Method and system for approximately solving Hamiltonian ground state energy and electronic equipment

Abstract

The invention discloses a method and system for approximately solving Hamiltonian ground state energy and electronic equipment, and the method comprises the steps: obtaining and preprocessing a to-be-solved Hamiltonian, and obtaining a cluster set corresponding to the Hamiltonian; according to the cluster set, generating a Clifford operation sequence, and acting the Clifford operation sequence on the cluster set to obtain a new Hamiltonian; according to the new Hamiltonian, an energy expression containing variable parameters is generated; and optimizing variable parameters in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian quantity. According to the method, the cluster set is processed through the Clifford operation sequence, the new Hamiltonian is obtained, the energy expression containing the variable parameters is generated, and the approximate value of the ground state energy of the Hamiltonian is obtained by optimizing the variable parameters, so that the calculation efficiency is improved, the calculation time is shortened, and meanwhile, when a plurality of cluster dominants exist in the Hamiltonian, the calculation efficiency of the Hamiltonian is improved. And the approximation effect is improved, so that the error between the calculated approximate value and the accurate value can be ignored.

Description

Technical Field

[0001] This invention relates to the field of quantum technology, and in particular to an approximate method for solving the ground state energy of a Hamiltonian, a system for approximate solving the ground state energy of a Hamiltonian, and an electronic device. Background Technology

[0002] In the field of quantum computing, the problem of solving the ground state of the Hamiltonian is a core task for determining the minimum energy of a quantum system and its corresponding quantum state, and it permeates key scenarios such as quantum chemical simulation, materials design, combinatorial optimization, and quantum error correction. However, related research shows that even if the Hamiltonian is constrained to be a k-local Hamiltonian (k≥2, i.e., each Pauli term in the Hamiltonian contains at most k Pauli matrices that are not identity matrices), its ground state solution complexity is still QMA-hard, which means that even with a quantum computer, it would require superpolynomial time to solve.

[0003] This dilemma has prompted engineering practices to shift towards approximate solutions. One current approximate solution method utilizes stable substates to approximate the ground state. Stable substates are a special class of quantum states generated by mutually commuting Pauli operators (or stabilizers). Stable substates are uniformly distributed throughout Hilbert space, and at a specific distance scale, there must exist a stable substate that best approximates the true ground state. However, in some special cases, the performance of approximate solutions based on stable substates deteriorates. For example, in bond-breaking regions of chemical reactions, or near phase transition points in quantum many-body systems, the true ground state energy is often dominated by multiple non-commuting Pauli terms, leading to a decline in the effectiveness of stable substate methods that approximate the ground state using a single cluster. Summary of the Invention

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

[0005] Obtain and preprocess the Hamiltonian to be solved to obtain the clique set corresponding to the Hamiltonian.

[0006] Based on the clique set, a Clifford operation sequence is generated, and the Clifford operation sequence is applied to the clique set to obtain a new Hamiltonian.

[0007] Based on the new Hamiltonian, an energy expression containing variable parameters is generated.

[0008] By optimizing the variable parameters in the energy expression, an approximate value of the ground-state energy of the Hamiltonian can be obtained.

[0009] In one embodiment of the present invention, the Hamiltonian consists of multiple Pauli terms, and the step of preprocessing the Hamiltonian includes:

[0010] Determine whether the Pauli term in the Hamiltonian is divided into multiple groups.

[0011] If the clusters have already been divided, then the clusters are sorted from low to high energy to obtain the cluster set.

[0012] If the graph has not yet been partitioned, the Pauli term in the Hamiltonian is modeled as an undirected graph. Using graph theory, the undirected graph is partitioned into multiple cliques, and the cliques are sorted from low to high energy to obtain the clique set.

[0013] In one embodiment of the present invention, the step of modeling the Pauli term in the Hamiltonian as an undirected graph includes:

[0014] Each Pauli term is considered as a vertex in the graph.

[0015] An edge is added between the vertices corresponding to the two Pauli terms in the commutation to construct the undirected graph.

[0016] In one embodiment of the present invention, the step of generating a sequence of Clifford operations based on the clique set includes:

[0017] Obtain the second lowest energy clique from the clique set, and apply a first Clifford operation to the second lowest energy clique such that the second lowest energy clique contains only Z-type Pauli operators.

