Method and system for encoding expected matrix into quantum circuit

Abstract

A computer-implemented method for encoding an expected matrix into a quantum circuit, the method comprising obtaining an MPO representation of the expected matrix; determining an approximate rank of the expected matrix based on the MPO representation; determining an orthogonal approximation initial guess of the expected matrix in the form of a tensor network of equidistant sub-tensors with approximate ranks; iteratively optimizing, starting from the initial guess, an orthogonal approximation of the expected matrix based on an optimization algorithm that minimizes a cost function under isometric constraints of the isometric sub-tensor, where the cost function imparts a cost to the orthogonal approximation of the expected matrix based on the quality of the orthogonal approximation with respect to the expected matrix; and encoding the orthogonal approximation into the quantum circuit based on encoding the equidistant sub-tensor as a quantum gate.

Description

Technical Field

[0001] The present invention belongs to the field of encoding computational operations into quantum circuits. More precisely, the present invention relates to computer-implemented optimization of desired computational operations for encoding into quantum circuits. Background Art

[0002] Quantum computers provide a platform for controllable quantum mechanical systems whose states and interactions can be controlled to perform computations. Computations are performed through the deterministic evolution of controllable quantum mechanical systems (e.g., qubits, which are quantum analogs of classical bits), and the states of quantum mechanical systems can be measured to determine the results of the computations.

[0003] The control operations on these qubits are called quantum gates. Quantum gates can act coherently on qubits to induce changes in the state of a single qubit (so-called single-qubit gates) and on multiple qubits (so-called multi-qubit gates) (for example, in order to entangle the states of multiple qubits) and any combination thereof. For example, a single-qubit gate can induce the spin state of an electron to rotate to a selectable value (for example, π / 2). Multi-qubit gates can act coherently on two or more qubits, such as performing a coherent CNOT operation on the states of two qubits. Multiple quantum gates can be applied to the qubits of a quantum computer in parallel or sequentially to perform calculations. Ultimately, the state of the qubit can be repeatedly measured after applying a series of quantum gates to determine the probability of each possible calculation result.

[0004] Essentially, the operation of a quantum computer can be thought of as relying on encoding a starting state into the internal quantum state of a qubit, then multiplying the internal quantum state of the qubit by a matrix (which can be implemented through a combination of different quantum gates), and measuring the result.

[0005] To compute solutions to problems that are considered intractable on classical computers, quantum computers can exploit the special properties of quantum mechanical states (particularly superposition and entanglement of different quantum states) to find solutions with a relatively small number of computational steps. In addition, since quantum operations can create complex superposition states of qubits, quantum computers can in principle access large internal memories to process computational states.

[0006] However, as Shende et al. discovered ("Minimal universal two-qubit controlled-not-based circuits"; Physical Review A, 69:062321, June 2004), converting an arbitrary operation into a combination of single-qubit gates and two-qubit gates typically requires a number of operations that is exponential in the number of qubits.

[0007] Lubasch et al. ("Variational quantum algorithms for nonlinear problems") teach solving nonlinear differential equations by variational quantum computation. The outputs of quantum variational circuits are combined by quantum nonlinear processing units and compared to an MPS anatomy, where a Hamiltonian representable as MPS is encoded as a quantum circuit according to the MPS quantum circuit encoding scheme. Summary of the invention

[0008] However, current approaches remain limited due to the significant complexity involved in implementing the corresponding steps of encoding states in qubits, encoding matrices through quantum gates, and measuring. Specifically, the exponential complexity of factoring arbitrarily multi-qubit gates with respect to the number of qubits, combined with the need for full connectivity between qubits, presents a significant obstacle. Although algorithms exist to convert large matrices into combinations of single- and two-qubit operations (e.g., CNOT gates), the complexity of the resulting circuits typically scales exponentially with the number of qubits. Examples of applicable problems using Hamiltonian-limited algorithms that can be expressed as MPSs, while variational methods typically rely on many quantum circuit evaluations to optimize variational quantum circuits.

[0009] In view of this prior art, an object of the present invention is to provide a computer-implemented method and a corresponding system for determining a matrix encoding of an intended computational operation, wherein the encoded operation can closely approximate the desired computational operation and can be efficiently implemented on quantum hardware.

