CN114819166B - Evolution method and device of quantum system - Google Patents
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Abstract
The invention relates to an evolution method of a quantum system, which comprises the steps of converting a first quantum state in an open quantum system to be simulated into a second quantum state in a closed quantum system through a first conversion relation, wherein the first quantum state in the open quantum system evolves according to a first relation, and the second quantum state in the closed quantum system evolves according to a second relation; wherein the first relation depends on a first evolution matrix in the form of a non-hermite matrix and the second relation depends on the positive evolution of the unitary; preparing a quantum state initial state of the closed quantum system according to the second quantum state; the method can be used for equating an open quantum system which is complex in a quantum computer and difficult to evolve into a closed quantum system, and the evolution of the closed quantum system can be realized based on the evolution of a unitary matrix, so that the time is shortened.
Description
Technical Field
The present invention relates to the field of quantum system evolution technologies, and in particular, to a method and apparatus for quantum system evolution.
Background
Some theoretical schemes have been proposed and experimentally implemented for quantum simulation algorithms of open quantum systems. One of the algorithms is based on first or second order tervee expansion. The method does not require auxiliary qubits and the time complexity is either O (1/ε) or
Is increased in form of (a). The specific implementation steps are as follows:
1. and preparing a quantum state initial state.
2. The total evolution time T will be described in N shares. For each evolution, the evolution operator will be split into several multiplication of sub-operators by the terdriver's expansion. Although the total evolution operators cannot be directly implemented, we can implement the evolution corresponding to each sub-operator in turn on a quantum computer. As long as the fraction N is large enough, we can accurately model the whole evolution process.
3. And measuring the physical quantity of the quantum state last state.
4. Repeating the steps 1-3, and averaging the measurement results to obtain the expected value of the physical quantity to be measured.
Whereas the time complexity of the first and second expansion methods of tervey is expressed as O (1/epsilon) or as E with the accuracy
Increasing with time t of simulated evolution with O (t 2 ) Or O (t) 1.5 ) Form increases of (c) far from the theoretical optimum. Although for closed quantum systems, there are many techniques available to achieve better time complexityDegree, such as methods based on taylor expansion and quantum signal processing algorithms. However, these algorithms cannot be directly applied to the simulation of an open quantum system for the following reasons. The evolution of the closed system can be directly described by the unitary matrix, and the high-efficiency simulation algorithm can be only applied to the evolution based on the unitary matrix. However, the evolution operators of open systems cannot be directly described by unitary matrices.
Disclosure of Invention
The object of the present invention is to solve the above-mentioned problems occurring in the prior art.
In order to achieve the above object, the present invention provides an evolution method of a quantum system, which is characterized by comprising:
converting a first quantum state in an open quantum system to be simulated into a second quantum state in a closed quantum system through a first conversion relation, wherein the first quantum state in the open quantum system evolves according to a first relation, and the second quantum state in the closed quantum system evolves according to a second relation, and the first relation depends on a first evolution matrix in a non-hermite matrix form and the second relation depends on unitary evolution; preparing a quantum state initial state of the closed quantum system according to the second quantum state; and evolving the quantum state initial state of the closed quantum system through the second relation to obtain the quantum state final state of the closed quantum system.
According to the evolution method of the quantum system, an open quantum system which is complex and difficult to evolve in a quantum computer can be equivalent to a closed quantum system, and the evolution of the closed quantum system can be realized based on the evolution of a unitary matrix, so that the time is shortened.
Drawings
Fig. 1 is a flow chart of an evolution method of a quantum system according to an embodiment of the present invention;
FIG. 2 is a flow chart of one embodiment of a method of evolution of a quantum system according to an embodiment of the present invention;
fig. 3 is a schematic structural diagram of an evolution device of a quantum system according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a computing device according to an embodiment of the present invention.
Detailed Description
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
It should be noted that quantum computing can solve many difficult problems that classical computers cannot effectively compute.
A typical example is the kinetic simulation of a quantum multisystem. Quantum dynamics simulation is performed in classical computers, and the time complexity tends to increase exponentially with the number of particles in the system. However, if we were able to directly model the microscopic interactions of the quantum system using quantum computers, the computation time required would be greatly reduced.
In the prior art, quantum simulation algorithms generally consist of the following steps.
The first step is to prepare the initial state to be simulated and evolved in the quantum computer. This step can be implemented by standard algorithms for quantum state preparation.
