Linear complexity quantum state preparation method based on tensor decomposition and quantum circuit construction system

Abstract

The invention relates to the technical field of quantum computing, and provides a linear complexity quantum state preparation method based on tensor decomposition and a quantum circuit construction system.The high-dimensional tensor is decomposed into a series of low-rank core tensors through continuous singular value decomposition, each core tensor in a core tensor sequence is expanded into a unitary matrix, and the unitary matrix is used as a quantum circuit; the unitary matrix sequence is mapped to quantum lines coupled using adjacent qubits, and the quantum lines are run to prepare a target quantum state. Based on this, the line generated by the method can approximately or accurately prepare a target quantum state only by coupling adjacent quantum bits, and is perfectly adaptive to quantum chips of linear or grid topologies such as superconducting and semiconductor quantum dots and the like.

Description

Technical Field

[0001] This invention relates to the field of quantum computing technology, and more specifically, to a method for preparing linear complex quantum states based on tensor decomposition and a quantum circuit construction system. Background Technology

[0002] In applications such as quantum machine learning (QML), quantum chemical simulation, and solving linear equations (HHL algorithm), high-dimensional classical data are used. Loaded into the registers of the quantum computer (i.e., preparing the quantum state) The first step in the algorithm execution is called amplitude encoding.

[0003] The existing technology has the following main problems: (1) Exponential resource consumption: Traditional precise amplitude coding methods (such as the Mottonen algorithm) have high resource consumption for... One qubit typically requires Quantum gates. As the number of qubits increases, the line depth increases exponentially. In noisy medium-scale quantum (NISQ) devices, excessively deep lines can cause the quantum state to be distorted under decoherence. (2) Hardware topology limitations: Existing schemes often assume that there is all-to-all connectivity between qubits, requiring a large number of SWAP gates to achieve long-distance control. However, mainstream superconducting and semiconductor quantum chips only support nearest-neighbor coupling, which further reduces the efficiency of the actual compiled lines.

[0004] To address the aforementioned issues, existing technologies urgently need improvement. Summary of the Invention

[0005] The purpose of this application is to provide a method for preparing linear complex quantum states and a quantum circuit construction system based on tensor decomposition, which has the ability to achieve linear complexity. This method utilizes only the coupling of adjacent qubits to approximately or precisely prepare the target quantum state.

[0006] In a first aspect, this application provides a method for preparing linear complex metric states based on tensor decomposition, including: Data preprocessing steps: Determine the required number of qubits based on the input data vector; pad the input data vector with zeros and normalize it to obtain the target quantum state vector; Tensor decomposition steps: Iterative truncated singular value decomposition is performed on the target quantum state vector, wherein in each decomposition layer, the retained rank is truncated to no greater than 2, thereby obtaining a core tensor sequence equal to the number of qubits; Quantum gate construction steps: Expand each core tensor in the core tensor sequence into a unitary matrix to form a unitary matrix sequence; wherein, the core tensor with dimension 2×r is expanded into a 2-dimensional unitary matrix, the core tensor with dimension 4×r is expanded into a 4-dimensional unitary matrix, and r is the rank of the core tensor after truncation. Quantum circuit construction and execution steps: Map the unitary matrix sequence to a quantum circuit using adjacent qubit coupling, and run the quantum circuit to prepare the target quantum state.

[0007] Preferably, the tensor decomposition step further includes: For the k-th iteration, the current data vector is reshaped into a matrix; where, n is the total number of qubits; Perform singular value decomposition on the matrix and retain only the two largest singular values ​​and their corresponding left and right singular vectors; The matrix formed by the left singular vectors is used as the k-th core tensor; the product of the matrix formed by the retained singular values ​​and the matrix formed by the right singular vectors is used as the data vector for the next iteration. After the (n-1)th iteration, the data vector obtained from the last iteration is used as the nth core tensor.

[0008] Preferably, the quantum gate construction step further includes: By constructing unit vectors that are orthogonal to all existing column vectors in the core tensor, and adding the constructed unit vectors as supplementary column vectors to the core tensor, a full-rank square matrix is ​​formed. The square matrix is ​​normalized to satisfy the conditions of a unitary matrix, thereby forming a sequence of unitary matrices.