[0018] The evolution matrix is ​​obtained based on the Z-type Pauli operator.

[0019] 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.

[0020] The Clifford operation sequence is generated based on the first Clifford operation and the second Clifford operation.

[0021] In one embodiment of the present invention, the step of transforming the evolution matrix until each row of the evolution matrix has exactly one element 1 includes:

[0022] Traverse the evolution matrix column by column. During the traversal, if there is exactly one element 1 in the j-th column and exactly two elements 1 in the row containing element 1, then add a new element. The operation is performed, and the j-th column of the matrix is ​​bitwise XORed with the k-th column.

[0023] Repeat the above steps until each row of the evolution matrix has exactly one element 1.

[0024] Where k and j represent the column containing element 1, The gate represents the CX gate on the k-th and j-th qubits, where k points to the control bit and j points to the target bit.

[0025] In one embodiment of the present invention, the step of generating an energy expression containing variable parameters based on the new Hamiltonian includes:

[0026] Choose a separate measurement axis for each qubit.

[0027] The energy expression is generated based on the new Hamiltonian and the measurement axis.

[0028] In one embodiment of the present invention, the step of generating the energy expression based on the new Hamiltonian and the measurement axis includes:

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

[0030] Replace the Pauli X matrix on the a-th qubit with Pauli Z matrix replaced with Pauli Y matrix replaced with Thus, the energy expression is obtained.

[0031] in, Let represent the angle between the measurement axis of the a-th qubit and the Z-axis of the Bloch sphere. Let X represent the angle between the projection of the measurement axis of the a-th qubit onto the horizontal plane of the Bloch sphere and the X-axis. , , , where n represents the number of qubits.

[0032] In one embodiment of the present invention, the step of optimizing the variable parameters in the energy expression to obtain an approximate value of the ground-state energy of the Hamiltonian includes:

[0033] The variable parameters are optimized using a classical optimizer to minimize the value of the energy expression.

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

[0035] According to a second aspect of the present invention, an approximate solution system for the Hamiltonian ground state energy is provided, 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 used to obtain and preprocess the Hamiltonian to be solved, and obtain the clique set corresponding to the Hamiltonian.

[0037] The Clifford evolution module is used to generate a Clifford operation sequence based on the clique set, and to apply the Clifford operation sequence to the clique set to obtain a new Hamiltonian.

[0038] The measurement module is used to generate an energy expression containing variable parameters based on the new Hamiltonian.

[0039] The optimization module is used to optimize the variable parameters 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 invention, an electronic device is provided, including a memory, a processor, and a computer program stored in the memory, wherein when the computer program is executed by the processor, it implements any of the above-described methods for approximating the ground state energy of the Hamiltonian.

[0041] According to embodiments of the present invention, an approximate solution method, system, and electronic device for the ground-state energy of a Hamiltonian are provided. The method of the present invention processes a set of cliques using a sequence of Clifford operations to obtain a new Hamiltonian and generates an energy expression containing variable parameters. Then, by optimizing the variable parameters, an approximate value of the ground-state energy of the Hamiltonian is obtained, improving computational efficiency and shortening computation time. Furthermore, it improves the approximation effect even when multiple cliques dominate the Hamiltonian, making the error between the calculated approximate value and the exact value negligible. Moreover, Clifford operations can be simulated in polynomial time on a classical computer; therefore, the method of the present invention can be directly and efficiently implemented by a classical computer. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart illustrating an approximate method for solving the Hamiltonian ground state energy according to an embodiment of the present invention.

[0044] Figure 2 This is a schematic diagram of a TFIM-tree model provided in an embodiment of the present invention;

[0045] Figure 3A schematic diagram of an XZ model provided in an embodiment of the present invention;

[0046] Figure 4 This is a schematic diagram of the structure of an approximate solution for the Hamiltonian ground state energy system provided in an embodiment of the present invention;

[0047] Figure 5 This is a hardware structure block diagram of a computer terminal for an approximate solution of the Hamiltonian ground state energy method provided in an embodiment of the present invention. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of the present invention more apparent, exemplary embodiments according to the present invention will be described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are merely a part of the embodiments of the present invention, and not all of the embodiments of the present invention. It should be understood that the present invention is not limited to the exemplary embodiments described herein. Based on the embodiments of the present invention described herein, all other embodiments obtained by those skilled in the art without inventive effort should fall within the protection scope of the present invention.

