Method for solving degenerate ground state of high-dimensional quantum system

Abstract

The invention discloses a method for solving a degenerate ground state of a high-dimensional quantum system, and belongs to the field of quantum calculation and machine learning. The invention provides a technical scheme based on fusion of a variable component sub-algorithm and a depth generation model, aiming at the problems of calculation complexity and solution completeness in degenerate ground state solution of a high-dimensional quantum system. Through a symmetric coding technology, information of a high-dimensional quantum system is mapped to a quantum bit symmetric subspace, and parameters required by a parameterized quantum circuit are generated in combination with a variational generation optimization network, so that variational simulation of the high-dimensional quantum system is realized. According to the method, the expected energy value, the parameter similarity and the regularization term are jointly optimized through the composite loss function, and all degenerate ground states covering the high-dimensional quantum system in single operation are ensured. According to the method, existing quantum bit platform computing resources are fully utilized, universality and high efficiency are achieved, the hardware threshold of high-dimensional quantum simulation is lowered, and new technical support is provided for actual application of quantum computing and quantum simulation.

Description

Technical Field

[0001] The present invention belongs to the fields of quantum computing and machine learning, and particularly relates to a method for solving the degenerate ground state of a high-dimensional quantum system based on a variational quantum algorithm and a deep generative model. Background Art

[0002] In recent years, with the rapid development of quantum computing technology, the research on high-dimensional quantum systems has become one of the important directions in the field of quantum information science. Compared with traditional two-level quantum systems (such as qubits), high-dimensional quantum systems have more degrees of freedom, higher information capacity, and more complex state space structures, which endow them with unique advantages and application potential in the fields of quantum computing, quantum simulation, and quantum communication. However, the complexity of high-dimensional quantum systems also brings many challenges, one of which is how to efficiently solve the degenerate ground state of high-dimensional quantum systems. The degenerate ground state refers to the situation in a quantum system where there are multiple different quantum states with the same lowest energy value. This phenomenon is common in many physical systems, such as antiferromagnetic materials, spin liquids, and strongly correlated electron systems. Understanding the properties of these degenerate ground states is of great significance for revealing the physical nature of such systems and designing new quantum devices. However, due to the exponential growth of the Hilbert space of high-dimensional quantum systems with the system scale, traditional numerical methods (such as exact diagonalization, quantum Monte Carlo methods, etc.) face huge challenges in terms of computational complexity and storage requirements when dealing with large-scale systems.

[0003] In response to these challenges, researchers have started to explore new computational paradigms, and among them, variational quantum algorithms, as a hybrid algorithm that combines quantum computing and classical computing, have emerged and quickly become an important research direction in the field of quantum computing. By combining the parallelism of quantum computing with the efficiency of classical optimization algorithms, variational quantum algorithms provide new ideas for solving the computational problems of high-dimensional quantum systems and have attracted extensive attention in the fields of condensed matter physics, quantum chemistry, and materials science. Among them, the Variational Quantum Eigensolver (VQE) is one of the most representative algorithms. VQE uses a parameterized quantum circuit to prepare a trial state and adjusts the circuit parameters with the help of a classical optimizer to solve the lowest ground state energy of the target Hamiltonian and its corresponding ground state. However, although VQE has shown potential in noisy medium-scale quantum systems, it still faces significant challenges when dealing with degenerate high-dimensional quantum systems. First, current mainstream quantum computing platforms and numerical simulation tools are mainly based on the qubit system and are difficult to be directly applied to the variational simulation of high-dimensional quantum systems. Second, the gradient-based optimization algorithm relied on by VQE takes the energy expectation value as the single optimization target and is difficult to effectively distinguish and generate multiple mutually different degenerate ground states.