The second step is to simulate the evolution of the quantum system. Algorithms that are more commonly used include terode (Trotter) decomposition, taylor expansion, quantum signal processing, quantum random walk, etc. After this step we have obtained the final state of the quantum system.
The third step is to measure the last state after evolution. According to the quantum mechanics principle, the measurement results are generally random, and the expected value of the physical quantity to be measured can be obtained by repeating the first to third steps and averaging all the measurement results. Using the measured expected values, we can obtain various properties of the quantum system, such as energy spectrum, magnetism, etc.
The evolution of the closed quantum system can be described by unitary positive matrix, the mathematical form is relatively simple, and the corresponding quantum simulation algorithm is mature. However, in a real environment, a multi-body quantum system will interact with the surrounding thermal reservoir, at which time it will be in an entangled state with the thermal reservoir. For such an open quantum system, we cannot describe again a simple unitary matrix, but the evolution of a non-unitary matrix is often difficult to realize directly in an experimental system, so the quantum simulation becomes more complex.
While the evolution method using the open quantum system in part does not require auxiliary qubits and the time complexity is either O (1/epsilon) or with accuracy
Is increased in form of (a). However, in this method, the temporal complexity of the first and second expansion methods of terdold is expressed as O (1/ε) or +.>
Increasing with time t of simulated evolution with O (t 2 ) Or O (t) 1.5 ) Form increases of (c) far from the theoretical optimum. In short, the evolution method of the open system requires a long time.
Fig. 1 is a flowchart of an evolution method of a quantum system according to an embodiment of the present invention, and the evolution method of the quantum system according to the embodiment of the present invention is explained in detail below according to fig. 1.
According to an embodiment of the invention, an evolution method of a quantum system comprises the following steps:
s101: and converting a first quantum state in the open quantum system to be simulated into a second quantum state in the closed quantum system through a first conversion relation, wherein the first quantum state in the open quantum system evolves according to a first relation, the second quantum state in the closed quantum system evolves according to a second relation, the first relation depends on a first evolution matrix in a non-hermite matrix form, and the second relation depends on unitary evolution.
In this step, for the open quantum system that needs to evolve, the computer simulation conversion may be performed first, that is, the open quantum system is simulated and converted into an equivalent closed quantum system, so as to obtain a form of closed quantum system corresponding to the open quantum system.
Specifically, the conversion of an open quantum system into an equivalent closed quantum system is achieved by a first conversion relationship. Converting an open quantum system to a closed quantum system is converting a first quantum state in the open quantum system to a second quantum state in the closed quantum system.
In some embodiments of the invention, the first quantum state may be a density matrix, which may be represented using ρ. The ρ can be vectorized, i.e. each column of ρ is arranged sequentially, expressed as a density matrix v ρ . In this embodiment, the first quantum state evolves according to a first relation, the first relation being as follows:
in the first relation (1), ρ is an original density matrix characterizing an open first quantum state, v ρ In order to vectorize rho, i is an imaginary unit, L is a first evolution matrix, and the first evolution matrix L is in the form of a non-hermite matrix.
The evolution of the first relational expression cannot be directly simulated on a quantum computer. What can be simulated on a quantum computer is a second quantum state of the closed system, the evolution relationship of which depends on a second relationship, the second relationship being as follows:
in the second relational expression, i is an imaginary unit and M is an hermite matrix, as in the first relational expression. In the second relationship, v appears to be a representation of the converted second quantum state, and can be considered as v is a normalized vector. And a second quantum state which evolves depending on a second relation, wherein the second quantum state can be subjected to simulated evolution on the quantum computer by the second relation because M is an hermite matrix.
From the foregoing, it can be seen that the first quantum state belongs to an open quantum system, while the second quantum state belongs to a closed quantum system. Thus, in order to simulate a quantum computer, an open quantum system needs to be converted into a closed quantum system, and therefore, the conversion needs to be performed by a first conversion relationship.
In some embodiments of the present invention, the first conversion relationship may include an evolution matrix conversion relationship and a quantum state conversion relationship, in which the evolution matrix conversion relationship is:
The quantum state conversion relationship in the first conversion relationship is:
By means of the first transformation relationship, the evolution matrix transformation relationship can be obtained with respect to M, L and
the matrix relation and the quantum state conversion relation between the two can obtain v and v ρ The relationship between the two can be converted into a second relationship, namely the first quantum state of the open quantum system can be converted into the second quantum state of the closed quantum system, so that the simulated evolution on the quantum computer can be realized, namely the open quantum system is equivalent to the closed quantum system which can perform unitary evolution on the quantum computer.