[0049] In the following description, numerous specific details are set forth in order to provide a more thorough understanding of the invention. However, it will be apparent to those skilled in the art that the invention can be practiced without one or more of these details. In other instances, certain technical features well-known in the art have not been described in order to avoid obscuring the invention.

[0050] It should be understood that the invention can be embodied in various forms and should not be construed as being limited to the embodiments set forth herein. Rather, providing these embodiments will make the disclosure thorough and complete, and will fully convey the scope of the invention to those skilled in the art.

[0051] To fully understand this invention, a detailed structure will be presented in the following description to illustrate the technical solution proposed by this invention. Optional embodiments of the invention are described in detail below; however, in addition to these detailed descriptions, the invention may have other embodiments.

[0052] The following detailed description of some embodiments of the present invention is provided in conjunction with the accompanying drawings. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0053] For the problem of finding the ground state energy of the Hamiltonian, directly calculating its exact value results in a QMA-hard problem, meaning that even with a quantum computer, it would require superpolynomial time to solve. However, by using approximation, the computational complexity can be significantly reduced, achieving a solvable solution in polynomial time. Therefore, a greedy strategy can be applied: continuously selecting the term with the largest absolute value of its coefficient from all Pauli terms of the Hamiltonian, ensuring that new terms commutate pairwise with the selected set, eventually yielding a stable subset. The energy of the stable substates corresponding to this stable subset can then be obtained, serving as an approximate solution to the ground state energy. In some cases, the approximate solution is not significantly different from the exact value. However, in situations where multiple clusters jointly dominate the energy, the performance of the approximation method based on stable substates deteriorates. For example, in bond-breaking regions of chemical reactions, or near phase transition points in quantum many-body systems, the true ground state energy is often dominated by multiple non-commutative Pauli terms, leading to a decline in the effectiveness of the stable substate method approximating the ground state using a single cluster.

[0054] Therefore, in order to improve computational efficiency and reduce the error between approximate and exact values, the first aspect of this invention provides a method for approximating the ground-state energy of the Hamiltonian, such as... Figure 1 As shown, the approximate methods for solving the Hamiltonian ground state energy include:

[0055] S1: Obtain and preprocess the Hamiltonian to be solved to obtain the clique set corresponding to the Hamiltonian.

[0056] As an example, the Hamiltonian consists of multiple Pauli terms, and all the Pauli terms in the Hamiltonian are divided into several cliques.

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

[0058] S2. Generate a sequence of Clifford operations based on the clique set, and apply the sequence of Clifford operations to the clique set to obtain a new Hamiltonian.

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

[0060] S3, based on the new Hamiltonian, generates an energy expression containing variable parameters.

[0061] S4. Optimize the variable parameters in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian.

[0062] As an example, classical optimization algorithms can be used to find the parameter that minimizes the expected energy value in the energy expression. In this case, the expected energy value corresponding to the parameter is an approximation of the ground state energy.

[0063] The approximate solution for the ground-state energy of the Hamiltonian in this invention involves processing the clique set using a sequence of Clifford operations to obtain a new Hamiltonian and generating an energy expression with variable parameters. By optimizing these variable parameters, an approximate value for the ground-state energy of the Hamiltonian is obtained, improving computational efficiency and shortening computation time. Furthermore, it enhances the approximation effect even when multiple cliques dominate the Hamiltonian, making the error between the calculated approximation and the exact value negligible. Moreover, Clifford operations can be simulated in polynomial time on a classical computer; therefore, the method of this invention can be efficiently implemented directly on a classical computer.

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

[0065] S11, determine whether the Pauli term in the Hamiltonian is divided into multiple groups.

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

[0067] S12, if the clusters have already been divided, sort the clusters by energy from low to high to obtain the cluster set.