[0004] Regarding the simulation challenges of high-dimensional quantum systems, two main solutions have been proposed in existing research: one is to design and construct a dedicated simulation platform based on high-dimensional quantum systems to directly simulate high-dimensional quantum systems; the other is to map high-dimensional quantum systems into qubit systems through appropriate encoding methods, so as to utilize existing qubit platforms for simulation. However, the first method is limited by current technical conditions and still faces huge challenges in experimental implementation; in contrast, the second method relies on mature qubit platforms and can overcome the difficulties of high-dimensional quantum system simulation through efficient encoding strategies, providing a feasible technical solution for related research.

[0005] Regarding the problem of solving degenerate ground states, the booming development of machine learning technology in recent years has provided new ideas for solving this problem. Machine learning, with its powerful data analysis and pattern recognition capabilities, has opened up new research perspectives and tools for solving quantum physics problems. For example, neural networks have been successfully applied to key problems such as quantum state reconstruction, quantum phase transition detection, and the solution of the ground state energy of quantum systems. In particular, deep generative models, by learning the complex probability distributions of high-dimensional data and generating samples similar to the training data, have achieved remarkable results in fields such as image recognition and generation, natural language processing, and molecular design. These characteristics make deep generative models a powerful tool for exploring complex quantum problems. Therefore, developing a method for solving high-dimensional degenerate ground states that combines variational quantum algorithms and deep generative models not only has important theoretical significance but also provides new technical support for the practical applications of quantum computing and quantum simulation. Summary of the Invention

[0006] To solve the above technical problems, the present invention proposes a method for solving the degenerate ground state of a high-dimensional quantum system by integrating variational quantum algorithms and deep generative models, achieving breakthroughs through the collaborative innovation of the following technical levels: mapping the quantum state, Hamiltonian, and quantum gates of a d-dimensional quantum system to the d - 1 qubit symmetric subspace, and using Dicke states to represent the information of the high-dimensional system; generating the required parameters of a parameterized quantum circuit (PQC) based on a Variational Generative Optimization Network (VGON), and jointly optimizing the energy expectation value, parameter similarity, and regularization term through a composite loss function; realizing the variational simulation of the high-dimensional system on a qubit platform to ensure that a single run covers all degenerate ground states of the high-dimensional quantum system, breaking through the technical bottlenecks in computational complexity and solution completeness of the existing technology. The specific technical solutions are as follows:

[0007] A method for solving the degenerate ground state of a high-dimensional quantum system, comprising the following steps:

[0008] Step 1: Use symmetric encoding technology to map the quantum state, Hamiltonian, and quantum gates of a high-dimensional quantum system to the symmetric subspace of qubits. Utilize the encoded symmetry-preserving quantum bit gates to construct a parameterized quantum circuit PQC that maintains symmetry. The symmetry-preserving PQC has a Hamiltonian measurement module;

[0009] Step 2: Construct a variational generation optimization network based on a deep generative model. Input random initial parameters into the variational generation optimization network and optimize to obtain the optimal circuit parameters based on the symmetry-preserving PQC;

[0010] Step 3: According to the optimal circuit parameters, drive the symmetry-preserving PQC to prepare a complete set of degenerate ground states of the high-dimensional quantum system.

[0011] The specific symmetric encoding technology is as follows:

[0012] Any d-dimensional quantum state is represented as a linear superposition of computational basis vectors where α k is the normalized complex amplitude, is the computational basis vector of the d-dimensional quantum state, k = 0, 1, …, d - 1; Through symmetric encoding, the computational basis vector is mapped to the qubit symmetric state |S k >, which is called the Dicke state:

[0013]

[0014] In the formula is the combination number, ∑ perm is the traversal summation for all possible permutations of |1>, |0> in denotes the tensor product of k qubits |1>;

[0015] Thus, any d-dimensional quantum state is encoded as a d - 1 qubit symmetric state:

[0016]

[0017] Encode the high-dimensional Hamiltonian and the high-dimensional quantum gate into the following Dicke state representation forms respectively:

[0018]

[0019] where H ij , u ij are the elements in the i-th row and j-th column of the matrix forms of the high-dimensional Hamiltonian and quantum gate respectively, is the high-dimensional computational basis vector, where \(i,j\in\{0,1,\ldots,d - 1\}\); \(P\) is the projection operator, to ensure that the operations are restricted within the symmetric subspace, is 2 d-1 dimensional identity matrix.