[0010] This object is solved by a method and a system according to the independent claims. The dependent claims relate to preferred embodiments.

[0011] According to a first aspect, the present invention relates to a computer-implemented method for encoding an expected matrix into a quantum circuit. The method comprises obtaining a matrix product operator representation of the expected matrix, and determining an approximate rank of the expected matrix based on the MPO representation. The method further comprises determining an initial guess of an orthogonal approximation of the expected matrix in the form of a tensor network of isometric subtensors having an approximate rank, and starting from the initial guess, iteratively optimizing the orthogonal approximation of the expected matrix based on an optimization algorithm that minimizes a cost function under isometry constraints of the isometric subtensors. The cost function assigns a cost to the orthogonal approximation of the expected matrix based on the quality of the orthogonal approximation with respect to the expected matrix. The method further comprises encoding the orthogonal approximation into the quantum circuit based on encoding the isometric subtensors as quantum gates.

[0012] The expectation matrix may encode an expected multi-qubit operation that may act on the input state of the computational qubits of the quantum processing system. It should be noted that, although the expectation matrix is ​​regarded as a multi-qubit operation, for the method according to the first aspect, the expectation matrix is ​​not necessarily unitary. For example, a matrix often used in the context of quantum computing algorithms is the Laplace matrix M L , for example when solving partial differential equations. As another example, the diagonal matrix M D Can be used as part of an optimization algorithm.

[0013] A common property of these two exemplary expected matrices when viewed as quantum operators is that they have relatively small entanglement and / or that the matrices can be described by tensor networks with relatively small ranks (i.e., ranks of two and three for diagonal matrices and Laplacian operators, respectively).

[0014] The method according to the first aspect exploits this property by initially obtaining a tensor network representation of the expected matrix (particularly in the form of a tensor column, which may have a relatively small bond dimension). However, the matrix product operator representation / approximation of the general expected matrix does not have a direct correspondence with a quantum circuit that can be determined without exponential computational difficulty. Instead, an approximation rank is determined based on the matrix product operator representation of the expected matrix, which should be equal to or greater than the rank of the tensor column representation in order to approximate the expected matrix using an orthogonal approximation.

[0015] In a preferred embodiment, the approximation rank is greater than the rank of the matrix product operator representation of the expected matrix, wherein the rank of the matrix product operator representation is in particular the bond dimension of the matrix product operator representation.

[0016] Since the orthogonal approximation relies on the constraint of isometric subtensors, an increase in the rank of the orthogonal approximation can increase the degrees of freedom to accurately approximate the expected matrix.

[0017] It should be noted that the exact matrix product operator representation can be determined for certain matrices, but the matrix product operator representation can usually be obtained through an approximation algorithm (such as a cross approximation or an algorithm based on singular value decomposition) to approximate the expected matrix using MPO without significantly affecting the results of the method. The skilled person will further recognize that the matrix product operator representation / approximation may already be known, and it may be sufficient to receive the matrix product operator representation from a suitable source.

[0018] The MPO representation may include non-isometric subtensors and may be expressed with a certain error (which may be less than 1% or less than 10 -6 , e.g., depending on the application) approximates the expected matrix and may also be referred to as a non-orthogonal MPO approximation of the expected matrix. For example, for solving optimization problems, an approximation to the expected matrix with an error of about 1% may be sufficient, while for solving PDE problems, an approximation with an error of about 10 -5 The desired matrix can be approximated with an error of 100% or less. The skilled person will recognize that not every matrix can be represented by an MPO representation with arbitrarily small accuracy, but the skilled person may still choose to apply the method in view of the increased inaccuracy of the MPO approximation, for example if there is no other way to perform the calculation, or if a relatively low accuracy is sufficient for the corresponding computational problem.

[0019] Based on the key dimension (rank) represented by the matrix product operator, the approximate rank can be chosen to be the next power of two (i.e. 2 N , where N is an integer), or any power of two that is greater than the key dimension of the matrix product operator representation. As an example, if the rank of the matrix product operator representation of the intended matrix is ​​3, then the approximation rank may be 4 or 8. However, if the approximation rank is equal to or less than the rank of the matrix product operator representation of the intended matrix, then a suitable approximation can in principle be made, although an orthogonal approximation is more likely to be an inaccurate representation of the intended matrix.

