Two-qubit and three-qubit entanglement certification method based on variational quantum circuits

Abstract

The application discloses a two-bit and three-bit unknown state entanglement determination method based on a variational quantum circuit, and comprises the following steps: generating quantum states of a plurality of known state labels based on uniform sampling of a random matrix to obtain a quantum state data set containing a two-bit data set and a three-bit data set; extracting observable operator expectation values from the quantum state data set to generate a classical feature vector; encoding the classical feature vector to an initial quantum state through a parameterized quantum gate; constructing a variational quantum circuit, and making the initial quantum state evolve through the variational quantum circuit to obtain an output state; constructing an objective function based on quantum measurement results of the output state, and outputting an entanglement determination result of the quantum state data according to the objective function value. Compared with a quantum state tomography method which depends on complete density matrix reconstruction, the method only uses observable operator expectation values highly related to entanglement determination as input, and thus significantly reduces measurement resource consumption and classical post-processing burden.

Description

Technical Field

[0001] This invention relates to the field of quantum information, and specifically to a method for determining the entanglement of two-bit and three-bit unknown states based on variable quantum circuits. Background Technology

[0002] Quantum entanglement is a core nonlocal resource in quantum information processing, playing an irreplaceable role in several cutting-edge fields such as quantum communication, quantum key distribution, and quantum teleportation. Accurately determining the entanglement characteristics of unknown states in two-qubit and three-qubit quantum systems is a prerequisite for achieving efficient quantum information processing.

[0003] Currently, the most common method for determining whether unknown quantum states are entangled is full quantum state tomography (QST). This method reconstructs the density matrix by performing complete measurements on the quantum system, and then uses mathematical tools such as the partial transpose criterion (PPT) to make a determination. However, the number of measurements required for full quantum state tomography increases with the number of qubits. The increase is shown Exponential growth. For high-dimensional bit systems, the QST method requires enormous measurement resources and classical post-processing time, making it often difficult to implement or extremely inefficient in practical experiments.

[0004] With the development of artificial intelligence technology, existing research has attempted to use classical neural networks or machine learning models to learn the characteristics of unknown quantum states for entanglement classification. However, the Hilbert space dimension of quantum systems grows exponentially with the increase of the number of qubits (e.g., The density matrix dimension of each qubit is This leads to a situation where, as the dimensionality of a quantum system increases, classical computers face an insurmountable "curse of dimensionality" when simulating quantum state evolution, storing massive amounts of state parameters, and training neural networks. Computational complexity and memory consumption rise sharply, making it difficult to handle high-dimensional and complex quantum systems.

[0005] In contrast, the inherent quantum parallelism of quantum computing enables it to process information efficiently directly in Hilbert space without collapsing or mapping quantum states into massive classical data vectors. Therefore, using quantum algorithms to handle quantum state determination problems theoretically possesses exponential speedup potential and scalability unmatched by classical algorithms.

[0006] Therefore, there is an urgent need for a method that can avoid the high resource consumption of full tomography while efficiently and accurately determining whether two-qubit and three-qubit unknown quantum states are entangled. Summary of the Invention

[0007] To address the problems existing in the prior art, this invention provides a method for determining the entanglement of two-qubit and three-qubit unknown states based on variable quantum circuits. The aim is to overcome the curse of dimensionality in classical computing by utilizing the parallelism of quantum circuits, while avoiding the exponential resource consumption of full quantum state tomography by utilizing local feature measurements. A unified architecture for determining the entanglement of unknown quantum states compatible with two-qubit and three-qubit quantum systems is constructed, and high-precision entanglement determination of two-qubit and three-qubit quantum states is achieved using variable quantum algorithms with few samples and low feature dimensions.

[0008] To achieve the above objectives, this invention provides a method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits, comprising: Based on uniform sampling of a random matrix, several quantum states with known state labels are generated, resulting in a quantum state dataset containing two-bit and three-bit datasets. The expected values ​​of observable operators are extracted from the quantum state dataset to generate classical feature vectors that characterize the entanglement properties of quantum state data; The classical eigenvectors are encoded into an initial quantum state using parameterized quantum gates; Construct a variable quantum circuit, and let the initial quantum state evolve through the variable quantum circuit to obtain the output state; An objective function is constructed based on the quantum measurement results of the output state, and the entanglement determination result of the quantum state data is output according to the value of the objective function.

[0009] Preferably, the generation of quantum states with known state labels based on uniform sampling of a random matrix includes: A random density matrix was generated based on the Ginibre random matrix to obtain several single-bit quantum quanta; In the two-bit dataset, separable state data is generated by convex combination of the direct product of two single-bit quantum states, and entangled state data is obtained by filtering randomly generated two-bit quantum states based on a partial transpose criterion. In the three-bit dataset, the fully separable state data is generated by convex combination of the product states of three single-bit quantum states, while the bound entangled state data is constructed by obtaining the orthogonal complement states based on a non-expandable product basis containing four orthogonal states.

[0010] Preferably, the process of extracting the expected values ​​of observable operators from the quantum state dataset includes: For a two-bit dataset, four observable operators are constructed based on the CHSH inequality. The expected values ​​of the four observable operators are calculated for each operator, and the four expected values ​​are combined as the four-dimensional classical feature vector of the two-bit state. For a three-bit dataset, four observable operators are constructed based on the Mermin inequality and their expected values ​​are calculated to generate a four-dimensional classical eigenvector; eight observable operators are constructed based on the Svetlichny inequality and their expected values ​​are calculated to generate an eight-dimensional classical eigenvector; and twelve observable operators are constructed based on the CHSH-like inequality and their expected values ​​are calculated to generate a twelve-dimensional classical eigenvector.

[0011] Preferably, the step of encoding the classical feature vector into the initial quantum state through parameterized quantum gates using a symmetric scaling angle encoding strategy specifically includes: The statistical boundary of each dimension of the classical eigenvector in the quantum state dataset is determined based on the maximum and minimum values ​​of each dimension of the eigenvector. Linear rescaling based on the statistical boundary maps each dimension of the classical feature vector to a preset angle range, serving as the angle input parameter for the quantum gate.

