Approximate solution hamiltonian ground state energy method, system, and electronic device

Abstract

The application discloses a kind of approximate solving hamiltonian ground state energy method, system and electronic equipment, approximate solving hamiltonian ground state energy method includes: obtaining and preprocessing the hamiltonian to be solved, obtains the set of groups corresponding to hamiltonian;According to group set, generate Clifford operation sequence, and the new hamiltonian is obtained by the action of Clifford operation sequence to group set;According to new hamiltonian, generate energy expression containing variable parameter;Optimize variable parameter in energy expression, obtain the approximate value of the ground state energy of hamiltonian.The application is processed to group set by Clifford operation sequence, obtains new hamiltonian, and generates energy expression containing variable parameter, and then the approximate value of the ground state energy of hamiltonian is obtained by optimizing variable parameter, improve the calculation efficiency, shorten the calculation time, simultaneously, when multiple group dominants exist in hamiltonian, improve the approximate effect, so that the approximate value calculated and the precise value error can be negligible.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of quantum technology, in particular to a method for approximately solving ground state energy of a Hamiltonian, a system for approximately solving ground state energy of a Hamiltonian and an electronic device. BACKGROUND

[0002] In the field of quantum computing, the ground state solving problem of a Hamiltonian is a core task to determine the lowest energy of a quantum system and its corresponding quantum state, which runs through key scenarios such as quantum chemistry simulation, material design, combinatorial optimization and quantum error correction. However, relevant research shows that even if the Hamiltonian is constrained to be a k-local (k≥2, i.e. each Pauli term in the Hamiltonian contains at most k non-identity Pauli matrices) Hamiltonian, the complexity of its ground state solving is QMA-hard, which means that even with a quantum computer, it will take super-polynomial time to solve.

[0003] This dilemma prompts the engineering practice to turn to approximate solving schemes. One of the approximate solving methods in the related art is to use stabilizer states to approximate the ground state. A stabilizer state is a special quantum state generated by mutually commutative Pauli operators (or called stabilizers). Stabilizer states are uniformly distributed in the entire Hilbert space, and under a certain distance scale, there must be a stabilizer state closest to the true ground state. However, in some special cases, the performance of the approximate solving method based on stabilizer states will deteriorate. For example, in the bond breaking region of a chemical reaction, or near the phase transition point of a quantum many-body system, the true ground state energy is often dominated by multiple non-commuting Pauli terms, which will cause the stabilizer state method to approximate the ground state by a single cluster to deteriorate. SUMMARY

[0004] The present application is proposed to solve at least one of the above problems. According to a first aspect of the present application, a method for approximately solving ground state energy of a Hamiltonian is provided, the method comprising:

[0005] obtaining and preprocessing a Hamiltonian to be solved, to obtain a cluster set corresponding to the Hamiltonian.

[0006] generating a Clifford operation sequence according to the cluster set, and applying the Clifford operation sequence to the cluster set to obtain a new Hamiltonian.

[0007] generating an energy expression containing a variable parameter according to the new Hamiltonian.

[0008] optimizing the variable parameter in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian.

[0009] In an embodiment of the present application, the Hamiltonian consists of a plurality of Pauli terms, and the step of preprocessing the Hamiltonian comprises:

[0010] determining whether the Pauli terms in the Hamiltonian are partitioned into a plurality of cliques.

[0011] If the Hamiltonian is partitioned, each clique is sorted in ascending order of energy to obtain the clique set.

[0012] If the Hamiltonian is not partitioned, the Pauli terms in the Hamiltonian are modeled as an undirected graph, the undirected graph is partitioned into a plurality of cliques by graph theory, and each clique is sorted in ascending order of energy to obtain the clique set.

[0013] In an embodiment of the present application, the step of modeling the Pauli terms in the Hamiltonian as an undirected graph comprises:

[0014] each Pauli term is regarded as a vertex in the graph.

[0015] an edge is added between the vertices corresponding to two Pauli terms that commute to construct the undirected graph.

[0016] In an embodiment of the present application, the step of generating a sequence of Clifford operations according to the clique set comprises:

[0017] the second-lowest-energy clique in the clique set is obtained, and a first Clifford operation is applied to the second-lowest-energy clique so that only Z-type Pauli operators are contained in the second-lowest-energy clique.

[0018] an evolution matrix is obtained according to the Z-type Pauli operators.

[0019] the evolution matrix is transformed until each row of the evolution matrix has and only has one element 1, and a second Clifford operation used in the transformation process is recorded.

[0020] the sequence of Clifford operations is generated according to the first Clifford operation and the second Clifford operation.

[0021] In an embodiment of the present application, the step of transforming the evolution matrix until each row of the evolution matrix has and only has one element 1 comprises:

[0022] the evolution matrix is traversed by column, and if there is and only is one element 1 in the jth column and the row where the element 1 is located has and only has two elements 1, a new operation is added, and the jth column of the matrix is bitwise XORed to the kth column.

[0023] ​The above steps are repeated until each row of the evolution matrix has and only has one element 1.

[0024] where k and j represent the column where the element 1 is located, The gate represents a CX gate on the kth and jth qubits, k points to the control bit and j points to the target bit.

[0025] In an embodiment of the present application, the step of generating an energy expression containing variable parameters according to the new Hamiltonian includes:

[0026] A measurement axis is selected for each qubit separately.

