WO2023130918A1 - Method and apparatus for managing state of quantum system, device and medium - Google Patents

Abstract

Provided are a method and apparatus for managing state of a quantum system, a device and a medium. The quantum virtual time evolution process is used for realizing the process of transferring the quantum system from an initial state to a ground state, and the quantum virtual time evolution process comprises at least one evolution stage. In one method, during an evolutionary phase of the at least one evolution stage, a Hermitian operator associated with the evolutionary phase is determined on the basis of a Hamiltonian of the quantum system and a first intermediate state between the initial state and the ground state. Pauli decomposition is performed for the Hermitian operator to determine a plurality of Pauli operators associated with the evolutionary phase. A target Pauli operator is selected from the plurality of Pauli operator on the basis of importance sampling. A second intermediate state associated with the evolutionary phase is determined on the basis of the target Pauli operator and coefficients of the plurality of Pauli operators, the second intermediate state being closer to the ground state than the first intermediate state. The calculation amount of each evolutionary phase can be reduced, and the performance of the whole quantum virtual time evolution process is improved. Furthermore, the depth of a corresponding quantum circuit can be reduced.

Description

Method, device, apparatus and medium for managing the state of a quantum system

This application claims the priority of the Chinese invention patent application filed on January 10, 2022, entitled "Method, device, device and medium for managing the state of a quantum system", application number 202210022806.5, the entire disclosure of which is incorporated by reference Incorporated into this article.

technical field

Exemplary implementations of the present disclosure relate generally to quantum systems, and in particular to methods, apparatuses, apparatuses, and computer-readable storage media for managing states of quantum systems.

Background technique

Quantum mechanics is a discipline that describes the basic laws of microscopic quantum systems. Unlike classical computers that obey the laws of classical physics, quantum computing is based on microscopic quantum systems and is implemented by applying the laws of quantum mechanics. Quantum computing can involve a series of basic operations (called quantum gates), and complex quantum circuits can be built using these basic quantum gates. Quantum circuits can be used to achieve state transitions (eg, from an initial state to a ground state) of a quantum system. However, the state transfer of a quantum system involves a relatively large amount of calculation, which leads to an extremely large number of quantum gates included in the quantum circuit.

As more quantum gates are added, the accumulation of errors can lead to less accurate quantum computing. Furthermore, the interaction of the quantum system with the surrounding environment will also lead to the gradual loss of effective information. Therefore, it is expected to reduce the calculation amount involved in managing the state transition of the quantum system, and then manage the quantum system in a more efficient manner.

Contents of the invention

According to an exemplary implementation of the present disclosure, a scheme for managing the state of a quantum system based on a quantum virtual time evolution process is provided.

In a first aspect of the present disclosure, a method for managing the state of a quantum system based on the quantum virtual time evolution process is provided. The quantum virtual time evolution process is used to realize the process of transferring the quantum system from the initial state to the ground state. The quantum virtual time The evolution process includes at least one evolution stage. In the method, during an evolution stage in at least one of the evolution stages, a Hermitian operator associated with the evolution stage is determined based on a Hamiltonian of the quantum system and a first intermediate state between the initial state and the ground state. A Pauli decomposition is performed on the Hermitian operators to determine the number of Pauli operators associated with the stages of evolution. A target Pauli operator is selected from multiple Pauli operators based on importance sampling. Based on the target Pauli operator and the coefficients of the plurality of Pauli operators, a second intermediate state associated with the evolution stage is determined, the second intermediate state being closer to the ground state than the first intermediate state.

In a second aspect of the present disclosure, there is provided an electronic device comprising: at least one processing unit; and at least one memory coupled to the at least one processing unit and storing instructions for execution by the at least one processing unit, The instructions, when executed by at least one processing unit, cause the device to perform actions. This action is used to perform the method according to the first aspect of the present disclosure.

In a third aspect of the present disclosure, an apparatus for managing a state of a quantum system based on a quantum virtual time evolution process is provided. The apparatus comprises means for performing the steps in the method according to the first aspect of the present disclosure.

In a fourth aspect of the present disclosure, a computer readable storage medium is provided. A computer program is stored on the medium for performing the method according to the first aspect of the present disclosure when the program is executed by a processor.

It should be understood that what is described in the Summary of the Invention is not intended to limit the key or important features of the implementations of the present disclosure, nor is it intended to limit the scope of the present disclosure. Other features of the present disclosure will be readily understood through the following description.

Description of drawings

Hereinafter, the above and other features, advantages and aspects of various implementations of the present disclosure will become more apparent with reference to the following detailed description in conjunction with the accompanying drawings. In the drawings, identical or similar reference numerals denote identical or similar elements, wherein:

Fig. 1 shows a block diagram of Trotter decomposition according to a technical solution;

Fig. 2 shows the block diagram of the quantum imaginary time evolution process that utilizes quantum circuit to realize according to a technical scheme;

Fig. 3 shows a block diagram of realizing the evolution process of quantum virtual time by using a quantum circuit according to an exemplary implementation of the present disclosure;

FIG. 4 shows a flowchart of a method for managing the state of a quantum system based on a quantum virtual time evolution process according to an exemplary implementation of the present disclosure;

FIG. 5 shows a block diagram for realizing the quantum virtual time evolution process based on the enlarged objective Pauli operator according to some implementations of the present disclosure;

Fig. 6 shows a block diagram of realizing quantum virtual time evolution process by using quantum circuits according to some implementations of the present disclosure;

FIG. 7 shows a flowchart of a method for managing the state of a quantum system in an evolution stage in the quantum virtual time evolution process according to an exemplary implementation of the present disclosure;

8 shows a block diagram of an apparatus for managing the state of a quantum system based on a quantum virtual time evolution process according to an exemplary implementation of the present disclosure; and

Figure 9 shows a block diagram of a device capable of implementing various implementations of the present disclosure.

Detailed ways

Implementations of the present disclosure will be described in more detail below with reference to the accompanying drawings. Although certain implementations of the present disclosure are shown in the drawings, it should be understood that the disclosure may be embodied in various forms and should not be construed as limited to the implementations set forth herein; It is for a more thorough and complete understanding of the present disclosure. It should be understood that the drawings and implementation manners of the present disclosure are for exemplary purposes only, and are not intended to limit the protection scope of the present disclosure.