[0012] Preferably, the variable quantum circuit is configured with k data qubits and 1 auxiliary qubit for the k-dimensional input characteristics of the initial quantum state; The variable quantum circuit includes an angle encoding layer and a training layer; The angle encoding layer encodes each dimension of the classical feature vector into the corresponding data qubit through a Y-axis rotation gate based on the angle input parameter. The training layer includes two training sub-layers. In each training sub-layer, X-axis rotation gate, Z-axis rotation gate, and X-axis rotation gate are sequentially applied to the k+1 qubits of each initial quantum state, and IsingZZ coupling centered on the auxiliary qubit is applied.

[0013] Preferably, the objective function constructed based on the quantum measurement results of the output state includes: Repeated sampling measurements are performed on the auxiliary qubit, and the probability distribution of the auxiliary qubit is estimated based on the frequency of the target ground state in the measurement results. The target function value is then obtained based on the probability distribution.

[0014] Preferably, the objective function is constructed using a dynamic probability flipping mechanism, including: When the true label of the quantum state data corresponds to the first category, the probability of the auxiliary qubit being in the first ground state is used as the probability of successful prediction; the first category represents the entanglement state of the quantum state data. When the true label of the quantum state data corresponds to the second category, the difference between the probability of 1 and the probability of the auxiliary quantum bit being in the first ground state is used as the probability of successful prediction. The quantum gate parameters of the variable quantum circuit are iteratively trained based on the predicted success probability by constructing a negative log-likelihood loss function.

[0015] Preferably, the method further includes: using a parameter translation method to translate the quantum gate parameters in the variable quantum circuit in positive and negative directions and calculating the difference of the objective function to obtain the gradient information of the quantum gate parameters, so as to update the parameters of the variable quantum circuit.

[0016] Preferably, the method further includes: employing a measurement-time annealing strategy during iterative training to gradually increase the number of samplings of the auxiliary qubits with the number of iterations, and combining a mini-batch stochastic gradient descent strategy and a learning rate adaptive decay strategy to iteratively optimize the variable quantum circuit.

[0017] Preferably, when the quantum state data is a two-bit dataset, a separable state is output when the objective function value is greater than a preset judgment threshold; otherwise, an entangled state is output. When the quantum state data is three bits, the output is a fully separable state if the objective function value is greater than the preset judgment threshold; otherwise, the output is a bound entangled state.

[0018] The present invention provides a method for determining the entanglement of two-bit and three-bit unknown states based on variable quantum circuits, which has the following beneficial effects: 1. This invention utilizes only the expected values ​​of observable operators highly correlated with entanglement determination as direct inputs to the variable quantum circuit for processing, significantly reducing measurement resource consumption and classical post-processing burden, without requiring quantum state tomography to reconstruct the complete density matrix. The optimal parameters of the quantum circuit can be found by optimizing the loss function. Compared to quantum state tomography, this invention avoids the exponential growth in resource consumption with the number of qubits, significantly reducing quantum resource consumption and time costs during the determination process.

[0019] 2. This invention leverages the inherent quantum parallelism of quantum computing to directly evolve and process high-dimensional quantum states in Hilbert space, avoiding the mapping of quantum states to massive classical data vectors. For two-qubit quantum systems, this invention uses the expectation value feature of the CHSH operator to determine separable and entangled states. For three-qubit quantum systems, this invention supports feature construction schemes based on various observable operators such as Mermin, Svetlichny, and CHSH-like operators, enabling the classification of fully separable and bound entangled states within a unified framework, enhancing the system's adaptability and application flexibility. Compared to classical neural network methods, this invention overcomes the "curse of dimensionality" and classical computing bottlenecks that arise with increasing qubit count, providing a scalable technical path for handling higher-dimensional multi-qubit entanglement determination problems.

[0020] 3. This invention constructs a unified decision architecture for entanglement of unknown quantum states in two-bit and three-bit systems. At the same time, through mechanisms such as parameter shifting, dynamic probability flipping, negative log-likelihood loss, measurement-time annealing, and adaptive decay of learning rate, it improves the trainability, convergence, and noise resistance of variable quantum circuits. It can flexibly adapt to the decision requirements of different two-bit and three-bit systems and has strong system compatibility and application flexibility. Attached Figure Description

[0021] Figure 1 This is a flowchart illustrating a two-bit and three-bit unknown state entanglement determination method based on variable quantum circuits according to the present invention. Figure 2 This is a schematic diagram of the quantum circuit structure for realizing different dimensional feature inputs according to the present invention; Figure 3 This is the present invention. Figure 2 A schematic diagram of the detailed decomposition structure of the composite quantum gate U1; Figure 4 This is a schematic diagram of the quantum circuit structure for achieving two-bit state differentiation according to the present invention; Figure 5 This is a schematic diagram illustrating the results of the entanglement determination method of the present invention for a two-bit entanglement determination task; Figure 6 This is a schematic diagram of the training results of the entanglement determination method of the present invention for a three-bit entanglement determination task under the condition of four-dimensional classical feature vector input. Figure 7 This is a schematic diagram of the training results of the entanglement determination method of the present invention for a three-bit entanglement determination task under the condition of eight-dimensional classical feature vector input. Figure 8 This is a schematic diagram illustrating the training results of the entanglement determination method of the present invention for a three-bit entanglement determination task under the condition of twelve-dimensional classical feature vector input. Detailed Implementation

[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] The terms “first,” “second,” “third,” “fourth,” etc. (if present) in this specification, claims, and accompanying drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of this application described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms “comprising” and “having,” and any variations thereof, are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0024] To facilitate understanding of this embodiment, the technical terms involved in this embodiment of the invention will first be explained: 1. Product state: In quantum mechanics, if a state consists of two subsystems (e.g., product state), then the product state is a state in which two subsystems (e.g., product state) are formed. and The state of a composite system composed of ( ) can be written as the tensor product of the states of its individual subsystems, in pure state form: ( (In tensor product or density matrix form): Then this state is a direct product state. Its physical meaning is: System and The systems are independent of each other and have no connection with each other (including quantum and classical connections).

[0025] 2. Partial transposition: for composite systems density matrix Select one of the subsystems (e.g.) Transpose.

[0026] 3. Separable state: A quantum state that can be decomposed into a classical probabilistic mixture of subsystem direct product states. In terms of physical operations, this state can be prepared solely through local operations and classical communication, without the need for prior sharing of entangled resources.

[0027] 4. Convex Combination: A set of objects Their convex combination is a summation with probability weights: The probability of and .

[0028] 5. Entangled State: When the quantum state of a many-body system cannot be written as a convex combination of the direct product states of its subsystems, it is called an entangled state. Entangled states possess nonlocal correlations that transcend classical physics.