[0027] The energy expression is generated according to the new Hamiltonian and the measurement axis.

[0028] In an embodiment of the present application, the step of generating the energy expression according to the new Hamiltonian and the measurement axis includes:

[0029] The expression structure and coefficients of the new Hamiltonian are retained.

[0030] The Pauli X matrix on the ath qubit is replaced by , the Pauli Z matrix is replaced by , and the Pauli Y matrix is replaced by to obtain the energy expression.

[0031] where represents the angle between the measurement axis of the ath qubit and the Z axis of the Bloch sphere, represents the angle between the projection of the measurement axis of the ath qubit on the horizontal plane of the Bloch sphere and the X axis, , and n represents the number of qubits.

[0032] In an embodiment of the present application, the step of optimizing the variable parameters in the energy expression to obtain an approximation of the ground state energy of the Hamiltonian includes:

[0033] The variable parameters are optimized by a classical optimizer so that the value of the energy expression is minimized.

[0034] The minimum energy value obtained is taken as an approximation of the ground state energy of the Hamiltonian.

[0035] According to the second aspect of the present application, there is provided a system for approximating the ground state energy of a Hamiltonian, characterized in that the system comprises a preprocessing module, a Clifford evolution module, a measurement module and an optimization module.

[0036] ​​The preprocessing module is configured to obtain and preprocess a Hamiltonian to be solved, and obtain a cluster set corresponding to the Hamiltonian.

[0037] The Clifford evolution module is configured to generate a Clifford operation sequence according to the cluster set, and apply the Clifford operation sequence to the cluster set to obtain a new Hamiltonian.

[0038] The measurement module is configured to generate an energy expression containing a variable parameter according to the new Hamiltonian.

[0039] The optimization module is configured to optimize the variable parameter in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian.

[0040] According to a third aspect of the present application, an electronic device is provided, which comprises a memory, a processor and a computer program stored in the memory, and when the computer program is executed by the processor, any of the above-mentioned approximate solution method of Hamiltonian ground state energy is implemented.

[0041] According to the approximate solution method of Hamiltonian ground state energy, the system and the electronic device provided by the embodiments of the present application, the approximate solution method of Hamiltonian ground state energy of the present application processes the cluster set through the Clifford operation sequence to obtain a new Hamiltonian, generates an energy expression containing a variable parameter, and then optimizes the variable parameter to obtain an approximate value of the ground state energy of the Hamiltonian, thereby improving the calculation efficiency and shortening the calculation time. At the same time, when there are multiple clusters in the Hamiltonian, the approximate effect is improved, so that the error between the approximate value and the exact value is negligible. Moreover, the Clifford operation can be simulated in polynomial time on a classical computer, so the method of the present application can be directly and efficiently implemented by a classical computer. BRIEF DESCRIPTION OF DRAWINGS

[0042] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings needed in the embodiment description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0043] Figure 1 The flowchart of the approximate solution method of Hamiltonian ground state energy provided by an embodiment of the present application is shown in the figure.

[0044] Figure 2 The schematic diagram of the TFIM-tree model provided by an embodiment of the present application is shown in the figure.

[0045] Figure 3A schematic diagram of an XZ model provided by an embodiment of the present application is shown in FIG. 1.

[0046] Figure 4 A structural schematic diagram of a system for approximately solving Hamiltonian ground state energy provided by an embodiment of the present application is shown in FIG. 2.

[0047] Figure 5 A hardware structure block diagram of a computer terminal for a method for approximately solving Hamiltonian ground state energy provided by an embodiment of the present application is shown in FIG. 3. DETAILED DESCRIPTION

[0048] In order to make the objects, technical solutions and advantages of the present application more obvious, the following will describe the example embodiments according to the present application in detail with reference to the drawings. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application, and it should be understood that the present application is not limited to the example embodiments described herein. Based on the embodiments of the present application described in the present application, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of the present application.

[0049] In the following description, a large number of specific details are given in order to provide a more thorough understanding of the present application. However, it should be apparent to those skilled in the art that the present application can be implemented without one or more of these details. In other instances, some well-known features are not described in detail in order to avoid obscuring the present application.

[0050] It should be understood that the present application can be implemented in different forms and should not be interpreted as being limited to the embodiments presented herein. On the contrary, the embodiments are provided to make the disclosure complete and full, and to fully convey the scope of the present application to those skilled in the art.

[0051] In order to thoroughly understand the present application, detailed structures will be presented in the following description in order to explain the technical solutions presented by the present application. The alternative embodiments of the present application are described in detail as follows, however, in addition to these detailed descriptions, the present application can also have other implementation manners.

[0052] The following will describe some embodiments of the present application in detail with reference to the drawings. In the case of no conflict, the following embodiments and features in the embodiments can be combined with each other.

[0053] For the ground state energy solving problem of Hamiltonian, if the exact value is directly calculated, the solving complexity is QMA-hard, which means that even with a quantum computer, it takes super-polynomial time to solve. By approximate solving, the computational complexity can be greatly reduced, and it can be solved in polynomial time. Therefore, the greedy strategy can be applied to constantly select the item with the largest absolute value from all the Pauli items of the Hamiltonian, and ensure that the new item is commutable with the selected set. Finally, a set of stabilizer states is obtained, and the energy of the stabilizer state corresponding to the stabilizer set is obtained. The energy of the stabilizer state is used as an approximate solution of the ground state energy. In some cases, the approximate solution is not much different from the exact value. However, in some cases, the performance of the approximate solving method based on the stabilizer state will deteriorate. For example, in the bond breaking region of chemical reaction, or near the phase transition point of quantum many-body system, the real ground state energy is often dominated by multiple non-commutable Pauli items, which will cause the stabilizer state method to approximate the ground state to deteriorate.