In other words, the first relation of the first quantum state depends on an evolution matrix in the form of hermite matrix which cannot evolve in the quantum computer, while the second relation of the second quantum state depends on the form in which unitary is evolving, so that evolution in the quantum computer can be achieved.
Therefore, it can be understood that in the evolution method of the quantum system in the embodiment of the invention, the evolution of the open system in the quantum computer can be realized by converting the open quantum system into the closed quantum system and converting the non-hermitian matrix into the unitary evolution form, so that the time complexity is reduced, and the method is convenient and quick.
S102: preparing a quantum state initial state of the closed quantum system according to the second quantum state;
in the step, according to the second quantum state of the closed quantum system obtained by the calculation in the previous step, a quantum state initial state is prepared by using the second quantum state, so that the quantum state initial state can be used as an evolved quantum state to carry out simulated evolution.
S103: and (3) evolving the quantum state initial state of the closed quantum system through a second relation to obtain the quantum state final state of the closed quantum system.
In the step, according to the initial state of the quantum state obtained in the previous step, the quantum state is subjected to simulated evolution through a second relational expression, so that a corresponding quantum state final state is obtained, and therefore, the evolution of the open quantum system is converted into the unitary evolution of the equivalent closed quantum system.
In other words, by the evolution method of the quantum system in the embodiment of the invention, the open quantum system can be converted into the closed quantum system, further, the quantum state end state is obtained by performing simulated evolution on the closed quantum system, and the expected value of the physical quantity to be measured of the open system is obtained by detecting the expected value of the quantum state end state, namely measuring the quantum state end state.
Also, as can be seen from the foregoing, the first quantum state evolves according to a first relationship that relies on an evolution matrix in the form of a non-hermite matrix that cannot evolve in a quantum computer. The second quantum state evolves according to a second relation that depends on the unitary evolving, so the second quantum state may evolve rapidly through the quantum computer. Compared with the first quantum state, the second quantum state can evolve in the quantum computer, and meanwhile, the evolution of the second quantum state depends on a second relation of unitary evolution, so that efficient simulated evolution is realized, the time complexity is reduced, the time is saved, and the evolution precision is also ensured.
In some embodiments of the present invention, the second relation includes a second evolution matrix M in the form of hermitian, and the evolution of the quantum state initial state of the closed quantum system by the second relation to obtain the quantum state final state may be the evolution of the second evolution matrix M which does not change with time.
If the second evolution matrix M does not change with time, i.e. is not time-dependent, performing a first operation method on the initial state of the quantum state to obtain the final state of the quantum state of the closed quantum system, wherein the first operation method is as follows:
wherein phi is a pending parameter.
The q operator is specifically:
Quantum bit
In non-time-dependent evolution, it is understood that the process from the initial state of the quantum state to the final state of the quantum state is not affected by time, and the specific value of the time effect is M.
In the non-time-dependent evolution process, the evolution can be achieved by introducing an additional qubit G, in particular, in the non-time-dependent evolution step V φ Wherein the value of M can be determined by the formula above
Obtained, and thus can be found in the formula
Obtain L parameters alpha l Is a numerical value of (2). The values of the parameters are carried into G to calculate G, and then G is carried into Q to obtain the value of Q, and then the step V can be obtained φ The operation flow of (1) is V φ The input quantum computer can be subjected to non-time-dependent evolution.
In this embodiment, a combination of a technique of equivalent of an open quantum system as unitary evolution of a closed quantum system and a quantum signal processing algorithm is employed. By superposing the operations for a plurality of times and reasonably setting the parameter phi, the high-precision approximation of the evolution to be simulated can be obtained. The algorithm based on quantum signal processing can reach the time complexity O (t+1/epsilon), and is close to the theoretical limit, so that the time is saved.
In still other embodiments of the present invention, the second relation includes a second evolution matrix M in the form of hermite, and the initial states of the quantum states of the closed quantum system are evolved by the second relation, so as to obtain the evolution of the end states of the quantum states over time, which may be the second evolution matrix M.
And if the evolution of the second evolution matrix M along the time change is the time-containing evolution, performing the quantum state initial state according to a second operation method to obtain the quantum state final state of the closed quantum system.