[0029] 6. Bound entangled state: The partial transpose of its density matrix is ​​still positive semi-definite, that is, it satisfies the PPT condition. Conventional PPT criteria will misclassify it as a separable state, but in fact it still has non-classical correlations that cannot be prepared by classical methods.

[0030] The following describes a method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits, as disclosed in an embodiment of the present invention. Figure 1 As shown, the method includes: Step S1: Generate several quantum states with known state labels based on uniform sampling of the random matrix, and obtain a quantum state dataset containing two-bit datasets and three-bit datasets; Step S2: Extract the expected values ​​of observable operators from the quantum state dataset to generate classical feature vectors that characterize the entanglement properties of quantum state data; Step S3: Encode the classical feature vector into the initial quantum state through parameterized quantum gates; Step S4: Construct a variable quantum circuit, and let the initial quantum state evolve through the variable quantum circuit to obtain the output state; Step S5: Construct an objective function based on the quantum measurement results of the output state, and output the entanglement determination result of the quantum state data according to the objective function value.

[0031] This invention constructs a complete and closed-loop entanglement determination method through a five-layer structure: data generation, feature extraction, quantum state encoding, variational evolution, measurement, and determination. First, in the data generation layer, a large amount of data on two-qubit separable / entangled states and three-qubit fully separable / bound entangled states with known labels is generated using mathematical methods such as the Ginibre matrix, PPT criterion, and UPB construction. In the feature extraction layer, this method does not perform quantum state tomography; it only extracts key physical quantities. By calculating the expected value of a set of observable operators constructed based on specific inequalities, the high-dimensional density matrix (4x4 or 8x8 complex matrix) is compressed into a low-dimensional real-valued feature vector (4D, 8D, or 12D). Determination can be completed with only a small number of measurements, avoiding the exponential measurement overhead of quantum state tomography. In the quantum encoding and variational evolution layers, the quantum encoding layer uses a rotation gate to map the classical feature vector to a quantum state through angular encoding; the variational evolution layer directly processes and evolves this quantum state in Hilbert space, ultimately outputting measurement results that can be directly used for classification. Finally, the measurement and decision layer can automatically determine the entanglement properties of unknown quantum states without reconstructing the complete density matrix, thereby reducing the overall cost of quantum measurement and the burden of classical post-processing.

[0032] The following section details each layer of this method. First, in the data generation layer, the system uses the Ginibre method to generate a large number of random quantum states that conform to physical laws, thereby constructing a high-quality labeled training set. Specifically, before generating quantum states of a specific category, a general generator needs to be defined. This can be understood as the state of a single qubit being determined by a 2×2 density matrix. A complete description is required, and three physical constraints must be satisfied: Hermitianness ( semi-definiteness (i.e., the eigenvalue is non-negative) and the trace is 1 ( ).

[0033] Therefore, step S1, which generates quantum states with known state labels based on uniform sampling of a random matrix, includes: Step S11: Generate a random density matrix based on the Ginibre random matrix to obtain several single-bit quantum states.

[0034] A Ginibre random matrix is ​​a random matrix whose elements all follow an independent standard Gaussian distribution. Its core idea is to construct a physically valid density matrix by generating random complex matrices that follow a Gaussian distribution, and the generated states statistically follow the Hilbert-Schmidt measure. Let... It is A random complex matrix of dimension (both real and imaginary parts follow a Gaussian distribution). When the dimension is... When a single-bit random density matrix is ​​specifically generated, the formula for generating the random density matrix is: in, for The conjugate transpose of; The trace of a matrix is ​​the sum of its diagonal elements. This is a random density matrix used to describe quantum states. Among them, the two-qubit quantum... A 4x4, three-qubit quantum The dimensions are 8x8. This method guarantees semi-positive definiteness and Hermitianness, ensuring that the generated quantum state data samples are uniform in Hilbert space, i.e., unbiased sampling.

[0035] Step S12, (1) In the two-bit dataset, the separable state data is generated by the direct product state composed of two single-bit quantum states through convex combination, and the entangled state data is obtained by screening the randomly generated two-bit quantum states according to the partial transpose criterion; Step S121, Obtain two-bit separable states: In this embodiment, the label of the separable states in the two-bit dataset is recorded as 0. By definition, a two-bit separable state is a convex combination of direct product states: , in, It is a two-qubit quantum state. It is the first system with a single-bit quantum state. It is a single-bit quantum state of the second system. These are probability weights.

[0036] Specifically, you can use `np.random.dirichlet` in the NumPy library to generate probability weights. The above random density matrix is ​​called repeatedly to generate single-bit states. , By expanding the tensor product np.kron and linearly superimposing it according to the weights, several two-bit separable states are obtained.

[0037] Step S122, Obtain the two-bit entangled state: In this embodiment, the label of the entangled separable state in the two-bit dataset is 1. According to the PPT criterion, for a two-bit system, the partial transpose matrix... The existence of negative eigenvalues ​​is a necessary and sufficient condition for entanglement.

[0038] It should be noted that the PPT criterion is: if If it is separable, then its partial transpose matrix is... , All are positive semi-definite matrices, meaning all eigenvalues ​​are non-negative. Conversely, if their partial transposes are... or If nonnegative eigenvalues ​​exist, then It must be in an entangled state.

[0039] Specifically, we introduce negativity: ,in, Represents eigenvalues. This represents the negative eigenvalues ​​of a partially transposed matrix. It involves globally sampling a random density matrix and calculating the sum of the absolute values ​​of the negative eigenvalues ​​of its partially transposed form. If... They are marked as entangled states and preserved.

[0040] After obtaining the two-bit separable and entangled states, data standardization is performed: the generated (N,4,4) dataset is flattened into a (N,16) feature matrix.

[0041] (2) In the three-bit dataset, the fully separable state data is generated by the convex combination of the product state composed of three single-bit quantum states, and the bound entangled state data is obtained by solving its orthogonal complement state from the non-expandable product basis containing four orthogonal states.

[0042] In a three-qubit quantum system, a fully separable state can be represented as a convex combination of the direct product of three single-qubit quantum states. Essentially, it only exhibits classical correlation and cannot provide quantum advantage in any task requiring entanglement. Bound entangled states, on the other hand, satisfy the PPT-mixture criterion and thus possess "weak" quantum correlation, exhibiting superior performance compared to separable states in secure key generation, state transitions / catalysis, and certain quantum metrology tasks. This distinction directly impacts detection methods: fully separable states can be verified using separability decomposition or simple criteria; while bound entangled states require more sophisticated tools such as entanglement witnessing, the PPT-mixture criterion, and semidefinite programming, because they satisfy the PPT criterion but remain entangled. Furthermore, in real physical processes (noise, thermalization), bound entanglement often appears as residual quantum correlation in decoherence processes, while full separability signifies that the system's quantum correlation has degenerated into purely classical correlation.

[0043] Step S123, Obtain the fully separable states of a three-bit dataset: In this embodiment, the label of the separable states in the three-bit dataset is 0. By definition, a fully separable three-bit state can also be written as a convex combination of direct product states: in, Representing a three-qubit quantum state, , , These are the single-bit quantum states of the first, second, and third systems, respectively. These are probability weights.

[0044] Step S124, obtain the three-bit bound entangled state: In this embodiment, the label of the bound entangled state in the three-bit dataset is recorded as 1.

[0045] This example constructs a bound entangled state based on a three-qubit system. It builds an unextendable product basis (UPB) containing four pairwise orthogonal states and selects its orthogonal complement state as the target bound entangled state. The product basis satisfies the non-extendable condition that there are no additional orthogonal product states in space.

[0046] These four non-extendable product bases are denoted as ,in: The normalized form of a three-bit bound entangled state is: Represents a three-bit bound entangled state. It is an identity matrix.

[0047] To generate a sufficient number of bound entangled states for training, this embodiment also employs a method of randomized local operations supplemented by classical communication. Specifically, this involves the direct product of three independent randomized invertible matrices: Will Acting on respectively Each qubit, that is: in , , This method is consistent with the generation of the single-bit density matrix in fully separable states. It achieves enhanced sampling at the distribution level by changing only the specific orientation and eigenstructure of the state without altering its entanglement type or PPT properties.

[0048] After obtaining the three-bit fully separable state and bound entangled state, data standardization is performed: the generated (N,8,8) dataset is flattened into a (N,64) feature matrix.

[0049] This embodiment employs convex combinations of direct product states, PPT criterion screening, and a bound entangled state construction method based on non-scalable direct product bases to ensure that training samples and objects to be identified have clear physical meaning and reliable label sources. This not only improves the targeting of training and judgment but also enables the model to cover more challenging three-qubit bound entanglement identification scenarios, significantly enhancing the system's applicability and ability to identify complex types of quantum entanglement.

[0050] Secondly, there's the feature extraction layer. The most common method for calculating entanglement criteria is to obtain complete information about the quantum states needed for the calculation through quantum state tomography. However, in quantum state tomography, we need to obtain all the information of the density matrix of a quantum state. The original density matrix contains a large number of independent unknowns (15 for 2-Qubit and 63 for 3-Qubit), which greatly increases the consumption of quantum resources. For quantum entanglement criteria, numerous theoretical calculations have proven that determining whether a quantum state is entangled does not require all the information of that quantum state, but only the expected values ​​of certain observable operators.

[0051] In this embodiment, the feature extraction layer is used to simulate the quantum measurement process, transforming the high-dimensional, abstract complex density matrix into a low-dimensional, observable real physical quantity, i.e., the expected value measured by observable operators.

[0052] Preferably, step S2, which involves extracting the expected values ​​of the observable operators from the quantum state dataset, includes: Step S21: For a two-bit dataset, construct four observable operators based on the CHSH inequality, calculate their expected values ​​in each two-bit quantum state, and combine them as the four-dimensional classical eigenvector of that two-bit state.

[0053] It should be noted that this step involves designing four specific physically observable operators to transform the high-dimensional abstract quantum state into classical data. The expected values ​​of these four observable operators are then obtained to form the input features of the quantum machine learning model.

[0054] It should be noted that the linear combination of the above four specific measurements constitutes the CHSH inequality, that is... .in , These represent the measurement directions of the first qubit. , These represent the measurement directions of the second qubit; Represents direction and The expected value of the combined measurement. These are CHSH polynomial values ​​used to determine whether classical locality is violated. For separable states, they must satisfy... For a given entangled state, this value can reach up to the quantum limit. Therefore, constructing observable operators based on CHSH-type inequalities can directly capture the degree to which quantum states violate classical local realism, thus providing a strong physical basis for entanglement determination.

[0055] Specifically, in quantum mechanics, observable operators correspond to actual measurement operations in experiments. For a two-bit system, the measurement directions of the two bits are first defined: the first bit uses... (Z-direction measurement) (X-direction measurement); the second bit uses (Measured along a 45-degree diagonal). (Measured along a 135-degree diagonal). Here... , This represents the Pauli matrix, which represents the physical spin measurement axis.

[0056] Combining the tensor products of the measurement directions of the two bits yields four observations: Having the observables and the density matrix of the quantum state to be determined, the feature extraction of the quantum state is to calculate the expected value of each two-qubit state under the four observables, which is also known as the trace operation: in, For any two unknown quantum states, , , , For observation purposes.

[0057] The four expected values ​​were finally calculated and combined together. This constitutes the four-dimensional classic feature vector used for training.

[0058] Step S22: For the three-bit dataset, extract the four-dimensional classical feature vector based on Mermin's inequality, extract the eight-dimensional classical feature vector based on Svetlichny's inequality, and extract the twelve-dimensional classical feature vector based on CHSH-like inequalities.

[0059] The Hilbert space dimension of a three-bit system is from leap to The entangled structure exhibits true multi-body entanglement (such as GHZ states and W states) and complex cases of bound entanglement. A single CHSH-type inequality is insufficient to fully describe the nonlocality and entangled structure of a three-qubit system. Therefore, this embodiment introduces three physical criteria at different levels of complexity. By calculating the expected values ​​of observable operators constructed based on Mermin's inequality, Svetlichny's inequality, and CHSH-like inequalities, an increasing feature dimension is designed as the input feature of the quantum circuit. This approach captures the weak quantum correlation signals of the three-qubit bound entangled state from different levels of physical criteria.

[0060] (1) For four-dimensional features, the Mermin-type inequality is used: Mermin's inequality is: ,in This is the Mermin value, used to detect multi-body entanglement. , It is the first bit observable operator; , It is the second-bit observable operator; , It is the third observable operator. This represents the expected value.

[0061] Four specific three-body correlation operators were selected based on Mermin's inequality: The measurement direction is set as follows: For the first bit, the following is used: , The second bit, using , The third bit uses , Based on these directions, we define four observable operators: For any quantum state Calculate the expected values ​​of these observable operators respectively, forming a four-dimensional eigenvector: .

[0062] That is, the classic eigenvector is Mermin's inequality is insensitive to certain bound entangled states and has a small feature space, which may limit classification accuracy (e.g., ...). Figure 6 The display accuracy is relatively low.

[0063] (2) For eight-dimensional features, the Svetlichny inequality is applied: in This is the Svetlichny value.

[0064] An observable operator is constructed based on Svetlichny-type inequalities. This involves iterating through the tensor products of all combinations of the first two bits and the third bit combination, resulting in... A complete observable operator: First consider the same set of measurement directions: Based on this, eight observables acting on the three subsystems can be defined: Furthermore, we define an eight-dimensional eigenvector: .

[0065] That is, the eight-dimensional classical feature vector is The 8-dimensional features provide more complete correlation information and can better distinguish between three-bit completely separable states and bound entangled states with weak entanglement properties (see [link]). Figure 7 (Results).

[0066] (3) For twelve-dimensional features, a CHSH-like inequality is used: In the construction of the twelve-dimensional feature, the construction method of the CHSH inequality is referenced. It can be understood that the CHSH inequality is a criterion for testing nonlocality in a two-body system. Its basic idea is to select two different measurement directions on each subsystem, thereby generating four observable operators. For a three-body system (A, B, C), if only the correlation between any two subsystems is considered, such as the correlation between A and B, then the measurement operators for the remaining subsystems can be regarded as the identity operator I. This yields four observation operators completely consistent with the two-body CHSH framework. However, to avoid the homogenization problem caused by fixed angle selection, random directions on the Bloch sphere are used in the construction of this embodiment. For unit vectors... The operator is defined as: Based on this, for the three-body system, any two bodies are selected as the subsystems to be tested: that is, the three-bit system is decomposed into three pairs of two-body relationships: AB, AC, and BC. For each pair of two-body relationships, four observable operators are constructed using the CHSH approach, for a total of 3 x 4 = 12 observation operators.

[0067] (i) AB: The four corresponding observations are: (ii) AC: The four corresponding observations are: (iii) BC: The four corresponding observations are: Next, define the feature vectors and their concatenation order: Thus, the complete twelve-dimensional classical eigenvector is obtained: Used to capture residual quantum correlations between pairwise subsystems, see [link / reference]. Figure 8 The results.

[0068] This embodiment refines and limits feature extraction, not relying on all information from the high-dimensional density matrix, but extracting low-dimensional features from the expected values ​​of observable operators closely related to entanglement properties. For two-bit systems, CHSH-type features are used; for three-bit systems, Mermin-type four-dimensional features, Svetlichny-type eight-dimensional features, and CHSH-like twelve-dimensional features are compatible. This design can preserve entanglement-related discriminative information as much as possible while compressing dimensions, reducing input redundancy, improving the physical interpretability and classification effectiveness of features, and allowing the same system to flexibly switch feature schemes according to different experimental conditions and accuracy requirements.

[0069] The next layer is the quantum encoding layer, which constructs a variable quantum classification model that uses parameterized quantum circuits to find the optimal hyperplane in Hilbert space. Since the core object processed by quantum computers is quantum states... Since the input data is typically classical feature vectors, a mapping mechanism from classical data to quantum parameters must be constructed. This embodiment employs an angle encoding strategy based on symmetric scaling to transform classical features into the rotation angle of a quantum rotation gate.

[0070] Specifically, firstly, the statistical boundary (maximum value) of each dimension of the quantum state dataset obtained above is calculated. and minimum value Then apply the linear transformation formula: in, For the input of the first The first of the classic eigenvectors Features in each dimension. The first linear transformation The first of the classic eigenvectors Features in each dimension, that is, the angular input parameters of the quantum gate.

[0071] when hour, ; when hour, The intermediate value is then linearly mapped proportionally.

[0072] The features are scaled using a linear transformation. This transformation strictly constrains all input features to... Within the closed interval, it is ensured that the encoded angle parameters can completely cover the periodic evolution space of the quantum rotation gate (such as Rx, Rz, etc.) (i.e., cover the complete unitary space), thereby maximizing the ability of the qubit to express feature information.

[0073] This embodiment employs a symmetric scaling angle encoding strategy to uniformly map classical feature vectors to a preset angle range, thereby enabling input features with different dimensions and amplitude ranges to drive the quantum rotation gate in a consistent manner. This avoids the encoding imbalance problem caused by excessive differences in the original feature distribution, and on the other hand, allows the encoding angle to more fully cover the periodic evolution space of the quantum rotation gate, improving the quantum state representation capability. Consequently, it enhances the stability and comparability of classical feature vectors when inputting to the quantum circuit, thereby improving the learning efficiency and robustness of the variable quantum circuit for different types of entanglement boundaries.

[0074] Next is the variational evolution layer, such as Figures 2-4 The diagram shows a schematic representation of the variable quantum circuit in this embodiment. Figure 4 A schematic diagram of the two-bit variable quantum circuit in this embodiment is given. In the diagram, n=4 / 8 / 12 represent the input dimensions, and the nth bit is an auxiliary bit.

[0075] The core of the model is parameterized quantum circuits. , These are the trainable parameters for the quantum gate. The model uses a "data qubit + auxiliary qubit" approach. For a k-dimensional input, k data qubits and 1 auxiliary qubit are used. The data qubits are numbered as follows: The total number of sub-bits is k+1.

[0076] The circuit includes: 1) Angle encoding, which maps the feature vector to a rotation angle around the Y-axis and encodes it onto k data qubits, i.e., each bit is subjected to RY once.

[0077] Y-axis angle encoding is used, that is, a value is applied to each data qubit. A revolving door, where the rotation angle is equal to the scaled feature value.

[0078] The initial quantum state Evolves into a superposition state: .angle Decision made and The probability amplitude distribution is used to encode classical feature information into the probability amplitude-phase relationship of the qubit.

[0079] It should be noted that The probability amplitude generated and All values ​​are real numbers to avoid introducing unnecessary imaginary phases, simplifying the initial encoded state and facilitating gradient propagation in subsequent parameterization circuits. In quantum machine learning, Angular encoding can smoothly map classical data to different basis vectors in Hilbert space, and is compatible with subsequent... , Gate combinations can express any single-bit unitary transform.

[0080] In this scheme, the feature dimension is equal to the number of data qubits. Processing 12-dimensional features requires only 12 data qubits, a significant reduction compared to the exponential resource consumption of quantum state tomography.

[0081] 2) Training layer: This embodiment includes two sub-training layers (e.g., ...). Figure 4 As shown in the diagram, training layer 1 and training layer 2 have identical structures, but their functional focuses differ. Both consist of a single-bit rotation gate sequence and a two-bit entanglement gate. The unitary transformation capability of a single training layer is limited; stacking a second layer allows the overall circuit to express a more complex family of quantum circuit functions, thus fitting a finer entanglement determination boundary. Simultaneously, the parameterized structure of the two layers provides parameter redundancy, making it easier for the optimization process to escape local minima and improving the convergence stability of the training.

[0082] Within each training layer, three single-bit rotation gates containing the parameters to be trained, namely RX, RZ, and RX, are sequentially applied to all (k+1) qubits, and IsingZZ coupling centered on the auxiliary qubit is applied.

[0083] It should be noted that, after the aforementioned There is no entanglement between the data bits of the encoded direct product state, nor between the data bits and the auxiliary bits; the information is stored locally only in the form of single-bit probability amplitudes.

[0084] The combination of single-bit rotation gates RX and RZ can express arbitrary single-bit unitary transforms, perform preliminary reshaping and combination of encoded angles, and provide nonlinear mapping capabilities for subsequent classification.

[0085] The IsingZZ coupling gate is used to establish quantum entanglement between the auxiliary bit and each data bit, diffusing local feature information to the entire many-body Hilbert space. Understandably, the variational quantum classification model achieves the following function: encoding classical data into a circuit, passing it through multiple quantum coupling gates, and measuring the auxiliary bits. probability Then run the parameter translation rule to shift the parameters. Fine-tuning Measure twice more to obtain the gradient. Then change to new parameters. In preparation for the next round of evolution.

[0086] This embodiment does not use traditional quantum state tomography because tomography is too complex and requires too many measurements. Growth. In this embodiment, the variable quantum circuit does not require knowledge of the complete density matrix inside the quantum state. It only needs to measure the output probability of one auxiliary bit and then feed the output probability to a classical computer for optimization.

[0087] This embodiment encodes the classical features of the input onto data qubits, and then uses auxiliary qubits as coupling centers to implement quantum entanglement and information convergence. This allows the discriminative information scattered in each data bit to form a measurable classification representation on the auxiliary bit. This structure combines input scalability and readout convenience, adapting to feature inputs of different dimensions while avoiding the need for complex multi-bit joint measurement schemes for output determination, thereby reducing measurement complexity and improving circuit feasibility.

[0088] As a preferred embodiment, the objective function constructed based on the quantum measurement results of the output state includes: Repeated sampling measurements are performed on the auxiliary qubit, and the probability distribution of the auxiliary qubit is estimated based on the frequency of the target ground state in the measurement results. The target function value is then obtained based on the probability distribution.

[0089] By repeatedly sampling and measuring auxiliary qubits and estimating the probability distribution using the frequency of the target ground state, the problem that single measurements can only obtain discrete labels and are difficult to use for continuous optimization and stability determination is solved. Employing a frequency-based approximation of probability provides a smooth and statistically analyzable output for the system while maintaining the feasibility of quantum measurements, allowing the objective function calculation to be based on probabilistic meaning rather than a single random event. This not only improves the statistical stability of classification results but also facilitates the effective integration of the variable quantum circuit output with subsequent loss function construction, threshold comparison, and trainable parameter update mechanisms.

[0090] As a preferred implementation, the objective function is constructed using a dynamic probability flipping mechanism, including: When the true label of the quantum state data corresponds to the first category, the probability of the auxiliary qubit being in the first ground state is used as the probability of successful prediction; the first category represents the entanglement state of the quantum state data. When the true label of the quantum state data corresponds to the second category, the difference between the probability of 1 and the probability of the auxiliary quantum bit being in the first ground state is used as the probability of successful prediction. The quantum gate parameters of the variable quantum circuit are iteratively trained based on the predicted success probability by constructing a negative log-likelihood loss function.

[0091] Considering the limitations of "one-shot" measurement of auxiliary bits, namely that the auxiliary bit is in a certain state after each sampling... or The state corresponds to the actual label 0 or 1, but the probability cannot be obtained, making it impossible to perform differentiation and update parameters. In this embodiment, repeated sampling is performed by setting "shots" and a dynamic probability flipping mechanism is introduced to uniformly define the "probability of successful prediction". Specifically, for quantum states... Through the measurement results The frequency of occurrence of the state (i.e., the first ground state mentioned above) can be used to approximate the probability corresponding to its probability amplitude. Correspondingly, The probability of a state satisfies the normalization condition. When the true label of the sample At that time, the model objective is to maximize The state probability is therefore set. Conversely, when the real label At that time, the goal is to maximize The state probability is therefore set. .

[0092] Furthermore, in constructing the loss function, this time the traditional mean squared error loss was abandoned in favor of the negative log-likelihood loss, which is more suitable for probabilistic output models. This design makes the loss function... It can impose an exponential penalty on misclassified samples, providing a steeper gradient compared to MSE, thus effectively accelerating the model's convergence process.

[0093] By introducing a dynamic probability flipping mechanism and constructing a negative log-likelihood loss function based on it, the measurement probability of the auxiliary qubit can adaptively correspond to the true label in the binary classification task. This embodiment avoids the training chaos caused by inconsistent output targets for samples of different categories, ensuring that the method is always optimized around the goal of improving the prediction success probability. Furthermore, by combining the negative log-likelihood loss to impose a more significant penalty on erroneous samples, a steeper and more effective optimization gradient can be formed during training, improving parameter update efficiency, shortening the convergence process, and enhancing the system's recognition sensitivity near the entangled and non-entangled boundaries.

[0094] As a preferred embodiment, the two-bit and three-bit unknown state entanglement determination method based on variable quantum circuits provided in this embodiment further includes: using a parameter translation method to translate the quantum gate parameters (i.e., trainable parameters) in the variable quantum circuit in positive and negative directions and calculating the difference of the objective function to obtain the gradient information of the quantum gate parameters, so as to update the parameters of the variable quantum circuit.