[0020] To determine the orthogonal approximation, the method includes determining an initial guess for an orthogonal approximation of the expected matrix, which initial guess may correspond to a tensor column of isometric subtensors of the approximate rank. The initial guess may be composed of random subtensors that satisfy an isometry condition.

[0021] In a preferred embodiment, the orthogonal approximation is a matrix product operator, and the orthogonal approximation is mathematically equivalent to a tensor network in which each intermediate isometric sub-tensor has two outer uncontracted indices and two inner indices that are contracted with adjacent isometric sub-tensors in a chained manner.

[0022] The unmerged indices may correspond to the dimensions of the input and output qubit states of each isometric subtensor, while the internal indices may be the approximate rank.

[0023] In a preferred embodiment, the approximate rank is the common bond dimension of all sub-tensors of the tensor network.

[0024] Subsequently, the initial guess is optimized based on an iterative optimization algorithm to minimize the difference between the orthogonal approximation and the expected matrix. The difference can be based on a norm of the difference, such as a Frobenius norm or a cosine norm of the difference between the orthogonal approximation and the expected matrix or a subtensor of the orthogonal approximation.

[0025] In a preferred embodiment, the cost function is based on a difference function of the expected matrix and a renormalized orthogonal approximation evaluated based on tensor network calculus, wherein the renormalized orthogonal approximation is based on the orthogonal approximation multiplied by a renormalization constant, wherein the cost function is particularly based on the Frobenius norm of the difference function evaluated based on tensor network calculus, and / or wherein the renormalization constant is selected such that dividing the expected matrix by the normalization constant does not increase the trace of the state on which the expected matrix acts, and / or such that the trace of the density matrix on which the renormalized orthogonal approximation acts is equal to or less than 1, regardless of the initial state.

[0026] The renormalization constant can preserve the constraint that quantum operations cannot increase the trace of the density matrix of the quantum state, that is, given any input density matrix ρ, the orthogonal approximation A should satisfy:

[0027] Given any expected matrix M, the orthogonal approximation A can be renormalized by the renormalization constant c, so that the cost function of the difference is expressed as C = ||cA-M|| 2 , where |||| represents the chosen difference norm, such as the Frobenius norm. In other words, the operation to be implemented in the quantum computer can be to renormalize the expected matrix c -1 Approximation of M.

[0028] In a preferred embodiment, according to Alternatively, an orthogonal approximation A of the expected matrix M is optimized according to a gradient-based optimization algorithm so as to obtain a renormalization constant c at each iterative step of minimizing the cost function.

[0029] For example, the renormalization constant may be updated using steepest descent on c to minimize the cost function.

[0030] Based on a cost function that quantifies the difference between the (renormalized) orthogonal approximation and the expected matrix, the elements of the orthogonal approximation can be updated to minimize the cost function.

[0031] In a preferred embodiment, the optimization algorithm comprises gradient descent, and in particular an optimization algorithm based on Riemannian gradient descent or derived therefrom.

[0032] In other words, the method may comprise (stochastically) determining the gradient of the difference between the orthogonal approximation and a matrix product operator representation of the expected matrix, and subsequently updating the orthogonal approximation in accordance with the determined gradient.

[0033] The skilled person will recognize that in some embodiments, minimizing the difference between the matrix product operator representation of the orthogonal approximation and the expected matrix may be preferred because this can reduce the computational complexity of the optimization. For example, the difference between the sub-tensor of the orthogonal approximation and the matrix product operator representation of the expected matrix can be determined, and the corresponding isometric sub-tensor can be updated based on the difference. Any update to the isometric sub-tensor should preserve the isometry constraint, that is, the updated isometric sub-tensor should be isometric.

[0034] In a preferred embodiment, the orthogonal approximation is a matrix product operator, which can be expressed as

[0035]

[0036] Among them, V k is isometric, that is, for any index k of the isometric subtensor, And where each isometric subtensor V k Specifically, an iterative optimization is performed by alternating descent to optimize the orthogonal approximation.

