A low-noise multi-dimensional quantum fourier transform fast implementation method and system

Abstract

The application discloses a low-noise multi-dimensional quantum Fourier transform fast implementation method and system. The method faces the Fourier transform demand of two-dimensional matrix and multi-dimensional tensor, decomposes any dense input into a matrix / tensor unit linear combination in a classical layer, and retains the coefficient. In a quantum layer, equivalent classes are divided according to the highest power factor of each coordinate direction basis number, a standard one-dimensional quantum Fourier transform is only performed on a representative element, and a non-representative element result is recovered through diagonal phase compensation. In the classical layer, a multi-dimensional tensor product is reconstructed to obtain a frequency domain matrix / tensor or specified frequency point information. The method avoids overall amplitude state preparation and repeated QFT calling, reduces line depth and noise exposure, and can be applied to quantum phase estimation, quantum simulation, quantum linear algebra, quantum image processing, quantum machine learning, material structure analysis, electromagnetic field sampling grid analysis and multi-dimensional scientific calculation as a quantum circuit implementation interface for multi-dimensional Fourier transform.

Description

Technical Field

[0001] This invention belongs to the field of quantum computing and quantum information processing technology, specifically relating to a method and system for fast implementation of low-noise multidimensional quantum Fourier transform. Background Technology

[0002] The Fourier transform is a fundamental tool in signal processing, communication, image analysis, computational electromagnetics, crystallography, materials structure analysis, and spectral methods for partial differential equations. For two-dimensional matrices and multidimensional tensors, the complete Fourier transform can reveal spatial frequencies, periodic structures, wave modes, and multi-degree-of-freedom coupling characteristics. However, as the dimension and grid size increase, the computational cost of the classical direct multidimensional discrete Fourier transform grows rapidly; even with the classical fast Fourier transform, explicit processing of all input and output frequency domain coefficients is still limited by data size, storage bandwidth, and parallel communication overhead.

[0003] The quantum Fourier transform is a core fundamental module in quantum computing, widely used in Shor's algorithm, quantum phase estimation, quantum simulation, quantum linear equation algorithms, and several quantum spectral analysis tasks. The standard quantum Fourier transform can coherently process the Fourier transform at the quantum state amplitude level, theoretically possessing low gate complexity. However, in two-dimensional and multi-dimensional scenarios, existing methods typically suffer from the following limitations.

[0004] First, existing implementations of multidimensional quantum Fourier transforms often assume that the input state satisfies low rank and can be written as a tensor product of several one-dimensional states, or that the input data can be prepared into quantum amplitude states at low cost. For arbitrary dense two-dimensional matrices or multidimensional tensors, this assumption often fails, and the preparation of complex amplitude-encoded states can incur high quantum gate overhead (when solving d-dimensional QFT problems, if the number of basis vectors in each direction is...). Then the quantum gate complexity required for initial state preparation reaches This results in significant noise exposure.

[0005] Second, many existing studies mainly optimize the number of gates, the number of controlled phase gates, the circuit depth, or the hardware connection mapping of the standard quantum Fourier transform itself, but still use the interface logic of "first pushing the input into the quantum state, and then performing the corresponding transformation". This type of technique is very effective for the standard one-dimensional quantum Fourier transform, but it cannot directly solve the problem of state preparation and multi-directional repeated calls of arbitrary multi-dimensional input matrices. These repeated calls will lead to deeper quantum circuits, larger control overhead, and more serious noise accumulation.

[0006] Therefore, a quantum circuit method suitable for two-dimensional and multi-dimensional quantum Fourier transforms is needed, enabling arbitrary matrices or tensors to enter a unified processing flow without satisfying special separable structures, while reducing repeated calls to the standard quantum Fourier transform and replacing complex permutation gates with quantum hardware-friendly phase gates. This invention addresses these problems. Summary of the Invention

[0007] The purpose of this invention is to provide a low-noise method and system for rapidly implementing multidimensional quantum Fourier transform, in order to solve the problems of existing multidimensional quantum Fourier transforms, such as strong dependence on input structure, high cost of preparing arbitrary dense input states, numerous calls to the standard one-dimensional quantum Fourier transform, and complex permutation recovery circuits.

[0008] The technical solution to achieve the objective of this invention is as follows: Firstly, this invention provides a method for rapidly implementing low-noise multidimensional quantum Fourier transform, comprising the following steps:

[0009] The first step is to obtain the input data to be subjected to the multidimensional quantum Fourier transform, wherein the input data is a two-dimensional coefficient matrix or a multidimensional coefficient tensor;

[0010] The second step is to decompose the input data into a linear combination of matrix units or tensor units in the classical preprocessing layer, and save the coefficients corresponding to each matrix unit or tensor unit.

[0011] The third step is to establish a corresponding one-dimensional Hilbert space for each coordinate direction of the input data, divide the calculation basis vector of the direction into several equivalence classes according to the exponent of the highest power factor in the basis number, and select representative elements for each equivalence class.

[0012] The fourth step is to prepare the ground state vector of each equivalence class representative element in each coordinate direction in the quantum computing layer and perform a standard one-dimensional quantum Fourier transform on it to obtain the one-dimensional quantum Fourier transform result of the representative element vector.

[0013] The fifth step is to recover the one-dimensional quantum Fourier transform result of the non-representative element vector in the same equivalence class by applying the diagonal phase compensation operator to the one-dimensional quantum Fourier transform result of the corresponding representative element.

[0014] The sixth step is to construct the matrix unit or tensor unit after undergoing a multidimensional quantum Fourier transform based on the one-dimensional quantum Fourier transform results in each coordinate direction.

[0015] The seventh step involves linearly superimposing the multidimensional quantum Fourier transform results of each matrix unit or tensor unit in the classical reconstruction layer according to the coefficients saved in the second step, to obtain the target frequency domain matrix, target frequency domain tensor, target frequency domain coefficient subset, or spectral peak information.