[0068] S13. If the graph has not yet been partitioned, the Pauli term in the Hamiltonian is modeled as an undirected graph. Using graph theory, the undirected graph is partitioned into multiple cliques, and the cliques are sorted from low to high energy to obtain a clique set.

[0069] As an example, given a Hamiltonian consisting of multiple Pauli terms on n qubits, if all Pauli terms are not explicitly divided into several cliques, then we can model it as an undirected graph and use graph theory methods to divide it into s (s≥2) cliques, the set of cliques denoted as . The groups are arranged in ascending order of energy.

[0070] Specifically, the clique partitioning operation can be performed by using the minimum clique cover algorithm in graph theory to divide all vertices in the graph into s cliques, which means dividing all Pauli terms into s cliques.

[0071] In this embodiment, graph theory is used to divide all Pauli terms into multiple cliques, providing an optimal starting point for subsequent steps. At the same time, the automatic division by graph theory replaces manual operation and can be adapted to large-scale Pauli terms.

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

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

[0074] S132, add an edge between the vertices corresponding to the two Pauli terms in the commutation to construct an undirected graph.

[0075] It should be noted that "ease" refers to two Pauli terms P and Q satisfying... ,Right now .

[0076] As an example, each Pauli term is considered as a vertex in a graph. If two Pauli terms commute, an edge is added between the corresponding vertices, thus modeling all Pauli terms in the Hamiltonian as an undirected graph.

[0077] In this embodiment, the abstract commutation relationships between Pauli terms are transformed into an intuitive graph structure to facilitate understanding and subsequent processing.

[0078] In some embodiments, the step of generating a sequence of Clifford operations based on a set of cliques includes:

[0079] S21, obtain the second lowest energy clique in the clique set, and apply the first Clifford operation to the second lowest energy clique so that the second lowest energy clique contains only Z-type Pauli operators.

[0080] It should be noted that the optimal choice of the second lowest energy cluster in the cluster set is for the following reasons: (1) The lowest energy cluster and the second lowest energy cluster are the two clusters that contribute the most to the ground state energy; (2) When optimizing the measurement axis angle later, the optimizer will naturally take the lowest energy cluster as the main body, while the second lowest energy cluster becomes the Pauli Z operator on a single qubit after the Clifford transformation. The energy of the second lowest energy cluster is only related to these single-bit operators and is separated from the complex coupling with other clusters; (3) The optimizer will naturally prioritize the integration of the energy of these easily adjustable single-bit operators into the energy optimization of the entire system; thus, while optimizing the energy of the lowest energy cluster, the energy of the second lowest energy cluster can be optimized as much as possible to obtain the energy of the two clusters that contribute the most, and finally obtain the ground state energy more approximately.

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

[0082] As an example, if the group contains or Then, through the H operation, it becomes possible to... Become , Become 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 utilizing the characteristic that Clifford evolution does not change the energy spectrum of the Hamiltonian, an appropriate Clifford evolution is applied to the Hamiltonian, and combined with an adaptive strategy for subsequent individual measurement of qubits, multiple dominant terms in the Hamiltonian are ultimately covered, resulting in an approximation closer to the exact value. This effectively solves the problem of poor approximation results 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 exactly one element 1 includes:

[0091] S221, traverse the evolution matrix column by column. During the traversal, if the j-th column contains exactly one element 1 and the row containing element 1 contains exactly two elements 1, then add... The operation is performed, and the j-th column of the matrix is ​​bitwise XORed with the k-th column.

[0092] S222, repeat step S221 until each row of the evolution matrix has exactly one element 1.

[0093] Where k and j represent the column containing element 1, The gate represents the CX gate on the k-th and j-th qubits, where k points to the control bit and j points to the target bit.

[0094] Next, with only or (1≤a,b≤n), or only containing or or only contains or Let's take a group, specifically one containing only generators, as an example. , , , , Similar to the definition The details will not be elaborated here. The specific definition of the Pauli X matrix is... The Pauli Y matrix is ​​specifically defined as follows: The specific steps are as follows:

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

[0096] A2, if the group only contains or Then, an H operation is added to the sequence CL, and the H operation is applied to all Pauli terms in the clique, such that... Become , Become After action, the group contains only or If the group only contains or Then, H operations and S operations are added sequentially to the sequence CL, and H operations and S operations are applied sequentially to all Pauli terms in the clique, such that... Become , Become After action, the group contains only or Ultimately, we obtained only... or The Z-type Pauli operator.

