CN114511093B - Simulation method of boson subsystem - Google Patents

Simulation method of boson subsystem Download PDF

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CN114511093B
CN114511093B CN202011281177.5A CN202011281177A CN114511093B CN 114511093 B CN114511093 B CN 114511093B CN 202011281177 A CN202011281177 A CN 202011281177A CN 114511093 B CN114511093 B CN 114511093B
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李德胜
吴伟
钟鸣
吴春旺
陈平形
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Abstract

The invention discloses a simulation method of a boson subsystem, which assumes that the maximum occupation number cutoff of the boson subsystem is 2 n -1, the method comprising: establishing n-bit qubit basis vectors and 2 n A code mapping relation of a space base vector of the Hilbert is maintained; will 2 n The generated operators after the space interception of the Hilbert are expressed by Pauli matrixes; pauli matrix representation to generate operators from 2 n Vehicrt space recurrence to 2 n+1 The Hilbert space is maintained. The invention uses 2 brought by the quantum bit entanglement and state superposition principle n The space complexity of the simulation of the boson subsystem is reduced to a logarithmic relationship by the aid of the space characteristics of the Hilbert, so that the simulation method can be used for a quantum computer, saves quantum bit computing resources, reduces computing difficulty, can be used for environments such as a quantum virtual machine of a classical computer, and solves the problem of memory index growth of the simulation boson subsystem of the classical computer.