[0020] The encoded quantum gate \(U\) remains unitary for any parameter, and the state \(U|\psi\rangle\) after acting on \(|\psi\rangle\) is still a symmetric state of qubits. Based on the symmetry-preserving quantum gate \(U\), a parameterized quantum circuit that preserves symmetry can be designed, abbreviated as symmetry-preserving PQC.

[0021] The variational generation optimization network includes the following modules:

[0022] Encoder: Maps randomly initialized parameters to a low-dimensional latent space to generate a latent distribution;

[0023] Latent space: Samples latent variables from the latent distribution through reparameterization;

[0024] Decoder: Decodes the latent variables into output parameters with the same structure as the initial parameters and inputs them into the symmetry-preserving PQC as the angular parameters of the parameterized quantum gates;

[0025] Symmetry-preserving PQC: Prepares a symmetric trial state using the output parameters, and then measures the energy expectation value of the symmetrically encoded Hamiltonian.

[0026] The specific implementation steps of step 2 are as follows:

[0027] Step 2.1: Input a batch of randomly initialized parameters \(\Theta_0\) into the encoder in the VGON, and obtain the mean \(\mu\) of the normal distribution in the latent space through the encoder mapping ω and variance to obtain the latent distribution

[0028] Step 2.2: Calculate the regularization term between the latent distribution and the standard normal distribution

[0029] Step 2.3: Use the reparameterization trick to sample the latent variable from the latent distribution where \(\epsilon\) is a random variable sampled from the standard normal distribution ; Input the latent variable into the decoder in the VGON to obtain the output parameter \(\Theta\);

[0030] Step 2.4: Based on the batch training mechanism, calculate the similarity between all output parameters within the current batch Let all the output parameters within a batch be where θ i (i = 1, 2, …, n b ) are the output parameters of the i-th group of samples in this batch, and n b is the batch size;

[0031] Step 2.5: The output parameter Θ is input into the symmetry-preserving PQC as the angular parameter of the parametric quantum gate. Through the trial quantum state |ψ(θ i )> = U(θ i )|0> corresponding to all the parameters within the batch, the energy expectation value of the Hamiltonian is measured:

[0032]

[0033] where U(θ i ) is the symmetry-preserving unitary evolution operator generated by the i-th group of parameters;

[0034] Step 2.6: Construct a composite loss function by weighted summation of the above regularization term similarity and the energy expectation value E(Θ):

[0035]

[0036] where the weights λ E , λ S , λ R are adjustable hyperparameters; use classical optimization algorithms to dynamically adjust the weights of the encoder and decoder to minimize the loss function;

[0037] Step 2.7: Repeat the above training and optimization process of the encoder and decoder weights until the loss function reaches the convergence condition and then stop the iteration;

[0038] Step 2.8: After the training is completed, obtain the optimal circuit parameters generated by the decoder.

[0039] Combining the above symmetric coding technology with the variational generation optimization network (VGON) can effectively solve the problem of solving the degenerate ground state of high-dimensional quantum systems. VGON realizes the collaborative optimization of the following objectives by constructing a composite loss function that weights and linearly combines multiple loss terms:

[0040] 1. The energy expectation value of the Hamiltonian of the high-dimensional quantum system By constructing a measurement module of the Hamiltonian in the symmetry-preserving PQC, measure the energy of the trial state under the action of the symmetrically encoded Hamiltonian, and push the trial state closer to the ground state;

[0041] 2. The similarity metric between all the generated parameters of the decoder within a batch (such as Euclidean distance and cosine similarity, etc.) to ensure the diversity of the generated parameters, avoid convergence to a single solution, and thus cover all possible degenerate ground states;

[0042] 3. Regularization term of the VGON model It can regulate the distribution of the latent space, prevent parameter overfitting, and improve the training stability and model generalization ability, such as the Kullback-Leibler (KL) divergence between the latent distribution and the standard normal distribution.