[0054] Therefore, in order to improve the calculation efficiency and reduce the error between the approximate value and the exact value, the first aspect of the present application provides an approximate solving method for the ground state energy of Hamiltonian, as shown in Figure 1 The approximate solving method for the ground state energy of Hamiltonian comprises:

[0055] S1, obtaining and preprocessing the Hamiltonian to be solved to obtain a set of clusters corresponding to the Hamiltonian.

[0056] As an example, the Hamiltonian is composed of a plurality of Pauli items, and all the Pauli items in the Hamiltonian are divided into a plurality of clusters.

[0057] Preferably, the clusters in the cluster set are arranged in ascending order of energy or descending order of energy.

[0058] S2, generating a Clifford operation sequence according to the cluster set, and applying the Clifford operation sequence to the cluster set to obtain a new Hamiltonian.

[0059] As an example, all the Pauli items in the second-lowest-energy cluster in the cluster set can be diagonalized into a Pauli Z matrix, and the Clifford operations performed in the transformation process are recorded to obtain a Clifford operation sequence. The remaining Pauli items in the Hamiltonian are updated through the Clifford operation sequence, and a new Hamiltonian is output.

[0060] S3, generating an energy expression containing a variable parameter according to the new Hamiltonian.

[0061] S4, optimizing the variable parameter in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian.

[0062] As an example, the parameter that minimizes the energy expectation value in the energy expression can be found by a classical optimization algorithm, and the energy expectation value corresponding to the parameter is the approximation of the ground state energy.

[0063] The method for approximating the ground state energy of a Hamiltonian in the embodiments of the present application processes the cluster set through a sequence of Clifford operations to obtain a new Hamiltonian, generates an energy expression containing a variable parameter, and then obtains an approximation of the ground state energy of the Hamiltonian by optimizing the variable parameter, thereby improving the calculation efficiency and shortening the calculation time. Meanwhile, the approximation effect is improved when there are multiple clusters in the Hamiltonian, so that the error between the calculated approximation value and the exact value is negligible. Moreover, the Clifford operation can be simulated in polynomial time on a classical computer, so the method of the present application can be efficiently implemented directly by a classical computer.

[0064] In some embodiments, the step of preprocessing the Hamiltonian comprises:

[0065] S11, determining whether the Pauli terms in the Hamiltonian are divided into multiple clusters.

[0066] It should be noted that a cluster is a concept in graph theory, representing a complete subgraph of an undirected graph.

[0067] S12, if the clusters have been divided, sorting the clusters in ascending order of energy to obtain a cluster set.

[0068] S13, if the clusters have not been divided, modeling the Pauli terms in the Hamiltonian as an undirected graph, dividing the undirected graph into multiple clusters by graph theory, and sorting the clusters in ascending order of energy to obtain a cluster set.

[0069] As an example, given a Hamiltonian composed of multiple Pauli terms on n qubits, if all the Pauli terms are not explicitly divided into several clusters, the Hamiltonian is modeled as an undirected graph, and the graph theory method is used to divide it into s (s≥2) clusters, and the cluster set is denoted as , and each cluster is sorted in ascending order of energy.

[0070] Specifically, the cluster division operation can be: using the minimum cluster cover algorithm in graph theory to divide all the vertices in the graph into s clusters, which means that all the Pauli terms are divided into s clusters.

[0071] In this embodiment, all the Pauli terms are divided into multiple clusters by the graph theory method, providing an optimal starting point for the subsequent steps. At the same time, the automatic division by graph theory replaces manual operation and can adapt to large-scale Pauli terms.

[0072] In some embodiments, the step of modeling the Pauli terms in the Hamiltonian as an undirected graph comprises:

[0073] S131, each Pauli term is regarded as a vertex in the graph.

[0074] S132, an edge is added between the vertices corresponding to the two Pauli terms that commute, and a graph is constructed.

[0075] It should be noted that commutation refers to two Pauli terms P and Q satisfying , that is, .

[0076] As an example, each Pauli term is regarded as a vertex in the graph, and if two Pauli terms commute, an edge is added between the corresponding vertices, so that all Pauli terms in the Hamiltonian are modeled as an undirected graph.

[0077] In this embodiment, the commutation relationship between Pauli terms is converted into an intuitive graph structure, so as to facilitate understanding and subsequent processing.

[0078] In some embodiments, the step of generating a sequence of Clifford operations according to the cluster set comprises:

[0079] S21, obtaining the second-lowest-energy cluster in the cluster set, and applying a first Clifford operation to the second-lowest-energy cluster so that only Z-type Pauli operators are included in the second-lowest-energy cluster.