[0016] Furthermore, the two-dimensional coefficient matrix Decomposed into

[0017]

[0018] in, Let A be the size of matrix A. Indicates the length of the row direction. Indicates the column length;

[0019] matrix The Middle Line 1 The elements of the column are ;

[0020] For matrix units, the first Line 1 The elements of the column are All other elements are , represented as ;

[0021] Represent a A column vector of dimension u, where only the u-th element is 1 and the rest are zero;

[0022] Represent a A column vector of dimension v, where only the v-th element is 1 and the rest are zero;

[0023] for The dual vector is computationally equivalent to... The conjugate transpose of;

[0024] The decomposition does not require It possesses low rank, separability, or can be represented as the tensor product of two low-dimensional vectors; similarly, it generalizes to multidimensionality, for input... dimensional coefficient tensor

[0025] ;

[0026] Generate a set of one-dimensional quantum Fourier transform results in the i-direction respectively. , Then, take the tensor product of the results in the corresponding direction for each tensor unit and accumulate them according to the original coefficients to output the multidimensional frequency domain coefficient tensor. The length of each coordinate direction is ; Let i be the set of one-dimensional quantum Fourier transform results in the i-direction; In a d-dimensional tensor, the tensor is numbered in each direction as follows: The coefficient; Represent a dimensional vector, only the 3rd dimension One element is 1, and the rest are zero; Indicates that for those who have The basis vectors in the i-direction of the basis vectors The one-dimensional QFT results.

[0027] Furthermore, regarding length The one-dimensional direction, i.e., the direction has N basis vectors, calculate the basis number. Classified into equivalence classes

[0028] ,

[0029] To meet Natural numbers, and set zero class. ,in The representative element is , The representative element is ;

[0030] Same equivalence class If the calculated basis vectors within the one-dimensional quantum Fourier transform summation are arranged in a fixed frequency domain with the order of natural numbers, then the phases preceding the basis vectors have the same period, and their minimum positive period is only related to the phases preceding the basis vectors. Related to odd factors This is irrelevant, thus enabling the one-dimensional quantum Fourier transform result of the non-representative element to be recovered from the result of the corresponding representative element.

[0031] Furthermore, regarding equivalence classes... vectors in Its one-dimensional QFT results can be obtained by analyzing equivalence classes. The representative element basis vector Results of one-dimensional QFT Diagonal phase compensation is performed to obtain the basis vectors of the representative elements. Results of one-dimensional QFT Perform diagonal phase compensation Non-representative elements can be obtained The result of a one-dimensional QFT The diagonal phase compensation operator for

[0032] ;

[0033] Its satisfaction

[0034] ;

[0035] To represent the basis vectors of the primitives Results of one-dimensional QFT Transform into The result of a one-dimensional QFT The unitary operator q is an odd number; Represents a one-dimensional quantum Fourier transform operator with respect to N basis vectors under the negative exponent convention; For simple basis vectors, except for the first one... One element is 1, and the rest are 0; for Secondary unit root ;

[0036] Frequency domain index Write in binary form (in Let k be the value of the r-th bit in binary. After being set to 0 or 1, the diagonal phase compensation operator Decomposed into single-qubit phase gates tensor product

[0037] ;

[0038] The diagonal phase compensation circuit Compilation optimization is performed based on quantum hardware gate sets, connection topology, noise model, and fault tolerance threshold, including omitting identity phase gates, merging adjacent phase gates, truncating small-angle phase gates, and caching phase angle parameters.

[0039] When the corresponding phase angle is When the value is an integer multiple of the value or meets a preset approximate threshold, the corresponding phase gate is omitted.

[0040] Furthermore, the quantum circuit involved in the method of this invention includes a representative initial state preparation sub-circuit, a standard one-dimensional quantum Fourier transform sub-circuit, and a diagonal phase compensation sub-circuit; wherein the representative initial state preparation sub-circuit starts from the initial state. Departure, for representative element At most one qubit flip gate is applied; the standard one-dimensional quantum Fourier transform subcircuit is only executed on the representative element vector set; the diagonal phase compensation subcircuit consists of a single qubit phase gate and is used to recover the result of the one-dimensional quantum Fourier transform of the non-representative element vectors.

[0041] Furthermore, for a two-dimensional coefficient matrix In the two-dimensional coefficient tensor representation, the matrix unit Two-dimensional quantum Fourier transform results Constructed as

[0042] ;

[0043] The target frequency domain matrix B is constructed as follows

[0044] ;

[0045] B is the frequency domain matrix of the matrix A to be determined by a two-dimensional QFT; For matrix units The two-dimensional quantum Fourier transform results;

[0046] In matrix operator representation, based on the convention of positive and negative exponents in the coordinate direction, the above transpose is replaced with conjugate transpose or an equivalent row and column direction transformation.

[0047] Furthermore, when at least one coordinate direction has a length that is not an integer power of two, a set of one-dimensional quantum Fourier transform results for that direction is generated using zero-element embedding, quantum Bluestein subroutines, or quantum Fourier transform subroutines of arbitrary length. In the case where multiple coordinate directions have the same length or can be embedded into the same power-of-two length space, the equivalence class partitioning results, representative quantum Fourier transform circuit structures, and phase angle parameter tables for the same direction are reused to reduce the compilation of repeated circuits and the calculation of repeated parameters in the multidimensional case.

[0048] Furthermore, the method reduces quantum gate resource consumption, circuit depth, state preparation pressure, and noise exposure by retaining input coefficients in the classical layer, avoiding the preparation of arbitrary dense input overall amplitude states, reducing the number of repetitive standard quantum Fourier transform sub-circuits, and replacing general frequency domain permutation synthesis with diagonal phase gates. The classical reconstruction layer can output a complete frequency domain matrix or frequency domain tensor, or it can output only a specified frequency point, a specified frequency domain region, spectral peak, threshold screening results, or frequency domain interface data for subsequent quantum algorithms.

[0049] In a second aspect, the present invention provides a multidimensional quantum Fourier transform system for implementing the method described in the first aspect, comprising a classical preprocessing module, an equivalence class indexing module, a quantum basis transformation module, a phase compensation module, a tensor product assembly module, a classical reconstruction module, and an application interface module; the classical preprocessing module is used to perform matrix unit or tensor unit decomposition and save coefficients; the equivalence class indexing module is used to generate a representative element set and a phase angle table; the quantum basis transformation module is used to prepare representative elements, calculate the ground state, and perform a standard one-dimensional quantum Fourier transform; the phase compensation module is used to perform diagonal phase compensation on non-representative elements; the tensor product assembly module is used to form the multidimensional transformation results of each unit element; the classical reconstruction module is used to output a target frequency domain matrix, a frequency domain tensor, or specified frequency domain coefficients; and the application interface module is used to provide frequency domain input for quantum phase estimation, quantum simulation, quantum linear algebra, quantum image processing, or multidimensional scientific computing tasks.