[0043] The mathematical expression of this composite loss function can be written as: where the weights λ E , λ S , λ R are adjustable hyperparameters.

[0044] Finally, use classical optimization algorithms (such as optimizers like SGD, Adam, etc.) to dynamically optimize the weights of the encoder and decoder to minimize the loss function, generate the optimal circuit parameters, and drive the symmetry-preserving PQC to prepare a complete set of degenerate ground states of the high-dimensional quantum system.

[0045] The beneficial effects of the present invention are as follows:

[0046] Compared with directly using a dedicated platform based on high-dimensional quantum systems to simulate high-dimensional quantum systems, mapping high-dimensional quantum systems to the qubit system for variational simulation through symmetric encoding technology can make full use of the existing qubit computing platform resources. In addition, the symmetric encoding algorithm used in the present invention is universal and can efficiently encode high-dimensional quantum systems of any dimension without specifically constructing specific hardware or algorithms for different dimensions, thus significantly reducing the hardware threshold and development cost for implementing complex high-dimensional quantum simulations. This method not only improves the flexibility of the existing quantum computing platform but also provides an extensible and efficient solution for variational simulation of high-dimensional quantum systems.

[0047] Moreover, existing variational quantum algorithms (such as the variational quantum eigensolver VQE) that update circuit parameters based on gradient optimizers can only obtain one ground state in one solution process. To prepare all ground states, multiple rounds of solutions are required, but this method cannot ensure that the ground states obtained in each round are different from each other, so it is difficult to find a complete set of degenerate ground states. In contrast, the present invention uses VGON to generate the optimal circuit parameters, and can solve a complete set of degenerate ground states with computational resources similar to those used by VQE. Description of the Drawings

[0048] Figure 1 It is a schematic flowchart of the method for solving degenerate ground states of high-dimensional quantum systems provided by an embodiment of the present invention. Detailed Embodiments

[0049] To better understand the purpose, structure, and function of the present invention, the following further describes in detail a method for solving the degenerate ground state of a high-dimensional quantum system according to the present invention with reference to the accompanying drawings.

[0050] In an embodiment of the present invention, a quantum computing simulator on a classical computer (such as Qiskit, Pennylane, or MindQuantum) is used as a verification platform to simulate the process of solving the degenerate ground state of a high-dimensional quantum system, as Figure 1 shown, and the specific steps are as follows:

[0051] Step 1: Select a d = 3-dimensional quantum system with a degenerate ground state (such as a three-level system), use the symmetric encoding technique to map the quantum state, Hamiltonian, and quantum gates of the 3-dimensional quantum system to the symmetric subspace of qubits, and construct a symmetry-preserving parameterized quantum circuit (Parameterized Quantum Circuit, PQC) using the encoded symmetry-preserving qubit gates. The symmetry-preserving PQC has a corresponding Hamiltonian measurement module;

[0052] The specific symmetric encoding algorithm is as follows:

[0053] The computational basis vectors of the 3-dimensional quantum state are mapped to the qubit symmetric state Dicke state |S k >

[0054]

[0055] The high-dimensional quantum state The high-dimensional Hamiltonian and the high-dimensional quantum gates are respectively mapped to the qubit form according to the above formula and where α k is the normalized complex amplitude, H ij , u ij are respectively the elements in the i-th row and j-th column of the matrix forms of the Hamiltonian and the quantum gate (i, j = 0, 1, 2), and the projection operator is used to ensure that the operation is limited within the symmetric subspace, and is the 4-dimensional identity matrix.