In recent years, quantum computing has developed rapidly, and in the future it is possible to reshape the technical fields of simulation, encryption, and solution combinatorial optimization of microscopic quantum systems. In the following, first refer to Table 1 to describe the basic concepts involved in the quantum system.

Table 1 Basic concepts of quantum systems

Quantum computing currently involves a large amount of calculation and is limited by many factors such as the number of qubits, noise, system coherence time, and the fidelity of executing quantum gates. In quantum computing, as the number of quantum gate operations increases (that is, the depth of the quantum circuit increases), the accumulation of errors will lead to a decrease in the accuracy of quantum computing, and at the same time, due to the interaction between the quantum system and the surrounding environment, it will also lead to effective information gradually lost. Therefore, how to reduce the amount of calculation involved in the state transition, and then use shallower quantum circuits to simulate the state transition process of the quantum system has become a current research hotspot.

In the field of quantum system simulation, determining the ground state of the quantum system is the key to solving subsequent problems. In the following, more details of an exemplary implementation according to the present disclosure will be described by taking the state transition process of a quantum system as an example. In quantum computing, the Hamiltonian H is the basic quantity describing the energy of a quantum system, and its corresponding unitary evolution can be mapped to a quantum circuit. Specifically, the Hamiltonian H can be converted into a Pauli expansion form, that is, the sum of multiple Pauli terms:

Here, H _j is a Pauli term, and h _j is a coefficient of the Pauli term, and L is the number of Pauli expansion terms. According to the Schrödinger equation, the ideal time evolution of a quantum system can be expressed as the following form (called unitary transformation): U(t)=e ^-iHt , where i is the imaginary unit.

will understand that single

The form of can be manipulated by quantum circuits in order to transform into basic single-bit gates and two-bit gates. However, the exponent part of e in the above evolution form U(t)=e ^-iHt includes the summation of multiple Pauli terms, which cannot be directly implemented on quantum circuits. At this point, the above evolutionary form needs to be transformed into

(that is, the exponential part of e includes only a single Pauli term) to operate in quantum gates.

At this point, U(t) can be converted to:

form. It will be appreciated that each Pauli term in the above formula is not a number but an operator (or matrix). So in general, H _i H _j ≠ H _j H _i for different Pauli terms H _i and H _j . This phenomenon is known as H _i and H _j non-commutation. Therefore, the following inequalities will exist in the evolution process:

Trotter decomposition can be used to solve the above-mentioned incompatibility problem. Hereinafter, an outline about Trotter decomposition will be described with reference to FIG. 1 . 1 shows a block diagram 100 of Trotter decomposition according to a technical solution, and each stage may correspond to a Trotter step. Assuming that the total time of the entire unitary evolution process is t, when the time t is large, the difference between the two sides of the above inequality will be very significant. The entire evolution process can be divided into a plurality of stages 110 , 112 , . . . and 114 . That is, decompose t into multiple smaller time units Δt (eg, decompose t into K parts, t=KΔt). Fig. 1 shows the circuit diagram after the quantum virtual time evolution has been performed, and H can be equal to A ⁽¹⁾ ,...,A ^(K) in each stage. At this time, each Δt is very small, so there is the following approximate relationship for each Δt:

Thus, the evolution during the entire time t can be approximately expressed as: U(t)≈U′(Δt) ^K . Each Δt can be called a Trotter step, and the entire evolution period includes K Trotter steps. At this time, in each of the stages 110 ^, 112 , .

At present, the technical scheme of quantum imaginary time evolution (QITE) has been proposed to calculate the ground state energy of quantum systems. The technical solution simulates the virtual time evolution process by constructing a unitary quantum circuit. When the approximate error and numerical error are small enough, the virtual time evolution process can converge to the ground state energy. However, the existing quantum virtual time evolution process involves a large amount of calculation, which leads to a relatively high depth of the corresponding quantum circuit. Here, a large amount of computation is involved during each Trotter step, which leads to a line depth of quantum imaginary time evolution proportional to the number of Pauli operators of the quantum system's Hamiltonian and the number of Trotter steps.

The imaginary time evolution is used to calculate the evolution of the energy ground state. The t in U(t) can be multiplied by the imaginary unit (that is, replaced by it), and the evolution is transformed into e ^-Ht . In virtual-time evolution, as long as the constructed initial state partially coincides with the ground state, the higher-order excited states will decay exponentially, and finally the main components of the ground state will be obtained. However, this transformation is non-unitary and cannot take advantage of quantum gate operations. QITE provides that by solving a system of linear equations, the unitary evolution e ^-iAΔt is found, and its effect on a given initial state |ψ> is approximately the same as the imaginary time evolution e ^-HΔt . To achieve the above-mentioned unitary evolution in quantum circuits, a Trotter decomposition can be performed. For the jth Trotter step, the following relationship exists:

(The parentheses superscripted here denote the jth Trotter step).

In the following, more details about the evolution of quantum virtual time are described with reference to FIG. 2 . FIG. 2 shows a block diagram 200 of a quantum virtual time evolution process realized by using a quantum circuit according to a technical solution. The various stages 110, 112, ..., and 114 may be implemented using quantum circuits. FIG. 2 only uses the first stage 110 (that is, the first Trotter step) as an example for description, and the processing for other stages is similar and thus will not be repeated here. As shown in FIG. 2 , the quantum circuit 320 can be used to realize the

A Pauli decomposition can be performed to obtain the expanded form of the evolution:

here,

represents the Pauli operator obtained after Pauli decomposition. At this point, the unfolded form can be realized on quantum circuits.

Specifically, quantum gate 220 can be used to realize

can be implemented using quantum gate 222

And can utilize quantum gate 220 to realize

At this point, each stage involves a lot of computation and needs to be implemented using M quantum gates. Further, for the entire quantum virtual time evolution process including K stages, at least M*K quantum gates need to be used. At this time, the depth of the quantum circuit is proportional to the number of Pauli operators of the Hamiltonian of the quantum system and the number of Trotter steps.

In order to at least partly solve the defects in the above technical solutions, according to an exemplary implementation of the present disclosure, a technical solution DRIFT-QITE for managing the state of a quantum system based on the quantum imaginary time evolution process is provided. Here, the quantum virtual time evolution process can realize the transfer from the initial state of the quantum system to the ground state of the quantum system, and the quantum virtual time evolution process includes at least one evolution stage. According to an exemplary implementation of the present disclosure, the state transition of the quantum system can be managed based on the target Pauli operator and the amplification step size, so as to reduce the calculation amount in each evolution stage, so that the calculation amount in each evolution stage does not depend on The number of Pauli operators on the Hamiltonian. Operations performed in each evolution stage may be similar. Hereinafter, an outline of an exemplary implementation according to the present disclosure will be described with reference to FIG. 3 .