[0009] Furthermore, the 2-dimensional unitary matrix corresponds to a quantum gate acting on a single qubit, and the 4-dimensional unitary matrix corresponds to a quantum gate acting on a pair of adjacent qubits.

[0010] Preferably, the quantum circuit construction and execution steps further include: The unitary matrix sequence is mapped to a ladder-like circuit structure, wherein the first unitary matrix acts on the first qubit, the last unitary matrix acts on the last qubit, and the kth unitary matrix in the middle acts on the (k-1)th and kth qubits; The unitary matrix sequence is mapped onto a quantum circuit to form a quantum circuit operation sequence; wherein the first unitary matrix acts on the first qubit, the last unitary matrix acts on the last qubit, and each unitary matrix in between acts on the corresponding pair of adjacent qubits. The 4-dimensional unitary matrix acting on adjacent qubit pairs is decomposed into a combination of CNOT gates and single-qubit rotation gates.

[0011] Preferably, determining the required number of qubits based on the input data vector includes: calculating the required number of qubits. Where N is the original length of the input data vector; The step of padding the input data vector with zeros includes adding zero elements to the end of the vector to make its total length equal to 2. n .

[0012] Secondly, this embodiment also provides a quantum circuit construction system for implementing the method described in the first aspect, the system comprising: The preprocessing module is used to determine the required number of qubits based on the input classical data vector, and to pad the data vector with zeros and normalize it to obtain the target quantum state vector; The tensor decomposition module is used to perform iterative truncated singular value decomposition on the target quantum state vector, wherein in each decomposition layer, the retained rank is truncated to no greater than 2, thereby obtaining a core tensor sequence equal to the number of qubits. A quantum gate construction module is used to expand each core tensor in the core tensor sequence into a unitary matrix to form a unitary matrix sequence; wherein, a core tensor with dimension 2×r is expanded into a 2-dimensional unitary matrix, a core tensor with dimension 4×r is expanded into a 4-dimensional unitary matrix, and r is the rank of the truncated core tensor. A circuit construction module is used to map the unitary matrix sequence into a quantum circuit using adjacent qubit coupling.

[0013] Furthermore, the system also includes a circuit execution module for controlling the quantum processor to run the quantum circuit to prepare the target quantum state.

[0014] Thirdly, this application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method for preparing linear complex metric states based on tensor decomposition as described in the first aspect above.

[0015] Fourthly, an electronic device comprising: one or more processors, and a memory for storing one or more computer programs; the computer programs being configured to be executed by the one or more processors, the programs including methods for performing the linear complex metric state preparation method based on tensor decomposition as described in the first aspect above.

[0016] As described above, this application provides a method for preparing linear complex quantum states and a quantum circuit construction system based on tensor decomposition. It decomposes a high-dimensional tensor into a series of low-rank core tensors through continuous singular value decomposition. Each core tensor in the core tensor sequence is expanded into a unitary matrix to form a unitary matrix sequence. This unitary matrix sequence is then mapped to a quantum circuit using adjacent qubit coupling, and the quantum circuit is run to prepare the target quantum state. Based on this, the circuit generated by this method only requires adjacent qubit connections, eliminating the need for additional SWAP gates. This method is perfectly compatible with quantum chips with linear or lattice topologies such as superconducting and semiconductor quantum dots. For vectors with low entanglement, this method can approximate them with extremely high fidelity. For full-rank random vectors, this method provides a standard and accurate preparation process. The core tensors can directly correspond to the parameterized circuit layers in variational quantum algorithm (VQE / QAOA), facilitating gradient calculation and optimization. Attached Figure Description

[0017] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a flowchart illustrating the steps of the method for preparing linear complex metric quantum states based on tensor decomposition disclosed in an embodiment of the present invention. Figure 2 This is a flowchart illustrating the tensor decomposition steps disclosed in an embodiment of the present invention. Figure 3 This is a schematic diagram of the quantum circuit structure disclosed in an embodiment of the present invention; Figure 4 This is a schematic diagram of the quantum circuit construction system disclosed in an embodiment of the present invention; Detailed Implementation Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which these embodiments belong; the terminology used herein and in the specification of the application is for the purpose of describing particular embodiments only and is not intended to limit these embodiments; the terms "comprising" and "having," and any variations thereof, in the specification of these embodiments and the foregoing drawings, are intended to cover non-exclusive inclusion. The terms "first," "second," etc., in the specification of these embodiments and the foregoing drawings are used to distinguish different objects, not to describe a particular order.

[0019] The implementation details of the technical solution in this embodiment are described in detail below: This invention aims to propose a method for preparing tensor decomposition states based on the concept of Matrix Product State (MPS). This method decomposes high-dimensional tensors into a series of low-rank core tensors through continuous singular value decomposition (SVD), and maps them to a ladder-like quantum circuit.

[0020] Firstly, this application provides a method for preparing linear complex metric states based on tensor decomposition, such as... Figure 1 As shown, it includes: S101, Data preprocessing steps: Determine the required number of qubits based on the input data vector; pad the input data vector with zeros and normalize it to obtain the target quantum state vector.

[0021] Preferably, determining the required number of qubits based on the input data vector includes: calculating the required number of qubits. Where N is the original length of the input data vector; padding the input data vector with zeros includes adding zero elements to the end of the vector so that its total length is equal to 2. n .

[0022] Specifically, in this embodiment, data preprocessing and normalization are first performed. This is done by inputting a real vector of arbitrary length N. .

[0023] (1) Determine the required number of qubits: Calculate the required number of qubits For example, if ,but .

[0024] (2) Perform zero-filling: if Then in vector End of sentence Add zeros to obtain a length of vector .For example, After padding with zeros, it becomes .

[0025] (3) Perform modulus normalization: calculate the Euclidean norm of the vector. ,make Let the target quantum state vector ,satisfy .

[0026] S102, Tensor decomposition step: Perform iterative truncated singular value decomposition on the target quantum state vector, wherein in each decomposition, the retained rank is truncated to no greater than 2, thereby obtaining a core tensor sequence equal to the number of qubits.

[0027] Preferably, the tensor decomposition step, such as Figure 2 As shown, it also includes: S201, for the k-th iteration, reshape the current data vector into a matrix; where, n is the total number of qubits; S202, Perform singular value decomposition on the matrix and retain only the two largest singular values ​​and their corresponding left and right singular vectors; S203, the matrix formed by the left singular vectors is used as the k-th core tensor; the product of the matrix formed by the retained singular values ​​and the matrix formed by the right singular vectors is used as the data vector for the next iteration; S204, after the (n-1)th iteration, uses the data vector obtained from the last iteration as the nth core tensor.

[0028] Specifically, in this embodiment, the global vector is treated as a tensor network, and extracted by peeling away layers one by one. One core tensor The specific operating steps are as follows: (1) Initialization: Let the current processing vector ; (2) Regarding the first qubits ( The iterative process of ) is as follows: a. Reshaping: Restore the remaining data Remodeling into a matrix .

[0029] Specifically, in this embodiment, the matrix dimension is set as follows: the number of rows is... The number of columns is ,in, , The rank to be retained. Here, the row index corresponds to the current qubit and the previous entanglement dimension, and the column index corresponds to the remaining qubit system. When The elements are divided equally, with the first half being the first row and the second half being the second row. ; b. Singular Value Decomposition (SVD): [This part is incomplete and likely refers to a separate process.] Decompose, decompose .

[0030] c. Rank truncation: In this embodiment, to adapt to the two-bit gate, the number of singular values ​​is forcibly preserved. ; If the rank obtained by SVD is greater than 2, then the two largest singular values ​​are retained and the rest are discarded.

[0031] d. Extract and update the core tensor. The core tensor... : Take matrix The former column, at this time The dimensions are: ; Remaining vector (i.e.) update: Let (Right now The former (Row / column product), passed to the next level.

[0032] (3) Closing: The vector byte corresponding to the last qubit is used as the last core tensor. .

[0033] S103, Quantum gate construction steps: Expand each core tensor in the core tensor sequence into a unitary matrix to form a unitary matrix sequence; wherein, the core tensor with dimension 2×r is expanded into a 2-dimensional unitary matrix, the core tensor with dimension 4×r is expanded into a 4-dimensional unitary matrix, and r is the rank of the core tensor after truncation. Preferably, the quantum gate construction step further includes: constructing unit vectors that are orthogonal to the original column vectors in the core tensor, and adding the constructed unit vectors as supplementary column vectors to the core tensor to form a full-rank square matrix; normalizing the square matrix to satisfy the conditions of a unitary matrix to form a unitary matrix sequence.

[0034] Furthermore, the 2-dimensional unitary matrix corresponds to a quantum gate acting on a single qubit, and the 4-dimensional unitary matrix corresponds to a quantum gate acting on a pair of adjacent qubits.

[0035] Specifically, in this embodiment, the core tensor obtained by SVD Typically, these are rectangular matrices (orthogonal columns, but not necessarily square matrices), which cannot be directly used as quantum gates. They need to be extended to unitary matrices. The specific method is as follows: (1) Input: Core tensor matrix (Dimension m×r, where ).

[0036] (2) Processing algorithm: a. Orthogonal Completion: Utilizing Gram-Schmidt or QR decomposition methods. By constructing an auxiliary vector set (such as standard basis vectors), find the corresponding vectors. The existing column vectors are all orthogonal unit vectors; add these new vectors as columns. Continue in this manner until an m×m square matrix is ​​formed.

[0037] b. Output: Obtain a full-rank unitary matrix. .like , corresponding to a single-bit gate; if This corresponds to a two-bit gate.

[0038] S104, Quantum circuit construction and execution steps: Map the unitary matrix sequence to a quantum circuit using adjacent qubit coupling, and run the quantum circuit to prepare the target quantum state.

[0039] Preferably, the quantum circuit construction and execution steps further include: The unitary matrix sequence is mapped to a ladder-like circuit structure, wherein the first unitary matrix acts on the first qubit, the last unitary matrix acts on the last qubit, and the kth unitary matrix in the middle acts on the (k-1)th and kth qubits; The unitary matrix sequence is mapped onto a quantum circuit to form a quantum circuit operation sequence; wherein the first unitary matrix acts on the first qubit, the last unitary matrix acts on the last qubit, and each unitary matrix in between acts on the corresponding pair of adjacent qubits. The 4-dimensional unitary matrix acting on adjacent qubit pairs is decomposed into a combination of CNOT gates and single-qubit rotation gates.

[0040] Specifically, in this embodiment, the specific quantum circuit construction and execution steps include generating a unitary matrix sequence. Mapped onto quantum circuits, they form a ladder-like structure.

[0041] (1) Line topology. Wherein: Level 1: In Top-operated single-bit gate Typically, in order to introduce entanglement, the first operation needs to be designed in conjunction with subsequent steps; Intermediate layer: in adjacent quantum bit pairs Last time a double-bit gate was applied ; End: at Apply a single-bit gate .

[0042] (2) Gate decomposition. Specifically, the KAK decomposition technique is used to decompose any... unitary matrix Decomposed into: ,in It is a unitary operator that operates on a single quantum bit, typically corresponding to combinations of rotary gates and NOT gates in physical circuits. The entanglement capability encompassing all quantum states has the following diagonal form: , , , Let x, y, and z be the Pauli matrix, where x, y, and z are real-valued parameters that determine the required degree of entanglement.

[0043] This transforms a logically large matrix into a CNOT gate and a single-bit rotation gate supported by a physical quantum chip.

[0044] In some embodiments, for example, the input vector: , .

[0045] The first step is data preprocessing. This involves determining the number of qubits in the system. ;because No need to pad with zeros; Modulus normalization: , .

[0046] The second step is tensor decomposition based on truncated SVD. Initialization. .

[0047] (1) First-level decomposition (for By Remodeling matrix: ;right Perform SVD decomposition and extract the left matrix U as the core matrix. ,Right now .

[0048] Therefore, the remaining matrix, which is also the right matrix in this embodiment, Pass it to the next loop.

[0049] (2) Second-level decomposition (for ).Will Remodeling matrix: ;right Perform SVD decomposition and extract the matrix with rank 2 from the left matrix U as the core matrix. ,Right now The remaining matrix, also known as the right matrix in this embodiment. ; (3) The last qubit (for ).Will orthogonalization ,Right now .

[0050] The third step is unitary matrix embedding (constructing quantum gates). This is achieved by obtaining the core matrix in the second step. Embedded into quantum gates : , ; therefore, ; For non-square matrices, the Gram-Schmidt method is first used to find the orthogonal complement, resulting in... ;Will Normalization yields .

[0051] The fourth step is to construct quantum circuits.

[0052] Among them, such as Figure 3 As shown, the quantum circuit consists of a 3-quantum-state system. The initial state is... For the first quantum state implement (Flip gate); for the first and second quantum states implement ,right Perform KAK decomposition. For the third quantum state implement (Entrance to the workplace).

[0053] Step 5: Circuit execution. Running the quantum circuit constructed in step 4 yields the quantum state: .

[0054] Based on this, the beneficial effects of this embodiment include: (1) Hardware-friendly: The circuit generated by this method only requires the connection of adjacent qubits, without the need for additional SWAP gates, and is perfectly compatible with quantum chips with linear or grid topologies such as superconducting and semiconductor quantum dots. (2) Universality: Applicable to any real vector. For vectors with low entanglement (i.e., vectors whose singular values ​​decay rapidly after SVD decomposition), this method can approximate with extremely high fidelity; for full-rank random vectors, this method provides a standard and accurate preparation process. (3) Parameterization capability: Core tensor It can directly correspond to the parameterized circuit layer in the Variable Quantum Algorithm (VQE / QAOA), which facilitates gradient calculation and optimization.

[0055] Secondly, this embodiment also provides a quantum circuit construction system for implementing the method described in the first aspect, such as... Figure 4 As shown, the system includes: The preprocessing module 401 is used to determine the required number of qubits based on the input classical data vector, and to pad the data vector with zeros and normalize it to obtain the target quantum state vector; Tensor decomposition module 402 is used to perform iterative truncated singular value decomposition on the target quantum state vector, wherein in each decomposition layer, the retained rank is truncated to no greater than 2, thereby obtaining a core tensor sequence equal to the number of qubits. Quantum gate construction module 403 is used to expand each core tensor in the core tensor sequence into a unitary matrix to form a unitary matrix sequence; wherein, a core tensor with dimension 2×r is expanded into a 2-dimensional unitary matrix, a core tensor with dimension 4×r is expanded into a 4-dimensional unitary matrix, and r is the rank of the truncated core tensor. The circuit construction module 404 is used to map the unitary matrix sequence into a quantum circuit using adjacent qubit coupling.

[0056] Furthermore, the system also includes a circuit execution module 405, which controls the quantum processor to run the quantum circuit to prepare the target quantum state.

[0057] This system can be used to execute the linear complex metric state preparation method based on tensor decomposition described in the first aspect, which will not be elaborated further here.

[0058] Thirdly, this application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method for preparing linear complex metric states based on tensor decomposition as described in the first aspect above.

[0059] Fourthly, an electronic device comprising: one or more processors, and a memory for storing one or more computer programs; the computer programs being configured to be executed by the one or more processors, the programs including methods for performing the linear complex metric state preparation method based on tensor decomposition as described in the first aspect above.

[0060] The above description is merely an embodiment of this application and is not intended to limit the scope of protection of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A method for preparing linear complex metric states based on tensor decomposition, characterized in that, include: Data preprocessing steps: Determine the required number of qubits based on the input data vector; The input data vector is padded with zeros and normalized to obtain the target quantum state vector; Tensor decomposition steps: Iterative truncated singular value decomposition is performed on the target quantum state vector, wherein in each decomposition layer, the retained rank is truncated to no greater than 2, thereby obtaining a core tensor sequence equal to the number of qubits; Quantum gate construction steps: Expand each core tensor in the core tensor sequence into a unitary matrix to form a unitary matrix sequence; wherein, the core tensor with dimension 2×r is expanded into a 2-dimensional unitary matrix, the core tensor with dimension 4×r is expanded into a 4-dimensional unitary matrix, and r is the rank of the core tensor after truncation. Quantum circuit construction and execution steps: Map the unitary matrix sequence to a quantum circuit using adjacent qubit coupling, and run the quantum circuit to prepare the target quantum state.