Description

Simulation method of boson subsystem
Technical Field
The invention belongs to the field of quantum computation, and particularly relates to a boson subsystem simulation method.
Background
Quantum simulation algorithms are an important class of quantum algorithms, with fermi and boson quantum simulation algorithms being particularly important. For the fermi simulation algorithm, there is a well-known Jordan-Wigner transform and a modified Bravyi-Kitaev transform. For quantum simulation of fermi or bosons, the Pauli matrix expression generating operators can be used for constructing quantum circuits for simulating related problems by using the existing method so as to further calculate the whole set of quantum simulation algorithm of the fermi or bosons. The Jordan-Wigner transform, the Bravyi-Kitaev transform, is Pauli matrix expression that gives two different Fermi-generating operators. In the prior art, a boson simulation algorithm generally adopts an algorithm based on one-to-one correspondence between quantum bits and occupied basis vectors. When the algorithm is executed on a quantum computer, the required quantum bit number is O (N) (N is the maximum occupancy number of bosons), and on a quantum virtual machine of a classical computer, the required calculation resource grows exponentially along with the increase of the occupancy number, so that the classical computer cannot complete the calculation when the occupancy number is large.
Disclosure of Invention
Based on the above, the invention aims to solve the problem of large consumption of computing resources in the boson simulation algorithm in the prior art by establishing a new space mapping relation between the quantum bits and Hilbert and defining a Pauli matrix expression of a new boson generating operator on the basis of the space mapping relation.
The invention aims at realizing the following technical scheme: a simulation method of a boson subsystem, wherein the maximum occupation number cutoff of the boson subsystem is 2 n -1, the method comprising:
establishing n-bit qubit basis vectors and 2 n A code mapping relation of a space base vector of the Hilbert is maintained;
the 2 n The generated operators after the space interception of the Hilbert are expressed by Pauli matrixes;
expressing the Pauli matrix of the generated operator from 2 n Vehicrt space recurrence to 2 n+1 The Hilbert space is maintained.
Specifically, the n-bit qubit basis vector is combined with 2 n The coding mapping relation of the space base vector of the Hilbert is as follows
|φ 0 >＝|0 1 ,0 2 ,...,0 n-1 ,0 n >,
|φ 1 >＝|0 1 ,0 2 ,...,0 n-1 ,1 n >,
|φ 2 >＝|0 1 ,0 2 ,...,1 n-1 ,0 n >,
|φ 3 >＝|0 1 ,0 2 ,...,1 n-1 ,1 n >,
...
Figure BDA0002780835320000029
wherein ,|φi >(i＝0,1,2,...,2 n -1) is represented at 2 n Basic vector with the number of bosons in Hilbert space being i, i 0>And |1>Respectively representing two bases of the qubit.
Specifically, said 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000021
wherein ,
Figure BDA0002780835320000022
is said 2 n Generating operators after space truncation of the Hilbert>
Figure BDA0002780835320000023
Is the effect operator of the ith item on the 1 st to nth quantum bits, and the effect operator is a tensor product formed by certain arrangement and combination of elements in the Pauli combination matrix;
the Pauli combined matrix is Pauli matrix sigma x ,σ y ，σ z The Pauli combination matrix includes at least four elements:
Figure BDA0002780835320000024
specifically, the generation algorithmPauli matrix expression of symbols from 2 n Vehicrt space recurrence to 2 n+1 The method for maintaining Hilbert space is that
Will be
Figure BDA0002780835320000025
The qubit position acted by each quantum gate is added by 1
Figure BDA0002780835320000026
1 st to 2 nd n -1 is written as
Figure BDA0002780835320000027
Will be 2 n The term is written as
Figure BDA0002780835320000028
Will be 2 n +1 to 2 n+1 -1 is written as
Figure BDA0002780835320000031
Said 2 n+1 The generation operator after the space truncation of the Hilbert is the addition of the above items
Figure BDA0002780835320000032
Specifically, when n=1, the ratio of 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000033
specifically, when n=2, the 2 n Dimension HilThe resulting operator after the bert space truncation is expressed as a Pauli matrix:
Figure BDA0002780835320000034
similarly, the technical scheme of the invention can write Pauli matrix expression of any truncated quantum bit number to generate an operator.
The technical scheme disclosed by the invention has the following beneficial effects:
1) The method reduces the number of qubits required for the simulation calculation of the boson subsystem with the maximum occupation number truncated to N from O (N) to O (log) 2 N)；
2) The method can be simultaneously applied to quantum simulation and classical numerical simulation based on a quantum virtual machine.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the following detailed description of the present invention will be made with reference to examples. It should be understood that the examples described herein are for illustrative purposes only and are not intended to limit the scope of the present invention.
The invention discloses a simulation method of a boson subsystem, which assumes that the maximum occupation number cutoff of the boson subsystem is 2 n -1, the method comprising:
s001, establishing n-bit quantum bit basic vector and 2 n A code mapping relation of a space base vector of the Hilbert is maintained;
s002. will 2 n The generated operators after the space interception of the Hilbert are expressed by Pauli matrixes;
s003, expressing Pauli matrix for generating operator from 2 n Vehicrt space recurrence to 2 n+1 The Hilbert space is maintained.
Specifically, in embodiments of the present invention, an n-bit qubit basis vector is combined with 2 n The coding mapping relation of the space base vector of the Hilbert is as follows
|φ 0 >＝|0 1 ,0 2 ,...,0 n-1 ,0 n >,
|φ 1 >＝|0 1 ,0 2 ,...,0 n-1 ,1 n >,
|φ 2 >＝|0 1 ,0 2 ,...,1 n-1 ,0 n >,
|φ 3 >＝|0 1 ,0 2 ,...,1 n-1 ,1 n >,
...
Figure BDA0002780835320000045
wherein ,|φi >(i＝0,1,2,...,2 n -1) is represented at 2 n Basic vector with the number of bosons in Hilbert space being i, i 0>And |1>Respectively representing two bases of the qubit.
In particular, 2 n The matrix form of the generated operators after the dimensional Hilbert spatial truncation can be written as:
Figure BDA0002780835320000041
wherein N＝2n -1。
Specifically, in the embodiment of the present invention, 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000042
wherein ,
Figure BDA0002780835320000043
is said 2 n Generating operators after space truncation of the Hilbert>
Figure BDA0002780835320000044
Is the effect operator of the ith item on the 1 st to nth quantum bits, and the effect operator is a tensor product formed by certain arrangement and combination of elements in the Pauli combination matrix;
pauli combined matrix is Pauli matrix sigma x ,σ y ，σ z The Pauli combination matrix includes at least four elements:
Figure BDA0002780835320000051
three Pauli matrices sigma x ,σ y ，σ z The following relationship is satisfied:
σ x σ y ＝iσ z ,σ y σ z ＝iσ x ,σ z σ x ＝iσ y ,σ x σ y ＝-σ y σ x ,σ y σ z ＝-σ z σ y ,σ z σ x ＝-σ x σ z ,σ x σ x ＝σ y σ y ＝σ z σ z ＝I；
wherein I is a 2×2 identity matrix
Figure BDA0002780835320000052
Pauli matrix and identity matrix I form a closed algebraic system, and can be used as the basis of hermite 2X 2 matrix space.
Specifically, in embodiments of the present invention, pauli matrix expression to generate operators is from 2 n Vehicrt space recurrence to 2 n+1 The method for maintaining Hilbert space is that
Will be
Figure BDA0002780835320000053
The qubit position acted by each quantum gate is added by 1
Figure BDA0002780835320000054
1 st to 2 nd n -1 is written as
Figure BDA0002780835320000055
Will be 2 n The term is written as
Figure BDA0002780835320000056
Will be 2 n +1 to 2 n+1 -1 is written as
Figure BDA0002780835320000057
2 n+1 The generation operator after the space truncation of the Hilbert is the addition of the above items
Figure BDA0002780835320000058
Specifically, in the embodiment of the present invention, when n=1, 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000061
when n=2, 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000062
when n=3, 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000063
when n=4, said 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000064
when n=5, said 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000065
/>
Figure BDA0002780835320000071
when n=6, said 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure BDA0002780835320000072
/>
Figure BDA0002780835320000081
the truncated generating operators given by the Pauli combination matrix can be simplified and the same class items can be obtained to obtain Pauli matrix expression of the generating operators. For example when n=3,
Figure BDA0002780835320000082
/>
Figure BDA0002780835320000091
in particular, according to the recurrence relation disclosed in the present invention and the Pauli expression of the generating operator after n=1 truncation, in principle, any Pauli matrix expression of the generating operator after n-value truncation and the corresponding annihilation operator can be written, and will not be described in detail herein. Specifically, in quantum simulation calculations, any expression of an operator combined by producing annihilation operators can be deduced from Pauli expressions that produce annihilation operators. In an embodiment of the present invention, some common operators combined by generating annihilation operators when n=3 are given below.
Truncated particle arithmetic symbols are expressed as
Figure BDA0002780835320000092
The square expression of truncated particle arithmetic is
Figure BDA0002780835320000093
Other common operator members
Figure BDA0002780835320000094
Finally, it should be noted that the above-mentioned embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications and equivalents may be made to the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention, and all such modifications and equivalents are intended to be encompassed in the scope of the claims of the present invention.