Fig. 3 shows a block diagram 300 of implementing quantum virtual time evolution process by using quantum circuits according to an exemplary implementation of the present disclosure. According to an exemplary implementation of the present disclosure, the quantum virtual time evolution process may include evolution stages 310, 312, ..., 314. Here, the entire quantum virtual time evolution process can transfer the quantum system from the initial state to the ground state, and each evolution stage involves a part of Δt in the total time t of the entire quantum virtual time evolution process. Each evolution stage can be transferred from a first intermediate state between the initial state and the ground state to a second intermediate state, and the second intermediate state is closer to the ground state. In other words, multiple evolution stages can be performed serially to gradually transform the initial state to the ground state.

Each evolution stage can be realized in a similar manner, taking the first evolution stage 310 as an example, the evolution stage 310 can transfer the first intermediate state (i.e., the initial state |ψ ₀ > of the quantum system) to the second intermediate state (i.e., The state of the quantum system |ψ ₁ >). Specifically, the Hermitian operator associated with the evolution stage 310 can be determined

Further, Pauli decomposition can be performed on the Hermitian operator to determine multiple Pauli operators associated with the evolution stage 310 . Here, the result of the Pauli decomposition can be expressed as

Among them, M represents the number of Pauli operators,

denote the jth Pauli operator among the Pauli operators during the first evolution stage, and

Indicates the coefficient of the jth Pauli operator.

According to an exemplary implementation of the present disclosure, multiple Pauli operators can be sampled based on importance

Select the target Pauli operator in

Furthermore, based on the objective Pauli operator

and the coefficients of several Pauli operators

A second intermediate state associated with the evolution stage 310 is determined, the second intermediate state being closer to the ground state than the first intermediate state. The target Pauli operator can be

The coefficients of are scaled up to the coefficients based on the individual Pauli operators

A certain order of magnitude in order to reduce the amount of computation in the evolution stage 310 .

During each evolution stage (that is, each Trotter step), when there are M Pauli operators, assuming that the amount of calculation associated with each Pauli operator is amount, then each evolution using the existing QITE The total amount of computation involved in the stage is amount*M. Different from the existing QITE, using the exemplary implementation of the present disclosure, by selecting the target Pauli operator

And amplify the corresponding coefficients, the calculation amount of each evolution stage is only with a single Pauli operator

The associated computational amount (ie, amount). In this way, the amount of calculation related to each evolution stage can be reduced to 1/M of the existing QITE. It will be appreciated that quantum virtual-time evolution processes typically include a large number of evolution stages (eg, tens, hundreds, or even more). Using the exemplary implementation of the present disclosure, the calculation amount of each evolution stage can be greatly reduced, thereby reducing the overall calculation amount of the entire quantum virtual time evolution process, thereby reducing the corresponding calculation resource and time resource overhead.

Continuing to refer to FIG. 3 , according to an exemplary implementation of the present disclosure, the evolution stage 310 may be implemented with quantum gates. At this point, only one quantum gate 322 is needed to complete the evolution stage 310 . In this way, the depth of the quantum circuit 320 used to implement each evolution stage can be greatly reduced. It will be understood that the operation of each quantum gate may be disturbed by a variety of external noises, and by reducing the depth of the quantum gates, the adverse effects of the external noise of the quantum circuit can be greatly reduced, thereby improving the accuracy of the quantum circuit. When the whole quantum virtual time evolution process is realized by quantum circuits, only one quantum gate is needed for each evolution stage. In this way, the depth of the overall quantum circuit for realizing the entire quantum virtual time evolution process can be greatly reduced. Therefore, the quantum virtual time evolution process can be realized with higher efficiency and precision.

The outline of an exemplary implementation according to the present disclosure has been described above, and hereinafter, more details of an exemplary implementation according to the present disclosure will be described with reference to FIG. 4 . Fig. 4 shows a flowchart of a method 400 for managing the state of a quantum system based on a virtual time evolution process according to an exemplary implementation of the present disclosure. As shown in Figure 4, at block 410, the Hamiltonian of the quantum system can be determined

and initial state. Here, the Hamiltonian is a Hermitian operator used to describe the energy of the system, and the Hamiltonian of a quantum system can be obtained based on various methods that are currently known and/or will be developed in the future

and initial state. According to an exemplary implementation of the present disclosure, the quantum system can go through multiple evolution stages in the quantum virtual time evolution process, so as to transfer from the initial state to the ground state. Here, each evolution stage can be used as a part of the quantum virtual time evolution process to perform state transition step by step.

According to an exemplary implementation of the present disclosure, multiple evolution stages may be determined based on Trotter decomposition. Specifically, the time steps of multiple evolution stages may be determined based on the time length and predetermined accuracy for performing the quantum virtual time evolution process. Generally speaking, the time length of the quantum virtual time evolution process may involve several seconds (for example, T=5, 10 or other values). The time step Δt can be set to 0.01, 0.05, and/or other values based on desired accuracy. It will be appreciated that the time step size Δt is inversely proportional to the precision, the smaller the Δt the higher the precision. According to an exemplary implementation of the present disclosure, the number of evolution stages can also be specified, for example, it can be specified that the quantum virtual time evolution process is decomposed into 100 (or other numerical values) evolution stages.

The Trotter decomposition process can be implemented based on currently known and/or technologies to be developed in the future, and the details will not be described in detail below. Each evolution stage can involve its own input state and output state, that is, each evolution stage can realize the process of transferring from an input state to an output state. Here, the input state of the first evolution stage among the plurality of evolution stages is the initial state of the quantum system, as shown in block 410, the input state of the current evolution stage may be set as the initial state of the quantum system. Further, the output state of the last evolution stage among the plurality of evolution stages is the ground state of the quantum system. For other evolution stages, the input state and output state of each evolution stage are intermediate states between the initial state and the base state. According to an exemplary implementation manner of the present disclosure, the input state and the output state of the current evolution stage may also be respectively referred to as a first intermediate state and a second intermediate state associated with the evolution stage.