[0097] A3, based on the Z-type Pauli operator, yields the evolution matrix. In this system, each column corresponds to a qubit, and each row corresponds to a Pauli term in the clique. If the j-th element of the i-th Pauli term is a Pauli Z matrix, then... The value in the i-th row and j-th column is 1, otherwise it is 0. Meanwhile... It also reflects the Clifford evolution of the Pauli term in the cluster.

[0098] A4, traverse the evolution matrix column by column. If element 1 appears only once in column j, and its row contains only two elements 1, and assuming the other element 1 is in column k, then the sequence CL will have an additional element 1. The operation involves controlling the k-th qubit and targeting the j-th qubit, while simultaneously manipulating the evolution matrix. XOR the j-th column bitwise with the k-th column and update. This also means the Clifford evolution result of the Pauli term in the updated cluster.

[0099] A5, repeat step A4 until the evolution matrix is ​​obtained. Each row contains only one element, 1, resulting in the complete Clifford operation sequence CL. Sequence CL operates sequentially on the Pauli terms in the clique, diagonalizing them into... or a subset thereof, The final form reflects the result of diagonalization.

[0100] In some embodiments, the step of generating an energy expression containing variable parameters based on the new Hamiltonian includes:

[0101] S31 selects a separate measurement axis for each qubit.

[0102] S32, based on the new Hamiltonian and the measurement axis, generates the energy expression.

[0103] In this embodiment, by performing an appropriate Clifford evolution on the Hamiltonian, and then selecting the most suitable measurement axis for each qubit individually and generating an energy expression, a prerequisite is provided for the subsequent approximate solution of the ground state energy, so that the subsequent solution can obtain results closer to the accurate value, and the problem of poor approximation effect when multiple clusters jointly dominate the energy is solved.

[0104] In some embodiments, the step of generating an energy expression based on the new Hamiltonian and the measurement axis includes:

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

[0106] S322, replace the Pauli X matrix on the a-th qubit with... Pauli Z matrix replaced with Pauli Y matrix replaced with The energy expression is obtained. .

[0107] in, Let represent the angle between the measurement axis of the a-th qubit and the Z-axis of the Bloch sphere. Let X represent the angle between the projection of the measurement axis of the a-th qubit onto the horizontal plane of the Bloch sphere and the X-axis. , , , where n represents the number of qubits.

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

[0109] In this embodiment, by replacing the Pauli matrix with trigonometric function values ​​containing variable parameters, quantum energy calculation is transformed into a classical parameter optimization problem, thereby 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 includes:

[0111] S41, optimizing variable parameters using a classic optimizer. This makes the energy expression The value is the smallest.

[0112] S42, the minimum energy value obtained is used as an approximation of the ground state energy of the Hamiltonian.

[0113] Next, this application takes the cross-field Ising model on a tree graph (TFIM-tree for short) as an example. The general form of the Hamiltonian of TFIM-tree is:

[0114]

[0115] Where J is the coupling strength, This represents the edge between adjacent nodes a and b in the tree graph. denoted as , where is the magnetic field strength.

[0116] Specifically, for example Figure 2 The corresponding Hamiltonian of the TFIM-tree in the example is:

[0117]

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

[0119] The specific process of approximating the ground state energy of the Hamiltonian using the method of this application is as follows:

[0120] B1, Hamiltonian All Pauli terms have been clearly divided into two groups, namely and The groups are arranged in ascending order of energy.

[0121] B2, for groups with the second lowest energy. Applying an appropriate Clifford operation sequence CL will The Pauli term is diagonalized into Or a subset thereof, the specific steps are as follows:

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

[0123] B22, Initialize the evolution matrix :

[0124]

[0125] B23, traverse by column In column 4, element 1 appears only once, and in row 3, there are only two elements 1; the other element 1 is in column 2. A new element 1 is added to sequence CL. The operation involves using the second qubit as the control bit and the fourth qubit as the target bit. Simultaneously, [the operation will...] XOR column 4 with column 2, then update. :

[0126]

[0127] The change in line 3 also means The third Pauli term is... Become .