[0056] Step 2: Input a batch of random initial parameters into the VGON, and obtain the optimal circuit parameters based on the symmetry-preserving PQC;

[0057] Among them, VGON, as a deep generative model, is composed of the following modules: The encoder maps randomly initialized parameters to a low-dimensional latent space to generate a specific latent distribution; the latent space samples latent variables from the latent distribution through the reparameterization trick to support the backpropagation of gradients during the training process; the decoder decodes the latent variables into output parameters with the same structure as the initial parameters and inputs them into the symmetric-preserving PQC as the angular parameters of parametric quantum gates (such as R x , R y , R z , etc.); the symmetric-preserving PQC prepares a symmetric trial state using circuit parameters and then measures the energy expectation value of the symmetrically encoded Hamiltonian. The specific implementation method is as follows:

[0058] Step 2.1: Input a batch of randomly initialized parameters Θ0 into the encoder in VGON, and map them through the encoder ε ω to obtain the mean μ ω of the normal distribution in the latent space and the variance thus obtaining the latent distribution

[0059] Step 2.2: Calculate the KL divergence between the latent distribution and the standard normal distribution ;

[0060] Step 2.3: Use the reparameterization trick to sample latent variables from the latent distribution where ε is a random variable sampled from the standard normal distribution . Input the latent variable into the decoder of VGON to obtain the output parameter Θ;

[0061] Step 2.4: Based on the batch training mechanism, calculate the cosine similarity between all output parameters within the current batch. Let all output parameters within the batch be where θ i (i = 1, 2,..., n b ) is the output parameter of the i-th group of samples in this batch, and n b is the batch size.

[0062] The cosine similarity between all output parameters is where ‖θ i ‖ is the 2-norm of the parameter θ i , is the set composed of all possible combinations of two elements in the set {1, 2,..., n b}, and

[0063] Step 2.5: The output parameter Θ is input into the symmetry-preserving PQC as the angular parameter of a parametric quantum gate (such as R x , R y , R z , etc.). Through the trial quantum states |ψ(θ i )> = U(θ i )|0> corresponding to all parameters within a batch, the energy expectation value of the Hamiltonian is measured as follows:

[0064]

[0065] where U(θ i ) is the symmetry-preserving unitary evolution operator generated by the i-th set of parameters.

[0066] Step 2.6: Construct a composite loss function by weighted summation of the above KL divergence, cosine similarity, and energy expectation value:

[0067]

[0068] where the weight is an adjustable hyperparameter. The Adam optimizer is used to jointly optimize the weights of the encoder and decoder to minimize the loss function.

[0069] Step 2.7: Repeat the above training and optimization process for the weights of the encoder and decoder until the loss function reaches the convergence condition, then stop the iteration.

[0070] Step 2.8: After the training is completed, the optimal circuit parameter Θ * ;

[0071] Step 3: According to the optimal circuit parameter Θ * , a complete degenerate ground state |ψ(Θ * )> = U(Θ * )|0> of a 3-dimensional quantum system is prepared through the symmetry-preserving PQC U(Θ * ).

[0072] It can be understood that the present invention is described through some embodiments. Those skilled in the art know that, without departing from the spirit and scope of the present invention, various changes or equivalent replacements can be made to these features and embodiments. Additionally, under the teaching of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application belong to the scope protected by the present invention.

Claims

1. A method for solving the degenerate ground state of a high-dimensional quantum system, characterized in that: The following steps are involved: Step 1: Use symmetric coding technology to map the quantum state, Hamiltonian and quantum gate of the high-dimensional quantum system to the symmetric subspace of the quantum bit, and use the encoded symmetry-preserving quantum bit gate to construct a symmetry-preserving parameterized quantum circuit PQC. The symmetry-preserving PQC has a Hamiltonian measurement module. Step 2: Construct a variational generative optimization network based on a deep generative model, input random initial parameters into the variational generative optimization network, and obtain the optimal circuit parameters based on symmetry-preserving PQC optimization; Step 3: According to the optimal circuit parameters, drive the symmetry-preserving PQC to prepare a complete set of degenerate ground states of the high-dimensional quantum system.