It should be noted that the second operation method depends on the dyson expansion and the unitary positive matrix linear combination, specifically, the second operation method is as follows:
wherein M (t) j ) Is the second evolution matrix M at t j And a unitary matrix of time, K is a fixed positive integer.
In this embodiment, the process of the initial state evolution of the quantum state into the final state of the quantum state is affected by time, so that the time-dependent evolution is the evolution affected by time, and the time-affected is M.
Specifically, in this embodiment, simulations are performed using the dyson expansion and unitary positive matrix linear combination technique, each of which, as described above, M (t j ) Are unitary matrices, so that M (t jk )…M(t j1 ) As well as unitary matrices. K is a fixed positive integer. The evolution process corresponding to the U can be realized on a quantum computer by utilizing unitary positive matrix linear combination technology and amplitude amplification technology.
For the time-dependent evolution, the traditional quantum signal processing technology and the Taylor expansion technology cannot be directly popularized to the time-dependent evolution equation. The first-order and second-order tervee unfolding methods can be popularized to the time-containing evolution situation, but have poorer time complexity. In the time-containing evolution of the evolution method of the quantum system, the adopted Dyson expansion method can achieve time complexity superior to that of known methods such as special Luo De expansion and the like.
S104: the quantum state end state of the closed quantum system is measured to obtain the expected value data.
In this step, the desired value can be obtained by measuring the quantum state end state of the closed quantum system.
In some embodiments of the invention, projection measurements may be performed on the quantum state end states while desired value measurements are performed on the physical quantities of the quantum state end states.
In some embodiments of the present invention, the above steps may be repeated to obtain the obtained quantum state end states multiple times, so that after projection measurement is performed on each of the plurality of quantum state end states, an average value is obtained, thereby improving the evolution precision of the evolution method of the quantum system of the present invention, and improving the precision of the expected value of the measured physical quantity.
An embodiment of the evolution method of the quantum system of the present invention will be described with reference to fig. 2.
Firstly, calculating an open quantum system to obtain a quantum state of a closed quantum system to be equivalent, and preparing the quantum state as a quantum state initial state.
The prepared quantum state initial state is evolved, time-containing evolution or non-time-containing evolution can be performed for the evolution selection, and the selection is specifically needed for whether the evolution of the quantum state is affected by time or not.
The evolution in time can be performed according to the second operation. Accordingly, when the non-inclusive evolution is performed, the evolution can be performed according to the first operation.
The corresponding evolved quantum state end state can be obtained through non-time-containing evolution or time-containing evolution, and the expected value of the physical quantity can be obtained through projection measurement of the quantum state end state.
The invention is not limited to this, and in one practical application of the invention, the evolution method of the quantum system of the invention can be applied to the research of the steady state problem of the multi-body system, and in particular, the invention can be practically applied to the simulated evolution process under the incoherent influence of the hessian model, and the evolution equation can be used as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,
is the unitary evolving part of the system. />
A Brix matrix representing the ith spin, h i Representing i self-energies of choice, J i,j Indicating the interaction strength of spins i and j. The latter two terms represent dissipative terms coupled to the environment and are part of the non-unitary positive evolution. By the quantum simulation algorithm described above, simulation can be performedThe enclosed quantum system evolves with or without time. When the evolution time is long enough, the stable state of the system can be obtained, and various properties of the system in the stable state can be obtained through measurement, including magnetization properties, locality, entanglement and the like.
By using the evolution method of the quantum system, an open quantum system which is complex in a quantum computer and difficult to evolve can be equivalent to a closed quantum system, and the evolution of the closed quantum system can be realized based on the evolution of a unitary matrix, so that the time is shortened. And the evolution of the closed quantum system is divided into non-time-containing evolution based on the combination of the unitary positive evolution technology and the quantum signal processing algorithm and time-containing evolution based on the combination of the unitary positive evolution technology and the Dyson expansion technology, so that the time complexity is low, and the simulation precision after the evolution is high.