[0095] This step employs a parameter translation method for gradient calculation, effectively avoiding the numerical truncation error introduced by the finite difference method. For the gradient of a parameterized quantum gate, this can be achieved by shifting the parameters... Translate in the positive and negative directions respectively And obtain it by calculating the difference between their expected values, that is: in, Describe the objective function Quantum gate parameters gradient, , These represent positive and negative shifts in the quantum gate parameters, respectively.

[0096] Specifically, the system projects the auxiliary bits to probability of state As the objective function By using this rule, backpropagation is used to accurately calculate the gradient of the rotation parameters of each layer, thereby ensuring the accuracy of the model parameter update direction.

[0097] The parameters that need to be trained include the rotation angle of the single-bit rotation gate and the coupling strength of the two-bit entanglement gate (i.e., the IsingZZ coupling gate). In each training layer, three rotation gates (RX, RZ, RX) are sequentially applied to k+1 qubits (k data bits + 1 auxiliary bit), with the rotation angle of each gate being an independent trainable parameter. In each training layer, an IsingZZ coupling gate is applied between the auxiliary bit and each data bit. , among them These are also independent trainable parameters. All of these trainable parameters are updated using the Adam optimizer in conjunction with the parameter translation gradient calculation rule.

[0098] It should be noted that this step precisely calculates the gradient of the quantum model by changing the difference between the results of the two runs of the quantum circuit, thereby realizing a trainable quantum neural network.

[0099] This embodiment introduces a parameter translation method into the gradient calculation of trainable parameters, enabling the direct acquisition of analytical gradient information for parameterized quantum gates without relying on traditional finite difference approximations. Compared to numerical difference methods, parameter translation reduces truncation errors and estimation biases, improving the accuracy of gradient directions and the reliability of parameter updates. For variable quantum circuits, this directly improves training stability and convergence quality, helping to maintain high optimization efficiency even in the presence of quantum measurement noise, thereby enhancing the overall trainability and final entanglement determination performance of the system.

[0100] As a preferred implementation, the two-bit and three-bit unknown state entanglement determination method based on variable quantum circuits provided in this embodiment further includes: adopting a measurement number annealing strategy during iterative training to gradually increase the number of samplings of the auxiliary quantum bits with the number of iterations, and combining a mini-batch stochastic gradient descent strategy and a learning rate adaptive decay strategy to iteratively optimize the variable quantum circuit.

[0101] To address common noise interference and "barren plateau" problems in quantum neural network training, this embodiment integrates measurement-based annealing, mini-batch stochastic gradient descent, and adaptive learning rate decay training strategies. First, measurement-based annealing is introduced, intentionally using fewer measurements in the early stages of training. The random perturbations introduced by larger statistical noise help the model quickly escape local minima and improve computational efficiency. In the later stages of training, the number of samples is linearly increased to suppress noise and ensure high precision in parameter fine-tuning. Second, mini-batch stochastic gradient descent is employed. Gradients are calculated by randomly sampling a subset of samples in each iteration, significantly reducing computational resource overhead while enhancing the model's generalization ability. Furthermore, combined with the adaptive learning rate decay strategy, the system monitors the validation set accuracy in real time. Once performance stagnation is detected, the learning rate is automatically reduced, helping the model to perform more refined searches and convergence near the global optimum.

[0102] Step S5, which determines whether the quantum state corresponding to the data to be determined is a separable or entangled state based on the value of the objective function, includes: if the quantum state to be determined is a two-qubit system, it is determined to be a separable state if the value of the objective function is greater than the preset determination threshold; otherwise, it is determined to be an entangled state. If the quantum state to be determined is a three-qubit system, it is determined to be a completely separable state if the value of the objective function is greater than the preset determination threshold; otherwise, it is determined to be a bound entangled state.

[0103] By extending the unified objective function and preset threshold comparison mechanism to different quantum systems, multi-scenario compatible judgments can be achieved within the same system framework, avoiding the need to build independent classification architectures for different systems. This design not only maintains the consistency of the overall system structure but also enhances the adaptability to complex entanglement scenarios involving two-qubit and three-qubit quantum entanglements, thereby improving the system's versatility, module reusability, and practical deployment value.

[0104] It should be understood that although the steps in the flowcharts of the embodiments described above are shown sequentially according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some steps in the flowcharts of the embodiments described above may include multiple steps or multiple stages. These steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential, but can be performed alternately or in turn with other steps or at least some of the steps or stages of other steps.

[0105] The experimental verification process of the two-bit and three-bit unknown state entanglement determination method based on variable quantum circuits proposed in this invention is described below.

[0106] Experimental Environment: An entanglement determination model based on the two-qubit and three-qubit unknown state entanglement determination method of this invention, using a Python programming environment, was constructed. The core quantum computing logic was implemented using the PennyLane framework. At the model architecture level, to find the optimal balance between limited quantum resource consumption and model expressive power, we captured quantum correlations in the data by alternately laying single-qubit rotation gates and two-qubit entanglement gates, and set the circuit depth to 2 layers (DEPTH=2). This circuit design not only effectively reduces the interference of quantum noise on the calculation results, but also ensures that the model has sufficient parameter space to fit complex entanglement feature boundaries.

[0107] Hyperparameter settings: For parameter optimization and training, the Adam optimizer was chosen as the core optimizer, utilizing its adaptive moment estimation capability to address the instability of quantum gradients. The training process lasted for 200 iterations (STEPS), employing a mini-batch stochastic gradient descent strategy with a batch size of BATCH=128 to enhance the randomness and generalization ability of the training process. To ensure the model converges to the global optimum, a dynamic learning rate decay mechanism was introduced: the initial learning rate was set to LR_INIT=0.02, and the validation set performance was monitored in real time; once the validation accuracy did not show a significant improvement within LR_DECAY_PATIENCE=30 consecutive cycles, the model automatically decayed the learning rate by LR_DECAY_FACTOR=0.5. This strategy allows the model to quickly search the parameter space in the early stages of training, and then converge more finely near the optimum with smaller step sizes in the later stages.

[0108] Furthermore, considering the unique measurement properties of quantum computing, a measurement-time annealing strategy was introduced during the training phase to overcome the common problem of getting trapped in local minima in variational algorithms. The number of samples is not fixed, but increases linearly from an initial value of TRAIN_SHOTS_START=50 to a final value of TRAIN_SHOTS_END=200, with the entire annealing process lasting ANNEAL_STEPS=80 steps. This dynamic adjustment mechanism utilizes the high statistical noise resulting from the fewer samples in the early stages of training as a regularization method, helping the model "escape" local extrema. As training progresses, the number of samples is gradually increased to improve the accuracy of gradient estimation, thereby achieving high-precision fine-tuning. In the final model evaluation phase, the number of samples is fixed at 201, which ensures both the statistical significance of the evaluation results and realistically simulates the operating behavior of current medium-scale quantum devices with limited measurement resources and in the presence of noise.