At block 420, the Hermitian operator A of the current evolution stage may be determined according to the obtained Hamiltonian and the input state. It will be appreciated that the Hermitian operator A may have different values at different stages of evolution. For example, the Hermitian operator in the first evolution stage can be expressed as A ⁽¹⁾ , the Hermitian operator in the second evolution stage can be expressed as A ⁽²⁾ , and so on. Here, the numbers in the superscripts of the Hermitian operators indicate the numbers of evolutionary stages. The processing in each evolution stage may be similar, and in the following, only one evolution stage will be used as an example for description.

At block 430, the Hermitian operators may be subjected to Pauli decomposition to determine a number of Pauli operators associated with the current evolution stage. Specifically, the Hermitian operator A can be decomposed into

where M represents the number of multiple Pauli operators, A _j represents the j-th Pauli operator, and a _j represents the coefficient of the j-th Pauli operator. Further, at block 440, a target Pauli operator _{Aj_s} may be selected from a plurality of Pauli operators based on importance sampling. It will be appreciated that the target Pauli operator _{Aj_s} can be selected here based on a number of ways.

According to an exemplary implementation of the present disclosure, a target Pauli operator A _{j_s} may be randomly selected from a plurality of Pauli operators A _j (j=1, . . . , M). For example, in the first evolution stage, the fifth Pauli operator can be randomly selected; in the second evolution stage, the third Pauli operator can be randomly selected, and so on. Using the exemplary implementation of the present disclosure, the randomly selected target Pauli operator A _{j_s} can be used to represent the state transition trend of the quantum system at the current evolution stage. In this way, the calculation amount of the evolution stage can be greatly reduced, and a single quantum gate can be used to realize an evolution stage.

According to an exemplary implementation of the present disclosure, based on the comparison of the coefficients a _j (where j=1,...,M) of multiple Pauli operators A _j (where A=1,...,M), from multiple Select the target Pauli operator in Pauli operator. Specifically, a Pauli operator with a larger coefficient (for example, the largest coefficient, the second largest coefficient, etc.) may be selected as the target Pauli operator A _{j_s} . In this way, the Pauli operator with the largest contribution in the evolution stage can be selected to represent the evolution trend of the evolution stage. In this way, the accuracy of the evolution process can be improved.

Continuing to refer to FIG. 4 , at block 450 , a unitary transformation associated with an evolution stage may be generated based on the target Pauli operator and the coefficients of the plurality of Pauli operators. According to an exemplary implementation of the present disclosure, the target Pauli operator A _{j_s} can be amplified to construct a unitary transformation, and then determine the output state associated with the evolution stage. It will be appreciated that the output state here is closer to the ground state than the first intermediate state. That is to say, in each evolution stage, the evolution is carried out toward the direction of transferring from the initial state of the quantum system to the ground state. When constructing a unitary transformation, an output state associated with an evolution stage can be determined based on a target Pauli operator and coefficients of a plurality of Pauli operators.

According to an exemplary implementation of the present disclosure, a sum of absolute values of coefficients of multiple Pauli operators may be determined, and a unitary transformation may be determined based on a target Pauli operator, the sum of coefficients of the target Pauli operator. Specifically, based on the probability

to select the target Pauli operator A _{j_s} , and amplify the coefficient of the target Pauli operator A _{j_s} to

Thus, it is possible to use

to replace e ^-iA△t , and then realize the current evolution stage based on the unitary transformation.

In the current evolution stage, although the evolution accuracy obtained by using the enlarged target Pauli operator may be lower than the evolution accuracy of the existing QITE, however, the amount of calculation involved in the simulation process based on the enlarged target Pauli operator will be It will be much smaller than the calculation amount of the existing QITE (for example, reduced to the original 1/M). Furthermore, when the calculation results of multiple evolution stages are considered comprehensively, the experiment shows that the accuracy of the ground state obtained by using the enlarged target Pauli operator is still high, and the difference from the existing QITE technical scheme is very subtle. In this way, the number of computing resources and time resources involved in each evolution stage can be greatly reduced, thereby improving the performance of the entire quantum virtual time evolution process.

According to an exemplary implementation manner of the present disclosure, the loop process shown in FIG. 4 may be repeated continuously until the total evolution time reaches a predetermined length of time. Specifically, in the first evolution stage among the plurality of evolution stages, the input is the initial state, at this time, the output state of the first evolution stage can be fed to the second evolution stage as the input state of the second evolution stage. Further, the output state of the second evolution stage can be fed to the third evolution stage as the input state of the third evolution stage, and so on.

Continuing with the example described above, in the first evolution stage, after the output state associated with the current evolution stage has been determined, this output state can be used as input to the next evolution stage (ie, the second evolution stage) state (for example, perform a set operation as follows: input state = output state). Further, at block 460, it may be determined whether the total time of evolutions that have been performed exceeds a predetermined length of time. If the judging result is "yes", the method 400 ends at this time, that is, all Trotter steps of the quantum virtual time evolution process have been completed at this time. If the determination is "No", at this point the method 400 returns to block 420 to perform the next evolution stage in a similar manner.

It will be appreciated that FIG. 4 illustrates by way of example only the process of determining whether method 400 is complete based on a time comparison. According to an exemplary implementation of the present disclosure, the number of iterations may also be specified in advance, for example, the method 400 may be stopped when the predetermined number of iterations is reached. According to an exemplary implementation of the present disclosure, predetermined convergence conditions may be specified. For example, the method 400 stops when the difference between the input state and the output state associated with an evolution stage (or the difference between the output states of two successive evolution stages) satisfies the convergence condition.

According to an exemplary implementation of the present disclosure, quantum gates may be utilized to perform unitary transformations to determine the output state associated with the current evolutionary stage. Using the exemplary implementation of the present disclosure, the process that originally needs to be performed by M quantum gates can be transferred to be implemented by a single quantum gate. In other words, only a single quantum gate is needed to complete one evolution stage using the exemplary implementations of the present disclosure. In this way, the amount of calculations involved in each evolution stage can be greatly reduced, thereby reducing the depth of the corresponding quantum circuits.