Fig. 3 is a schematic structural diagram of an evolution device of a quantum system according to an embodiment of the present invention, and an evolution device 100 of a quantum system includes:
the conversion module 10 converts a first quantum state in an open quantum system to be simulated into a second quantum state in a closed quantum system through a first conversion relation, wherein the first quantum state in the open quantum system evolves according to a first relation, and the second quantum state in the closed quantum system evolves according to a second relation; wherein the first relation depends on a first evolution matrix in the form of a non-hermite matrix and the second relation depends on the positive evolution of the unitary;
the preparation module 20 prepares a quantum state initial state of the closed quantum system according to the second quantum state;
the evolution module 30 evolves the quantum state initial state of the closed quantum system through the second relation to obtain the quantum state final state of the closed quantum system.
Fig. 4 is a schematic structural diagram of a computing device 200 according to an embodiment of the present invention, including a memory 40 and a processor 50, wherein executable code is stored in the memory, and the processor implements the method according to any of claims 1-7 when executing the executable code.
The foregoing detailed description of the invention has been presented for purposes of illustration and description, and it should be understood that the invention is not limited to the particular embodiments disclosed, but is intended to cover all modifications, equivalents, alternatives, and improvements within the spirit and principles of the invention.
Claims (7)
1. A method of evolving a quantum system, comprising:
converting a first quantum state in an open quantum system to be simulated into a second quantum state in a closed quantum system through a first conversion relation, wherein the first quantum state in the open quantum system evolves according to a first relation, and the second quantum state in the closed quantum system evolves according to a second relation, and the first relation depends on a first evolution matrix in a non-hermite matrix form and the second relation depends on unitary evolution;
preparing a quantum state initial state of the closed quantum system according to the second quantum state;
evolving the quantum state initial state of the closed quantum system through the second relation to obtain a quantum state final state of the closed quantum system;
wherein the first relation is as follows:
wherein ρ is an original density matrix characterizing the first quantum state, v ρ In order to vectorize rho, i is an imaginary unit, and L is the first evolution matrix;
the second relation is as follows:
where M is an hermitian matrix and v is a representation of the converted second quantum state.
2. The method as recited in claim 1, further comprising:
the quantum state end states of the closed quantum system are measured to obtain desired value data.
3. The method of claim 2, wherein the first transformation relationship comprises an evolution matrix transformation relationship and a quantum state transformation relationship,
the evolution matrix conversion relation is as follows:
the quantum state conversion relation is as follows:
4. The method according to claim 1, characterized in that said second relation comprises a second evolution matrix M in the form of hermite;
the evolution of the quantum state initial state of the closed quantum system through a second relation to obtain the quantum state final state of the closed quantum system comprises the following steps:
if the second evolution matrix M does not change with time, evolving the quantum state initial state according to a first operation method to obtain a quantum state final state of the closed quantum system, wherein the first operation method is as follows:
wherein phi is a parameter to be determined, and the Q operator specifically comprises:
/>
wherein H is l Is a linear superposition of the second evolution matrix M expressed as several unitary positive matrix components
Unitary positive matrix component, qubit->
5. The method according to claim 1, characterized in that said second relation comprises a second evolution matrix M in the form of hermite; the evolution of the quantum state initial state of the closed quantum system through a second relation to obtain the quantum state final state of the closed quantum system comprises the following steps:
if the second evolution matrix M changes with time, the quantum state initial state is evolved according to a second operation method to obtain a quantum state final state of the closed quantum system, wherein the second operation method is as follows:
wherein M (t) j ) Is the second evolution matrix M at t j And a unitary matrix of time, K is a fixed positive integer.
6. An evolution device of a quantum system, comprising:
the conversion module is used for converting a first quantum state in an open quantum system to be simulated into a second quantum state in a closed quantum system through a first conversion relation, wherein the first quantum state in the open quantum system evolves according to a first relation, and the second quantum state in the closed quantum system evolves according to a second relation; wherein the first relation depends on a first evolution matrix in the form of a non-hermite matrix and the second relation depends on the positive evolution of unitary;
the preparation module is used for preparing a quantum state initial state of the closed quantum system according to the second quantum state;
the evolution module is used for evolving the quantum state initial state of the closed quantum system through the second relation to obtain a quantum state final state of the closed quantum system;
wherein the first relation is as follows:
wherein ρ is an original density matrix characterizing the first quantum state, v ρ In order to vectorize rho, i is an imaginary unit, and L is the first evolution matrix;
the second relation is as follows:
where M is an hermitian matrix and v is a representation of the converted second quantum state.
7. A computing device comprising a memory and a processor, wherein the memory has executable code stored therein, which when executed by the processor, implements the method of any of claims 1-5.
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