Claims (8)

1. A simulation method of a boson subsystem, wherein the maximum occupation number cutoff of the boson subsystem is 2 n -1, the simulation method comprising, on a quantum computer or a quantum virtual machine of a classical computer, the following steps:
generating an operator using Pauli matrix construction;
building a quantum circuit with the generated operators; and
applying the quantum circuit to a boson initial quantum state to simulate evolution of the boson system; wherein the method for generating operators by Pauli matrix construction comprises the following steps:
establishing n-bit qubit basis vectors and 2 n Constructing 2 by using the coding mapping relation of the space basic vectors of the Hilbert n A dimension Hilbert space;
the 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure FDA0004181132960000011
expressing the Pauli matrix of the generated operator from 2 n Vehicrt space recurrence to 2 n+1 A dimension Hilbert space;
wherein ,
Figure FDA0004181132960000012
is said 2 n Generating operators after space truncation of the Hilbert>
Figure FDA0004181132960000013
Is the effect operator of the ith item on the 1 st to nth quantum bits, and the effect operator is a tensor product formed by certain arrangement and combination of elements in the Pauli combination matrix;
the Pauli combined matrix is Pauli matrix sigma x ,σ y ，σ z The Pauli combination matrix includes at least four elements:
Figure FDA0004181132960000014
the Pauli matrix expression of the generated operator is from 2 n Vehicrt space recurrence to 2 n+1 The method for maintaining the Hilbert space comprises the following steps:
will be
Figure FDA0004181132960000015
The qubit position acted by each quantum gate is added by 1
Figure FDA0004181132960000016
1 st to 2 nd n -1 is written as
Figure FDA0004181132960000017
Will be 2 n The term is written as
Figure FDA0004181132960000018
Will be 2 n +1 to 2 n+1 -1 is written as
Figure FDA0004181132960000021
Said 2 n+1 The generation operator after the space truncation of the Hilbert is the addition of the above items
Figure FDA0004181132960000022
2. The boson system simulation method of claim 1, wherein the n-bit qubit basis vector is 2 n The coding mapping relation of the space base vector of the Hilbert is as follows
|φ 0 >＝|0 1 ,0 2 ,...,0 n-1 ,0 n >,
|φ 1 >＝|0 1 ,0 2 ,...,0 n-1 ,1 n >,
|φ 2 >＝|0 1 ,0 2 ,...,1 n-1 ,0 n >,
|φ 3 >＝|0 1 ,0 2 ,...,1 n-1 ,1 n >,
...
Figure FDA0004181132960000023
wherein ,|φi >(i＝0,1,2,...,2 n -1) is represented at 2 n Basic vector with the number of bosons in Hilbert space being i, i 0 j> and |1j >Each represents two bases of a j-th bit (j=1, 2,., n) qubit.
3. The boson system simulation method according to any one of claims 1-2, wherein when n=1, the ratio of 2 is n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure FDA0004181132960000024
4. a boson system simulation method according to any of claims 1-2, characterized in that when n = 2, the 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure FDA0004181132960000025
5. a boson system simulation method according to any of claims 1-2, characterized in that when n=3, the ratio of 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure FDA0004181132960000026
6. the boson system simulation method according to any one of claims 1-2, wherein when n=4, the ratio of 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure FDA0004181132960000031
7. the boson system simulation method according to any one of claims 1-2, wherein when n=5, the ratio of 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure FDA0004181132960000032
Figure FDA0004181132960000041
8. the boson system simulation method according to any one of claims 1-2, wherein when n=6, the ratio of 2 n The resulting operators after the dimensional Hilbert spatial truncation are expressed as Pauli matrices:
Figure FDA0004181132960000042
/>
Figure FDA0004181132960000051
/>
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