2. A method for solving the degenerate ground state of a high-dimensional quantum system according to claim 1, characterized in that: The symmetric encoding technology is specifically as follows: Any d-dimensional quantum state is represented as a linear superposition of computational basis vectors Among them, α k is the normalized complex amplitude, is the computational basis vector of the d-dimensional quantum state, k = 0, 1, ..., d-1; the computational basis vector is encoded by symmetry Mapped to quantum bit symmetry state |S k >, called the Dicke state: In the formula is the number of combinations, ∑ perm To traverse and sum All possible permutations of |1>,|0>, represents the tensor product of k qubits |1>; Thus, any d-dimensional quantum state is encoded into a d-1 quantum bit symmetric state: The high-dimensional Hamiltonian And high-dimensional quantum gates They are respectively encoded into the following Dicke state representations: Among them, H ij ,u ij are the i-th row and j-th column elements in the form of high-dimensional Hamiltonian and quantum gate matrix, respectively. is the high-dimensional computing basis, i,j∈{0,1,…,d-1}; P is the projection operator, To ensure that the operation is confined to the symmetric subspace, For 2 d-1 dimensional identity matrix. The encoded quantum gate U maintains unitary property under arbitrary parameters, and the state U|ψ> after acting on |ψ> is still a quantum bit symmetric state. Based on the symmetry-preserving quantum gate U, a parameterized quantum circuit that maintains symmetry can be designed, which is referred to as symmetry-preserving PQC.

3. A method for solving the degenerate ground state of a high-dimensional quantum system according to claim 2, characterized in that: The variational generation optimization network includes the following modules: Encoder: maps randomly initialized parameters to a low-dimensional latent space to generate a latent distribution; Latent space: Latent variables are sampled from the latent distribution through reparameterization; Decoder: decodes the latent variables into output parameters with the same structure as the initial parameters, which are input into the symmetry-preserving PQC as the angle parameters of the parameterized quantum gate; Symmetry-preserving PQC: The output parameters are used to prepare symmetric trial states, and then the energy expectation value of the symmetric encoded Hamiltonian is measured.

4. A method for solving the degenerate ground state of a high-dimensional quantum system according to claim 3, characterized in that: The specific implementation steps of step 2 are as follows: Step 2.1: Input a batch of random initial parameters Θ0 into the encoder in VGON, and obtain the mean μ of the normal distribution in the latent space through the encoder mapping ω and variance So we get the potential distribution Step 2.2: Calculate the latent distribution With standard normal distribution The regularization term between Step 2.3: Use the reparameterization trick to extract the latent distribution The latent variable z is sampled from ω +εσ ω , where ε is derived from a standard normal distribution A random variable sampled from ; The latent variable z is input to the decoder D of VGON φ The output parameter Θ is obtained from Step 2.4: Based on the batch training mechanism, calculate the similarity between all output parameters in the current batch Suppose all output parameters in the batch are where θ i (i=1,2,…,n b ) is the output parameter of the i-th group of samples in the batch, n b is the batch size; Step 2.5: The output parameter θ is input into the symmetry-preserving PQC as the angle parameter of the parameter-containing quantum gate. The test quantum states |ψ(θ i )>=U(θ i )|0>, and measure the energy expectation value of the Hamiltonian: Among them U(θ i ) is the symmetry-preserving unitary evolution operator generated by the i-th group of parameters; Step 2.6: Add the above regularization term Similarity The composite loss function is constructed by weighted summation of the expected value of energy E(Θ): The weight λ E ,λ S ,λ R is an adjustable hyperparameter; the weights of the encoder and decoder are dynamically adjusted using classic optimization algorithms to minimize the loss function; Step 2.7: Repeat the above training and optimization process of the encoder and decoder weights until the loss function reaches the convergence condition and then stop the iteration; Step 2.8: After training is completed, the optimal circuit parameters generated by the decoder are obtained.

5. A method for solving the degenerate ground state of a high-dimensional quantum system according to claim 4, characterized in that: The regularization term is calculated using KL divergence.

6. A method for solving the degenerate ground state of a high-dimensional quantum system according to claim 5, characterized in that: The similarity is specifically cosine similarity or Euclidean distance.

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