[0050] Thirdly, the present invention provides a quantum computing device, including a classical controller, a quantum processor, and a memory, wherein the memory stores instructions for enabling the classical controller and the quantum processor to collaboratively execute the method described in the first aspect; wherein the quantum processor executes the representative standard quantum Fourier transform and diagonal phase compensation gate, and the classical controller executes input decomposition, equivalence class index calculation, phase angle calculation, quantum circuit scheduling, result caching, and output reconstruction.

[0051] Compared with the prior art, the significant advantages of this invention are:

[0052] (1) Any two-dimensional matrix or multi-dimensional tensor that needs to be subjected to multidimensional Fourier transform does not need to satisfy low rank, separability or special tensor product structure and can be uniformly processed.

[0053] (2) The overall quantum gate complexity of the algorithm is Using an innovative one-dimensional QFT method, only one operation is needed for each direction. The standard QFT of the first-level representative vectors is invoked, and then the QFT results of the remaining basis vectors are recovered through diagonal phase compensation;

[0054] (3) Implement multidimensional QFT as a novel quantum-classical interface of “classical unit decomposition, quantum direction basis transformation, phase reuse recovery, and classical linear reconstruction” to reduce quantum gate resource consumption, circuit depth and noise exposure;

[0055] (4) Based on the fact that one-dimensional QFT needs to be performed on the basis vectors in each direction, the proposed innovative one-dimensional QFT method has strong multidimensional extensibility. The total subgate complexity is linearly related to the dimension. Especially when the number of basis vectors in multiple directions is the same or can be embedded into a space containing the same power of two basis vectors, the property of isomorphism of finite-dimensional Hilbert space can be used to reuse the equivalence class partition, representative element line and phase angle parameter in the same direction. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the overall technical route of the present invention.

[0057] Figure 2 This is a schematic diagram comparing the differences and innovations between the present invention and existing methods.

[0058] Figure 3 This is a schematic diagram of the key quantum circuit and phase multiplexing recovery of the present invention.

[0059] Figure 4 This is a diagram of the overall implementation architecture of the hybrid quantum-classical system of this invention.

[0060] Figure 5 This is a schematic diagram of the technical steps, results implementation, and output path of the present invention.

[0061] Figure 6 This is a schematic diagram illustrating the future application prospects and expansion directions of the present invention.

[0062] Figure 7 In this invention Verification diagram of time phase compensation recovery.

[0063] Figure 8 This is a comparison chart of the complexity trends of the present invention and existing processing methods.

[0064] Figure 9 This is a comparison chart of the noise exposure trends of the present invention and existing treatment methods.

[0065] Figure 10 This is a spectrum verification diagram of the one-dimensional input signal in this invention.

[0066] Figure 11 This is a residual verification diagram of the one-dimensional input signal in this invention.

[0067] Figure 12 This is a spectrum verification diagram of the two-dimensional analytical signal in this invention.

[0068] Figure 13 This is the two-dimensional analytical signal residual heatmap in this invention.

[0069] Figure 14 This is a summary diagram of multiple sets of programming verification residuals in this invention. Detailed Implementation

[0070] This invention proposes a fast method for multidimensional quantum Fourier transform based on matrix unit decomposition, equivalence class phase multiplexing, and low-noise quantum circuits, belonging to the fields of quantum computing, quantum information processing, quantum circuit synthesis, hybrid quantum-classical computing, and multidimensional spectrum analysis. This method addresses the requirements for Fourier frequency domain transforms of two-dimensional matrices and multidimensional tensors. Arbitrary input data is decomposed into linear combinations of matrix units or tensor units at the classical layer, and each unit element is represented as a tensor product combination of multiple one-dimensional Hilbert space basis vectors. At the quantum layer, the basis vectors in each coordinate direction are divided into equivalence classes according to the exponent of the highest quadratic factor in the basis number. Only the representative elements of each equivalence class are prepared in their ground states and subjected to a standard one-dimensional quantum Fourier transform. Then, the basis vectors of the non-representative elements in the same equivalence class are recovered through a diagonal phase compensation circuit composed of single-qubit phase gates after the one-dimensional quantum Fourier transform. Finally, at the classical computing layer, the multidimensional Fourier transform results of each unit element are linearly superimposed according to the original coefficients to obtain the target frequency domain matrix, target frequency domain tensor, specified frequency domain coefficients, or spectral peak information.

[0071] The breakthrough of this invention lies in simultaneously altering the quantum-classical interface and quantum circuit organization of multidimensional Fourier transforms: arbitrary dense matrices or tensors no longer need to be prepared as complex quantum amplitude states; the quantum processor only handles simple computation of the ground state, performs standard quantum Fourier transforms on representative elements, performs phase compensation on non-representative element basis vectors, and combines tensor products of vectors in each direction. Thus, the initial state preparation for each representative element vector requires only a single quantum gate, and the number of standard one-dimensional quantum Fourier transform calls required for each direction is reduced from linearly increasing with the number of ground states to increasing with the number of qubits. Furthermore, the phase recovery operation for non-representative element basis vectors employs a quantum hardware-friendly diagonal phase gate. This invention reduces state preparation pressure, quantum gate resource consumption, circuit depth, and noise exposure, and can serve as a quantum circuit implementation interface for multidimensional Fourier transforms, applicable to quantum phase estimation, quantum simulation, quantum linear algebra, quantum image processing, quantum machine learning, material structure analysis, electromagnetic field sampling grid analysis, and multidimensional scientific computing. Programming verification results show that the proposed method is consistent with the standard Fourier transform results in tests involving one-dimensional signals, two-dimensional analytic signals, random dense matrices, and non-power-zero-padded embeddings, with residuals within the order of floating-point rounding errors.

