Method and model for generating quantum state set

Abstract

The invention discloses a method and a model for generating a quantum state set, and the method comprises the following steps: adding depolarization noise to any target quantum state through a non-unitary forward process, and converting the target quantum state into a completely mixed state; and de-noising the complete mixed state by using a trainable reverse process, and recovering the complete mixed state to the target quantum state. According to the method, more stable convergence performance is achieved based on the convex optimization method, the pure state and the mixed state can be effectively generated, excellent fidelity is achieved, and the huge potential in the aspect of coping with complex quantum generation tasks is shown.

Description

Technical Field

[0001] The present invention relates to the field of quantum machine learning, and in particular to a method and model for generating a set of quantum states. Background Art

[0002] Generative models have attracted much attention due to their wide application and remarkable success in the industrial field. Especially in the past two years, large models such as CHATGPT and STABLE DIFFUSION have performed well in human-machine dialogue and image generation, showing extraordinary creativity. They have achieved technological breakthroughs and have had a significant impact on our daily lives. The essence of generative models is to learn the probability distribution behind the data from a given dataset. Once trained, these models can generate new data following this distribution, thereby demonstrating their understanding of the underlying patterns in the data.

[0003] Quantum generative models, as the counterparts of classical generative models in quantum machine learning, have attracted great interest in the research community. These models combine the advantages of quantum computing and classical generative models, providing new possibilities for data generation. At present, various explorations of these models have been carried out in the field, including quantum generative adversarial networks (QGAN), quantum circuit Born machines (QCBM), quantum variational autoencoders (QVAE), and quantum Boltzmann machines (QBM). Among them, as a quantum version of the classical generative adversarial network (GENERATIVEADVERSARIAL NETWORK), QGAN has attracted widespread attention for its outstanding applications in generating quantum states, image generation, and fitting classical distributions. It provides researchers with a way to effectively simulate and explore complex quantum systems, especially in generating and processing data related to quantum states.

[0004] Although QGAN performs well in a variety of applications, it inherits the convergence problem of GAN. QGAN may suffer from model collapse, causing the losses of the generator and discriminator to oscillate and fail to converge to the global optimum. In addition, in the quantum realm, they face a unique challenge of failing to achieve convergence when generating mixed states. Specifically, according to the Helstrom measurement theory, under the minimax optimization strategy, the optimal analytical output of the generator is a pure state, which indicates that it cannot generate the target mixed state. The above challenges make QGAN difficult to train, limiting its widespread application in quantum state generation tasks. Summary of the invention

[0005] The present invention aims to provide a method and model for generating a set of quantum states, which can not only transform any target quantum state into a completely mixed state through a non-unitary forward process, but also recover the target state from the completely mixed state through a trainable reverse process, while having more stable convergence performance.

[0006] According to one aspect of the present invention, a method for generating a set of quantum states is provided, comprising the following steps: adding depolarization noise to an arbitrary target quantum state through a non-unitary forward process to convert it into a completely mixed state; and denoising the completely mixed state using a trainable inverse process to restore it to the target quantum state.

[0007] Preferably, depolarization noise is added using a depolarization channel according to the following formula: ρ t =ε(ρ t-1 ,t)=(1-α t )I / d+α t ρ t-1 Among them, ρ t is the quantum state with depolarization noise added, ρ t-1 is the quantum state at time step t-1, ε(ρ t-1 ,t) is the depolarization channel, α t is a scalar related to the time step, α t ∈[0,1], d represents the dimension of the Hilbert space, and I / d means that the quantum state is evenly distributed in the entire Hilbert space.

[0008] Preferably, the non-unitary forward process further comprises: For any t∈{1,…,T},ρ0 and ρ t The relationship between them is: Among them, ρ0 is the target quantum state, because It means that for all i.

[0009] Preferably, denoising the fully mixed state comprises a time step embedding circuit and a trainable denoising circuit U(θ) to form the function f Θ (ρ t ,t), where Θ={ω,θ}, ω and θ represent the trainable parameters in the embedding circuit at this time step, and t′=tπ / T is the result of mapping the time step t to the range [0,π]; The output state of the embedding circuit at this time step is: Among them, τ t is a state containing time information, Embed circuits for time steps The Hermitian conjugation of is used to recover the time information from the quantum state, |0><0| represents the projection operator of the ground state |0>, N is the ground state |0> time overlapping; The output state of the denoising circuit is: in, To represent the quantum state restored at time step t-1, For a composite system, tr B (·) is the partial trace operation on the subsystem, represents the Hermitian conjugation of U(θ).

[0010] Preferably, the training objective is to maximize the quantum fidelity between the predicted denoised state and the true value, and the loss function is defined as follows: in, is the overall loss function, and λ is used to balance the loss and The hyperparameters of represents a uniform distribution from 2 to T.

[0011] Preferably, the quantum fidelity function is defined as: F(·,·) is the quantum fidelity function used to measure the distance between two quantum states ρ and σ, and Tr(·) is the trace operation.

[0012] According to another aspect of the present invention, a model for generating a set of quantum states is provided, comprising: a noise adding module for adding depolarization noise to an arbitrary target quantum state through a non-unitary forward process and converting it into a completely mixed state; and a denoising recovery module for denoising the completely mixed state using a trainable inverse process to recover the target quantum state.

[0013] Preferably, depolarization noise is added using a depolarization channel according to the following formula: ρ t =ε(ρ t-1 ,t)=(1-α t )I / d+α t ρ t-1 Among them, ρ t is the quantum state with depolarization noise added, ρ t-1is the quantum state at time step t-1, ε(ρ t-1 ,t) is the depolarization channel, α t is a scalar related to the time step, α t ∈[0,1], d represents the dimension of the Hilbert space, and I / d means that the quantum state is evenly distributed in the entire Hilbert space.

[0014] Preferably, the model includes: A low-dimensional quantum data representation is learned from the original high-dimensional Hilbert space through parameterized circuits, thereby reducing the number of qubits required in the denoising process.

[0015] Preferably, the output state of the denoising process is: in, represents the quantum state recovered at time step t-1, tr4(·) is the trace operation, For a composite system, represents the Hermitian conjugation of U2(θ2); ρ′ t is t The quantum state obtained by deviating the first two qubits through the parameterized circuit U1(θ1), tr 12 (·) is the trace operation, ρ t represents the quantum state with depolarization noise added, represents the Hermitian conjugation of U1(θ1).