[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 is approximately -11.5046 (rounded to four decimal places), while the exact solution is -11.5366 (rounded to four decimal places).

[0141] To illustrate the accuracy of the calculation results in this application, existing methods for solving the results of Example 1 are presented. For the original Hamiltonian... The relevant technique (an approximation technique based on stable substates) continuously selects the Pauli term with the largest absolute value of its coefficient and ensures that it commutes pairwise with the selected set of Pauli terms. Thus, this method will... As the only cluster, assigning 1 to all its generators yields an approximate energy. Obviously, this result is far less accurate than the result calculated by the method in this invention, which is -11.5046, and closer to the exact solution -11.5366.

[0142] Next, this application takes the XZ model on a tree diagram as an example. The general form of its Hamiltonian is:

[0143]

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

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

[0146]

[0147] Let the coupling constant .

[0148] The specific process of approximating the ground state energy of the Hamiltonian using the method of this invention is as follows:

[0149] C1, Hamiltonian All Pauli terms have been clearly divided into two groups, namely and The groups are arranged in ascending order of energy.

[0150] C2, for the second lowest energy group Applying an appropriate Clifford operation sequence CL will The Pauli term is diagonalized into Or a subset thereof, the specific steps are as follows:

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

[0152] C22, only contains Therefore, the H operation is added to the sequence CL, and... All Pauli terms in the equation undergo the H operation, making become .

[0153] C23, as can be seen from the above embodiments, the complete Clifford operation sequence is as follows: Acting sequentially on In the Pauli term, it can be diagonalized into .

[0154] C3 applies the Clifford operations in sequence CL sequentially to the remaining Pauli terms of the Hamiltonian, i.e. All Pauli terms become after the first H operation. Then, using a similar evolutionary matrix The method for initializing the evolution matrix :

[0155]

[0156] Each The operation will XOR the k-th column of the parity matrix X with the j-th column. After applying the remaining Clifford operations in sequence CL in turn, It becomes:

[0157]

[0158] express Pauli term becomes .

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

[0160]

[0161] C4, selecting a measurement axis for each qubit, yields an energy expression containing variable parameters:

[0162]

[0163] in, .

[0164] C5 optimizes 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 is approximately -6.3343 (rounded to four decimal places), while the exact solution is -6.3396 (rounded to four decimal places). Similarly, by using related techniques (approximation techniques based on stable substates) to find the minimum value of -6, it is clear that this result is far less accurate than the result calculated by the method in this invention, which is -6.3343 and is closer to the exact solution -6.3396.

[0165] In addition, this invention also provides an approximate solution for the Hamiltonian ground state energy system, such as... Figure 4 As shown, the approximate solution for the Hamiltonian ground state energy system includes: 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, and obtain the clique set corresponding to the Hamiltonian.

[0167] Clifford evolution module 20 is used to generate a sequence of Clifford operations based on the clique set, and to apply the sequence of Clifford operations to the clique set to obtain a new Hamiltonian.

[0168] The measurement module 30 is used to generate an energy expression containing variable parameters based on 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] Other specific implementations of the approximate solution of the Hamiltonian ground state energy system in the embodiments of the present invention can be found in the specific implementations of the approximate solution of the Hamiltonian ground state energy method in the above embodiments of the present invention.

[0171] The approximate solution for the Hamiltonian ground-state energy system in this invention involves processing the clique set using a sequence of Clifford operations to obtain a new Hamiltonian and generating an energy expression with variable parameters. By optimizing the variable parameters, an approximate value for the Hamiltonian's ground-state energy is obtained, improving computational efficiency and shortening computation time. Furthermore, it enhances the approximation effect even when multiple cliques dominate the Hamiltonian, making the error between the calculated approximation and the exact value negligible. Moreover, Clifford operations can be simulated in polynomial time on a classical computer; therefore, the system of this invention can be efficiently implemented directly on a classical computer.