[0072] The following section combines formulas and derivations... Figures 1 to 14 The present invention will be further described in detail below with reference to specific embodiments. It should be understood that the following embodiments are for illustrative purposes only and are not intended to limit the scope of protection of the present invention. Equivalent substitutions made to the number of dimensions, transformation length, quantum Fourier transform approximation accuracy, quantum hardware connection method, phase compensation compilation method, classical reconstruction strategy, and application interface form without departing from the core idea of ​​the present invention shall all fall within the scope of protection of the present invention.

[0073] The core problem addressed by this invention is that existing multidimensional QFT techniques often compress any dense input into a quantum amplitude state when performing Fourier frequency domain transformations on two-dimensional matrices and multidimensional tensors, or assume the input can be separated into a tensor product of several one-dimensional states. This leads to difficulties in state preparation, repeated deep circuit calls, and increased noise exposure. This invention does not directly prepare the entire dense matrix into a quantum amplitude state. Instead, it retains the input coefficients in the classical layer and shrinks the task of the quantum layer to the generation of reusable one-dimensional basis vector QFT results. The following explanation uses a two-dimensional matrix as an example because it represents the smallest nontrivial scenario in multidimensional Fourier transforms. It clearly demonstrates the complete logic of "two coordinate directions, one-dimensional basis transformation, and matrix unit reconstruction," and can be naturally generalized to any matrix through tensor units. Dimensional tensors. Therefore, the two-dimensional embodiments are not a limitation on the dimensions of the invention, but rather examples of basic implementations for multi-dimensional scenarios.

[0074] Figure 1This invention demonstrates the overall process of obtaining a frequency domain matrix or tensor from an arbitrary two-dimensional matrix or multi-dimensional tensor input, through classical matrix or tensor unit decomposition, establishment of one-dimensional Hilbert spaces in each direction, equivalence class partitioning, representative element standard QFT, diagonal phase compensation, directional result set, tensor product assembly, and classical linear reconstruction. The figure also illustrates the core technical effects of this invention: eliminating the need for overall amplitude state preparation, reducing the number of standard QFT calls, and reducing noise exposure through diagonal phase gates.

[0075] Figure 2 This paper compares existing research or conventional approaches with the approach of this invention in terms of input structure requirements, state preparation pressure, number of standard QFT calls, non-representative recovery methods, quantum circuit complexity and noise, and multidimensional expansion capabilities, highlighting the innovation and breakthrough of this invention in quantum-classical interface, equivalence class representative reuse, and diagonal phase compensation.

[0076] The technical solution of the present invention will be described in detail below.

[0077] The present invention provides a method for implementing multidimensional quantum Fourier transform based on matrix unit decomposition, equivalence class phase multiplexing, and low-noise quantum circuits, comprising the following steps:

[0078] The first step is to obtain the input data to be processed and determine the dimension parameters.

[0079] Let the input to be performed on the multidimensional QFT be... dimensional tensor

[0080] ;

[0081] Among them, the first The length of each coordinate direction (the number of basis vectors contained therein) is ;

[0082] The d-dimensional tensor is numbered sequentially in each direction as follows: The coefficient;

[0083] For ease of derivation, we first take a two-dimensional input. , where the matrix The Middle Line 1 The elements of the column are Two-dimensional input can be an image matrix, an electromagnetic field sampling matrix, a material structure sampling matrix, a two-dimensional signal array, or a coefficient matrix in a quantum algorithm.

[0084] The second step is to perform matrix unit or tensor unit decomposition in the classical preprocessing layer.

[0085] For a two-dimensional input matrix The classic preprocessing layer decomposes it into a linear combination of matrix units:

[0086]

[0087] in, Let A be the size of matrix A. Indicates the length of the row direction. This indicates the column length, and both also represent the number of basis vectors in the x and y directions;

[0088] If we consider the matrix unit as the tensor unit in two directions x and y, then it can be written as

[0089]

[0090] For matrix units, the first Line 1 The elements of the column are All other elements are It can be represented as ;

[0091] ( ) represents a A column vector of dimension u, where only the u-th element is 1 and the rest are zero;

[0092] ( ) represents a A column vector of dimension v, where only the v-th element is 1 and the rest are zero;

[0093] for The dual vector is computationally equivalent to... The conjugate transpose of;

[0094] It is a vector in the x-direction;

[0095] The y-direction vector The conjugate transpose of;

[0096] Therefore, it can be seen that the present invention does not require It has a low-rank structure and does not require All the complexity of arbitrary dense matrices is preserved in classical coefficients. In this process, the quantum processing layer only needs to process the direction basis vectors. and The one-dimensional QFT result. This step is the first "quantum-classical interface gate" that distinguishes this invention from the direct amplitude encoding route.

[0097] The third step is to establish a one-dimensional Hilbert space and QFT conventions for each coordinate direction.

[0098] For the two-dimensional case, establish

[0099]

[0100] in, and Let Hilbert space be defined in the x and y directions, respectively. and They are respectively and The number of basis vectors contained therein;

[0101] For general When using a dimensional tensor as input, establish...

[0102]

[0103] Let i be the Hilbert space along the direction i, where i has basis vectors; ,……, It is the Hilbert space in the i-direction. A set of basis vectors;

[0104] Using the negative exponent convention of the quantum Fourier transform, the one-dimensional quantum Fourier transform is:

[0105]

[0106] in Root of unity of power N ; This represents a one-dimensional quantum Fourier transform operator for N basis vectors under the negative exponent convention.

[0107] All subsequent one-dimensional transformations in all directions will follow this convention. If a positive exponential convention is used, only the phase sign needs to be adjusted accordingly.

[0108] The fourth step is to divide the equivalence classes and perform standard one-dimensional QFT only on the representative elements.

[0109] for In the one-dimensional direction, the non-zero computational basis number is written as

[0110]

[0111] in It is an odd number. According to the binary factor order Equivalence class partitioning

[0112]

[0113] And set class zero For each Select representative elements Select representative elements for the zero class The quantum computing layer only applies to the representative set.