[0109] like Figures 5-8 The figures show the experimental results of the two-bit and three-bit unknown state entanglement determination methods based on variable quantum circuits provided in this embodiment on two-bit entanglement determination tasks and three-bit entanglement determination tasks, respectively. As can be seen from the figures: In the task of determining two-bit entanglement, the model demonstrated efficient and accurate learning capabilities. For example... Figure 5 As shown, the loss curve drops rapidly in the early stages of training and eventually stabilizes, with the highest classification accuracy on the test set being 87.1%.

[0110] according to Figure 8 The experimental results show that, with twelve feature inputs, the model's discrimination accuracy exceeds 85%, reaching a maximum of 86.9% in the 195th iteration. Meanwhile, the monotonically converging trend of the loss function fully validates the effectiveness of the adopted quantum circuit structure design and the corresponding training algorithm.

[0111] This invention is not limited to the specific embodiments described above. Any modifications made by those skilled in the art based on the above concept without creative effort are within the scope of protection of this invention.

Claims

1. A method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits, characterized in that, Includes the following steps: Based on uniform sampling of a random matrix, several quantum states with known state labels are generated, resulting in a quantum state dataset containing two-bit and three-bit datasets. The expected values ​​of observable operators are extracted from the quantum state dataset to generate classical feature vectors that characterize the entanglement properties of quantum state data; The classical eigenvectors are encoded into an initial quantum state using parameterized quantum gates; Construct a variable quantum circuit, and let the initial quantum state evolve through the variable quantum circuit to obtain the output state; An objective function is constructed based on the quantum measurement results of the output state, and the entanglement determination result of the quantum state data is output according to the value of the objective function.

2. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 1, characterized in that, The quantum states generated by uniform sampling based on a random matrix, which have several known state labels, include: A random density matrix is ​​generated based on the Ginibre random matrix to obtain several single-bit quantum states; In the two-bit dataset, separable state data is generated by convex combination of the direct product of two single-bit quantum states, and entangled state data is obtained by screening two-bit quantum states randomly generated based on single-bit quantum states according to a partial transpose criterion. In the three-bit dataset, fully separable state data is generated by convex combination of the product states of three single-bit quantum states, while bound entangled state data is constructed by obtaining the orthogonal complement states of a non-expandable product basis containing four orthogonal states.

3. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 1, characterized in that, The extraction of the expected value of the observable operator from the quantum state dataset includes: For a two-bit dataset, four observable operators are constructed based on the CHSH inequality. The expected value of each two-bit state under the four observable operators is calculated, and the four expected values ​​are combined as the four-dimensional classical eigenvector of the two-bit quantum state. For a three-bit dataset, four observable operators are constructed based on the Mermin inequality, and the expected value of each three-bit state under the four observable operators is calculated to generate a four-dimensional classical feature vector; eight observable operators are constructed based on the Svetlichny inequality, and the expected value of each three-bit state under the eight observable operators is calculated to generate an eight-dimensional classical feature vector; twelve observable operators are constructed based on CHSH-like inequalities, and the expected value of each three-bit state under the twelve observable operators is calculated to generate a twelve-dimensional classical feature vector.

4. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 3, characterized in that, The step of encoding the classical feature vector into the initial quantum state using a parameterized quantum gate employs a symmetric scaling angle encoding strategy, specifically including: The statistical boundary of each dimension of the classical eigenvector in the quantum state dataset is determined based on the maximum and minimum values ​​of each dimension of the eigenvector. Linear rescaling based on the statistical boundary maps each dimension of the classical feature vector to a preset angle range, serving as the angle input parameter for the quantum gate.

5. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 4, characterized in that, For the k-dimensional input characteristics of the initial quantum state, the variable quantum circuit is configured with k data qubits and 1 auxiliary qubit. The variable quantum circuit includes an angle encoding layer and a training layer; The angle encoding layer encodes each dimension of the classical feature vector into the corresponding data qubit through a Y-axis rotation gate based on the angle input parameter. The training layer includes two training sub-layers. In each training sub-layer, X-axis rotation gates, Z-axis rotation gates, and X-axis rotation gates are applied sequentially to k+1 qubits, and IsingZZ coupling centered on the auxiliary qubits is applied.

6. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 5, characterized in that, The objective function constructed based on the quantum measurement results of the output state includes: Repeated sampling measurements are performed on the auxiliary qubit, and the probability distribution of the auxiliary qubit is estimated based on the frequency of the target ground state in the measurement results. The target function value is then obtained based on the probability distribution.

7. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 6, characterized in that, The objective function is constructed using a dynamic probability flipping mechanism, including: When the true label of the quantum state data corresponds to the first category, the probability of the auxiliary qubit being in the first ground state is used as the probability of successful prediction; the first category represents the entanglement state of the quantum state data. When the true label of the quantum state data corresponds to the second category, the difference between the probability of 1 and the probability of the auxiliary quantum bit being in the first ground state is used as the probability of successful prediction. The quantum gate parameters of the variable quantum circuit are iteratively trained based on the predicted success probability by constructing a negative log-likelihood loss function.

8. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 1, characterized in that, The method further includes: using a parameter translation method to translate the quantum gate parameters in the variable quantum circuit in positive and negative directions and calculating the difference of the objective function to obtain the gradient information of the quantum gate parameters, so as to update the parameters of the variable quantum circuit.

9. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 7, characterized in that, The method further includes: employing a measurement-time annealing strategy during iterative training to gradually increase the number of samplings of the auxiliary qubits with the number of iterations, and combining a mini-batch stochastic gradient descent strategy and a learning rate adaptive decay strategy to iteratively optimize the variable quantum circuit.

10. The method for determining entanglement of two-bit and three-bit unknown states based on variable quantum circuits according to claim 1, characterized in that, When the quantum state data is two bits, it is determined to be a separable state if the objective function value is greater than the preset judgment threshold, otherwise it is determined to be an entangled state. When the quantum state data is three bits, it is determined to be a completely separable state if the objective function value is greater than the preset judgment threshold; otherwise, it is determined to be a bound entangled state.

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