (b),
Figure RE-FDA00035742341600001012
the result of the final N-bit quantum Fourier transform circuit is obtained.
CN202210112495.1A 2022-01-29 2022-01-29 Dividing method for multi-quantum bit quantum Fourier transform line Pending CN114529003A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202210112495.1A CN114529003A (en) 2022-01-29 2022-01-29 Dividing method for multi-quantum bit quantum Fourier transform line

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202210112495.1A CN114529003A (en) 2022-01-29 2022-01-29 Dividing method for multi-quantum bit quantum Fourier transform line

Publications (1)

Publication Number Publication Date
CN114529003A true CN114529003A (en) 2022-05-24

Family

ID=81623249

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202210112495.1A Pending CN114529003A (en) 2022-01-29 2022-01-29 Dividing method for multi-quantum bit quantum Fourier transform line

Country Status (1)

Country Link
CN (1) CN114529003A (en)

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115358407A (en) * 2022-08-16 2022-11-18 北京中科弧光量子软件技术有限公司 Approximate quantum compiling method and system based on tensor network and electronic equipment
CN116050529A (en) * 2022-11-30 2023-05-02 北京百度网讯科技有限公司 Quantum circuit diagram error correction method, device, apparatus, storage medium and program product
CN116451797A (en) * 2023-04-24 2023-07-18 西安电子科技大学 Quantum circuit segmentation method based on quantum process chromatography

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115358407A (en) * 2022-08-16 2022-11-18 北京中科弧光量子软件技术有限公司 Approximate quantum compiling method and system based on tensor network and electronic equipment
CN116050529A (en) * 2022-11-30 2023-05-02 北京百度网讯科技有限公司 Quantum circuit diagram error correction method, device, apparatus, storage medium and program product
CN116451797A (en) * 2023-04-24 2023-07-18 西安电子科技大学 Quantum circuit segmentation method based on quantum process chromatography

Similar Documents

Publication Publication Date Title
CN114529003A (en) Dividing method for multi-quantum bit quantum Fourier transform line
CN110533183B (en) Task placement method for heterogeneous network perception in pipeline distributed deep learning
EP3540652B1 (en) Method, device, chip and system for training neural network model
Jain et al. Gems: Gpu-enabled memory-aware model-parallelism system for distributed dnn training
DE112019003529T5 (en) Data representation for dynamic accuracy in the cores of neural networks
CN114764549B (en) Quantum circuit simulation calculation method and device based on matrix product state
CN111563599A (en) Quantum line decomposition method and device, storage medium and electronic device
DE102020113791A1 (en) GLEITKOMMA SCALAR PRODUCT HARDWARE WITH WIDE MULTIPLIER ADDER TREE FOR MACHINE LEARNING ACCELERATORS
CN112073221A (en) Method and device for realizing network node sequencing
He et al. A low-complexity quantum principal component analysis algorithm
CN113935489A (en) Variational quantum model TFQ-VQA based on quantum neural network and two-stage optimization method thereof
CN114358216B (en) Quantum clustering method based on machine learning framework and related device
CN113297127A (en) Parameter updating method and platform system for large-scale distributed training cluster
CN111277710A (en) Telephone traffic prediction method, system, computer device and storage medium
CN114444723A (en) Model for data reconstruction
CN111931939A (en) Single-amplitude quantum computation simulation method
CN113361279A (en) Medical entity alignment method and system based on double neighborhood map neural network
CN109522630B (en) Power system transient stability simulation parallel computing method based on diagonal edge adding form
CN112906291B (en) Modeling method and device based on neural network
US11270208B2 (en) Neural network batch normalization optimization method and apparatus
Li et al. Analysis of a splitting approach for the parallel solution of linear systems on GPU cards
CN116451797B (en) Quantum circuit segmentation method based on quantum process chromatography
Petrosino et al. P-AFLC: a parallel scalable fuzzy clustering algorithm
CN117291271A (en) Efficient quantum circuit simulation method based on distributed system
CN116385839B (en) Training method and device for image pre-training model, electronic equipment and storage medium

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination