Variational quantum algorithm hierarchical training method based on lie algebra

Abstract

The application discloses a Lie algebra-based variable quantum algorithm hierarchical training method, through a hierarchical activation-freezing training schedule, first optimization and freezing of ZX on a generating element set G 1 ={XX, YY, XY}, barren plateaus are significantly alleviated and trainability is improved, instantaneous reachable algebraic dimension is reduced, gradient variance is increased and optimization stability is improved; after meeting switching criteria, ZX is introduced to expand to G2={XX, YY, XY, ZX}, and final expression capability is maintained and restored; switching is consistent with dynamic Lie algebra structure, and is jointly triggered by gradient statistics and subalgebra size indexes, so that the training process is schedulable, interpretable, and convenient for auditing and reproduction.

Description

Technical Field

[0001] This invention relates to the fields of quantum computing and quantum machine learning, specifically to a hierarchical training method for variational quantum algorithms based on Lie algebras. Background Technology

[0002] Variational quantum algorithm (VQA) minimizes the task-related loss function using a parameterized unitary circuit. In engineering practice, a ring circuit is often used. -Local spin chain topology: Assume that qubits are ordered by mode Numbered, the adjacent edge set is;

[0003] ;

[0004] On each edge Above, from the generator set Construct a 2-local Pauli interaction stacked circuit, whose dynamic Lie algebra is denoted as ;

[0005] ;

[0006] remember;

[0007] ;

[0008] in For input state, For observable measurement, For the generators mentioned above in the edge set The parameterized circuit obtained by stacking. If the circuit distribution satisfies the approximation... -If the design or equivalent fully mixed assumption is adopted, then the same as Compatible hierarchical (representation) decomposition Give the variance expression for parameter dependence:

[0009] ;

[0010] in Operator In a subspace (or subalgebra) The Hilbert–Schmidt projection energy on the basis can be obtained from any orthogonal basis. Defined as;

[0011] ;

[0012] The above formula shows that the gradient and loss statistics during training are determined by three types of structural quantities: first, the hierarchical dimension. This can be seen as a dimension of instantaneous expressive ability; secondly, the input state Projected intensity at each level Thirdly, observable measurement Projection intensity .when Larger and , When the variance is smaller, the contribution of the corresponding term to the variance decreases, and the Barren Plateau (BP) phenomenon is more likely to occur during training.

[0013] Existing mitigation approaches for backpropagation (BP) include: improving variance by limiting correlation length through shallowing and local loss; reducing redundant degrees of freedom by pruning the generator set based on problem symmetry; improving optimization geometry through small-angle initialization, natural gradients, or Fisher information metrics; and reducing the instantaneous search dimensionality by employing block-based or layer-by-layer training (freezing / thawing subsets of parameters). The closest existing technique is block-based / layer-by-layer training, which typically divides the parameters based on circuit geometry, gate assignment, or parameter topology, with switching criteria often based on fixed rounds or validation errors.

[0014] However, the above scheme does not address the variance decomposition process. , , The training phase was not organized as an explicit scheduling metric, nor was it based on a fixed set of generators. With ring Within the framework of Lie algebras in local topology, an feasible process of "restriction-expansion based on the induced subalgebras of the generator set" is presented. Therefore, it is still necessary to schedule and switch the training phase in a manner consistent with the Lie algebra hierarchical structure, without changing the circuit topology and generator set, in conjunction with the aforementioned variance decomposition, in order to improve variance and optimization feasibility in the early stages and restore complete expressive power in the later stages. Summary of the Invention

[0015] The purpose of this invention is to propose a hierarchical training method for variational quantum algorithms based on Lie algebras. This method employs a hierarchical activation-freeze training schedule, first training a subset of generators... Optimize and freeze This significantly alleviates the barrenness of the plateau and improves trainability; it is introduced only after the switching criteria are met. Extended to the complete set of generators It maintains and restores the final expressive power; the switching is based on the same criteria as the dynamic Lie algebra structure and is jointly triggered by gradient statistics and sub-algebra size indicators, making the training process schedulable, interpretable and easy to audit and reproduce.

[0016] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0017] The hierarchical training method for variational quantum algorithms based on Lie algebras includes the following steps:

[0018] S1. Task and Circuit Setup: Determine the target task and loss function. Construct a ring-shaped 2-local spin chain quantum circuit and a generator set {XX, YY, XY, ZX};

[0019] S2. Generator subset partitioning and parameter grouping:

[0020] Divide the generator set into generator subsets With the complete set of generators And divide the circuit parameters into subsets and subsets ;

[0021] S3. Parameter and Hyperparameter Initialization:

[0022] right Perform random initialization at a preset angle. Set to zero or add perturbation; set the learning rate, stage round limit T1 / T2, gradient variance threshold, loss improvement threshold, and measurement shot value;

[0023] S4, Phase I:

[0024] S41, only in the subset of generators Training: Freeze Only for subsets Perform gradient estimation and update, calculate and record the hierarchical variance proxy V1 and the relative improvement rate of loss ΔL1;

[0025] S42, Decision 1: Determine whether to proceed to the next step. If yes, proceed to step S5; otherwise, return to step S41.

[0026] S5, Phase II:

[0027] S51, Extending to the complete generator set Training: Freeze Activate the parameters corresponding to ZX In the complete set of generators The loss is calculated and the gradient is estimated on the constructed circuit, but only for... Perform gradient updates, calculate and record V2 and the relative improvement rate of loss ΔL2;

[0028] S52, Decision 2: Determine whether to proceed to the next step. If yes, proceed to step S6; otherwise, return to step S51.

[0029] S6, Joint Fine-tuning Phase:

[0030] thaw and Reduce the learning rate in the complete generator set The joint optimization yields the optimal final parameters. ;

[0031] S7. Result Output and Recording:

[0032] Output the optimal parameters and corresponding quantum circuit performance indicators, and save the reproducibility information.

[0033] As a preferred technical solution of the present invention, step S1 is specifically as follows:

[0034] Let the adjacent edge set E be:

[0035] ;

[0036] Where j is the index of the qubit, and its value ranges from 1 to 10. N represents the total number of qubits in the system; adjacency relationships are based on modulus. The values ​​form a closed-loop topology, that is It is also an adjacent edge;

[0037] The allowed 2-local generators on each edge are:

[0038] ;

[0039] in, , , These represent the actions of Pauli operators X, Y, and Z on the first... Tensor operators on qubits;

[0040] Therefore, the generator set is obtained as .

[0041] As a preferred technical solution of the present invention, step S2 is specifically as follows:

[0042] S21. The specific partitioning of generator subsets is as follows:

[0043] Define and classify two families of circuits and their induced dynamic Lie algebras:

[0044] , ;

[0045] , ;

[0046] in, , Each represents the set of allowed two-bit generators; , Let be the dynamic Lie algebras spanned by the corresponding sets over the circular adjacency set via Lie bracket closures, respectively, at the same depth and bit size. The upper bound of the dimension is not less than the subset. ,

[0047] This constructs a subset of generators. To the complete set of generators The foundation of layered training;

[0048] S22. Parameter grouping is as follows:

[0049] Fixed in Hierarchical training is performed within this framework, with parameters grouped by generators:

[0050] ;

[0051] in, correspond , correspond .

[0052] As a preferred technical solution of the present invention, step S3 is as follows:

[0053] Given a differentiable loss function In approximation Under the design assumption, its parameter dependency variance satisfies:

[0054] ;

[0055] Indicates in the parameter Under random initialization and measurement statistics, the loss function The variance; Or its task-related monotonic transformations, For the input quantum state, It is observable; For all trainable parameters, where correspond rotation angle, correspond Rotation angle of generator, To and Compatible hierarchical decomposition, Number of levels; For the first Hierarchical dimension Operator In subspace The Hilbert–Schmidt projection energy on the surface can be expressed using any orthogonal basis. Written as:

[0056] ;

[0057] The above formula reveals that the gradient / loss variance is determined by the hierarchical dimension. Input state With observable Projected energy at each level Joint decision,

[0058] in To and Compatible hierarchical decomposition, The Hilbert–Schmidt projection energy.

[0059] As a preferred technical solution of the present invention, step S41 is specifically as follows:

[0060] Phase I: Activate and optimize only ,freeze At this point, the effective reachable algebra is limited to The specific steps of Phase I:

[0061] S411, Initialization: , Keep frozen, usually set to 0 or the value from the previous stage; the learning rate is... The maximum number of rounds is The number of measurements was ,in The anchor value represents the second set of parameters frozen in stage I: taken in the first iteration. or After that, each round takes the stage II of the previous round in a fixed manner. The result obtained by optimization under the given conditions, i.e. ;

[0062] S412. Gradient estimation, using the parametric offset method, with Pauli rotation as the default offset. :

[0063] right Each scalar parameter ,remember Let its standard basis vectors be ,

[0064]

[0065] Each expectation is derived from Subsampling estimation;

[0066] S413, Optimization and Update: Employs optimizers such as Adam / SGD. Gradient clipping and weight decay can be selected;

[0067] S414. Switching Criterion: If the hierarchical variance proxy is satisfied... or relative improvement rate or reach If the upper limit of the round is reached, then phase I ends;

[0068] S415, Product: Obtained And preserve the optimizer's internal state for continuous training in subsequent stages.

[0069] As a preferred technical solution of the present invention, step S51 is as follows:

[0070] Phase II: Freeze Only optimization This can be equivalently derived from algebra. Expand to Gradually restore complete The specific steps of Stage II: (This section discusses) the development of expressive abilities.

[0071] S511, Initialization: Keep frozen. Or set to zero; the learning rate is... The maximum number of rounds is The number of measurements was ;

[0072] S512. Gradient estimation, using the parameter offset method:

[0073] right Each scalar parameter ,

[0074]

[0075] Number of samplings The rest are in the same stage I;

[0076] S513, Optimization and Update: ;

[0077] S514, Switching / Termination: When or or reach When the upper limit is reached, Phase II ends.

[0078] As a preferred technical solution of the present invention: Stage I / II , , , Window length , Use the same diameter.

[0079] As a preferred technical solution of the present invention: threshold These are settable parameters.

[0080] As a preferred technical solution of the present invention, step S6 is specifically as follows:

[0081] The joint fine-tuning is as follows:

[0082] Unfreeze Simultaneously participate in training Rounds, the learning rate decays to Record the final parameters And indicators.

[0083] As a preferred technical solution of the present invention: in step S7, the reproducibility information includes training logs, random seeds, number of shots, and gradient variance and loss curves at each stage.

[0084] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0085] This invention is applicable to various variable quantum tasks under toroidal 2-local structures: in VQE in chemistry and materials science for ground state / low-excited state energy estimation; in combinatorial optimization and QAOA-type circuits for approximate solutions to local coupling problems such as Ising / Max-Cut; and in quantum machine learning (QML) for quantum feature mapping and classification / regression models to achieve more stable convergence, through a hierarchical activation-freeze training schedule, first in a subset of generators. Optimize and freeze This significantly alleviates the barren plateau and improves trainability (reducing the instantaneous reachability algebra dimension, increasing gradient variance, and improving optimization stability); it is introduced only after the switching criterion is met. Extended to the complete set of generators It maintains and restores the final expressive power; the switching is based on the same criteria as the dynamic Lie algebra structure and is jointly triggered by gradient statistics and sub-algebra size indicators, making the training process schedulable, interpretable and easy to audit and reproduce. Attached Figure Description

[0086] Figure 1 This is the overall flowchart of the present invention.

[0087] Figure 2 Based on generator set A schematic diagram of a ring-shaped 2-body stacked structure.

[0088] Figure 3 yes , With layering Comparison of gradient variance under different numbers of qubits.

[0089] in Figure 3 (a) is a comparison of the variance of the loss function of the model with different numbers of circuit layers for different qubits. Figure 3 (b) is a comparison of the variance of the loss function of the model with different numbers of circuit layers for different qubits. Figure 3 (c) is a comparison of the variance of the loss function of the model with different numbers of circuit layers for different qubits.

[0090] Figure 4 This is a comparison of test accuracy based on MedMNIST: stratified. , , . Detailed Implementation

[0091] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0092] like Figure 1 As shown, the hierarchical training method for variational quantum algorithms based on Lie algebras proposed in this invention includes the following steps:

[0093] S1. Task and Circuit Setup: Determine the target task and loss function. Construct a ring-shaped 2-local spin chain quantum circuit and a generator set {XX, YY, XY, ZX};

[0094] Step S1 is as follows:

[0095] Let the adjacent edge set be:

[0096] ;

[0097] Where j is the index of the qubit, and its value ranges from 0 to 1. N is the total number of qubits in the system; adjacency relationships are based on modulus. The values ​​form a closed-loop topology, that is It is also an adjacent edge;

[0098] The allowed 2-local generators on each edge are:

[0099] ;

[0100] in, , , These represent the actions of Pauli operators X, Y, and Z on the first... Tensor operators on qubits;

[0101] Therefore, the generator set is obtained as .

[0102] S2. Generator subset partitioning and parameter grouping:

[0103] Divide the generator set into generator subsets With the complete set of generators And divide the circuit parameters into subsets and subsets ;

[0104] Step S2 is as follows:

[0105] S21. The specific partitioning of generator subsets is as follows:

[0106] Define and classify two families of circuits and their induced dynamic Lie algebras:

[0107] , ;

[0108] , ;

[0109] in, , Each represents the set of allowed two-bit generators; , Let be the dynamic Lie algebras spanned by the corresponding sets over the circular adjacency set via Lie bracket closures, respectively, at the same depth and bit size. The upper bound of the dimension is not less than the subset. ,

[0110] This constructs the subset To the complete set of generators The foundation of layered training;

[0111] like Figure 2 As shown, in the ring -Under conditions of local structure and full mixing, as depth and bit number increase, The reachable transformation space can approximate a high-dimensional Lie algebra, with an upper bound close to This tends to produce variance dilution and is relatively limited to a smaller subset of generators. Induced It can provide higher observable gradient variance in the early stages, but its expressive power is limited.

[0112] S22. Parameter grouping is as follows:

[0113] Fixed in Hierarchical training is performed within this framework, with parameters grouped by generators:

[0114] ;

[0115] in, Corresponding generative subset , correspond .

[0116] S3. Parameter and Hyperparameter Initialization:

[0117] right Perform random initialization at a preset angle. Set to zero or add perturbation; set the learning rate, stage round limit T1 / T2, gradient variance threshold, loss improvement threshold, and measurement shot value;

[0118] Step S3 is as follows:

[0119] Given a differentiable loss function In approximation Under the design assumption, its parameter dependency variance satisfies:

[0120] ;

[0121] Indicates in the parameter Under random initialization and measurement statistics, the loss function The variance; Or its task-related monotonic transformations, For the input quantum state, It is observable; For all trainable parameters, where correspond rotation angle, correspond Rotation angle of generator, To and Compatible hierarchical decomposition, Number of levels; For the first Hierarchical dimension Operator In subspace The Hilbert–Schmidt projection energy on the surface can be expressed using any orthogonal basis. Written as:

[0122] ;

[0123] The above formula reveals that the gradient / loss variance is determined by the hierarchical dimension. Input state With observable Projected energy at each level Joint decision,

[0124] in To and Compatible hierarchical decomposition, The Hilbert–Schmidt projected energy.

[0125] S4, Phase I:

[0126] S41, only in the subset of generators Training: Freeze Only for subsets Perform gradient estimation and update, calculate and record the hierarchical variance proxy V1 and the relative improvement rate of loss ΔL1;

[0127] Step S41 is as follows:

[0128] Phase I: Activate and optimize only ,freeze At this point, the effective reachable algebra is limited to The specific steps of Phase I:

[0129] S411, Initialization: , Keep frozen, usually set to 0 or the value from the previous stage; the learning rate is... The maximum number of rounds is The number of measurements was ,in The anchor value represents the second set of parameters frozen in stage I: taken in the first iteration. or After that, each round takes the stage II of the previous round in a fixed manner. The result obtained by optimization under the given conditions, i.e. ;

[0130] S412. Gradient estimation, using the parametric offset method, with Pauli rotation as the default offset. :

[0131] right Each scalar parameter ,remember Let its standard basis vectors be ,

[0132]

[0133] Each expectation is derived from Subsampling estimation;

[0134] S413, Optimization and Update: Employs optimizers such as Adam / SGD. Gradient clipping and weight decay can be selected;

[0135] S414. Switching Criterion: If the hierarchical variance proxy is satisfied... or relative improvement rate (or reach) If the upper limit of the round is reached, then phase I ends;

[0136] S415, Product: Obtained And preserve the optimizer's internal state for continuous training in subsequent stages;

[0137] S42, Decision 1: Determine whether to proceed to the next step. If yes, proceed to step S5; otherwise, return to step S41.

[0138] S5, Phase II:

[0139] S51, Extending to the complete generator set Training: Freeze Activate the parameters corresponding to ZX In the complete set of generators The loss is calculated and the gradient is estimated on the constructed circuit, but only for... Perform gradient updates, calculate and record V2 and the relative improvement rate of loss ΔL2;

[0140] Step S51 is as follows:

[0141] Phase II: Freeze Only optimization This can be equivalently derived from algebra. Expand to Gradually restore complete The specific steps of Stage II: (This section discusses) the development of expressive abilities.

[0142] S511, Initialization: Keep frozen. Or set to zero; the learning rate is... The maximum number of rounds is The number of measurements was ;

[0143] S512. Gradient estimation, using the parameter offset method:

[0144] right Each scalar parameter ,

[0145]

[0146] Number of samplings The rest are in the same stage I;

[0147] S513, Optimization and Update: ;

[0148] S514, Switching / Termination: When or or reach When the upper limit is reached, Phase II ends;

[0149] S52, Decision 2: Determine whether to proceed to the next step. If yes, proceed to step S6; otherwise, return to step S51.

[0150] Phase I / II , , , Window length , Use the same diameter.

[0151] threshold These are settable parameters.

[0152] S6, Joint Fine-tuning Phase:

[0153] thaw and Reduce the learning rate in the complete generator set The joint optimization yields the optimal final parameters. ;

[0154] Step S6 is as follows:

[0155] The joint fine-tuning is as follows:

[0156] Unfreeze Simultaneously participate in training Rounds, the learning rate decays to Record the final parameters and indicators

[0157] S7. Result Output and Recording:

[0158] Output the optimal parameters and corresponding quantum circuit performance indicators, and save the reproducibility information.

[0159] The reproducibility information includes training logs, random seeds, number of shots, and gradient variance and loss curves for each stage.

[0160] The technical solution of this invention, without changing the circuit topology and generator set, explicitly controls the instantaneous sub-algebra scale through only a phased activation-freeze process consistent with the dynamic Lie algebra structure: improving gradient variance and optimization stability in the early stages of training with lower-dimensional sub-algebras, and subsequently expanding to the complete scale. This would restore expressive power, thereby theoretically alleviating the variance dilution and barren plateau problems caused by circuit depth and bit size.

[0161] Specifically: This invention compares three ring-shaped 2-local structures with the same aperture under the same circuit and layered organization: , And training the first three generators first, then training the next three generators... Sequential training Quantum bits by mode Numbering, the nearest neighbor set is written using the modulo method:

[0162]

[0163] Each layer applies a 2-local Pauli rotation to all edges, first covering even edges, then odd edges, and finally closing the loop edges, to reduce conflicts within the same layer and ensure parallelism. The 2-local rotation used in the circuit is defined as...

[0164]

[0165] Symbol interpretation (adjacency and two-bit rotation):

[0166] Total number of qubits; : Quantum bit number ( ).

[0167] : Circular nearest neighbor edge set, by modulo closed loop, It is also an adjacent edge.

[0168] Pauli operator; Operator Only applies to the first On each bit.

[0169] Tensor product; Imaginary unit.

[0170] : in the A two-bit interactive rotating gate on each bit; Let it be the angle of rotation.

[0171] This invention relates to four categories of doors:

[0172] ,

[0173] ,

[0174] ,

[0175] .

[0176] The laying order of each layer is as follows: first cover even edges, then cover odd edges, and finally close loop edges, in order to reduce conflicts within the same layer and ensure parallelism.

[0177] This invention relates to Four types, the whole of which is:

[0178]

[0179] Symbol interpretation (overall unit and depth):

[0180] : All trainable parameter vectors; For the first Layers, edges Door-to-door type The rotation angle.

[0181] : Layer index; This represents the total depth of the circuit.

[0182] : No. Door type label for the layer.

[0183] The order follows a bottom-up, left-to-right arrangement; different edges within the same layer are implemented in parallel / serial order of "even edge → odd edge → closed loop edge".

[0184] Regular segmentation: Divide the four categories into two segments, first in... Training (corresponding to A1), then introducing And jointly fine-tuned (extended to A2). For comparison, A1, A2 and... Take a uniform depth ( (The number of layers for each type of gate in a round).

[0185] Layered version Regular block division is adopted;

[0186]

[0187] Indicates continuity The layer consists of only two-bit gates. The layers that make up the structure; , , Similar. Each layer has an edge set. The layers are laid out in the order of "even edges → odd edges → closed loop edges" to reduce conflicts within the same layer and ensure parallelism. A round of A2-Layered regularization consists of four layers connected in sequence: . : The number of layers of each type of gate in one round; thus the total depth of one round is . For fair comparison, the total depth of A1 (only ) and A2 ( full parameters) is uniformly taken as . If the -th layer gate type is denoted as and its parameter is , then the overall unitary is written as

[0188]

[0189] where , corresponding to .

[0190] Comparative experiments are carried out, so and have the same depth and .

[0191] Experiment 1: Gradient variance comparison

[0192] The initial state is taken as , and the observable , with

[0193] , as the objective function;

[0194] Symbol interpretation (task and loss):

[0195] : Input quantum state; is the all-zero state of

[0196] : Task-related observable (e.g., the tensor product of several );

[0197] : Hermitian conjugate; : Trace operation.

[0198] : Expectation value type loss (can be replaced by other differentiable losses such as cross entropy, and this application uses this form to unify the notation).

[0199] Used to measure the gradient and no parameter update is performed. For a given , under small variance random initialization, it is independently repeated times, and the The gradient vector initialized at this time is The sample statistics for each parameter are as follows

[0200]

[0201] The overall index is obtained by uniformly summarizing the parameters.

[0202]

[0203] : No. The gradient vector obtained from each independent random initialization; For its first Each component. : The number of independent initializations (each time it is expected that a fixed number of shots can be used to ensure statistical comparability). : No. Each component in The sample mean in this experiment; Its unbiased sample variance. : Parameter dimension (the total number of trainable angles). Given a number of bits With depth Below, the variance of each parameter will be... The overall index obtained by taking the arithmetic mean is used to compare the gradient variance levels of different circuit structures and depths.

[0204] Traverse multiple groups Three structures were obtained – Curves, such as Figure 3 As shown, where Figure 3 (a) is a comparison of the variance of the loss function of the model with different numbers of circuit layers for different qubits. Figure 3 (b) is a comparison of the variance of the loss function of the model with different numbers of circuit layers for different qubits. Figure 3 (c) is a comparison of the variance of the loss function of the model with different numbers of circuit layers for different qubits: The decay rate increases with depth; Slower decay; stratification Maintaining above in the medium to low depth range The variance level is reflected in the fact that it starts with a low-dimensional subalgebra and then introduces... Promotes trainability.

[0205] Experiment 2: Comparison of Classification Task Accuracy

[0206] The readout vector is taken as the bits of each bit. expect The model is then used for four-class classification using a classic linear algorithm, with the loss being cross-entropy. The comparison model is... , (Full Parameter) and Layering Layered training alternates between two phases: Phase I only updates... Parameters (Frozen) Phase II (Updated Only) (freeze The two phases alternate with fixed window lengths, and can be unfrozen and fine-tuned jointly at the end. All three models use the same optimizer and total number of rounds. Test set accuracy is as follows: Figure 4 Layering Superior to direct training And significantly better The variance trend is consistent with that of Experiment 1.

[0207] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any modifications or equivalent changes made based on the technical essence of the present invention shall still fall within the scope of protection claimed by the present invention.

Claims

1. A hierarchical training method for variational quantum algorithms based on Lie algebras, characterized in that, Includes the following steps: S1. Task and Circuit Setup: Determine the target task and loss function. Construct a ring-shaped 2-local spin chain quantum circuit and a generator set {XX, YY, XY, ZX}; S2. Generator subset partitioning and parameter grouping: Divide the generator set into generator subsets With the complete set of generators And divide the circuit parameters into subsets and subsets ; S3. Parameter and Hyperparameter Initialization: right Perform random initialization at a preset angle. Set to zero or add a perturbation; Set the learning rate, the maximum number of stages T1 / T2, the gradient variance threshold, the loss improvement threshold, and the measurement shot value; S4, Phase I: S41, only in the subset of generators Training: freeze Only for subsets Perform gradient estimation and update, calculate and record the hierarchical variance proxy V1 and the relative improvement rate of loss ΔL1; S42, Decision 1: Determine whether to proceed to the next step. If yes, proceed to step S5; otherwise, return to step S41. S5, Phase II: S51, Extending to the complete generator set train: freeze Activate the parameters corresponding to ZX In the complete set of generators The loss is calculated and the gradient is estimated on the constructed circuit, but only for... Perform gradient updates, calculate and record V2 and the relative improvement rate of loss ΔL2; S52, Decision 2: Determine whether to proceed to the next step. If yes, proceed to step S6; otherwise, return to step S51. S6, Joint Fine-tuning Phase: thaw and Reduce the learning rate in the complete generator set The joint optimization yields the optimal final parameters. ; S7. Result Output and Recording: Output the optimal parameters and corresponding quantum circuit performance indicators, and save the reproducibility information.

2. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 1, characterized in that, Step S1 is as follows: Let the adjacent edge set E be: ； Where j is the index of the qubit, and its value ranges from 1 to 10. N represents the total number of qubits in the system; adjacency relationships are based on modulus. The values ​​form a closed-loop topology, that is It is also an adjacent edge; The allowed 2-local generators on each edge are: ； in, , , These represent the actions of Pauli operators X, Y, and Z on the first... Tensor operators on qubits; Therefore, the generator set is obtained as .

3. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 2, characterized in that, Step S2 is as follows: S21. The specific partitioning of generator subsets is as follows: Define and classify two families of circuits and their induced dynamic Lie algebras: , ； , ； in, , Each represents the set of allowed two-bit generators; , Let be the dynamic Lie algebras spanned by the corresponding sets over the circular adjacency set via Lie bracket closures, respectively, at the same depth and bit size. The upper bound of the dimension is not less than the subset. , This constructs a subset of generators. To the complete set of generators The foundation of layered training; S22. Parameter grouping is as follows: Fixed in Hierarchical training is performed within this framework, with parameters grouped by generators: ； in, Corresponding generative subset , correspond .

4. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 3, characterized in that, Step S3 is as follows: Given a differentiable loss function In approximation Under the design assumption, its parameter dependency variance satisfies: ； Indicates in the parameter Under random initialization and measurement statistics, the loss function The variance; Or its task-related monotonic transformations, For the input quantum state, It is observable; For all trainable parameters, where correspond rotation angle, correspond Rotation angle of generator, To and Compatible hierarchical decomposition, Number of levels; For the first Hierarchical dimension Operator In subspace The Hilbert–Schmidt projection energy on the surface can be expressed using any orthogonal basis. Written as: ； The above formula reveals that the gradient / loss variance is determined by the hierarchical dimension. Input state With observable Projected energy at each level Joint decision, in To and Compatible hierarchical decomposition, The Hilbert–Schmidt projection energy.

5. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 4, characterized in that, Step S41 is as follows: Phase I: Activate and optimize only ,freeze At this point, the effective reachable algebra is limited to The specific steps of Phase I: S411, Initialization: , Keep frozen, usually set to 0 or the value from the previous stage; the learning rate is... The maximum number of rounds is The number of measurements was ,in The anchor value represents the second set of parameters frozen in stage I: taken in the first iteration. or After that, each round takes the stage II of the previous round in a fixed manner. The result obtained by optimization under the given conditions, i.e. ; S412. Gradient estimation, using the parametric offset method, with Pauli rotation as the default offset. : right Each scalar parameter ,remember Let its standard basis vectors be , ； Each expectation is derived from Subsampling estimation; S413, Optimization and Update: Employs the Adam / SGD optimizer. ; Gradient clipping and weight decay can be used; S414. Switching Criterion: If the hierarchical variance proxy is satisfied... or relative improvement rate or reach If the upper limit of the round is reached, then phase I ends; S415, Product: Obtained And preserve the optimizer's internal state for continuous training in subsequent stages.

6. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 5, characterized in that, Step S51 is as follows: Phase II: Freeze Only optimization This can be equivalently derived from algebra. Expand to Gradually restore complete The specific steps of Stage II: (This section discusses) the development of expressive abilities. S511, Initialization: Keep frozen. Or set to zero; the learning rate is... The maximum number of rounds is The number of measurements was ; S512. Gradient estimation, using the parameter offset method: right Each scalar parameter , ； Number of samplings The rest are in the same stage I; S513, Optimization and Update: ; S514, Switching / Termination: When or or reach When the upper limit is reached, Phase II ends.

7. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 6, characterized in that, Phase I / II , , , Window length , Use the same diameter.

8. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 6, characterized in that, threshold These are settable parameters.

9. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 1, characterized in that, Step S6 is as follows: The joint fine-tuning is as follows: Unfreeze Simultaneously participate in training Rounds, the learning rate decays to Record the final parameters And indicators.

10. The hierarchical training method for variational quantum algorithms based on Lie algebras according to claim 1, characterized in that, In step S7, reproducibility information includes training logs, random seeds, number of shots, and gradient variance and loss curves for each stage.

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