[0114]

[0115] The ground state is prepared and calculated, and a standard one-dimensional QFT is performed. The corresponding representative frequency domain results are as follows:

[0116]

[0117] The result of the zero representation element is

[0118]

[0119] After step four, manual calculation can be performed for verification. For example, Each equivalence class is

[0120]

[0121] At this point, all that's needed is... The standard QFT is performed on four representative elements instead of on each of the eight computational ground states. Figure 7 Given The visualization verification of the temporal phase compensation recovery relationship demonstrates that the frequency domain phase structure of the representative element can be reused for non-representative elements in the same equivalence class.

[0122] The fifth step is to construct the diagonal phase compensation operator and restore the non-representative element results.

[0123] For the same equivalence class target element in Defined by one-dimensional QFT

[0124]

[0125] Compare it with the representative element results:

[0126]

[0127] achievable

[0128]

[0129] Therefore, a diagonal phase compensation operator is defined.

[0130]

[0131] To convert the representative element basis vector the result of the one-dimensional QFT into (where q is odd) the result after one-dimensional QFT the unitary operator

[0132] satisfies

[0133]

[0134] Further, the frequency domain index is written in binary form

[0135]

[0136] then can be decomposed into the tensor product of single-qubit phase gates:

[0137]

[0138] where

[0139]

[0140] This derivation shows that the present invention does not need to synthesize a general frequency domain permutation matrix, but converts the task of "moving the position of the frequency domain basis vector" into the task of "fixing the order of the frequency domain basis states and compensating the phase". For quantum hardware, diagonal phase gates are more suitable for compilation, caching and approximate truncation than unstructured permutation gates, so this step is the key to reducing noise exposure. Figure 3 presents a schematic diagram of the key quantum circuit and phase multiplexing recovery of the present invention, showing the connection relationship between the representative element preparation, the standard one-dimensional QFT sub-circuit and the diagonal phase compensation sub-circuit. In the figure represents the diagonal phase compensation operator, which is used to restore the frequency domain result of the representative element to the frequency domain result of the non-representative element under the condition of fixing the order of the frequency domain basis states, thereby avoiding general permutation circuit synthesis.

[0141] After the fifth step, the innovative one-dimensional QFT method can be verified. Taking a length one-dimensional signal

[0142]

[0143] Comparing the frequency spectrum obtained by the equivalence class phase multiplexing method of the present invention with the result of the standard fast Fourier transform, the maximum error is about , and the relative error is about . Figure 10 and Figure 11The overlapping cases of one-dimensional spectra and pointwise residuals are given respectively, demonstrating that the innovative one-dimensional QFT method keeps the standard Fourier results unchanged at the numerical level, and the error mainly comes from floating-point rounding. Figure 10 This invention demonstrates the overlap between the one-dimensional spectrum obtained by the equivalence class phase multiplexing method and the standard FFT reference result, which is used to verify that the innovative one-dimensional QFT method still maintains the standard Fourier transform result after representative element multiplexing and phase compensation. Figure 11 Display the absolute residuals of a one-dimensional signal at each frequency point. The residuals are at... to The magnitude indicates that the error mainly comes from floating-point rounding rather than algorithm structure error.

[0144] Step 6: Construct multidimensional QFT results in matrix units or tensor units based on the one-dimensional results in each direction.

[0145] For the unit of a two-dimensional matrix The one-dimensional QFT results in the two directions are as follows: and Therefore, the matrix unit frequency domain result under the two-dimensional coefficient tensor representation is:

[0146]

[0147] If matrix operators are used, then according to the exponent conventions in the row and column directions, it can be written in equivalent left and right Fourier action forms, and the target frequency domain matrix can be equivalently represented as follows:

[0148]

[0149] It refers to having A one-dimensional Fourier transform matrix in the space of basis vectors;

[0150] It refers to having The transpose of a one-dimensional Fourier transform matrix in the space of basis vectors;

[0151] The focus of this invention is the technical approach of multidimensional assembly of one-dimensional basis transformation results in each direction through matrix units or tensor units.

[0152] for dimensional tensor unit

[0153]

[0154] Its multidimensional QFT results are

[0155]

[0156] This represents the d-dimensional tensor unit, numbered sequentially in each direction as follows: The coefficient of is 1, and the coefficients of the others are 0; d-dimensional tensor unit The result after d-dimensional QFT; Indicates that for those who have The basis vectors in the i-direction of the basis vectors One-dimensional QFT results;

[0157] Therefore, this invention, based on an innovative one-dimensional QFT method, possesses inherent multi-dimensional scalability: each dimension only needs to independently generate a one-dimensional result set. Multidimensional results are combined using tensor product structures. When multiple directions have the same number of basis vectors or are embedded into directions with the same power of two basis vectors, equivalence class indices, representative element lines, and phase angle parameter tables can be reused, thereby avoiding repetitive line design in multidimensional cases.

[0158] Step 7: Linearly superimpose the original coefficients in the classical reconstruction layer and output the frequency domain results.

[0159] For a two-dimensional matrix, the target frequency domain matrix is ​​constructed as follows:

[0160]

[0161] B is the result of A after undergoing a two-dimensional QFT; at this point, both A and B are matrices.

[0162] for The construction of the target frequency domain tensor is as follows:

[0163]

[0164] B is the result of A after undergoing a two-dimensional QFT. At this point, both A and B are d-dimensional tensors.

[0165] Depending on the actual task requirements, the classic reconstruction layer can output the complete frequency domain matrix, the complete frequency domain tensor, or only the specified frequency point, the specified frequency domain region, the threshold filtering results, or the spectral peak information. Figure 5 The technical steps, result implementation, and output path are illustrated in the diagram, demonstrating the input data, unit decomposition, QFT result generation in each direction, non-representative element recovery, multidimensional result assembly, frequency domain result output, and optional frequency point query or post-processing flow. The result implementation box on the right illustrates that this invention can output a complete two-dimensional spectrum matrix. 1D frequency domain tensor, local frequency domain coefficients, and spectral peak information.

[0166] The two-dimensional signal results can be verified after step seven. Two-dimensional real signals are used as input signals.