[0080] It should be noted that the second-lowest-energy cluster in the cluster set is optimal in the present application, for the following reasons: (1) the lowest-energy cluster and the second-lowest-energy cluster are the two clusters that contribute most to the ground state energy; (2) when optimizing the axis angle, the optimizer will naturally take the lowest-energy cluster as the main body, and the second-lowest-energy cluster will be transformed into a Pauli Z operator on a single qubit after Clifford transformation, and the energy of the second-lowest-energy cluster is only related to these single-bit operators, which are separated from the complex coupling with other clusters; (3) the optimizer will naturally prefer to integrate the energy of these single-bit operators that are easy to adjust into the energy optimization of the entire system; thus, while optimizing the energy of the lowest cluster, the energy of the second-lowest cluster can be optimized as much as possible, and the energies of the two clusters that contribute most to the ground state energy are obtained, and finally the ground state energy is more accurately obtained.

[0081] It should also be noted that the Clifford operation specifically includes one or more of the CX operation, the H operation, and the S operation, and the operations in the Clifford operation are commonly used operations in quantum computing.

[0082] As an example, if the cluster contains or , then by means of the H operation, the is changed to , is changed to If the group includes or Then, through the H operation and the S operation, we can make Become , Become If the group includes or If so, the Clifford operation will not take effect.

[0083] S22, the evolution matrix is ​​obtained according to the Z-type Pauli operator.

[0084] S23, transform the evolution matrix until each row of the evolution matrix has exactly one element 1, and record the second Clifford operation used in the transformation process.

[0085] S24, Generate the Clifford operation sequence based on the first Clifford operation and the second Clifford operation.

[0086] As an example, for the second lowest energy cluster Appropriate Clifford operation sequence ,Will The Pauli term is diagonalized into or a subset thereof.

[0087] in, (1≤a≤n) is The abbreviation indicates that the a-th qubit is a Pauli Z matrix, and the remaining qubits are Pauli I matrices (i.e., second-order identity matrices). The symbol is... The Pauli Z matrix represents the stratigraphic operation of matrices. Clifford Operations (1≤d≤m) works by transforming the Pauli term P into a new Pauli term. ,symbol Represents matrix multiplication, symbol This represents the conjugate transpose operation of a matrix.

[0088] Next, the Clifford operations in sequence CL can be applied sequentially to the remaining Pauli terms of the Hamiltonian to obtain the new Hamiltonian after Clifford evolution.

[0089] In this embodiment, by using the characteristics of Clifford evolution that does not change the energy spectrum of the Hamiltonian, applying appropriate Clifford evolution on the Hamiltonian, and combining the subsequent adaptive strategy of single qubit measurement, the dominant terms in the Hamiltonian are eventually covered, and the approximate value closer to the exact value is obtained, effectively solving the problem of poor approximation effect in the case of multiple dominant groups.

[0090] In some embodiments, the step of transforming the evolution matrix until each row of the evolution matrix has and only has one element 1 comprises:

[0091] S221, traversing the evolution matrix by column, and if there is and only one element 1 in the jth column and the row where the element 1 is located has and only two elements 1, then adding operation, and performing bitwise XOR between the jth column and the kth column of the matrix.

[0092] S222, repeating step S221 until each row of the evolution matrix has and only has one element 1.

[0093] wherein k and j represent the columns where the element 1 is located, the gate represents the CX gate on the kth and jth qubits, k points to the control bit, and j points to the target bit.

[0094] Next, take a group containing only or (1≤a,b≤n), or only or , or only or , and only containing generators as an example. Among them, , , , , The definitions are similar to , which will not be repeated here, wherein the specific definition of the Pauli X matrix is , and the specific definition of the Pauli Y matrix is . The specific steps are as follows:

[0095] A1, initialize an empty Clifford operation sequence CL.

[0096] A2, if the group contains only or , then add H operation to the sequence CL, and apply H operation to all Pauli terms in the group, so that becomes , becomes , and after the operation, the group contains only or ; if the cluster only contains or , then the H operation and the S operation are added to the sequence CL in turn, and the H operation and the S operation are applied to all the Pauli terms in the cluster in turn, so that becomes , becomes , and after the operations, the cluster only contains or . Finally, the Z-type Pauli operator containing only or is obtained.

[0097] A3, according to the Z-type Pauli operator, the evolution matrix is obtained. Each column corresponds to a quantum bit, and each row corresponds to a Pauli term in the cluster. If the jth element of the ith Pauli term is the Pauli Z matrix, then the jth column of the ith row of is 1, otherwise it is 0, and can also reflect the Clifford evolution result of the Pauli term in the cluster.

[0098] A4, traverse the evolution matrix by column. If the element 1 in the jth column appears only once, and the row only contains two elements 1, and the other element 1 is in the kth column, then the operation is added to the sequence CL, the control bit is the kth quantum bit, and the target bit is the jth quantum bit, and the jth column of the evolution matrix is XORed to the kth column, and is updated, which also means that the Clifford evolution result of the Pauli term in the cluster is updated.

[0099] A5, repeat step A4 until each row of the evolution matrix only has one element 1, and finally obtain the complete Clifford operation sequence CL. The sequence CL is applied to the Pauli terms in the cluster in turn, which can diagonalize them to or a subset of , and the final form of reflects the diagonalization result.

[0100] In some embodiments, the step of generating an energy expression containing a variable parameter according to the new Hamiltonian includes:

[0101] S31, select a measurement axis for each quantum bit.

[0102] S32, generate an energy expression according to the new Hamiltonian and the measurement axis.