[0016] The present invention discloses a method and model for generating a set of quantum states, the method comprising the following steps: adding depolarization noise to any target quantum state through a non-unitary forward process to convert it into a completely mixed state; using a trainable reverse process to denoise the completely mixed state and restore it to the target quantum state. The method based on convex optimization of the present invention achieves more stable convergence performance, and can effectively generate pure states and mixed states, achieving excellent fidelity, showing great potential in coping with complex quantum generation tasks. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0018] Figure 1 A schematic diagram of the core concept of QGDM according to an embodiment of the present invention;

[0019] Figure 2A framework diagram of a quantum generation diffusion model (QGDM) according to an embodiment of the present invention;

[0020] Figure 3 Graphs of numerical simulation results according to the limitations of the design method according to an embodiment of the present invention, wherein (a), (b), and (c) are respectively graphs of training loss corresponding to the number of training times; a graph of the Hilbert-Schmidt (HS) distance between the input state and the output state of the denoising circuit U(θ); and a graph of the generation fidelity of the trained QGDM at different time steps;

[0021] Figure 4 A visualization diagram of a generated density matrix and a target density matrix according to an embodiment of the present invention;

[0022] Figure 5 A framework diagram of a denoising process according to an embodiment of the present invention;

[0023] Figure 6 According to an embodiment of the present invention, N τ =Architecture diagram of the time-step embedding circuit for 4 qubits;

[0024] Figure 7 4 is a schematic diagram of a 4-qubit denoising circuit U(θ) according to an embodiment of the present invention;

[0025] Figure 8 A design diagram of a reverse process in a resource efficient quantum generation diffusion model (RQGDM) according to an embodiment of the present invention;

[0026] Fig. 9 The QGDM in N under different quantum bit conditions when generating the target pure state according to an embodiment of the present invention is τ Aspect-generated fidelity graph;

[0027] Fig.10 A graph showing the pure state generation result according to an embodiment of the present invention, wherein circles represent outliers, i.e., data points that significantly deviate from other data;

[0028] Fig.11 A selected example diagram of a single-bit time-step embedded state according to an embodiment of the present invention;

[0029] Fig.12 A result graph of a mixed state generation according to an embodiment of the present invention;

[0030] Fig.13 A 4-bit circuit architecture diagram for preparing a target pure state according to an embodiment of the present invention;

[0031] Fig.14 is an indicator diagram in the training and generation process of QGDM according to an embodiment of the present invention;

[0032] Fig.15 is an indicator diagram (N=2, 3, 4, 5) in the training and generation process of RQGDM according to an embodiment of the present invention;

[0033] Fig.16 Graphs (N=6, 7, 8) of indicators in the training and generation process of RQGDM according to an embodiment of the present invention. DETAILED DESCRIPTION

[0034] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present application can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0035] Reference will now be made in detail to various embodiments of the present invention, examples of which are shown in the accompanying drawings and described below. For ease of interpretation and precise definition in the appended claims, the terms "upper", "lower", "inner" and "outer" are used to describe features of the exemplary embodiments shown in the drawings with reference to their positions.

[0036] The present invention protects a model for generating a set of quantum states, which can also be called a device for generating a set of quantum states.

[0037] Figure 1 , 2 , 3, and 4 respectively show the schematic diagram of the core idea of ​​QGDM according to an embodiment of the present invention; the framework diagram of the quantum generation diffusion model (QGDM); the training loss diagram corresponding to the number of training times, the Hilbert-Schmidt (HS) distance diagram between the input state and the output state of the denoising circuit U(θ), ​​the generation fidelity diagram of the trained QGDM at different time steps; and the visualization diagram of the generation density matrix and the target density matrix. As shown in the figure, an embodiment of the present invention provides a method for generating a set of quantum states, comprising the following steps: adding depolarization noise to any target quantum state through a non-unitary forward process to convert it into a completely mixed state; using a trainable inverse process to denoise the completely mixed state and restore it to the target quantum state.

[0038] In the prior art, the Quantum Generative Adversarial Network (QGAN) will encounter model collapse, resulting in oscillations in the losses of the generator and the discriminator, and it cannot converge to the global optimum. In addition, in the quantum field, convergence cannot be achieved when generating mixed states, and it cannot generate target mixed states. The embodiment of the present invention proposes a version of the quantum diffusion model, called QGDM. Any target quantum state can be diffused into a completely mixed state through a forward process. Then, through a learnable reverse process, the former can be recovered from the latter. Since the convex optimization-based method adopted by the proposed model can achieve more stable convergence, QGDM exhibits better convergence performance than QGAN. The comparison results with QGAN show that QGDM can effectively generate pure states and mixed states. In terms of generating mixed states, the fidelity of QGDM is 53.02% higher than that of QGAN.

[0039] According to an embodiment of the present invention, depolarization noise is added by using a depolarization channel according to the following formula: ρ t =ε(ρ t-1 ,t)=(1-α t )I / d+α t ρ t-1 Among them, ρ t is the quantum state with depolarization noise added, ρ t-1 is the quantum state at time step t-1, ε(ρ t-1 ,t) is the depolarization channel, α t is a scalar related to the time step, α t ∈[0,1], d represents the dimension of the Hilbert space, and I / d means that the quantum state is evenly distributed in the entire Hilbert space.

[0040] In the embodiment of the present invention, the depolarized channel model is introduced to quantify the evolution of the quantum state, making the influence of noise controllable, thereby improving the stability of the system and the accuracy of the output state. The formula shows that the input quantum state ρ t-1 With α t The probability remains unchanged, but changes to 1-α t The probability of being replaced by the completely mixed state I / d.

[0041] According to an embodiment of the present invention, the non-unitary forward process further includes: For any t∈{1,…,T},ρ0 and ρ t The relationship between them is: Among them, ρ0 is the target quantum state, because It means that for all i.

[0042] The embodiment of the present invention optimizes the efficiency of quantum state transfer, allowing direct transfer from the initial state to the target state at any time step, improving the efficiency of the generation process and saving computing resources. This formula allows direct transfer from ρ0 to the quantum state ρ at any selected time step. t It becomes possible to improve the efficiency of quantum state transfer at any time step.

[0043] Figure 5 , 6 7 respectively show the framework diagram of the denoising process according to an embodiment of the present invention, and τ = The architecture diagram of the time step embedding circuit of 4 quantum bits and the architecture diagram of the 4 quantum bit denoising circuit U(θ). As shown in the figure, according to an embodiment of the present invention, denoising the fully mixed state includes a time step embedding circuit and a trainable denoising circuit U(θ) to form the function f Θ (ρ t ,t), where Θ={ω,θ}, ω and θ represent the trainable parameters in the embedding circuit at this time step, and t′=tπ / T is the result of mapping the time step t to the range [0,π]; The output state of the embedding circuit at this time step is: Among them, τ t is a state containing time information, Embed circuits for time steps The Hermitian conjugation of is used to recover the time information from the quantum state, |0><0| represents the projection operator of the ground state |0>, N is the ground state |0> time overlapping; The output state of the denoising circuit is: in, To represent the quantum state restored at time step t-1, For a composite system, tr B (·) is the partial trace operation on the subsystem, represents the Hermitian conjugation of U(θ).