[0172] The following detailed explanation uses a computer terminal as an example. Figure 5 This is a hardware structure block diagram of a computer terminal for an approximate method of solving the Hamiltonian ground state energy, provided in an embodiment of the present invention. Figure 5 As shown, a computer terminal may 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] This application also provides a computer program product, which includes a non-transitory computer-readable storage medium storing a computer program operable to cause a computer to perform some or all of the steps of any of the methods described in the above method embodiments. The computer program product may be a software installation package, and the computer may include an electronic device.

[0176] Although exemplary embodiments have been described herein with reference to the accompanying drawings, it should be understood that the above exemplary embodiments are merely illustrative and are not intended to limit the scope of this application. Various changes and modifications can be made therein by those skilled in the art without departing from the scope and spirit of this application. All such changes and modifications are intended to be included within the scope of this application as claimed in the appended claims.

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

[0178] In the several embodiments provided in this 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 instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another device, or some features may be ignored or not executed.

[0179] Numerous specific details are set forth in the specification provided herein. However, it will be understood that embodiments of this application may be practiced without these specific details. In some instances, well-known methods, structures, and techniques have not been shown in detail so as not to obscure the understanding of this specification.

[0180] Similarly, it should be understood that, in order to streamline this application and aid in understanding one or more of the various inventive aspects, features of this application may sometimes be grouped together in a single embodiment, figure, or description thereof in the description of exemplary embodiments of this application. However, this approach should not be construed as reflecting an intention that the claimed application requires more features than are expressly recited in each claim. Rather, as reflected in the corresponding claims, its inventive point lies in solving the corresponding technical problem with features fewer than all features of a single disclosed embodiment. Therefore, the claims following the detailed description are hereby expressly incorporated into that detailed description, wherein each claim itself is a separate embodiment of this application.

[0181] Those skilled in the art will understand that, apart from the mutual exclusion of features, all features disclosed in this specification (including the accompanying claims, abstract, and drawings) and all processes or elements of any method or apparatus so disclosed may be combined in any combination. Unless otherwise expressly stated, each feature disclosed in this specification (including the accompanying claims, abstract, and drawings) may be replaced by an alternative feature that serves the same, equivalent, or similar purpose.

[0182] Furthermore, those skilled in the art will understand that although some embodiments described herein include certain features included in other embodiments but not others, combinations of features from different embodiments are intended to be within the scope of this application and form different embodiments. For example, in the claims, any one of the claimed embodiments can be used in any combination.

[0183] The various component embodiments of this application can be implemented in hardware, or as software modules running on one or more processors, or a combination thereof. Those skilled in the art will understand that microprocessors or digital signal processors (DSPs) can be used in practice to implement some or all of the functions of some modules according to the embodiments of this application. This application can also be implemented as an apparatus program (e.g., a computer program and computer program product) for performing part or all of the methods described herein. Such an implementation of this 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, provided on a carrier signal, or provided in any other form.

[0184] It should be noted that the above embodiments are illustrative of this application and not restrictive, and that those skilled in the art can devise alternative embodiments without departing from the scope of the appended claims. In the claims, any reference signs placed between parentheses should not be construed as limiting the claims. The word "comprising" does not exclude the presence of elements or steps not listed in the claims. The word "a" or "an" preceding an element does not exclude the presence of a plurality of such elements. This application can be implemented by means of hardware comprising several different elements and by means of a suitably programmed computer. In the unit claims enumerating several means, several of these means may be embodied by the same item of hardware. The use of the words first, second, and third, etc., does not indicate any order. These words can be interpreted as names.

[0185] The above description is merely a specific embodiment or illustration of the embodiments of this application. The scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. The scope of protection of this application shall be determined by the scope of the claims.

Claims

1. A method for approximating the ground-state energy of the Hamiltonian, characterized in that, The method includes: Obtain and preprocess the Hamiltonian to be solved to obtain the clique set corresponding to the Hamiltonian; Based on the clique set, a Clifford operation sequence is generated, and the Clifford operation sequence is applied to the clique set to obtain a new Hamiltonian; Based on the new Hamiltonian, generate an energy expression containing variable parameters; By optimizing the variable parameters in the energy expression, an approximate value of the ground-state energy of the Hamiltonian can be obtained.