According to an exemplary implementation of the present disclosure, the calculation amount of each evolution stage can be further reduced based on importance sampling. Specifically, in the initial state, it can be based on the Hamiltonian of the quantum system

(j=1, . . . , L, where L is the number of Pauli expansion terms of the Hamiltonian), multiple energy Pauli operators for the Hamiltonian are determined. It will be appreciated that here the Hamiltonian H can be decomposed by Pauli into a sum of terms h _j H _j . Similar to the operations performed in each evolution stage, importance sampling can be performed based on the coefficients h _j of the above-mentioned multiple terms, so as to select a target energy Pauli operator H _{j_sh} from a plurality of energy Pauli operators H _j . Here, the coefficient h _{j_sh} of the target energy Pauli operator H _{j_sh} may have the largest absolute value among the plurality of coefficients h _j .

Further, the Hermitian operator associated with the evolution stage may be determined based on the target energy Pauli operator H _{j_sh} and the amplification factor ||h|| obtained based on the coefficients of a plurality of energy Pauli operators. Specifically, the coefficient h _{j_sh} of H _{j_sh} can be amplified to

At this time, during the evolution phase, it is not necessary to perform calculations on all the data in the Hamiltonian, but only need to perform subsequent calculations on the amplified energy Pauli operator _Hj therein. In this way, the calculation amount involved in the evolution stage can be further reduced, thereby reducing the calculation resource and time resource overhead of the entire quantum virtual time evolution process.

Hereinafter, more details of selecting a target Pauli operator will be described with reference to FIG. 5 . FIG. 5 shows a block diagram 500 for implementing a quantum virtual time evolution process based on an amplified objective Pauli operator, according to some implementations of the present disclosure. In the context of the present disclosure, the evolution process can be processed based on the DRIFT-QITE principle. For ease of description, the evolution process only includes 3 evolution stages, and each evolution stage includes a series of evolution U ₁ U _{2 .} . . U _M .

Specifically, in one evolution stage, one item from U ₁ U ₂ . . . U _M can be selected to replace the entire evolution stage. In other words, V-evolution can be constructed to approximate the entire evolution stage. At this time, the e-exponent part of V only includes a single Pauli operator, so the depth is still 1 when implemented with quantum gates. Further, the coefficient of the selected target Pauli operator can be amplified so as to achieve the effect of reducing the depth.

At this time, the entire quantum virtual time evolution process is shown as the curve 510 in FIG. 5 . Curve 510 includes 3 evolution stages, and each evolution stage includes a corresponding plurality of parts. Specifically, the first evolutionary stage involves

The second evolutionary stage involves

and the 3rd evolutionary stage involves

At this point, each part takes a small step in its own direction, for example, the first part in the 1st evolutionary stage goes towards

Forward, the second part of the first evolutionary stage towards

etc.

Using an exemplary implementation of the present disclosure, during each evolution stage, a large step can be taken along the direction V with the largest coefficient (||a||, where

That is, the sum of the absolute values of the coefficients of multiple Pauli operators after Pauli decomposition). According to an exemplary implementation of the present disclosure, based on the probability

to select the target Pauli operator A _{j_s} , and use

to replace e ^-iA△t . In this way, the amount of computation involved in each evolution stage can be greatly reduced.

As shown in FIG _. 5 , in each evolution stage, each line segment U ₁ , U ₂ , . Further, the length of the selected line segment can be amplified to the sum of the lengths of the line segments U ₁ , U ₂ , . . . U _M in the evolution stage. That is, in each evolution stage, a large step is taken in the direction of the selected line segment. At this time, the three evolutionary stages (that is, three Trotter steps) shown in FIG. 5 only need three steps forward in three directions.

As shown by broken line 520 shown in dotted line in FIG. 5 , the M line segments in each evolution stage can be replaced with a single line segment. For example, in the first evolution stage you can use

Replaced by V ⁽¹⁾ , in the second evolution stage, the

is replaced by V ⁽²⁾ , and in the third evolution stage it is possible to replace

Replaced by V ⁽³⁾ . That is, the evolution path of the curve 510 including multiple small line segments is replaced by three polylines 520 with larger lengths.

As shown in Figure 5, at certain positions during the quantum virtual time evolution process, although there are differences between the broken line 520 and the curve 510, the changing trends of the broken line 520 and the curve 510 remain consistent throughout the quantum virtual time evolution process . Each line segment in polyline 520 may correspond to a quantum circuit with a depth of one. At this time, the depth of the existing QITE realized based on the curve 510 is 3M (including 3 evolution stages, and the depth of each evolution stage is M), while the depth of the DRIFT-QITE of the present disclosure based on the broken line 520 is only 3. In this way, the depth of the quantum circuit can be greatly reduced, and the deviation of the evolution path can be controlled within the error range.

According to an exemplary implementation of the present disclosure, a quantum virtual time evolution process may be performed using a quantum circuit including a plurality of quantum gates. Hereinafter, more details about implementing the quantum virtual time evolution process using a quantum circuit including a plurality of quantum gates will be described with reference to FIG. 6 . FIG. 6 shows a block diagram 600 of implementing quantum virtual time evolution process using quantum circuits according to some implementations of the present disclosure. As shown in FIG. 6, the number of multiple quantum gates can be determined based on the number of multiple evolution stages. Specifically, a quantum gate can be used to implement a unitary transformation associated with an evolution stage. For example, evolution stage 310 may be implemented using quantum gate 322 , evolution stage 312 may be implemented using quantum gate 610 , . . . , and evolution stage 314 may be implemented using quantum gate 620 . At this time, the quantum circuit 610 may include K quantum gates for simulating a quantum virtual-time evolution process including K evolution stages.

Using the exemplary implementation of the present disclosure, in each evolution stage of the quantum virtual time evolution process, the target Pauli operator is selected based on importance sampling and by amplifying the coefficient of the target Pauli operator, it is possible to reduce the amount of computation involved. When the quantum circuit is used to realize the quantum virtual time evolution process, the number of quantum gates used in each evolution stage can be greatly reduced, thereby reducing the depth of the quantum circuit and improving the overall performance of the quantum virtual time evolution process.

It will be understood that although FIG. 6 shows that the DRIFT-QITE scheme according to the present disclosure is used in each stage 310, 312, ..., and 314 to reduce the amount of computation, according to an exemplary implementation of the present disclosure, the The DRIFT-QITE protocol is applied in some of the above multiple stages. At this time, the performance of the evolution stage applying the DRIFT-QITE scheme will be significantly improved.