[0044] The embodiment of the present invention combines a denoising circuit and a time step embedding circuit, which enhances the model's ability to utilize time information, makes the denoising process more accurate, and improves the quality of reconstructed quantum states. Among them, a time step embedding circuit is used to generate a quantum state containing time information, which improves the expression ability of time information and effectively improves the performance and application potential of the generated state in quantum computing; a denoising circuit is used to process the contaminated quantum state and the time step embedded state to generate a denoised quantum state. By processing a composite system containing time information, the quantum state of time step t-1 is extracted, thereby effectively removing noise, restoring the target quantum state, and enhancing the reliability and accuracy of quantum information.

[0045] According to an embodiment of the present invention, the training goal is to maximize the quantum fidelity between the predicted denoised state and the true value, and the loss function is defined as follows: in, is the overall loss function, and λ is used to balance the loss and The hyperparameters of represents a uniform distribution from 2 to T.

[0046] The embodiments of the present invention have clear training objectives and loss function designs, which can effectively guide model learning, improve training efficiency, and achieve higher quality quantum state generation.

[0047] According to an embodiment of the present invention, the quantum fidelity function is defined as: F(·,·) is the quantum fidelity function used to measure the distance between two quantum states ρ and σ, and Tr(·) is the trace operation.

[0048] The quantum fidelity in the embodiments of the present invention provides an effective metric for quantifying the similarity between two quantum states.

[0049] According to another embodiment of the present invention, a model for generating a set of quantum states is provided, comprising: a noise adding module for adding depolarization noise to an arbitrary target quantum state through a non-unitary forward process and converting it into a completely mixed state; and a denoising recovery module for denoising the completely mixed state using a trainable inverse process to recover the target quantum state.

[0050] According to an embodiment of the present invention, depolarization noise is added by using a depolarization channel according to the following formula: ρ t =ε(ρ t-1 ,t)=(1-α t )I / d+α t ρ t-1 Among them, ρ t is the quantum state with depolarization noise added, ρ t-1 is the quantum state at time step t-1, ε(ρ t-1 ,t) is the depolarization channel, α t is a scalar related to the time step, α t ∈[0,1], d represents the dimension of the Hilbert space, and I / d means that the quantum state is evenly distributed in the entire Hilbert space.

[0051] Figure 8 A reverse process design diagram in a resource-efficient quantum generation diffusion model (RQGDM) according to an embodiment of the present invention is shown. As shown in the figure, the model also includes: learning a low-dimensional quantum data representation from the original high-dimensional Hilbert space through parameterized circuits, thereby reducing the number of quantum bits required in the denoising process.

[0052] The embodiments of the present invention reduce the number of quantum bits by learning low-dimensional representation, thereby improving resource utilization efficiency and reducing the complexity of the model, making it more suitable for practical applications.

[0053] According to an embodiment of the present invention, the output state of the denoising process is: in, represents the quantum state recovered at time step t-1, tr4(·) is the trace operation, For a composite system, represents the Hermitian conjugation of U2(θ2); ρ′ t is t The quantum state obtained by deviating the first two qubits through the parameterized circuit U1(θ1), tr 12 (·) is the trace operation, ρ t represents the quantum state with depolarization noise added, represents the Hermitian conjugation of U1(θ1).

[0054] The embodiment of the present invention provides an efficient denoising process design, which makes the output state closer to the target state, improves the accuracy and reliability of the generation process, and promotes the development of quantum generation technology.

[0055] Inspired by the classical diffusion model, this paper proposes a novel quantum generation model, called QGDM, for the quantum state generation task. Figure 1 The core idea is that on the Bloch sphere, any target quantum state (ρ0) can be transformed into a completely mixed state (ρ T); subsequently, a trainable inverse (denoising) process can be used to obtain T Recover ρ0.

[0056] The symbols are summarized in Table 1. Table 1 Symbols

[0057] In quantum computing, the state of an N-qubit can be represented by the Hilbert space C d A unit vector in C, denoted by |ψ>∈C d , where the ket symbol |> represents a column vector, d = 2 N is the dimension of the Hilbert space. represents a row vector, represents the conjugate transpose. The evolution of the pure state can be expressed as: |ψ′>=U|ψ>, where U satisfies The unitary operator of , I is the identity matrix.

[0058] A mixed state of a quantum system is a probabilistic mixture of pure states. It can be represented by a density matrix (or density operator): ρ = ∑ i p i |ψ i ><ψ i |, where the set of pure states {p i ,|ψ i >} indicates that the system has probability p i In state |ψ i >. The density matrix description is particularly useful when the exact state of the system is unknown or when considering subsystems of entangled pairs. The evolution of the mixed state is described as ρ′ =

[0059] The Hilbert space of a combined system consisting of two (or more) different quantum systems is the tensor product of the Hilbert spaces of the subsystems. Specifically, if the state of the subsystem is a pure state |ψ A > and |ψ B >, or mixed state ρ A and ρ B , then the state of the combined system can be expressed as or The tensor product operation allows us to describe the complete state of a combined system in terms of the individual states of its constituent parts.

[0060] Suppose there are two quantum systems, A and B. Their composite system state is represented by the density matrix ρ AB The deviation operation produces the reduced density matrix of subsystem A, which is calculated as ρ A =tr B (ρAB ).

[0061] Figure 2 The framework of QGDM is presented. It consists of a forward process (or diffusion process ε(ρ t-1 ,t) and a backward process (or denoising process f Θ (ρ t ,t).ε(ρ t-1 ,t) with N quantum bits of quantum state ρ t-1 and a time step scalar t as input, producing a quantum state ρ with added noise t Similar to the classical diffusion model, in QGDM, a “time step” refers to a discrete step in the model process, where noise is sequentially added (forward process) or subtracted (backward process) from the quantum state. In the present invention, ε(ρ t-1 ,t) is modeled as a depolarized channel. When t=0, ρ0 represents the target quantum state. When t=T, ρ T is a completely mixed state. The denoising process uses a model f with trainable parameters Θ Θ (ρ t ,t), the model is used to calculate the input noise quantum state ρ t De-noising to produce a cleaner quantum state ρ t-1 By T Application Θ After applying the (·,·) function T times, the target quantum state ρ0 can be restored.

[0062] In nature, diffusion is governed by nonequilibrium thermodynamics, driving the system to evolve toward an equilibrium state. As the system evolves, its entropy increases until it reaches a maximum value at equilibrium, indicating that evolution has stopped. The classical diffusion model was inspired by this diffusion mechanism. In quantum information theory, depolarized channels are considered a "worst case scenario." Under their influence, the quantum state gradually loses its original information and eventually transforms into a completely mixed state. This state is characterized by maximum entropy and represents the peak of randomness in the quantum system. At this point, the degree of unknown in the system is also at its maximum. A completely mixed state of N quantum bits is mathematically represented as This means that each computational basis state |i> has equal probability This uniform probability distribution across all computational ground states indicates that the state is maximally random, similar to the uniform distribution in classical data. At time t, the diffusion process of QGDM to an N-qubit quantum state ρ t-1 Add depolarization noise: ρ t =ε(ρ t-1 ,t)=(1-α t )I / d+α t ρ t-1,(1)

[0063] Where d = 2 N represents the dimension of Hilbert space, α t ∈[0,1] is a scalar associated with the time step.

[0064] Equation (1) shows that the input quantum state ρ t-1 With α t The probability remains unchanged, but changes to 1-α t The probability that the state is replaced by the fully mixed state I / d indicates the loss of the original information in the quantum system. Therefore, the depolarization channel is non-unitary because information is lost when the quantum state passes through the channel, making the process irreversible. By setting a suitable α t Sequence, a series of depolarization channel sequences acting on the quantum state ρ0 can finally obtain the output state ρ T ≈I / d.

[0065] Formula (1) mathematically describes a single-step diffusion process. However, the ε operation is repeatedly applied to obtain the quantum state ρ at a specific time step. t is inefficient. In order to improve the speed from the initial state ρ0 to the specific state ρ t The transfer efficiency of , introduce Theorem 1.

[0066] Theorem 1: For any t∈{1,…,T}, ρ0 and ρ t The relationship between can be described as:

[0067] in Therefore, due to

[0068] prove:

[0069] Therefore, formula (2) allows a direct transfer from ρ0 to the quantum state ρ at any chosen time step t become possible.

[0070] Select α t Must ensure And α t ∈[0,1] holds for all t. The present invention uses the cosine noise scheduling in the classical diffusion model to select α t :

[0071] in

[0072] s is a small bias hyperparameter. There are other noise schedules, such as linear noise schedules and noise schedules via learning.

[0073] In QGDM, the goal of the inverse process is to transform ρ T Restore ρ0. In the design process, two factors need to be considered.

[0074] First, since the denoising process is essentially the inverse of the non-unitary diffusion process, it must also be non-unitary for effective denoising. Second, in principle, if the denoising process is modeled as In the form of , T different denoising processes need to be trained to cope with T different denoising requirements. In order to reduce the number of trainable parameters in QGDM, the denoising process shares the same parameters by introducing the time step t as one of the inputs. This parameter sharing scheme is very useful for constructing f Θ (ρ t ,t) is crucial because it allows the model to effectively exploit temporal information to accurately estimate ρ t-1 , while keeping the number of parameters small.

[0075] In order to give the above two characteristics to the denoising process, the non-unitary characteristic is realized by offsetting part of the subsystem. In addition, the present invention uses parameterized circuits to embed time information into the quantum system. The present invention provides a denoising process f Θ (ρ t ,t) The design scheme is as follows Figure 5 As shown. Θ (ρ t ,t) consists of two modules: a time-step embedding circuit parameterized by ω where t′ = tπ / T, and a denoising circuit U(θ) parameterized by θ, so the trainable parameters are θ = {ω,θ}. Time step embedding circuit In the case of N τ The operation is performed on an auxiliary register of qubits to generate a state τ containing time information. t Then, the composite system After that, all qubits are regrouped into two registers: A (with N qubits) and B (with N τ The present invention deviates B to obtain the predicted denoised quantum state

[0076] When designing the denoising process, a key consideration is how to encode time information into the quantum state. This is a challenge from classical to quantum coding, and methods such as quantum bit encoding or probability amplitude encoding can be used to embed the time step t into the quantum system. However, the quantum bit encoding method only uniformly maps the time step sequence to a ring on the Bloch sphere, which lacks expressiveness and flexibility. The number of gates required by the amplitude encoding method is exponential to the number of quantum bits. In order to improve the expressiveness and flexibility of time step embedding while maintaining an appropriate gate complexity, the present invention uses a quantum embedding method to obtain a gate with N τ The time-step embedded state τ of the qubit t ,Right now:

[0077] where the scalar t′=tπ / T is the result of mapping the time step t to the range [0,π], and ω represents a trainable parameter in the time step embedding circuit.

[0078] Architectural details about the time-step embedding circuit and denoising circuit, with N τ = The architecture of the time step embedding circuit with 4 qubits is as follows Figure 6 As shown, the present invention uses the traditional ansatz for time step embedding with a trainable parameter ω, given the parameter t′=tπ / T. The circuit within the dotted line repeats times, among which The circuit includes single-qubit gates Rx and Ry, and a two-qubit gate The method proposed in the present invention maps time information to an appropriate position on the Bloch sphere (see the demonstration content in the numerical simulation for details), while maintaining the linear complexity of the number of quantum gates, which is sufficient to meet the requirements of the present invention.

[0079] The architecture of the 4-qubit denoising circuit U(θ) is as follows Figure 7 Figure 1 shows the architecture of the 4-qubit denoising circuit U(θ), ​​where θ represents a trainable parameter. To enhance its learning ability, the part within the dashed box is repeated L times, where l∈{1,…,L}. The circuit includes single-qubit gates Rz and Rx, and a two-qubit gate use This circuit architecture of the two-qubit gate may affect the efficiency of the method. However, to ensure that the effectiveness of the evaluation method is not affected by the circuit design, the present invention chooses a more complex architecture despite its increased gate complexity. This decision allows the results to accurately represent the potential of the present method. In addition, it is not necessary to maintain a consistent circuit architecture in all time steps. Adaptive circuits that change the architecture in different time steps can reduce the total number of gates in QGDM.

[0080] The training goal of QGDM is to maximize the predicted denoised state and the true value ρ t-1 The quantum fidelity between . The loss function is defined as follows:

[0081] Where λ is used to balance the loss and A smaller λ can improve the generation effect of QGDM. represents a uniform distribution from 2 to T. Therefore:

[0082] where F(·,·) is the quantum fidelity function, which is used to measure the distance between two quantum states ρ and σ:

[0083] The training of QGDM is implemented by Algorithm 1. In each iteration, the present invention calculates and expected value To update the trainable parameters until the training loss converges or the maximum number of iterations is reached. The expected value is estimated by the Monte Carlo method.

[0084] Once the parameters are trained, ω can be used * and θ * By iterative process from the fully mixed state ρ T Get the target state ρ0. Then, generate ρ0 through Algorithm 2. The algorithm starts from the fully mixed state and uses the trained denoising process to generate the target state through T steps. Therefore, its complexity is

[0085] The number of quantum bits required for the QGDM denoising process is N τ + N. In order to reduce the number of auxiliary qubits, the present invention proposes a new denoising process design, called RQGDM.

[0086] Inspired by quantum data compression methods, the core idea of ​​RQGDM is to first use a parameterized quantum circuit to learn a low-dimensional representation of quantum data from the original high-dimensional Hilbert space for the denoising process. This low-dimensional representation uses fewer quantum bits than the input state, thereby reducing the total number of quantum bits required for the denoising process. In RQGDM, only N+1 quantum bits are needed to generate a quantum state of N quantum bits, which improves its efficiency in resource utilization. Figure 8 An example of the denoising process circuit for a 3-qubit target state RQGDM is shown. t The information is compressed into the last qubit by parameterizing the circuit U1(θ1), where θ1 is its training parameter. This qubit contains the condensed information and thent Another circuit U2(θ2) acts on this composite state. After applying U2(θ2) to N+1 qubits, the last qubit of the system is discarded, and the reduced density matrix of the remaining N qubits is the predicted output state.

[0087] where τ t Given by formula (5), ρ t By deviating the first two qubits output from line U1, we get:

[0088] Therefore, the denoising process of RQGDM Θ (ρ t ,t) is Θ={ω,θ1,θ2}. QGDM and RQGDM share the same diffusion process, training and generation process, but differ in the denoising process.

[0089] The following is a detailed analysis of the denoising process of QGDM, and numerical simulation results are presented to support the analysis results.

[0090] An intuitive denoising process design f Θ (ρ t ,t) can use a trainable circuit U(θ) to train the composite system Then, τ t-subsystem (i.e. τ t All quantum bits of ) are deviated as follows:

[0091] However, the noise added in the diffusion step has a negligible effect on the quantum state, resulting in ρ t and ρ t-1 There is only a small difference between them. In this case, if the deviation track τ t registers and As the goal, it may mislead the optimizer to adjust the circuit U(θ) to produce an output close to the input state, thereby minimizing the training loss. However, this adjustment deviates from the intended role of the denoising circuit.

[0092] Figure 3 Numerical simulation results are presented that demonstrate the limitations of this design approach. Figure 3 (a) shows the decreasing trend of training loss with the number of training times. Figure 3(b) shows the Hilbert-Schmidt (HS) distance between the input and output states of the denoising circuit U(θ) during training. As a measure used in quantum information to quantify the distance between two quantum states, the HS distance is defined as Figure 3 (c) shows the fidelity of the QGDM generated state over time step t. The solid orange line represents the fidelity between the generated state and the target state, while the dashed blue line represents the theoretical fidelity curve that the model should achieve. This result indicates that the model cannot change the input state during the generation process.

[0093] Figure 3 (a)-(b) show that as the number of training times increases, the training loss and HS distance decrease rapidly. The decrease in HS distance indicates that the input state and output state The difference between becomes negligible and eventually converges to zero. At this point, for the t The deviation of the subsystem will be close to ρ t result. Figure 3 (c) shows that QGDM cannot change the input state during the generation process. The denoising process only outputs the input quantum state. Therefore, the fidelity between the generated quantum state and the target state remains unchanged.

[0094] Figure 4 The resulting quantum state is shown (first row) compared to ρ0 (second row). Given that the elements in the density matrix are complex numbers, the real and imaginary parts are plotted separately, with the real part on the left and the imaginary part on the right. This visualization clearly shows Still equal to the completely mixed state ρ T , indicating that U(θ) substantially reproduces the input state. This highlights the importance of a well-designed denoising process in QGDM that effectively integrates the noisy input state ρ t and the time step embedded state τ t information between them, especially when the differences in quantum states at adjacent time steps are small.

[0095] Finally, observe that U(θ) does not necessarily optimize to the identity matrix I every time, even though it seems straightforward to use I as a simple way to keep the input and output states consistent. The learning algorithm does not always choose to do this. This is because different sets of quantum states can be transformed into each other via U(θ) while maintaining the same density matrix. There may be multiple transformations in the hypothesis space that satisfy this condition. After training, which transformation is chosen depends on the position of U(θ) in the hypothesis space when the parameters θ are initialized.

[0096] The following numerical simulations evaluate the proposed model by generating various random quantum states, covering both pure and mixed states, and performing comparative analysis on the proposed model, QuGAN, and EQ-GAN. The implementation of QuGAN follows the original framework in Pennylane, while EQ-GAN is built using the source code in .

[0097] For details on the hyperparameter settings for model training, see Table 2 and Table 3 below. Table 2 Hyperparameter settings of QGDM model Table 3 Hyperparameter settings of the RQGDM model

[0098] Regarding the preparation of the target state, Fig.13 The circuit for obtaining the target pure state |ψ〉 is shown. All parameters in the target state generation circuit are uniformly sampled from the range [0,π].

[0099] The quantum mixed state is obtained by ρ = ∑ i p i |ψ i ><ψ i |Generate, where {|ψ i >} is passed Fig.13 The circuit shown is randomly generated. The probability vector {p i} is obtained by uniform sampling from (0,1] and normalized by a softmax operation to ensure that any index i satisfies ∑ i p i =1 and p i > 0. In this paper, the target mixed state is generated by probabilistically mixing two random pure states.

[0100] Regarding the training and generation curves of this model, Fig.14 The metrics during QGDM training and generation are shown, including the loss value and fidelity of the generated quantum states. Figure 15-16 The indicators of RQGDM are shown.

[0101] Each scenario (number of quantum bits, pure state or mixed state) includes two sub-graphs: the left sub-graph shows the trend of the training loss function value as the number of epochs increases, while the right sub-graph shows the fidelity as the time step changes. The red dotted line indicates the change in fidelity between the noise quantum state and the target state as the time step increases (indicated by the red arrow from right to left) during the diffusion process. The blue solid line indicates the trend of the fidelity between the generated quantum state and the target state during the generation process (indicated by the blue arrow from left to right). The shaded area in the figure indicates the range of the standard deviation.

[0102] The loss curves show that as the number of training epochs increases, the loss functions of QGDM and RQGDM converge to close to zero, and the number of epochs required to converge to zero increases with the increase of N, regardless of whether the target state is a pure state or a mixed state. In addition, generating mixed states requires more iterations than generating pure states, especially when N increases. This shows that generating mixed states is more challenging than generating pure states.

[0103] Regarding the fidelity boundary between any quantum state and a completely mixed state, from the quantum state ρ and the completely mixed state We begin with the definition of quantum fidelity F, where I is the identity matrix and d is the dimension of the Hilbert space. The fidelity is defined as:

[0104] There are three different situations to consider at this point:

[0105] (1) If ρ is a completely mixed state, the fidelity reaches its maximum value:

[0106] (2) If ρ is a mixed state that is not a completely mixed state, then the eigenvalue {λ k} are not necessarily equal, but are both non-negative and sum to 1. This results in:

[0107] Therefore, if ρ is a mixed state that is not a completely mixed state, the fidelity satisfies

[0108] (3) If ρ is a pure state (ρ = |ψ><ψ|), then ρ has a unique eigenvalue of 1, and its corresponding eigenvector is |ψ>. Therefore

[0109] In summary, any quantum state ρ and the completely mixed state The fidelity boundary between

[0110] Therefore, the fidelity of any quantum state ρ with the fully mixed state is For pure states, the fidelity between the target state and the fully mixed state is 1 / d. Therefore, the standard deviation of all diffusion fidelity curves (red dashed lines) for pure states is zero. For mixed states, the lower limit of fidelity is As N increases, random quantum states become more similar, resulting in a decrease in the variance of fidelity. Therefore, at t = T (i.e., the far left of the horizontal axis), both the fidelity value and the standard deviation decrease as N increases.

[0111] In all scenarios, the fidelity curve (solid blue line) during the generation process increases as the time step t decreases, eventually approaches 1, and has the smallest standard deviation. As the number of qubits increases, the overlap between the generation fidelity curve and the diffusion fidelity curve decreases. As the number of qubits increases, the difference between the generation fidelity curve and the diffusion fidelity curve increases. In addition, the generation fidelity curve (solid blue line) becomes steeper as N increases.

[0112] The present invention uses the Tensorcircuit framework to simulate quantum circuits, and uses the Tensorflow framework for parameter optimization, with Adam as the optimizer. During the training process, a larger learning rate may make it difficult for the model to converge, while setting a too small learning rate will make the model converge too slowly. In order to better optimize the model, the cosine decay method is used to dynamically adjust the learning rate.

[0113] The numerical simulations covered the number of qubits from 1 to 8. QGDM requires 2N qubits to generate a target state of N qubits. Therefore, in our simulations, QGDM can achieve the results for the case of N≤4. EQ-GAN can generate mixed states for N≤7, because the task of simulating N=8 requires an unrealizable amount of computer memory. When N=1, RQGDM degenerates to QGDM. Therefore, the following does not include the results of QGDM generating N>4 mixed states, the results of RQGDM when N=1, and the results of EQ-GAN when N=8. All simulations were repeated 10 times, and different random seeds were used to obtain the statistical characteristics of the results.

[0114] This paper first studies the τ The effect on the generation of pure state of QGDM. The results are as follows Fig. 9 As shown. τ ≤N, the generation fidelity increases with N τ The increase of N τ =N, it reaches its maximum value.

[0115] However, when N τWhen N > N, except for the case of N = 1, the fidelity decreases slightly in other cases. The reason may be the existence of a redundant qubit, which makes the denoising process complicated. The purpose of the denoising circuit is to predict the denoised state by integrating information from τ t and ρ t and then project its prediction onto the first N qubits of the τ state. These qubits are labeled as register A in t . In the case of N > N, there is one more qubit in the state τ Figure 5 than in ρ τ . This extra qubit may be redundant, thus complicating the process of combining and outputting the prediction. t This extra qubit may be redundant, thus complicating the process of combining and outputting the prediction. t When N < N, register A contains a part of the qubits of ρ

[0116] . However, the overlap between register A and the qubits of ρ τ may weaken the effectiveness of U(θ) in this case because part of the information of ρ t has already been stored in A. The above-mentioned content once mentioned an intuitive but defective denoising design, which maximizes the overlap between A and the qubits of ρ t by transferring all qubits from ρ t to the output register. This extreme case verifies the hypothesis and illustrates the potential impact. t This extreme case verifies the hypothesis and illustrates the potential impact. t Based on the conclusions in

[0117] , subsequent simulations are default set to N Fig. 9 = N because this setting can obtain the best generation results in the method proposed in our invention. τ

[0118] Fig.10 In , QGDM and RQGDM are compared with QuGAN and EQ-GAN to evaluate their fidelity in generating the target pure state. The results show that as the number of qubits in the target state increases, the fidelity of all models decreases.

[0119] ​At lower qubit numbers, QuGAN maintains high fidelity. However, as the number of qubits increases, its median fidelity decreases while the variance increases. For the case of N ≥ 7, QuGAN's fidelity is below 0.99 in all 10 experiments, indicating its limitations in dealing with more complex quantum states. EQ-GAN achieves perfect fidelity of 1.0 when the system has only one qubit. However, as the number of qubits increases, its fidelity decreases, showing the largest variance among the four models. Starting from 3 qubits, EQ-GAN's fidelity is always below 0.99, highlighting its challenges in generating pure states. The fidelity of QGDM and RQGDM remains basically above 0.99 for all qubit numbers, and the data distribution is more concentrated, indicating that they have better scalability than QGAN and EQ-GAN. In most cases, RQGDM surpasses the other three models in median fidelity and shows greater stability than its counterparts at higher qubit numbers. In summary, QGDM excels in smaller quantum systems, while RQGDM provides a more scalable solution across a wider range of qubit configurations.