[0037] The alternating descent may be implemented as a stochastic gradient descent of each sub-tensor, for example in a predetermined or random order, such that each equidistant sub-tensor of the orthogonal approximation approaches a sub-tensor represented by a matrix product operator of the expected matrix.

[0038] In a preferred embodiment, the gradient of the cost function is determined based on multiple partial derivatives of the sub-tensors of the orthogonal approximation (e.g., by determining the partial derivative of the cost function with respect to the value of each sub-tensor, and updating the orthogonal approximation based on the partial derivatives of multiple sub-tensors (e.g., all sub-tensors). The skilled person will understand that the optimization may include stochastic optimization, for example, partial derivatives need not be determined for all parameters of the orthogonal approximation at each iteration, but rather a randomly determined subset of parameters may be updated, such as in stochastic gradient descent and related optimization algorithms.

[0039] However, the updated subtensors should preserve the isometries, for example by updating the isometries according to Riemannian gradient descent. Riemannian gradient descent can preserve the isometries, for example by projecting the update vectors and / or the updated subtensors onto the Stiefel manifold of isometries.

[0040] For example, optimizing the orthogonal approximation under the isometry constraint may include determining the partial derivative of the cost function or the optimization vector obtained thereby at the point V kStiefel manifolds in m×p isometric matrices Projection on the space tangent

[0041] For example, the subtensor V k An optimization algorithm similar to steepest descent optimization can be used according to The update is performed by , where α is the step size. However, the partial derivatives may also be used initially to determine the search direction, for example based on adaptive moment estimates, and the search direction may be projected onto the space tangent of the Stiefel manifold.

[0042] In a preferred embodiment, optimizing the orthogonal approximation under the isometry constraint comprises determining the subtensor V based on partial derivatives of the cost function and optionally based on information from a previous iteration step of the optimization algorithm. k The search direction p k ; and determine the search direction p k At point V k Stiefel manifolds with m×p isometric matrices Projection on tangent space Among them, when reshaped into a two-dimensional matrix, the isometric subtensor V k The dimension is m×p.

[0043] The partial derivatives and / or the search direction may then be used to determine an update direction, and the isometric subtensor and / or the orthogonal approximation may be modified based on the update direction. For example, the modified subtensor and / or the modified orthogonal approximation may be retracted onto the Stiefel manifold to obtain an updated subtensor and / or an updated orthogonal approximation.

[0044] In some embodiments, optimizing the orthogonal approximation under the isometry constraint includes: projecting the modified subtensor and / or the modified orthogonal approximation onto a Stiefel manifold of the subtensor and / or the orthogonal approximation, respectively, wherein the modified subtensor and / or the modified orthogonal approximation is modified based on the search direction and / or the update direction; and / or includes determining the subtensor V k The update to the retraction on the manifold To update the sub-tensor V k , where the retraction is based on the update direction The update direction is based on the partial derivative of the cost function with respect to the subtensor V k Encodes the direction of the update.

[0045] Retracting can update the vector Projected onto the product of the Stiefel manifold and the isometric submatrices, for example, according to Thus, isometry constraints are enforced in all iteration steps. For example, the updated intermediate subtensor can be determined based on the projection of the updated gradient on the space tangent of the Stiefel manifold, and the updated intermediate subtensor can be retracted onto the Stiefel manifold.

[0046] The skilled person will appreciate that this technique can be extended to an optimization technique (which may include momentum), such that the update vector can be a function of the projection of the partial derivatives onto the tangent of the Stiefel manifold and the calculation results from the previous iteration step. For example, the iterative optimization can be based on an adaptive moment based optimization algorithm. Therefore, more generally, the update of the isometric subtensors can be based on an update function that is based on the projection of the partial derivatives of the cost function onto the tangent of the Stiefel manifold, wherein the update function determines the intermediate updates; and / or the update of the isometric subtensors can be based on the retraction of the intermediate updates on the Stiefel manifold, so that the resulting subtensors / orthogonal approximations are isometric.

[0047] In a preferred embodiment, optimizing the orthogonal approximation under the isometry constraint includes updating the orthogonal approximation by updating all isometric subtensors, wherein each isometric subtensor V k All pass through point V k Stiefel manifold of m×p isometric matrices along the update direction The retraction is performed on the update, where, when reshaped into a two-dimensional matrix, the isometric subtensor V k The dimension of is m×p, and the update direction is the direction that minimizes the cost function based on the partial derivative of the cost function, and in particular uses the projection p k,tangent As used to update the isometric subtensor V k Update direction.