According to one exemplary implementation of the present disclosure, the DRIFT-QITE scheme described above can be applied in quantum systems including _BeH2 molecules. Near the molecular equilibrium position, within the allowable error range, using DRIFT-QITE requires only fewer steps (based on different configurations, such as 52 steps or 37 steps) to reach the ground state of the quantum system with the desired accuracy. At this time, the number of required quantum gates is 52 or 37. However, in the existing QITE, a single Trotter step requires 252 quantum gates, and in the quantum virtual time evolution process including dozens of Trotter steps, the existing QITE requires at least thousands of quantum gates. It can be seen that by using the DRIFT-QITE scheme of the present disclosure, the calculation amount of the quantum virtual time evolution process can be greatly reduced, and then implemented with a smaller number of quantum gates. In this way, the depth of the quantum circuit can be greatly reduced without loss of precision, the interference of environmental noise can be reduced, and the performance of the quantum virtual time evolution process can be improved.

example process

Fig. 7 shows a flowchart of a method 700 for managing the state of a quantum system in an evolution stage in the quantum virtual time evolution process according to an exemplary implementation of the present disclosure. Here, the quantum virtual time evolution process is used to realize the process of transferring the quantum system from the initial state to the ground state, and the quantum virtual time evolution process includes at least one evolution stage. As shown in FIG. 7, at block 710, during an evolution stage in at least one of the evolution stages, an err associated with the evolution stage is determined based on the quantum system's Hamiltonian and a first intermediate state between the initial state and the ground state. meter operator.

At block 720, a Pauli decomposition is performed on the Hermitian operators to determine a number of Pauli operators associated with the stages of evolution. At block 730, a target Pauli operator is selected from a plurality of Pauli operators based on importance sampling. According to an exemplary implementation of the present disclosure, a target Pauli operator may be selected from a plurality of Pauli operators based on a comparison of coefficients of the plurality of Pauli operators.

At block 740, based on the target Pauli operator and coefficients of the plurality of Pauli operators, a second intermediate state associated with the evolution stage is determined, the second intermediate state being closer to the ground state than the first intermediate state. According to an exemplary implementation of the present disclosure, in order to determine the second intermediate state, a unitary transformation associated with an evolution stage may be generated based on a target Pauli operator and coefficients of a plurality of Pauli operators. In turn, a unitary transformation can be utilized to determine a second intermediate state associated with the evolution stage.

According to an exemplary implementation of the present disclosure, in order to generate a unitary transformation associated with an evolution stage, the sum of the absolute values of the coefficients of a plurality of Pauli operators may be determined. Further, the unitary transformation can be determined based on the objective Pauli operator and the sum.

According to an exemplary implementation of the present disclosure, in order to determine the second intermediate state using unitary transformation, quantum gates may be used to perform unitary transformation to determine the second intermediate state.

According to an exemplary implementation of the present disclosure, at least one evolution stage includes a plurality of evolution stages. According to an exemplary implementation of the present disclosure, in order to determine the multiple evolution stages, the time steps of the multiple evolution stages may be determined based on the time length and predetermined accuracy for performing the quantum virtual time evolution process. Trotter decomposition can then be performed on the quantum virtual-time evolution process according to the time step to determine multiple evolution stages.

According to an exemplary implementation manner of the present disclosure, in the first evolution stage among the plurality of evolution stages, the first intermediate state is the initial state. According to an exemplary implementation of the present disclosure, the second intermediate state may be output to the second evolution stage following the first evolution stage as the first intermediate state of the second evolution stage.

According to an exemplary implementation of the present disclosure, a quantum virtual time evolution process may be performed using a quantum circuit including a plurality of quantum gates, the number of the plurality of quantum gates is determined based on the number of evolution stages, and the plurality of quantum gates A given quantum gate in is used to perform the unitary transformation associated with a given evolutionary stage of the plurality of evolutionary stages.

According to an exemplary implementation of the present disclosure, a target Pauli operator may be randomly selected from a plurality of Pauli operators.

According to an exemplary implementation of the present disclosure, in order to determine the Hermitian operator associated with the evolution stage, multiple energy Pauli operators of the Hamiltonian may be determined based on the Hamiltonian and the initial state of the quantum system. Then, a target energy Pauli operator may be selected from a plurality of energy Pauli operators based on importance sampling. Further, the Hermitian operator associated with the evolution stage may be determined based on the target energy Pauli operator and the coefficients of the plurality of energy Pauli operators.

Example Apparatus and Equipment

Fig. 8 shows a block diagram of an apparatus 800 for managing the state of a quantum system based on a quantum virtual time evolution process according to an exemplary implementation of the present disclosure. Here, the quantum virtual time evolution process is used to realize the process of transferring the quantum system from the initial state to the ground state, and the quantum virtual time evolution process includes at least one evolution stage.

As shown in FIG. 8 , the apparatus 800 includes a determination module 810 , a decomposition module 820 , a selection module 830 and a status determination module 840 . According to an exemplary implementation of the present disclosure, the determination module 810 is configured to determine and evolve the quantum system based on the Hamiltonian of the quantum system and the first intermediate state between the initial state and the ground state during at least one of the evolution stages. Hermitian operators associated with the stages; the decomposition module 820 is configured to perform Pauli decomposition on the Hermitian operators to determine a plurality of Pauli operators associated with the stages of evolution; the selection module 830 is configured for importance-based sampling Selecting a target Pauli operator from the plurality of Pauli operators; and state determination module 840 configured to determine a second intermediate state associated with the evolution stage based on the target Pauli operator and coefficients of the plurality of Pauli operators , the second intermediate state is closer to the ground state than the first intermediate state. According to an exemplary implementation manner of the present disclosure, the apparatus further includes a module for performing other steps in the method 700 .

Hereinafter, an apparatus according to an exemplary implementation of the present disclosure is described with reference to FIG. 9 . FIG. 9 shows a block diagram of a device 900 capable of implementing various implementations of the present disclosure. It should be understood that the computing device 900 shown in FIG. 9 is exemplary only, and should not constitute any limitation as to the functionality and scope of the implementations described herein. For example, computing device 900 may be used to perform method 700 described above.

As shown in FIG. 9, computing device 900 is in the form of a general-purpose computing device. Components of computing device 900 may include, but are not limited to, one or more processors or processing units 910, memory 920, storage devices 930, one or more communication units 940, one or more input devices 950, and one or more output devices 960. The processing unit 910 may be an actual or virtual processor and can perform various processing according to programs stored in the memory 920 . In a multi-processor system, multiple processing units execute computer-executable instructions in parallel to increase the parallel processing capability of the computing device 900 .