[0103] In this embodiment, by performing appropriate Clifford evolution on the Hamiltonian, then selecting the most appropriate measurement axis for each qubit and generating an energy expression, the subsequent approximate solution of the ground state energy provides a premise, so that the subsequent solution can get a result closer to the exact value, solving the problem of poor approximation effect in the case of multiple groups jointly dominating the energy.

[0104] In some embodiments, according to the new Hamiltonian and the measurement axis, the step of generating an energy expression comprises:

[0105] S321, retaining the expression structure and coefficients of the new Hamiltonian.

[0106] S322, replacing the Pauli X matrix on the a-th qubit with , replacing the Pauli Z matrix with , and replacing the Pauli Y matrix with , to obtain an energy expression .

[0107] wherein, denotes the angle between the measurement axis of the a-th qubit and the Z axis of the Bloch sphere, denotes the angle between the projection of the measurement axis of the a-th qubit on the horizontal plane of the Bloch sphere and the X axis, , , , and n denotes the number of qubits.

[0108] It is worth mentioning that if the Hamiltonian only contains Pauli X matrix and Pauli Z matrix, then .

[0109] In this embodiment, by replacing the Pauli matrix with a trigonometric function value containing a variable parameter, the quantum energy calculation is converted into a classical parameter optimization problem, reducing the complexity of energy calculation.

[0110] In some embodiments, the step of optimizing the variable parameters in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian comprises:

[0111] S41, optimizing the variable parameters by a classical optimizer, so that the value of the energy expression is minimized.

[0112] S42, taking the obtained minimum energy value as an approximate value of the ground state energy of the Hamiltonian.

[0113] Next, the present application takes the transverse field Ising model (TFIM-tree for short) on the tree graph as an example, and the general form of the Hamiltonian of the TFIM-tree is:

[0114]

[0115] where J is the coupling strength, denotes the edge between adjacent nodes a and b in the tree graph, is the magnetic field strength.

[0116] Specifically, for example Figure 2 , the corresponding Hamiltonian is:

[0117]

[0118] Let the coupling strength J = 1, and the magnetic field strength .

[0119] The specific process of applying the method of the present application to approximate the ground state energy of the Hamiltonian is as follows:

[0120] B1, all the Pauli terms in the Hamiltonian have been explicitly divided into two groups, and and , respectively, and each group is arranged in ascending order of energy.

[0121] B2, apply an appropriate sequence of Clifford operations CL to the group with the second lowest energy, diagonalize the Pauli terms in to or a subset thereof, the specific steps are as follows:

[0122] B21, initialize an empty sequence of Clifford operations CL.

[0123] B22, initialize the evolution matrix :

[0124]

[0125] B23, traverse by column, the element 1 in the 4th column appears only once, and the 3rd row where it is located contains only two elements 1, the other element 1 is in the 2nd column, add operation to the sequence CL, the control bit is the 2nd qubit, the target bit is the 4th qubit, at the same time, XOR the 4th column of to the 2nd column, update :

[0126]

[0127] The change in the 3rd row also means that the 3rd Pauli term in changes from to .

[0128] B24, repeat step B23 until... Each row contains only one element, 1, thus yielding the complete Clifford operation sequence. ,sequence Acting in sequence Diagonalize the Pauli term in the equation to... or a subset thereof, Finally, it becomes:

[0129]

[0130] This indicates that after all Clifford operations in sequence CL, The Pauli term in the equation is diagonalized to .

[0131] B3, apply the Clifford operations in sequence CL sequentially to the remaining Pauli terms of the Hamiltonian, i.e. For all Pauli terms, a similar evolutionary matrix can be used. The method for initializing the evolution matrix :

[0132]

[0133] Each The operation will The k-th column is XORed with the j-th column. After performing all Clifford operations in sequence CL, It becomes:

[0134]

[0135] express Pauli term becomes .

[0136] Therefore, the original Hamiltonian evolves into a new Hamiltonian after passing through the CL sequence:

[0137]

[0138] B4, by selecting a measurement axis for each qubit, yields an energy expression containing variable parameters:

[0139]

[0140] B5, optimizing parameters using a classic optimizer. For example, you can use the optimizer in the Python third-party library scipy to calculate the energy. The minimum value of the above is about -11.5046 (retaining four decimal places), while the exact solution is -11.5366 (retaining four decimal places).

[0141] To facilitate the accuracy of the calculation results of the present application, the results of the existing method for solving embodiment 1 are given. For the original Hamiltonian , the related art (an approximation technique based on a stable state) will continuously select the Pauli term with the largest absolute value of the coefficient, and ensure that it is pairwise commutable with the selected Pauli term set. Therefore, the method takes as the only one group, assigns all the generators in it to 1, and obtains the approximate energy Obviously, this result is far from the result -11.5046 calculated by the method in the present application, which is close to the exact solution -11.5366.

[0142] Next, the present application takes the XZ model on the tree graph as an example, and the general form of its Hamiltonian is:

[0143]

[0144] wherein, represents the edge between adjacent nodes a and b in the tree graph, is a coupling constant.

[0145] Specifically, for example Figure 3 XZ model, the corresponding Hamiltonian is:

[0146]

[0147] Let the coupling constant .