2. The method for approximating the Hamiltonian ground state energy according to claim 1, characterized in that, The Hamiltonian consists of multiple Pauli terms. The steps for preprocessing the Hamiltonian include: Determine whether the Pauli term in the Hamiltonian is divided into multiple groups; If the clusters have already been divided, then the clusters are sorted from low to high energy to obtain the cluster set. If the graph has not yet been partitioned, the Pauli term in the Hamiltonian is modeled as an undirected graph. Using graph theory, the undirected graph is partitioned into multiple cliques, and the cliques are sorted from low to high energy to obtain the clique set.

3. The approximate solution method for the Hamiltonian ground state energy according to claim 2, characterized in that, The step of modeling the Pauli term in the Hamiltonian as an undirected graph includes: Treat each Pauli term as a vertex in the graph; An edge is added between the vertices corresponding to the two Pauli terms in the commutation to construct the undirected graph.

4. The approximate solution method for the Hamiltonian ground state energy according to claim 1, characterized in that, The step of generating the Clifford operation sequence based on the clique set includes: Obtain the second lowest energy clique in the clique set, and apply the first Clifford operation to the second lowest energy clique such that the second lowest energy clique contains only Z-type Pauli operators; The evolution matrix is ​​obtained based on the Z-type Pauli operator; 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; The Clifford operation sequence is generated based on the first Clifford operation and the second Clifford operation.

5. The approximate solution method for the Hamiltonian ground state energy according to claim 4, characterized in that, The step of transforming the evolution matrix until each row of the evolution matrix has exactly one element 1 includes: Traverse the evolution matrix column by column. During the traversal, if there is exactly one element 1 in the j-th column and exactly two elements 1 in the row containing element 1, then add a new element. The operation is performed, and the j-th column of the matrix is ​​bitwise XORed with the k-th column; Repeat the above steps until each row of the evolution matrix has exactly one element 1; Where k and j represent the column containing element 1, The gate represents the CX gate on the k-th and j-th qubits, where k points to the control bit and j points to the target bit.

6. The approximate solution method for the Hamiltonian ground state energy according to claim 1, characterized in that, The step of generating an energy expression with variable parameters based on the new Hamiltonian includes: Choose a separate measurement axis for each qubit; The energy expression is generated based on the new Hamiltonian and the measurement axis.

7. The approximate solution method for the Hamiltonian ground state energy according to claim 6, characterized in that, The step of generating the energy expression based on the new Hamiltonian and the measurement axis includes: The expression structure and coefficients of the new Hamiltonian are preserved; Replace the Pauli X matrix on the a-th qubit with Pauli Z matrix replaced with Pauli Y matrix replaced with Thus, the energy expression is obtained; in, Let represent the angle between the measurement axis of the a-th qubit and the Z-axis of the Bloch sphere. Let X represent the angle between the projection of the measurement axis of the a-th qubit onto the horizontal plane of the Bloch sphere and the X-axis. , , , where n represents the number of qubits.

8. The method for approximating the Hamiltonian ground state energy according to claim 1, characterized in that, The step of optimizing the variable parameters in the energy expression to obtain an approximate value of the ground-state energy of the Hamiltonian includes: The variable parameters are optimized using a classical optimizer to minimize the value of the energy expression; The minimum energy value obtained is used as an approximation of the ground state energy of the Hamiltonian.

9. A method for approximating the Hamiltonian ground-state energy of a system, characterized in that, The system includes: The preprocessing module is used to acquire and preprocess the Hamiltonian to be solved, and obtain the clique set corresponding to the Hamiltonian; The Clifford evolution module is used to generate a Clifford operation sequence based on the clique set, and to apply the Clifford operation sequence to the clique set to obtain a new Hamiltonian; The measurement module is used to generate an energy expression containing variable parameters based on the new Hamiltonian. An optimization module is used to optimize the variable parameters in the energy expression to obtain an approximate value of the ground state energy of the Hamiltonian.

10. An electronic device comprising a memory, a processor, and a computer program stored in the memory, characterized in that, When the computer program is executed by the processor, it implements the method for approximating the ground state energy of the Hamiltonian according to any one of claims 1-8.

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