2. The method for preparing linear complex metric states based on tensor decomposition according to claim 1, characterized in that, The tensor decomposition step further includes: For the k-th iteration, the current data vector is reshaped into a matrix; where, n is the total number of qubits; Perform singular value decomposition on the matrix and retain only the two largest singular values ​​and their corresponding left and right singular vectors; The matrix formed by the left singular vectors is used as the k-th core tensor; the product of the matrix formed by the retained singular values ​​and the matrix formed by the right singular vectors is used as the data vector for the next iteration. After the (n-1)th iteration, the data vector obtained from the last iteration is used as the nth core tensor.

3. The method for preparing linear complex metric states based on tensor decomposition according to claim 2, characterized in that, The quantum gate construction steps also include: By constructing unit vectors that are orthogonal to all existing column vectors in the core tensor, and adding the constructed unit vectors as supplementary column vectors to the core tensor, a full-rank square matrix is ​​formed. The square matrix is ​​normalized to satisfy the conditions of a unitary matrix, thereby forming a sequence of unitary matrices.

4. The method for preparing linear complex metric states based on tensor decomposition according to claim 1, characterized in that, The 2-dimensional unitary matrix corresponds to a quantum gate acting on a single qubit, and the 4-dimensional unitary matrix corresponds to a quantum gate acting on a pair of adjacent qubits.

5. The method for preparing linear complex metric states based on tensor decomposition according to claim 4, characterized in that, The quantum circuit construction and execution steps also include: The unitary matrix sequence is mapped to a ladder-like circuit structure, wherein the first unitary matrix acts on the first qubit, the last unitary matrix acts on the last qubit, and the kth unitary matrix in the middle acts on the (k-1)th and kth qubits; The unitary matrix sequence is mapped onto a quantum circuit to form a quantum circuit operation sequence; wherein the first unitary matrix acts on the first qubit, the last unitary matrix acts on the last qubit, and each unitary matrix in between acts on the corresponding pair of adjacent qubits. The 4-dimensional unitary matrix acting on adjacent qubit pairs is decomposed into a combination of CNOT gates and single-qubit rotation gates.

6. The method for preparing linear complex metric states based on tensor decomposition according to claim 1, characterized in that, Determining the required number of qubits based on the input data vector includes: calculating the required number of qubits. Where N is the original length of the input data vector; The step of padding the input data vector with zeros includes adding zero elements to the end of the vector to make its total length equal to 2. n .

7. A quantum circuit construction system for implementing the method as described in any one of claims 1 to 6, characterized in that, include: The preprocessing module is used to determine the required number of qubits based on the input classical data vector, and to pad the data vector with zeros and normalize it to obtain the target quantum state vector; The tensor decomposition module is used to perform iterative truncated singular value decomposition on the target quantum state vector, wherein in each decomposition layer, the retained rank is truncated to no greater than 2, thereby obtaining a core tensor sequence equal to the number of qubits. A quantum gate construction module is used to expand each core tensor in the core tensor sequence into a unitary matrix to form a unitary matrix sequence; wherein, a core tensor with dimension 2×r is expanded into a 2-dimensional unitary matrix, a core tensor with dimension 4×r is expanded into a 4-dimensional unitary matrix, and r is the rank of the truncated core tensor. A circuit construction module is used to map the unitary matrix sequence into a quantum circuit using adjacent qubit coupling.

8. The quantum circuit construction system according to claim 7, characterized in that, The system also includes a circuit execution module for controlling the quantum processor to run the quantum circuit to prepare the target quantum state.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method for preparing linear complex metric states based on tensor decomposition as described in any one of claims 1 to 6.

10. An electronic device, the electronic device comprising: One or more processors, a memory for storing one or more computer programs; characterized in that the computer programs are configured to be executed by the one or more processors, the programs comprising methods for performing linear complex metric state preparation based on tensor decomposition as described in any one of claims 1 to 6.

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