[0148] The specific process of applying the method of the present application to approximate the ground state energy of the Hamiltonian is as follows:

[0149] C1, all Pauli terms in the Hamiltonian have been explicitly divided into two groups, namely and , and each group is arranged in ascending order of energy.

[0150] C2, apply an appropriate sequence of Clifford operations CL to the group with the second lowest energy , diagonalize the Pauli terms in to or a subset thereof, and the specific steps are as follows:

[0151] C21, initialize an empty sequence of Clifford operations CL.

[0152] C22, contains only , then a H operation is added in sequence CL and the H operation is applied to all Pauli terms in , so that becomes .

[0153] C23, according to the above embodiment, the complete Clifford operation sequence is , which is applied to the Pauli terms in in turn, and can diagonalize it into .

[0154] C3, the Clifford operation in sequence CL is applied to the remaining Pauli terms of the Hamiltonian in turn, that is, all the Pauli terms in , which becomes after the first H operation, and then the evolution matrix is initialized by using a method similar to the evolution matrix :

[0155]

[0156] Each operation will bitwise XOR the kth column of the check X matrix to the jth column. After applying the remaining Clifford operations in sequence CL in turn, becomes:

[0157]

[0158] , which means that the Pauli term in becomes .

[0159] Therefore, the original Hamiltonian evolves into a new Hamiltonian after sequence CL:

[0160]

[0161] C4, a measurement axis is selected for each quantum bit, and an energy expression containing a variable parameter is obtained:

[0162]

[0163] , where .

[0164] C5, the parameter is optimized by using a classical optimizer, for example, the optimizer optimize in the third-party library scipy in Python, and the energy -6.3343 (retaining four decimal places), while the exact solution is -6.3396 (retaining four decimal places). Similarly, by using the related technology (an approximation technology based on stable substate), the minimum value is -6, which is obviously far from the result -6.3343 calculated by the method of the present application, and is far from the exact solution -6.3396.

[0165] In addition, the present application also provides an approximate solution system for solving Hamiltonian ground state energy, such as Figure 4 As shown in the figure, the approximate solution system for solving Hamiltonian ground state energy comprises a preprocessing module 10, a Clifford evolution module 20, a measurement module 30 and an optimization module 40.

[0166] The preprocessing module 10 is used to obtain and preprocess the Hamiltonian to be solved, so as to obtain a cluster set corresponding to the Hamiltonian.

[0167] The Clifford evolution module 20 is used to generate a Clifford operation sequence according to the cluster set, and apply the Clifford operation sequence to the cluster set to obtain a new Hamiltonian.

[0168] The measurement module 30 is used to generate an energy expression containing variable parameters according to the new Hamiltonian.

[0169] The optimization module 40 is used to optimize the variable parameters in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian.

[0170] The other specific embodiments of the approximate solution system for solving Hamiltonian ground state energy of the embodiments of the present application can refer to the specific embodiments of the approximate solution method for solving Hamiltonian ground state energy of the embodiments of the present application.

[0171] The approximate solution system for solving Hamiltonian ground state energy of the embodiments of the present application processes the cluster set by the Clifford operation sequence to obtain a new Hamiltonian, generates an energy expression containing variable parameters, and then optimizes the variable parameters to obtain an approximate value of the ground state energy of the Hamiltonian, thereby improving the calculation efficiency and shortening the calculation time. At the same time, when there are multiple cluster dominations in the Hamiltonian, the approximation effect is improved, so that the error between the calculated approximate value and the exact value is negligible. Moreover, the Clifford operation can be simulated in polynomial time on a classical computer, so the system of the present application can be directly and efficiently realized by a classical computer.

[0172] The following will be described in detail by taking a computer terminal as an example. Figure 5 A hardware structure block diagram of a computer terminal provided by the approximate solution method for solving Hamiltonian ground state energy of the embodiments of the present application is shown in the figure. Figure 5 As shown in the figure, the computer terminal can include one or more Figure 5Only one is shown in the diagram. A processor 501 (processor 501 may include, but is not limited to, a microprocessor MCU or a programmable logic device FPGA, etc.) and a memory 502 for storing data are also shown. Optionally, the computer terminal may further include a transmission device 503 for communication functions and an input / output device 504. Those skilled in the art will understand that... Figure 5 The structure shown is for illustrative purposes only and does not limit the structure of the computer terminal described above. For example, the computer terminal may also include components that are more complex than those described above. Figure 5 The more or fewer components shown, or having the same Figure 5 The different configurations shown.

[0173] The memory 502 can be used to store software programs and modules for application software, such as the program instructions / modules corresponding to the approximate solution of the Hamiltonian ground state energy method in this embodiment. The processor 501 executes various functional applications and data processing by running the software programs and modules stored in the memory 502, thereby implementing the above-described method. The memory 502 may include high-speed random access memory, and may also include non-volatile memory, such as one or more magnetic storage devices, flash memory, or other non-volatile solid-state memory. In some instances, the memory 502 may further include memory remotely located relative to the processor 501, and these remote memories can be connected to a computer terminal via a network. Examples of such networks include, but are not limited to, the Internet, corporate intranets, local area networks, mobile communication networks, and combinations thereof.