[0167]

[0168] Its normalized two-dimensional spectrum exhibits peaks only at four frequency domain positions. Verification results show that the peak positions and amplitudes obtained by the method of this invention are consistent with the analytical spectrum and the standard two-dimensional FFT, with a maximum error of approximately [missing information]. The relative error is approximately . Figure 12 and Figure 13 Two-dimensional spectrum comparison and residual heatmap are given respectively to illustrate that the matrix unit reconstruction and directional basis result assembly of the present invention do not change the two-dimensional Fourier transform result. Figure 12 The diagram compares the peak positions and amplitudes of the two-dimensional analytical signal under the standard FFT reference results and the results obtained by the method of this invention. The peak positions are consistent, indicating that the matrix unit reconstruction and directional basis assembly of the method correctly recover the two-dimensional frequency domain structure. Figure 13 The thermal distribution of the pointwise residuals in the two-dimensional frequency domain results is shown. The residuals are concentrated on the order of double-precision floating-point errors, verifying the numerical correctness of this invention in two-dimensional problems.

[0169] From a system perspective, this invention can be deployed as a hybrid quantum-classical architecture in which a classical preprocessing layer, a quantum processing layer, and a classical reconstruction layer work together. Figure 4 The connections between the input task layer, classical preprocessing layer, quantum processing layer, classical reconstruction layer, and output layer are illustrated. Compared to existing approaches that directly prepare arbitrary amplitude states, this invention retains the input complexity in the classical coefficient layer and uses the quantum processor for representative QFT and structured phase compensation, thus concentrating quantum resources on parts more suitable for hardware implementation.

[0170] In terms of complexity, let the length of a single direction be... Performing the standard QFT state-by-state requires... Sub-representation level standard QFT call; this invention only requires... For each representative element, the standard QFT is applied. For non-representative elements, the recovery operator is used. Composed of single-qubit phase gate tensor products; if a one-dimensional QFT result set of all calculated ground states in a given direction is generated, the total magnitude of phase compensation can be estimated as follows: It should be noted that if explicit output of the complete text is required... A frequency domain tensor, the output of which contains The coefficient represents a lower bound on data size that is unavoidable for any complete output method. This invention reduces the quantum gate costs associated with repetitive calls to the representative meta-standard QFT, complex state preparation, and the synthesis of unstructured permutations. Figure 8 A comparison of the complexity trends of this invention with existing processing methods is presented. This invention reduces the quantum gate cost required by the algorithm; if all frequency domain coefficients are fully output, the lower bound of the data size of the output tensor itself still exists.

[0171] Regarding noise, error sources in real quantum hardware include single-qubit gates, two-qubit gates, controlled phase rotation, readout errors, decoherence, and circuit compilation errors. If we consider the effective number of noisy gates... The surrogate model represents the circuit success probability approximation based on the independent error model.

[0172]

[0173] in To reduce the average error coefficient, reducing the number of repeated deep circuit executions and avoiding unstructured permutation synthesis can lower the error rate. The growth trend of noise exposure can improve noise exposure. Figure 9 A comparison of noise exposure trends is presented. Because this invention reduces repetitive standard QFT deep circuit calls and replaces complex permutation synthesis with structured diagonal phase gates, it exhibits a lower noise exposure trend as the scale increases.

[0174] To verify the case of arbitrarily dense input, The frequency domain results of the random complex matrix were calculated using the matrix unit reconstruction method of this invention and the standard two-dimensional FFT, respectively. The relative residuals of the random dense matrix remained at [value missing]. arrive The magnitude indicates that this invention does not depend on the sparsity, low rank, or separability of the input matrix. For non-power-two inputs, [the following is omitted as the text is incomplete and requires further context]. The input matrix is ​​zero-padded and embedded into The spatial calculations were performed using both the method of this invention and the standard FFT, and the experimental results showed a relative error of approximately [missing information]. The maximum error is approximately . Figure 14 The summary of multiple sets of verification residuals demonstrates that the numerical residuals of this invention are at the level of double-precision floating-point errors. Figure 14 This figure summarizes the relative residuals from tests including one-dimensional signals, two-dimensional analytic signals, random dense matrices, and non-power-padded zero embeddings. It is used to illustrate the correctness of the innovative one-dimensional QFT method in this invention and that the method does not depend on input sparsity, low rank, or separability. Furthermore, it is consistent with the standard Fourier transform reference results in multiple tests.

[0175] The basic equivalence class expression of this invention is as follows: Taking a qubit register as an example, for practical problems involving coordinate directions of non-power-two lengths, one of the following methods can be used: First, embed the non-power-two length input with zero elements into a power-two dimensional space of at least the original length, and then apply the method of this invention in the embedded space; Second, introduce a quantum Bluestein subroutine or other quantum Fourier transform subroutines of arbitrary length during the generation stage of the one-dimensional result set in the corresponding direction; Third, for tasks that only require partial frequency domain coefficients, extract only the effective frequency domain coefficients or target frequency points corresponding to the original dimensions in the classical reconstruction layer. The above processing methods do not change the core structure of this invention, that is, the input coefficients are retained in the classical layer, and the quantum layer is responsible for reusable one-dimensional basis transformation and phase compensation.

[0176] This invention can be embedded as a basic module for multidimensional frequency domain transformation into multiple quantum algorithms and scientific computing processes. Typical application scenarios and expansion directions are as follows: Figure 6 As shown, Figure 6 Centering on a multidimensional QFT quantum circuit interface, this invention demonstrates its potential applications in quantum phase estimation, quantum simulation, quantum linear algebra, quantum image processing, quantum machine learning, electromagnetic field and wave analysis, material structure and lattice spectrum analysis, multidimensional scientific computing, and PDE spectral methods. The extended directions below indicate that this invention can also be combined with approximate QFT truncation, arbitrary-length QFT or Bluestein embedding, hardware connectivity optimization, fault-tolerant quantum platforms, and fast lookup at specified frequencies. Specifically, in quantum phase estimation and quantum simulation, this invention can be used to construct frequency domain basis representations for multi-degree-of-freedom systems; in quantum linear algebra and partial differential equation solving, this invention can be used to transform discrete field variables or operator kernels into the frequency domain, making convolution, difference, and spectral methods more compact; in quantum image processing and quantum signal processing, this invention can be used for frequency domain feature extraction of two-dimensional images, three-dimensional volume data, and spatiotemporal signals; in material structures, lattices, metamaterials, nanosheets, electromagnetic field sampling, and wave system analysis, this invention can be used to extract multidimensional spatial frequency information; in hybrid quantum-classical machine learning workflows, this invention can serve as a frequency domain preprocessing module before and after quantum feature mapping, reducing the pressure of compressing all arbitrarily dense data into quantum amplitude states.