Computing device 900 typically includes a plurality of computer storage media. Such media can be any available media that is accessible by computing device 900, including but not limited to, volatile and nonvolatile media, removable and non-removable media. The memory 920 can be volatile memory (eg, registers, cache, random access memory (RAM)), nonvolatile memory (eg, read only memory (ROM), electrically erasable programmable read only memory (EEPROM) , flash memory) or some combination of them. Storage device 930 may be removable or non-removable media, and may include machine-readable media, such as flash drives, magnetic disks, or any other media that may be capable of storing information and/or data (e.g., training data for training ) and can be accessed within computing device 900.

Computing device 900 may further include additional removable/non-removable, volatile/nonvolatile storage media. Although not shown in FIG. 9, a disk drive for reading from or writing to a removable, nonvolatile disk (such as a "floppy disk") and a disk drive for reading from a removable, nonvolatile disk may be provided. CD-ROM drive for reading or writing. In these cases, each drive may be connected to the bus (not shown) by one or more data media interfaces. The memory 920 may include a computer program product 925 having one or more program modules configured to perform the various methods or actions of the various implementations of the present disclosure.

The communication unit 940 enables communication with other computing devices through communication media. Additionally, the functionality of the components of computing device 900 may be implemented in a single computing cluster or as a plurality of computing machines capable of communicating via communication links. Accordingly, computing device 900 may operate in a networked environment using logical connections to one or more other servers, a network personal computer (PC), or another network node.

Input device 950 may be one or more input devices, such as a mouse, keyboard, trackball, and the like. Output device 960 may be one or more output devices, such as a display, speakers, printer, or the like. The computing device 900 can also communicate with one or more external devices (not shown) through the communication unit 940 as needed, such as storage devices, display devices, etc., and one or more devices that enable the user to interact with the computing device 900 In communication, or with any device (eg, network card, modem, etc.) that enables computing device 900 to communicate with one or more other computing devices. Such communication may be performed via an input/output (I/O) interface (not shown).

According to an exemplary implementation of the present disclosure, there is provided a computer-readable storage medium on which computer-executable instructions are stored, wherein the computer-executable instructions are executed by a processor to implement the methods described above. According to an exemplary implementation of the present disclosure, there is also provided a computer program product tangibly stored on a non-transitory computer-readable medium and comprising computer-executable instructions, and the computer-executable instructions are executed by a processor to implement the method described above. According to an exemplary implementation of the present disclosure, there is provided a computer program product on which a computer program is stored, the program implementing the method described above when executed by a processor.

Aspects of the present disclosure are described herein with reference to flowchart illustrations and/or block diagrams of methods, apparatus, apparatus, and computer program products implemented according to the disclosure. It should be understood that each block of the flowcharts and/or block diagrams, and combinations of blocks in the flowcharts and/or block diagrams, can be implemented by computer-readable program instructions.

These computer-readable program instructions may be provided to a processing unit of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine such that when executed by the processing unit of the computer or other programmable data processing apparatus , producing an apparatus for realizing the functions/actions specified in one or more blocks in the flowchart and/or block diagram. These computer-readable program instructions can also be stored in a computer-readable storage medium, and these instructions cause computers, programmable data processing devices and/or other devices to work in a specific way, so that the computer-readable medium storing instructions includes An article of manufacture comprising instructions for implementing various aspects of the functions/acts specified in one or more blocks in flowcharts and/or block diagrams.

computer-readable program instructions can be loaded onto a computer, other programmable data processing apparatus, or other equipment, so that a series of operational steps are performed on the computer, other programmable data processing apparatus, or other equipment to produce a computer-implemented process, Instructions executed on computers, other programmable data processing devices, or other devices can thus implement the functions/actions specified in one or more blocks in the flowcharts and/or block diagrams.

The flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various implementations of the present disclosure. In this regard, each block in a flowchart or block diagram may represent a module, a program segment, or a portion of an instruction that contains one or more executable instruction. In some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks in succession may, in fact, be executed substantially concurrently, or they may sometimes be executed in the reverse order, depending upon the functionality involved. It should also be noted that each block of the block diagrams and/or flowchart illustrations, and combinations of blocks in the block diagrams and/or flowchart illustrations, can be implemented by a dedicated hardware-based system that performs the specified function or action , or may be implemented by a combination of dedicated hardware and computer instructions.

Having described various implementations of the present disclosure above, the foregoing description is exemplary, not exhaustive, and is not limited to the disclosed implementations. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described implementations. The choice of terminology used herein aims to best explain the principle of each implementation, practical application or improvement of technology in the market, or to enable other ordinary skilled in the art to understand the various implementations disclosed herein.

Claims (23)

1. A method for managing the state of a quantum system based on a quantum virtual time evolution process, the quantum virtual time evolution process is used to realize the process of transferring the quantum system from an initial state to a ground state, and the quantum virtual time evolution process includes at least An evolution stage, the method comprising:

   During an evolution stage of said at least one evolution stage, a Hermitian associated with said evolution stage is determined based on a Hamiltonian of said quantum system and a first intermediate state between said initial state and said ground state operator;

   performing Pauli decomposition on the Hermitian operator to determine a plurality of Pauli operators associated with the evolution stage;

   selecting a target Pauli operator from the plurality of Pauli operators based on importance sampling; and

   Based on the target Pauli operator and coefficients of the plurality of Pauli operators, determining a second intermediate state associated with the evolution stage, the second intermediate state being closer to the first intermediate state than the first intermediate state ground state.
2. The method of claim 1 , wherein selecting the target Pauli operator comprises selecting the target Pauli operator from the plurality of Pauli operators based on a comparison of the coefficients of the plurality of Pauli operators. Pauli operator.
3. The method of claim 1, wherein determining the second intermediate state comprises:

   generating a unitary transformation associated with the evolution stage based on the target Pauli operator and the coefficients of the plurality of Pauli operators; and

   The second intermediate state associated with the evolution stage is determined using the unitary transformation.
4. The method of claim 3, wherein generating the unitary transformation associated with the evolution stage comprises:

   determining the sum of the absolute values of the coefficients of the plurality of Pauli operators; and