[0174] The transmission device 503 is used to receive or send data via a network. Specific examples of the network described above may include a wireless network provided by a communication provider of the computer terminal. In one example, the transmission device 503 includes a Network Interface Controller (NIC), which can connect to other network devices via a base station to communicate with the Internet. In another example, the transmission device 503 may be a Radio Frequency (RF) module, used for wireless communication with the Internet. Embodiments of this application also provide a computer-readable storage medium storing a computer program for electronic data interchange, which causes a computer to perform some or all of the steps of any of the methods described in the above method embodiments, wherein the computer includes an electronic device.

[0175] The embodiment of the application further provides a computer program product, the computer program product comprising a non-transitory computer-readable storage medium storing a computer program, the computer program being operable to cause a computer to execute some or all of the steps of any of the methods described in the above method embodiments. The computer program product can be a software installation package, and the computer comprises an electronic device.

[0176] Although the example embodiments have been described herein with reference to the accompanying drawings, it is to be understood that the example embodiments are only exemplary and are not intended to limit the scope of the application. Those of ordinary skill in the art can make various changes and modifications without departing from the scope and spirit of the application. All such changes and modifications are intended to be included within the scope of the application as claimed in the appended claims.

[0177] Those of ordinary skill in the art can be aware that, in combination with the examples described in the embodiments disclosed in the present application, units and algorithm steps can be implemented by electronic hardware or a combination of computer software and electronic hardware. Whether the functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Those of ordinary skill in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered to be beyond the scope of the present application.

[0178] In several embodiments provided in the present application, it should be understood that the disclosed devices and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative, for example, the division of the units is only a logical function division, and actual implementation can have another division manner, for example, a plurality of units or components can be combined or integrated into another device, or some features can be omitted or not executed.

[0179] In the specification provided herein, a large number of specific details are described. However, it can be understood that the embodiments of the application can be practiced without these specific details. In some examples, well-known methods, structures and techniques are not described in detail in order not to obscure the understanding of the specification.

[0180] Similarly, it is to be understood that the features of the present application that are of a proprietary nature are set forth in the appended set of claims. If specific embodiments are described in detail in the specification, it is understood that the features of the application further comprise any combination of the features described in the specification including any single feature or sub-combination of features unless expressly identified otherwise. In addition, it is to be understood that features described herein are to be interpreted broadly and are not limited to the specific embodiments described in the specification. For example, the features of the application described in the specification can be used in any combination with the features of the application described in the claims. Similarly, it is to be understood that, in order to obviate obscuring the application, in the description of the example embodiments of the application, various features of the application are sometimes grouped together in a single embodiment, figure, or description of related features. However, the method of the application should not be construed to reflect an intention that the application requires more features than are explicitly recited in each claim. Rather, it is the intention that the application lies in less, where less means less than all of the features of any disclosed single embodiment. Accordingly, the claims that follow, which reflect the application as presently contemplated, are hereby expressly incorporated into this detailed description, with each claim acting as a separate embodiment of this application.

[0181] Those skilled in the art will appreciate that all features described in this specification (including the summaries of the application and the abstract) and / or annexed drawings can be taken in any combination with each other. Each feature disclosed in this specification (including the claims, abstract, and drawings) can be replaced by alternative features serving the same, equivalent or a similar purpose, unless expressly stated otherwise.

[0182] Furthermore, those skilled in the art will recognize that references in the specification to "one embodiment", "an embodiment", "an example embodiment", etc., mean that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the application. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment, nor are they necessarily referring to a single embodiment. Furthermore, the terms "comprises", "comprising", "includes", "including", "has", "having" and the like are to be open ended. For a term A comprising an element B means that an A can further include additional elements B or other elements not B. In other words, A can include other elements other than or in addition to B. In addition, the term "comprising" is used throughout the specification to mean including, but not limited to.

[0183] Embodiments of the application can be implemented in hardware, or as software modules running in one or more processors, or in combinations thereof. Skilled persons will appreciate that a microprocessor or digital signal processor (DSP) can be used in practice to implement some or all of the functionality of some of the modules according to embodiments of the application. The application can also be implemented as a program for executing part or all of the methods described herein on a computer program (for example, a computer program and a computer program product). Such a program implementing the application can be stored on a computer readable medium, or can be in the form of one or more signals. Such signals can be downloaded from an Internet website, or provided on a carrier medium, or in any other form.

[0184] It should be noted that the above-mentioned embodiments illustrate rather than limit the application, and that one skilled in the art will be able to design many alternative embodiments without departing from the scope of the appended claims. In the claims, any reference signs placed between parentheses shall not be construed as limiting the claim. The word "comprising" does not exclude the presence of elements or steps other than those listed in a claim. The word "a" or "an" preceding an element does not exclude the presence of a plurality of such elements. The application can be implemented by means of both hardware and software, and any combination thereof. In a unitary claim, several devices or sub-claims can be joined by means of the expression "and / or". The use of the term "at least" followed by a list of one or more items should be interpreted as including at least one of the items but it does not exclude the presence of others not listed. The use of the term "one" followed by a list of one or more items should be interpreted as including at least one of the items but it does not exclude the presence of others not listed. It is emphasized that the terms "comprises / comprising" when used in this specification are taken to specify the presence of stated features, integers, steps or components but do not preclude the presence or addition of one or more other features, integers, steps, components or groups thereof.