[0177] In practical implementation, either a precise quantum Fourier transform circuit or an approximate quantum Fourier transform circuit can be selected based on hardware conditions. For the small-angle controlled phase gate in the standard one-dimensional quantum Fourier transform, truncation can be performed according to the task's fault tolerance threshold. In quantum computing, single-qubit phase gates can pre-calculate the phase angle at the classical layer and compile it into a quantum hardware-supported rotation gate. For interconnected quantum chips, the representative quantum Fourier transform circuit and phase compensation gate can be jointly optimized to reduce the number of swapping gates. For quantum simulators, the representative state vectors can be directly saved and diagonal phases can be applied in batches to verify the correctness of the algorithm.

[0178] By adopting this invention, the organization of multidimensional quantum Fourier transform is changed from "directly preparing arbitrary high-dimensional amplitude states and performing multidimensional transform as a whole" to "the classical layer preserves arbitrary coefficients, the quantum layer generates one-dimensional basis results in each direction, and the classical layer reconstructs the frequency domain results according to the unit element." This method mathematically preserves the Fourier transform results and achieves the following engineering effects: First, it can handle dense matrices with arbitrary inputs, and the inputs do not need to satisfy a special tensor product structure, breaking through the dependence of many existing multidimensional quantum Fourier transform methods on the input state structure; Second, it can handle dense matrices containing... The quantum gate complexity required for a d-dimensional QFT of a d-dimensional tensor with d elements is reduced to 1. The number of independent standard one-dimensional quantum Fourier transform calls in each direction is determined by... Downgraded to First, the invention reduces the noise of quantum circuits by replacing a large number of repetitive QFT circuits with a combination of a small number of representative QFT circuits and quantum-friendly phase compensation circuits. Second, the diagonal phase compensation circuit is used to recover the QFT results of non-representative elements, reducing the complexity of circuit synthesis and minimizing unnecessary exposure to multi-qubit gates and deep circuit noise. Third, programming verification shows that the invention is consistent with the standard Fourier transform results in processing one-dimensional, two-dimensional, and arbitrarily input dense data and non-power-law zero-padded embedding scenarios, with residuals at the level of double-precision floating-point errors (see...). Figures 10 to 14 Fifth, this invention constitutes a quantum circuit implementation method for multidimensional Fourier transform, which can serve as a low-noise frequency domain interface for quantum phase estimation, quantum simulation, quantum linear algebra, multidimensional quantum image processing and scientific computing, providing an feasible path for the effective combination of classical computing and quantum computing.

[0179] In summary, this invention not only improves the resource organization of multidimensional quantum Fourier transforms, but also represents a systematic technological advancement in quantum-classical interfaces, quantum circuit call counts, noise control, engineering verification, and application expansion. Its scope covers key aspects such as input representation, equivalence class partitioning based on the highest quadratic power factor of basis vector numbers, QFT result reuse of basis vectors in different directions, standard QFT calls using representative vectors, diagonal phase compensation methods, matrix unit or tensor unit assembly, hybrid quantum-classical reconstruction, low-noise circuit compilation, partial frequency output, and multi-domain application interfaces. These aspects work together to form a low-noise quantum circuit implementation path for two-dimensional and multidimensional Fourier transforms. This invention reduces the pressure of state preparation for arbitrary dense inputs, the number of repeated standard QFT calls, quantum gate resource consumption, and noise exposure while maintaining the mathematical results of Fourier transforms. It also improves the engineering feasibility of multidimensional quantum Fourier transforms in scenarios such as quantum phase estimation, quantum simulation, quantum linear algebra, quantum image processing, and multidimensional scientific computing.

[0180] The above are merely preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention. All components not explicitly stated in this embodiment can be implemented using existing technology.

Claims

1. A method for fast implementation of low-noise multidimensional quantum Fourier transform, characterized in that, Includes the following steps: The first step is to obtain the input data for the multidimensional quantum Fourier transform to be performed, wherein the input data is a two-dimensional coefficient matrix or a multidimensional coefficient tensor; The second step is to decompose the input data into a linear combination of matrix units or tensor units in the classical preprocessing layer, and save the coefficients corresponding to each matrix unit or tensor unit. The third step is to establish a corresponding one-dimensional Hilbert space for each coordinate direction of the input data, divide the calculation basis vector of the direction into several equivalence classes according to the exponent of the highest power factor in the basis number, and select representative elements for each equivalence class. The fourth step is to prepare the ground state vector of each equivalence class representative element in each coordinate direction in the quantum computing layer and perform a standard one-dimensional quantum Fourier transform on it to obtain the one-dimensional quantum Fourier transform result of the representative element vector. The fifth step is to recover the one-dimensional quantum Fourier transform result of the non-representative element vector in the same equivalence class by applying the diagonal phase compensation operator to the one-dimensional quantum Fourier transform result of the corresponding representative element. The sixth step is to construct the matrix unit or tensor unit after undergoing a multidimensional quantum Fourier transform based on the one-dimensional quantum Fourier transform results in each coordinate direction. The seventh step involves linearly superimposing the multidimensional quantum Fourier transform results of each matrix unit or tensor unit in the classical reconstruction layer according to the coefficients saved in the second step, to obtain the target frequency domain matrix, target frequency domain tensor, target frequency domain coefficient subset, or spectral peak information.