[0120] To better evaluate the performance differences between the models, the generation fidelity values ​​for each model are given in Table 4. The fidelity of QuGAN decreases as the number of qubits increases, from 0.999±5e-5 when N=1 to 0.971±1e-1 when N=8. It is worth noting that for N≥6, its fidelity drops below 0.99, indicating that its performance decreases as the size of the quantum system increases. Although the fidelity of QuGAN from N=1 to 5 remains above 0.99, these values ​​are always lower than the level achieved by the proposed method. EQ-GAN achieves the best initial results at N=1 with a fidelity of 1. However, as the number of qubits increases, EQ-GAN shows the lowest average fidelity among the four models, while also having the highest variance. QGDM exhibits stable fidelity, with an average of 0.999 at N=1, 2, and 3, and 0.996±8e-3 at N=4. Importantly, EQ-GAN consistently exhibits high fidelity across all numbers of qubits, even reaching 0.99 at N = 8. This highlights the superior robustness and accuracy of RQGDM in pure state generation, underscoring the effectiveness and scalability of the proposed model. Table 4 Comparison of pure state generation effects Models N＝1 N＝2 N＝3 N＝4 N＝5 N＝6 N＝7 N＝8 QuGAN 0.999±5e-5 0.996±4e-3 0.995±4e-3 0.993±3e-3 0.990±4e-3 0.986±3e-3 0.978±6e-3 0.971±1e-1 EQ-GAN 1.00±0 0.978±5e-2 0.947±4e-2 0.896±8e-2 0.898±9e-2 0.872±9e-2 0.833±1e-1 0.824±8e-2 QGDM 0.999±3e-5 0.999±2e-4 0.999±2e-3 0.996±8e-3 / / / / QUR / 0.994±8e-3 0.999±6e-4 0.999±4e-4 0.997±4e-3 0.995±1e-2 0.994±4e-3 0.990±1e-2

[0121] Fig.11 Showing some trained sequences of single-qubit time-step embedded states Selected examples of , where different colors are used to represent the states at each time step. Each example shows that the position on the surface of the Bloch sphere changes consistently as . The density of states also varies. Fig.11 (a)-(b) show a more dispersed distribution on the surface of the Bloch sphere. Fig.11 (c)-(d) show a more concentrated state. Fig.11 (e)-(f) show a moderate case where the state sequence is moderately and uniformly distributed on the surface of the Bloch sphere. The method of the present invention allows the time-step embedded states to be adaptively distributed on the surface of the Bloch sphere while maintaining temporal consistency. This shows that the method of the present invention is expressive and flexible in encoding temporal information.

[0122] Compared to qubit encoding methods, our approach inspired by existing literature is more expressive and flexible in time-step embedding. The states generated by qubit encoding methods are uniformly arranged in a ring on the surface of the Bloch sphere. This restricts subsequent learning models to learn only from these fixed patterns. In contrast, our method allows the time-step embedded states to be adaptive, enhancing their versatility. This may reduce the difficulty of learning for subsequent models and improve the performance of the overall model.

[0123] Fig.12 The fidelity of four quantum generation models in generating mixed states is shown. The results show that the fidelity of QuGAN and EQ-GAN decreases as the number of qubits increases, while QGDM and RQGDM maintain high generation fidelity in multi-qubit tasks. Specifically, the fidelity of QuGAN continues to decrease with increasing N, with most fidelity points and medians dropping sharply from above 0.9 at N=1 to about 0.3 at N=8. EQ-GAN also shows a significant drop in fidelity, but maintains a better median and data point distribution compared to QuGAN. These findings indicate that EQ-GAN outperforms QuGAN in generating mixed states, but both have difficulty in multi-qubit mixed state generation tasks. In contrast, QGDM and RQGDM consistently show median fidelity above 0.99 in all scenarios. QGDM outperforms QuGAN and EQ-GAN at N=1, with most of its fidelity points above 0.99. RQGDM has a slight advantage over QGDM when N = 2, 3, and 4, and the data points of RQGDM are more concentrated. It is worth noting that when N ≥ 5, RQGDM significantly outperforms QuGAN and EQ-GAN, keeping all fidelity scores above 0.9.

[0124] The fidelity of the four methods is compared in detail in Table 5. The fidelity of QuGAN is 0.902±6e-2 when N=1, and decreases to 0.324±6e-2 when N=8. This trend highlights the limitation of QuGAN in scalability. EQ-GAN has a lower standard deviation in most cases (except for the case of N=3), and its average fidelity is generally higher than QuGAN. In contrast, QGDM and RQGDM maintain a fidelity greater than 0.99 in all cases, accompanied by a lower standard deviation, indicating their excellent scalability. It is worth noting that the method of the present invention achieves a fidelity of up to 0.999 in some cases, showing its excellent generation ability. In particular, RQGDM reaches 0.992±1e-2 when N=7 and maintains 0.993±1e-2 when N=8, indicating that the present model is able to maintain high fidelity generation on larger scale problems. Table 5 Mixed state generation results Models N＝1 N＝2 N＝3 N＝4 N＝5 N＝6 N＝7 N＝8 QuGAN 0.902±6e-2 0.694±1e-1 0.598±6e-2 0.477±9e-2 0.414±8e-2 0.439±9e-2 0.396±8e-2 0.324±6e-2 EQ-GAN 0.972±3e-2 0.867±8e-2 0.741±1e-1 0.676±7e-2 0.602±6e-2 0.555±6e-2 0.463±1e-1 / QGDM 0.995±7e-3 0.999±1e-3 0.992±1e-2 0.992±7e-3 / / / / QUR / 0.999±6e-4 0.999±5e-4 0.996±2e-3 0.993±4e-3 0.990±1e-2 0.992±2e-2 0.993±2e-2

[0125] In addition, the performance improvement of this model relative to the QGAN model can be quantified by calculating the relative changes of different model groups (i.e., this model and the QGAN-based model). By extracting the best fidelity value of each N in different model groups in Table 5 and calculating the average fidelity, the results show that this model shows a relative performance improvement of 53.02% compared to the QGAN model group.

[0126] The relative change is used below to quantify the improvement of the proposed model over the QGAN-based model in the mixed state generation task. First, the average of the best fidelity values ​​of the model at different N is calculated:

[0127] Then, the average of the best fidelity values ​​of the QGAN-based model at different N is calculated:

[0128] The relative performance improvement can be calculated as follows:

[0129] Compared with QuGAN and EQ-GAN, the method of the present invention shows significant advantages. The results show that QuGAN and EQ-GAN perform poorly in the task of generating mixed states. As the number of quantum bits increases, the fidelity decreases, revealing the limitations of these models in dealing with multi-bit tasks. This limitation may be due to the fact that the optimization strategy of QGAN relies on the maximum and minimum saddle point game, which is generally not as effective as using gradient descent for convex functions. It is well known that training classical GAN ​​models is prone to problems such as model collapse, instability, and non-convergence. With the exponential growth of the density matrix dimension, there may be higher requirements in the training phase, such as the need for a more appropriate learning rate. Under these conditions, the unstable training performance of QGANs may significantly affect the final generation results. In contrast, the training process of this model shows more friendly and stable training performance by minimizing convex functions using gradient descent. This may explain why this model outperforms QuGAN and EQ-GAN.