[0185] The above description is only specific embodiments of the present application or specific explanations of specific embodiments, and the protection scope of the present application is not limited thereto. Any person skilled in the art can easily think of changes or replacements within the technical range disclosed in the present application, and all of them should be covered in the protection scope of the present application. The protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for approximately solving ground state energy of a Hamiltonian, characterized in that, The method comprises: acquiring and preprocessing a Hamiltonian to be solved to obtain a cluster set corresponding to the Hamiltonian; generating a Clifford operation sequence according to the cluster set, and applying the Clifford operation sequence to the cluster set to obtain a new Hamiltonian; generating an energy expression containing a variable parameter according to the new Hamiltonian; optimizing the variable parameter in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian; wherein the step of generating a Clifford operation sequence according to the cluster set comprises: acquiring a cluster with the second lowest energy in the cluster set, and applying a first Clifford operation to the cluster with the second lowest energy so that only Z-type Pauli operators are contained in the cluster with the second lowest energy; obtaining an evolution matrix according to the Z-type Pauli operators; transforming the evolution matrix until each row of the evolution matrix has and only has one element 1, and recording the second Clifford operation used in the transformation process; generating the Clifford operation sequence according to the first Clifford operation and the second Clifford operation.

2. The method for approximating the ground state energy of a Hamiltonian according to claim 1, wherein, The Hamiltonian is composed of multiple Pauli terms, and the step of preprocessing the Hamiltonian comprises: determining whether the Pauli terms in the Hamiltonian are divided into multiple clusters; if the Hamiltonian has been divided, sorting each cluster according to energy from low to high to obtain the cluster set; if the Hamiltonian has not been divided, modeling the Pauli terms in the Hamiltonian as an undirected graph, dividing the undirected graph into multiple clusters through graph theory, and sorting each cluster according to energy from low to high to obtain the cluster set.

3. The method for approximately solving ground state energy of Hamiltonian according to claim 2, characterized in that, The step of modeling the Pauli terms in the Hamiltonian as an undirected graph comprises: regarding each Pauli term as a vertex in the graph; adding an edge between the vertices corresponding to two Pauli terms that commute to construct the undirected graph.

4. The method for approximating the ground state energy of a Hamiltonian according to claim 1, wherein, The step of transforming the evolution matrix until each row of the evolution matrix has and only has one element 1 comprises: traversing the evolution matrix by column, and in the traversal process, if there is and only is one element 1 in the jth column and the row where the element 1 is located has and only has two elements 1, then a CX(k, j) operation is added, and the jth column of the matrix is XORed to the kth column; repeating the above steps until each row of the evolution matrix has and only has one element 1; where k and j represent the column in which element 1 is located, CX kj The gate represents a CX gate on the kth and jth qubits, with k pointing to the control bit and j pointing to the target bit.

5. The method for approximating the ground state energy of a Hamiltonian according to claim 1, wherein, The step of generating an energy expression containing a variable parameter according to the new Hamiltonian comprises: selecting a measurement axis for each qubit separately; generating the energy expression according to the new Hamiltonian and the measurement axis.

6. The method for approximating the ground state energy of a Hamiltonian according to claim 5, wherein, The step of generating the energy expression according to the new Hamiltonian and the measurement axis comprises: retaining the expression structure and coefficients of the new Hamiltonian; replacing the Pauli X matrix on the a-th qubit by replacing the Pauli Z matrix by cos(θ a ), and the Pauli Y matrix by yields the energy expression; where θ a represents the angle between the measurement axis of the a-th qubit and the Z axis of the Bloch sphere, represents the angle between the projection of the measurement axis of the a-th qubit on the horizontal plane of the Bloch sphere and the X axis, 1≤a≤n, 0≤θ a ≤2π, n represents the number of qubits.

7. The method for approximately solving ground state energy of Hamiltonian according to claim 1, characterized in that, The step of optimizing the variable parameter in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian comprises: optimizing the variable parameter by a classical optimizer so that the value of the energy expression is minimized; taking the minimum energy value obtained as the approximate value of the ground state energy of the Hamiltonian.

8. A system for approximately solving ground state energy of a Hamiltonian, the system comprising: The system comprises: A preprocessing module configured to obtain and preprocess a Hamiltonian to be solved, to obtain a cluster set corresponding to the Hamiltonian; A Clifford evolution module configured to generate a Clifford operation sequence according to the cluster set, and apply the Clifford operation sequence to the cluster set to obtain a new Hamiltonian; A measurement module configured to generate an energy expression containing a variable parameter according to the new Hamiltonian; An optimization module configured to optimize the variable parameter in the energy expression to obtain an approximate value of a ground state energy of the Hamiltonian; In the process of generating the Clifford operation sequence according to the cluster set, the Clifford evolution module is configured to: obtain a cluster with the second lowest energy in the cluster set, and apply a first Clifford operation to the cluster with the second lowest energy, so that the cluster with the second lowest energy only contains Z-type Pauli operators; obtain an evolution matrix according to the Z-type Pauli operators; transform the evolution matrix until each row of the evolution matrix has and only has one element 1, and record the second Clifford operation used in the transformation process; generate the Clifford operation sequence according to the first Clifford operation and the second Clifford operation.

9. An electronic device comprising a memory, a processor, and a computer program stored on the memory, characterized in that, The computer program is executed by the processor to implement the method for approximately solving the ground state energy of the Hamiltonian according to any one of claims 1-7.

Images