[0048] The skilled person will appreciate that different deflations may be used in embodiments, such as the Cayley deflation or the singular value decomposition, to name just two examples, and that the skilled person may select a suitable deflation from a variety of known deflations.

[0049] During iterative optimization of the orthogonal approximation, the isometric subtensors may be reshaped into matrices based on isometric constraints on the subtensors, the matrices being square for intermediate subtensors of the orthogonal approximation and the reshaped matrices corresponding to the intermediate subtensors being unitary.

[0050] In a preferred embodiment, before the orthogonal approximation is expanded, when the isometric subtensors and intermediate tensors of the orthogonal approximation at the ends of the orthogonal approximation are reshaped into a two-dimensional matrix, the isometric subtensors are isometric and the intermediate tensors are unitary, so that after the isometric subtensors are reshaped into a square matrix, all the isometric subtensors are unitary.

[0051] To obtain a set of unitary matrices that can be implemented in a quantum circuit, auxiliary qubits can be used to extend the subtensors at the ends of the tensor chain (sometimes also called boundary cores), for example, by preparing an auxiliary qubit in the zero state |0> to multiply with the first isometric subtensor and by implementing a post-selection at the opposite end of the tensor chain.

[0052] Thus, the method may include expanding the orthogonal approximation so that after reshaping the isometric subtensors into square matrices, all isometric subtensors are unitary. Expanding the orthogonal approximation may be part of determining an initial guess for the orthogonal approximation, or may be performed at the beginning of an iterative optimization of the orthogonal approximation or after it has been iteratively optimized. For example, the initial guess may consist of subtensors that may be reshaped into unitary matrices, and the determination of the cost function may take into account an implicit expansion of the orthogonal approximation.

[0053] In some embodiments, the extended orthogonal approximation is based on providing one or more additional auxiliary qubits as input to the first end of the tensor network, wherein the method specifically provides at least log R additional auxiliary qubits, wherein R is the approximation rank.

[0054] In some embodiments, extending the orthogonal approximation includes measuring a qubit at the second end of the tensor network, wherein selecting a result having a predetermined measurement result of the qubit measured at the second end of the tensor network is part of implementing the expected matrix.

[0055] Based on the addition and post-selection of auxiliary qubits, all sub-tensors can be square after appropriate reshaping, and can thus form a set of unitary matrices that can be implemented in quantum circuits by decomposing the isometric sub-tensors into quantum gates of the corresponding quantum architecture (e.g., a combination of single-qubit rotations and CNOT gates).

[0056] The skilled person will recognize that the expansion can be performed before or after the iterative optimization of the orthogonal approximation. In other words, the initial guess can be defined based on the expanded matrix product operator, or the orthogonal approximation can be expanded after it has been iteratively optimized, for example by introducing and filling additional elements of the boundary core so that the boundary core obtained after appropriate reshaping is unitary.

[0057] In a preferred embodiment, the method further comprises implementing an orthogonal approximation of the expected matrix as a quantum circuit on quantum hardware based on encoding the isometries as quantum gates.

[0058] The decompositions of the corresponding isometric subtensors can be combined to implement an orthogonal approximation based on sequential application of the corresponding isometric subtensors, where the output state of log R qubits produced by the unitary operation corresponding to one of the subtensors can be used as input to the unitary operation corresponding to the next isometric subtensor in the tensor network.

[0059] In a preferred embodiment, implementing an orthogonal approximation of the expected matrix as a quantum circuit is based on sequentially applying isometries to a set of qubits and rearranging the unitary matrix into a quantum network.

[0060] As a result, the depth of a quantum circuit can grow only polynomially with the number of qubits, rather than exponentially as is typically done by factoring matrices into single-qubit and two-qubit gates.

[0061] According to a second aspect, the present invention relates to a processing system for encoding an expected matrix into a quantum circuit. The system is configured to determine the approximate rank of a matrix product operator representation of the expected matrix, and to determine an initial guess of an orthogonal approximation of the expected matrix in the form of a tensor network of isometric subtensors having the approximate rank. The system is further configured to iteratively optimize the orthogonal approximation of the expected matrix based on an optimization algorithm starting from the initial guess, the optimization algorithm minimizing a cost function under the isometry constraint of the isometric subtensors, wherein the cost function assigns a cost to the orthogonal approximation of the expected matrix based on the quality of the orthogonal approximation with respect to the expected matrix. The system is further configured to determine the implementation of the orthogonal approximation of the expected matrix in the quantum circuit based on encoding the isometric subtensors as quantum gates.

[0062] The system may implement the method according to the first aspect or any combination of the embodiments of the first aspect. In particular, the system according to the second aspect may also benefit from any features of the preferred embodiments of the first aspect.

[0063] The processing system may include a single processing unit, or may include multiple processing units that may be functionally connected. The processing unit may include a microcontroller, an ASIC, a PLA (CPLA), an FPGA, or other processing device (including a processing device that operates based on software, hardware, firmware, or a combination thereof). The processing device may include an integrated memory, or communicate with an external memory, or both, and may further include an interface for connecting to a sensor, device, apparatus, integrated logic circuit, other controller, etc., wherein these interfaces may be configured to receive or send signals such as electrical signals, optical signals, wireless signals, acoustic signals, etc.

[0064] The system may be configured to obtain an MPO representation of the expected matrix, for example by executing a decomposition algorithm on the expected matrix or by receiving the MPO representation from a suitable source, such as a database.

[0065] In some embodiments, the system is further configured to extend the orthogonal approximation so that after reshaping the isometric subtensors into square matrices, all isometric subtensors are unitary.

[0066] In a preferred embodiment, the system is further configured to transmit the implementation to a quantum computing system to perform the orthogonal approximation on quantum hardware.

[0067] According to a third aspect, the present invention relates to a hybrid quantum-classical computing system, the hybrid quantum-classical computing system comprising the system according to the second aspect and quantum computing hardware. The hybrid quantum-classical computing system is configured to receive the embodiment from the processing system, implement the embodiment in the quantum computing hardware, and receive the calculation result from the quantum computing hardware.

[0068] According to a fourth aspect, the invention relates to a non-transitory medium comprising machine-readable instructions which, when executed by a processing system, implement the method according to the first aspect and / or the system according to the second and / or third aspect. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] The features and numerous advantages of the method and system according to the present invention will be best understood from the detailed description of the preferred embodiments with reference to the accompanying drawings, in which:

[0070] Figure 1 An example of a quantum computing system is schematically illustrated;

[0071] Figure 2 An example of a computer-implemented method for encoding a desired matrix into a quantum circuit is illustrated;

[0072] Figure 3 schematically illustrates a flow chart for determining a matrix product operator representation based on a given expected matrix;

[0073] Figure 4 schematically illustrates a strategy for determining a quantum circuit encoding an approximation of an expected matrix according to an example;

[0074] Figure 5 and Figure 6 The diagram shows the combination Figure 4 The results obtained by applying the discussed strategy to two different expectation matrices; and

[0075] Fig. 7A and Figure 7B

[0046] Illustrated are experimental runtime results from implementing a method for determining a quantum circuit when approximating a diagonal matrix, according to an example. DETAILED DESCRIPTION

[0076] Figure 1 An example of a quantum computing system 10 is schematically illustrated. The system 10 includes a qubit register 12 including computational qubits, a state preparation module 14 for preparing the quantum state of the computational qubits to a predetermined quantum state, a quantum matrix multiplication module 16 for transforming the predetermined quantum state according to a multi-qubit operation, and a measurement module 18 for measuring the result of the quantum computation, for example, by performing a projection measurement on the state of some or all of the qubits. The processing system 20 may initialize the computational qubits in the qubit register 12, define the operations of the state preparation module 14 and / or the quantum matrix multiplication module 16, and may obtain the measurement results from the measurement module 18, for example, by determining a control operation and sending the control operation to the hardware of the quantum computing system 10.