2. The method according to claim 1, characterized in that, The two-dimensional coefficient matrix Decomposed into ； in, Let A be the size of matrix A. Indicates the length of the row direction. Indicates the column length; matrix The Middle Line number The elements of the column are ; For matrix units, the first Line 1 The elements of the column are All other elements are , represented as ; Represent a A column vector of dimension u, where only the u-th element is 1 and the rest are zero; Represent a A column vector of dimension v, where only the v-th element is 1 and the rest are zero; for The dual vector is computationally equivalent to... The conjugate transpose of; The decomposition does not require It possesses low rank, separability, or can be represented as the tensor product of two low-dimensional vectors; similarly, it generalizes to multidimensionality, for input... dimensional coefficient tensor ； Generate a set of one-dimensional quantum Fourier transform results in the i-direction respectively. , Then, take the tensor product of the results in the corresponding direction for each tensor unit and accumulate them according to the original coefficients to output the multidimensional frequency domain coefficient tensor. The length of each coordinate direction is ; Let i be the set of one-dimensional quantum Fourier transform results in the i-direction; In a d-dimensional tensor, the tensor is numbered in each direction as follows: The coefficient; Represent a dimensional vector, only the 3rd dimension One element is 1, and the rest are zero; Indicates that for those who have The basis vectors in the i-direction of the basis vectors The one-dimensional QFT results.

3. The method according to claim 2, characterized in that, For length The one-dimensional direction, i.e., the direction has N basis vectors, calculate the basis number. Classified into equivalence classes ， To meet Natural numbers, and set zero class. ,in The representative element is , The representative element is ; Same equivalence class If the calculated basis vectors within the one-dimensional quantum Fourier transform summation are arranged in a fixed frequency domain with the order of natural numbers, then the phases preceding the basis vectors have the same period, and their minimum positive period is only related to the phases preceding the basis vectors. Related to odd factors This is irrelevant, thus enabling the one-dimensional quantum Fourier transform result of the non-representative element to be recovered from the result of the corresponding representative element.

4. The method according to claim 3, characterized in that, For equivalence classes vectors in Its one-dimensional QFT results are obtained by analyzing equivalence classes. The representative element is the basis vector. Results of one-dimensional QFT Diagonal phase compensation is performed to obtain the basis vectors of the representative elements. Results of one-dimensional QFT Perform diagonal phase compensation Obtaining non-representative elements The result of a one-dimensional QFT The diagonal phase compensation operator for ， Its satisfaction ， in To represent the basis vectors of the primitives Results of one-dimensional QFT Transform into The result of a one-dimensional QFT The unitary operator q is an odd number; Represents a one-dimensional quantum Fourier transform operator with respect to N basis vectors under the negative exponent convention; For simple basis vectors, except for the first one... One element is 1, and the rest are 0; for Secondary unit root ; Frequency domain index Write in binary form Then, the diagonal phase compensation operator Decomposed into single-qubit phase gates tensor product ， ， in This represents the value of the r-th bit when k is written in binary. It can be 0 or 1; The diagonal phase compensation circuit Compilation optimization is performed based on the quantum hardware gate set, connection topology, noise model, and fault tolerance threshold, including omitting identity phase gates, merging adjacent phase gates, truncating small-angle phase gates, and caching phase angle parameters.

5. The method according to claim 4, characterized in that, Quantum circuits include a representative initial state preparation sub-circuit, a standard one-dimensional quantum Fourier transform sub-circuit, and a diagonal phase compensation sub-circuit; among which the representative initial state preparation sub-circuit starts from the initial state. Departure, for representative element At most one qubit flip gate can be applied; The standard one-dimensional quantum Fourier transform subcircuit is only executed on the representative element vector set; the diagonal phase compensation subcircuit, composed of single-qubit phase gates, is used to recover the result of the one-dimensional quantum Fourier transform of the non-representative element vectors.

6. The method according to claim 4, characterized in that, For a two-dimensional coefficient matrix In the two-dimensional coefficient tensor representation, the matrix unit Two-dimensional quantum Fourier transform results Constructed as ， The target frequency domain matrix B is constructed as follows ， Where B is the frequency domain matrix of the matrix A to be determined by a two-dimensional QFT. For matrix units The two-dimensional quantum Fourier transform results; In matrix operator representation, based on the convention of positive and negative exponents in the coordinate direction, the above transpose is replaced with conjugate transpose or an equivalent row and column direction transformation.

7. The method according to claim 1, characterized in that, When at least one coordinate direction has a length that is not an integer power of two, a set of one-dimensional quantum Fourier transform results for that direction is generated using zero-element embedding, quantum Bluestein subroutines, or quantum Fourier transform subroutines of arbitrary length. Furthermore, for multiple coordinate directions with the same length or that can be embedded into the same power-of-two length space, the equivalence class partitioning results, representative quantum Fourier transform circuit structures, and phase angle parameter tables for the same direction are reused to reduce the compilation of redundant circuits and the calculation of redundant parameters in multidimensional cases.

8. The method according to claim 1, characterized in that, The classical reconstruction layer outputs a complete frequency domain matrix or frequency domain tensor, or only outputs a specified frequency point, a specified frequency domain region, a spectral peak, a threshold filtering result, or frequency domain interface data for subsequent quantum algorithms.

9. A multidimensional quantum Fourier transform system implementing the method of any one of claims 1 to 8, characterized in that, The system includes a classical preprocessing module, an equivalence class indexing module, a quantum basis transformation module, a phase compensation module, a tensor product assembly module, a classical reconstruction module, and an application interface module. The classical preprocessing module performs matrix or tensor unit decomposition and stores the coefficients. The equivalence class indexing module generates a set of representative elements and a phase angle table. The quantum basis transformation module prepares the representative elements, calculates the ground state, and performs a standard one-dimensional quantum Fourier transform. The phase compensation module performs diagonal phase compensation on non-representative elements. The tensor product assembly module forms the multidimensional transformation results for each unit element. The classical reconstruction module outputs the target frequency domain matrix, frequency domain tensor, or specified frequency domain coefficients. The application interface module provides frequency domain input for quantum phase estimation, quantum simulation, quantum linear algebra, quantum image processing, or multidimensional scientific computing tasks.

10. A quantum computing device, characterized in that, The method includes a classical controller, a quantum processor, and a memory, wherein the memory stores instructions for enabling the classical controller and the quantum processor to collaboratively execute the method of any one of claims 1 to 8; wherein the quantum processor executes the representative standard quantum Fourier transform and diagonal phase compensation gate, and the classical controller executes input decomposition, equivalence class index calculation, phase angle calculation, quantum circuit scheduling, result caching, and output reconstruction.

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