   Based on the target Pauli operator and the sum, the unitary transformation is determined.
5. The method of claim 3, wherein using the unitary transformation to determine the second intermediate state comprises performing the unitary transformation with a quantum gate to determine the second intermediate state.
6. The method of claim 5, wherein the at least one evolution stage comprises a plurality of evolution stages, and the method further comprises determining the plurality of evolution stages based on:

   determining time steps for said plurality of evolution stages based on a length of time and a predetermined precision for performing said quantum virtual time evolution process; and

   Trotter decomposition is performed on the quantum virtual time evolution process according to the time step size to determine the plurality of evolution stages.
7. The method according to claim 6, wherein in a first evolution stage of the plurality of evolution stages, the first intermediate state is the initial state, and the method further comprises: converting the second intermediate state to A state is output to a second evolution stage following the first evolution stage as a first intermediate state of the second evolution stage.
8. The method according to claim 7, further comprising: performing the quantum virtual time evolution process using a quantum circuit comprising a plurality of quantum gates, the number of the plurality of quantum gates is based on the number of the plurality of evolution stages It is determined that a given quantum gate of the plurality of quantum gates is used to perform a unitary transformation associated with a given evolutionary stage of the plurality of evolutionary stages.
9. The method according to any one of claims 1 to 8, wherein selecting the target Pauli operator comprises randomly selecting the target Pauli operator from the plurality of Pauli operators.
10. The method of any one of claims 1 to 8, wherein determining the Hermitian operator associated with the evolution stage comprises:

    determining a plurality of energy Pauli operators of the Hamiltonian based on the Hamiltonian of the quantum system and the initial state;

    selecting a target energy Pauli operator from the plurality of energy Pauli operators based on importance sampling; and

    The Hermitian operator associated with the evolution stage is determined based on the target energy Pauli operator and coefficients of the plurality of energy Pauli operators.
11. An electronic device comprising:

    at least one processing unit; and

    at least one memory coupled to the at least one processing unit and storing instructions for execution by the at least one processing unit that, when executed by the at least one processing unit, cause the device Executing an action for managing the state of the quantum system based on a quantum virtual time evolution process, the quantum virtual time evolution process is used to realize the process of transferring the quantum system from the initial state to the ground state, and the quantum virtual time evolution process includes at least one In the evolution stage, the actions include:

    During an evolution stage of said at least one evolution stage, a Hermitian associated with said evolution stage is determined based on a Hamiltonian of said quantum system and a first intermediate state between said initial state and said ground state operator;

    performing Pauli decomposition on the Hermitian operator to determine a plurality of Pauli operators associated with the evolution stage;

    selecting a target Pauli operator from the plurality of Pauli operators based on importance sampling; and

    Based on the target Pauli operator and coefficients of the plurality of Pauli operators, determining a second intermediate state associated with the evolution stage, the second intermediate state being closer to the first intermediate state than the first intermediate state ground state.
12. The apparatus of claim 11 , wherein selecting the target Pauli operator comprises selecting the target Pauli operator from the plurality of Pauli operators based on a comparison of the coefficients of the plurality of Pauli operators. Pauli operator.
13. The apparatus of claim 11, wherein determining the second intermediate state comprises:

    generating a unitary transformation associated with the evolution stage based on the target Pauli operator and the coefficients of the plurality of Pauli operators; and

    The second intermediate state associated with the evolution stage is determined using the unitary transformation.
14. The apparatus of claim 13, wherein generating the unitary transformation associated with the evolution stage comprises:

    determining the sum of the absolute values of the coefficients of the plurality of Pauli operators; and

    Based on the target Pauli operator and the sum, the unitary transformation is determined.
15. The apparatus of claim 13, wherein determining the second intermediate state using the unitary transformation comprises performing the unitary transformation using a quantum gate to determine the second intermediate state.
16. The apparatus of claim 15, wherein the at least one evolution stage comprises a plurality of evolution stages, and the actions further comprise determining the plurality of evolution stages based on:

    determining time steps for said plurality of evolution stages based on a length of time and a predetermined precision for performing said quantum virtual time evolution process; and

    Trotter decomposition is performed on the quantum virtual time evolution process according to the time step size to determine the plurality of evolution stages.
17. The apparatus of claim 16, wherein in a first evolution stage of the plurality of evolution stages, the first intermediate state is the initial state, and the actions further comprise: converting the second intermediate state to A state is output to a second evolution stage following the first evolution stage as a first intermediate state of the second evolution stage.
18. The apparatus according to claim 17, wherein said actions further comprise: performing said quantum virtual time evolution process using a quantum circuit comprising a plurality of quantum gates, the number of said plurality of quantum gates being based on said plurality of evolution A given quantum gate of the plurality of quantum gates is used to perform a unitary transformation associated with a given evolutionary stage of the plurality of evolutionary stages.
19. The apparatus according to any one of claims 11 to 18, wherein selecting the target Pauli operator comprises randomly selecting the target Pauli operator from the plurality of Pauli operators.
20. The apparatus according to any one of claims 11 to 18, wherein determining the Hermitian operator associated with the evolution stage comprises:

    determining a plurality of energy Pauli operators of the Hamiltonian based on the Hamiltonian of the quantum system and the initial state;

    selecting a target energy Pauli operator from the plurality of energy Pauli operators based on importance sampling; and

    The Hermitian operator associated with the evolution stage is determined based on the target energy Pauli operator and coefficients of the plurality of energy Pauli operators.
21. A device for managing the state of a quantum system based on a quantum virtual time evolution process, the quantum virtual time evolution process is used to realize the process of transferring the quantum system from an initial state to a ground state, and the quantum virtual time evolution process includes at least One stage of evolution, the device includes:

    A determination module configured to determine, during an evolution stage of the at least one evolution stage, based on the Hamiltonian of the quantum system and a first intermediate state between the initial state and the ground state, the Hermitian operators associated with phases;

    a decomposition module configured to perform Pauli decomposition on the Hermitian operator to determine a plurality of Pauli operators associated with the evolution stage;

    a selection module configured to select a target Pauli operator from the plurality of Pauli operators based on importance sampling; and

    a state determination module configured to determine a second intermediate state associated with the evolution stage based on the target Pauli operator and the coefficients of the plurality of Pauli operators, the second intermediate state being larger than the The first intermediate state is closer to the ground state.
22. A computer-readable storage medium, on which a computer program is stored, and when the program is executed by a processor, the method according to any one of claims 1 to 10 is realized.
23. A computer program product on which is stored a computer program which, when executed by a processor, implements the method according to any one of claims 1 to 10.

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