[0130] Inspired by the classical diffusion model, the present invention proposes a quantum generative diffusion model (QGDM) as a novel quantum generative method for generating a collection of quantum states. The basic concept is that any target quantum state can be converted into a fully mixed state through a non-unitary forward process. Subsequently, a trainable reverse process is used to reconstruct the target state from the fully mixed state. In the forward process, a time-step-dependent depolarization channel gradually converts the input state into a most random state, namely the fully mixed state. The denoising process adopts the concept of parameter sharing and is a parameterized function consisting of a time-step embedding circuit and a denoising circuit. Specifically, the time-step embedding circuit embeds time information into the quantum state, while the denoising circuit processes the contaminated quantum state and the time-step embedded state to produce the denoised state. In order to minimize the auxiliary qubits in QGDM, the present invention introduces a resource-efficient version called RQGDM. In numerical simulations with up to 8 qubits, the model of the present invention is compared with QuGAN and EQ-GAN. The results show that both QGDM and RQGDM outperform QuGAN and EQ-GAN in generating pure and mixed quantum states. In particular, our model achieves 53.02% higher fidelity than QGAN in the mixed-state generation task. These results highlight its excellent performance and scalability.

[0131] The present invention has the following advantages:

[0132] (1) More stable convergence performance: The present invention establishes a convex objective function to minimize the distance between the target quantum state and the generated quantum state, and optimizes the convex function through gradient descent to bring better convergence characteristics, thereby overcoming the convergence problem faced by QGAN when generating mixed states and presenting better convergence performance.

[0133] (2) Higher fidelity: In numerical simulations with up to 8 qubits, the model of the present invention was compared with QuGAN and EQ-GAN. The results show that QGDM and RQGDM outperform QuGAN and EQ-GAN in generating pure and mixed quantum states. In particular, the fidelity in the mixed state generation task is 53.02% higher than that of QGAN, showing excellent performance and scalability.

[0134] (3) Better ability to generate complex quantum states: The present invention proposes a method for generating a set of quantum states. In the forward process, a non-unitary depolarized channel model is adopted, so that any target quantum state can be diffused into a completely mixed state. In the reverse process, a bias operation is introduced to ensure non-unitary property, and a parameter sharing strategy and time information are used as input to reduce the number of trainable parameters, thereby improving its overall applicability in quantum state generation tasks.

[0135] (4) More efficient problem-solving capabilities: This paper introduces a resource-efficient version of QGDM (RQGDM) to minimize the number of auxiliary qubits required in the reverse process. The core idea is to use parameterized quantum circuits to extract a low-dimensional representation of quantum data from the original high-dimensional Hilbert space. This compression is crucial for the reverse process because the low-dimensional representation uses far fewer qubits than the input state. Therefore, RQGDM enhances the model's ability to efficiently solve large-scale problems.

[0136] The above embodiments are only examples for clearly explaining the present invention, and are not intended to limit the embodiments of the present invention. For those skilled in the art, other different forms of changes or modifications may be made based on the following description, and these various changes, modifications, substitutions and deformations derived from the principles and spirit of the present invention still fall within the scope of protection of the present invention.

Claims

1. A method for generating a set of quantum states, characterized in that: The following steps are involved: Through the non-unitary forward process, any target quantum state is converted into a fully mixed state by adding depolarized noise. The fully mixed state is denoised using a trainable inverse process to restore the target quantum state.

2. The method according to claim 1, characterized in that Depolarized channels are used and depolarized noise is added according to the following formula: r t =e(r t-1 ,t)=(1-a t )I / d+a t r t-1 Among them, ρ t is the quantum state with depolarization noise added, ρ t-1 is the quantum state at time step t-1, ε(ρ t-1 , t) is the depolarization channel, α t is a scalar related to the time step, α t ∈[0, 1], d represents the dimension of the Hilbert space, and I / d means that the quantum state is evenly distributed in the entire Hilbert space.

3. The method according to claim 2, characterized in that The non-unitary forward process further comprises: For any t∈{1,...,T},ρ0 and ρ t The relationship between them is: Among them, ρ0 is the target quantum state, because It means that for all i.

4. The method according to claim 1, characterized in that Denoising the fully mixed state includes a time step embedding circuit and a trainable denoising circuit U(θ) to form the function f Θ (ρ t , t), where Θ = {ω, θ}, ω and θ represent trainable parameters in the time step embedding circuit, and t′ = tπ / T is the result of mapping the time step t to the range [0, π]; The output state of the time step embedding circuit is: Among them, τ t is a state containing time information, Embed circuits for time steps The Hermitian conjugation of is used to recover the time information from the quantum state, |0><0| represents the projection operator of the ground state |0>, N is the ground state |0> time overlapping; The output state of the denoising circuit is: in, To represent the quantum state restored at time step t-1, For a composite system, tr B (·) is the partial trace operation on the subsystem, represents the Hermitian conjugation of U(θ).

5. The method according to claim 4, characterized in that The training goal is to maximize the quantum fidelity between the predicted denoised state and the true value. The loss function is defined as follows: in, is the overall loss function, and λ is used to balance the loss and The hyperparameters of represents a uniform distribution from 2 to T.

6. The method according to claim 5, characterized in that The quantum fidelity function is defined as: F(·,·) is the quantum fidelity function used to measure the distance between two quantum states ρ and σ, and Tr(·) is the trace operation.

7. A model for generating a collection of quantum states, characterized in that: include: A noise adding module, used to add depolarization noise to any target quantum state through a non-unitary forward process and convert it into a fully mixed state; A denoising and recovery module is used to denoise the completely mixed state using a trainable inverse process to recover the target quantum state.

8. The model according to claim 7, characterized in that Depolarized channels are used and depolarized noise is added according to the following formula: r t =e(r t-1 ,t)=(1-a t )I / d+a t r t-1 Among them, ρ t is the quantum state with depolarization noise added, ρ t-1 is the quantum state at time step t-1, ε(ρ t-1 , t) is the depolarization channel, α t is a scalar related to the time step, α t ∈[0, 1], d represents the dimension of the Hilbert space, and I / d means that the quantum state is evenly distributed in the entire Hilbert space.

9. The model according to claim 7, characterized in that The model also includes: A low-dimensional quantum data representation is learned from the original high-dimensional Hilbert space by parameterizing the circuit, thereby reducing the number of qubits required in the denoising process.

10. The model according to claim 9, characterized in that The output state of the denoising process is: in, represents the quantum state recovered at time step t-1, tr4(·) is the trace operation, For a composite system, represents the Hermitian conjugation of U2(θ2); ρ′ t is t The quantum state obtained by deviating the first two qubits through the parameterized circuit U1(θ1), tr 12 (·) is the trace operation, ρ t represents the quantum state with depolarization noise added, represents the Hermitian conjugation of U1(θ1).

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