[0077] The state preparation module 14 and the quantum matrix multiplication module 16 may be similar in construction, and may implement a combination of single-qubit quantum gates and multi-qubit quantum gates to affect the state of the qubits of the qubit register 12. It should be noted that in some embodiments, the state preparation module 14 may not be required, and the combination of quantum gates determined by the quantum matrix multiplication module 16 may directly act on the quantum state of the qubits initialized to a certain initialization quantum state in the qubit register 12. The type of quantum gate implemented in the quantum circuit by the state preparation module 14 and / or the quantum matrix multiplication module 16 may depend on the architecture of the specific quantum hardware used, such as a native gate based on quantum hardware. The state preparation module 14 and the quantum matrix multiplication module 16 may calculate the quantum gate arrangement and / or send control operations to control the quantum hardware so that the quantum hardware implements the desired computational operation on a certain quantum state.

[0078] However, decomposing arbitrary computational operations into quantum circuits is generally a difficult computational problem, whose complexity usually grows exponentially with the number of qubits involved. In addition, for general operations, the depth of the circuit (i.e., the number of consecutive quantum gates used to implement the quantum operation) may also grow exponentially with the number of qubits. Although it is known that for certain types of problems, it is possible to efficiently encode problems as combinations of quantum gates, encoding many desired types of problems as quantum circuits may still be a significant obstacle to achieving quantum computing.

[0079] Figure 2An example of a computer-implemented method for encoding an expected matrix into a quantum circuit is illustrated. The method includes obtaining a matrix product operator representation of the expected matrix (S10), and determining an approximate rank of the expected matrix based on the MPO representation (S12). The method further includes determining an initial guess of an orthogonal approximation of the expected matrix in the form of a tensor network of isometric subtensors having approximate ranks (S14), and starting from the initial guess, iteratively optimizing the orthogonal approximation of the expected matrix based on an optimization algorithm (S16), the optimization algorithm minimizing a cost function under isometry constraints of the isometric subtensors. The method further includes encoding the orthogonal approximation into the quantum circuit based on encoding the isometric subtensors as quantum gates (S18).

[0080] The cost function assigns a cost to an orthogonal approximation of the expected matrix based on the quality of the orthogonal approximation with respect to the expected matrix. Thus, instead of directly implementing the expected matrix in a quantum circuit, the method comprises approximating the expected matrix using an orthogonal approximation constructed in the form of a tensor network with isometric subtensors (hereinafter also referred to as isometric cores).

[0081] The isometry constraints on the subtensors of the orthogonal approximation can be enforced at each step of the iterative optimization (e.g., based on a modified gradient descent method and / or scaling back any updated subtensors onto isometry subtensors) to cause the optimization algorithm to converge toward an approximation of the desired matrix with isometry subtensors. In general, the approximation rank should be at least equal to the rank of the matrix product operator representation of the desired matrix, so that the constraint of isometry subtensors can be at least partially compensated by greater freedom in the merge indexing of the orthogonal approximation.

[0082] For any subtensor between the ends of the tensor network (the so-called boundary cores), the isometric subtensors can be reshaped into square matrices, which can be unitary after appropriate reshaping. These subtensors that can be reshaped into unitary matrices can be directly implemented as quantum circuits using known decomposition algorithms, which can be efficiently computed due to the small size of the reshaped matrices / isometric subtensors. The skilled person will recognize that the subtensors can be implemented without explicitly reshaping the isometric subtensors into matrices.

[0083] To implement boundary cores (i.e., isometric subtensors arranged at the ends of a tensor network in the form of tensor columns), the orthogonal approximation can be extended so that all isometric subtensors can be reshaped into unitary matrices, for example by providing auxiliary qubits at one end of the tensor network and introducing measurements with post-selection at the other end so that the boundary cores can likewise be reshaped into square matrices. This extension can be pre-configured as part of the initial guess for the orthogonal approximation, or can be configured when the orthogonal approximation has been iteratively optimized, for example by completing the boundary cores with additional indices so that they are square matrices when the appropriate reshaping is applied, and filling in any missing matrix elements so that the resulting matrix is ​​unitary.

[0084] Therefore, the orthogonal approximation can be implemented in quantum circuits based on the decomposition of isometric subtensors that require less memory than the expected matrix. As a result, the circuit depth of the quantum circuit for the entire expected matrix can grow only polynomially with the number of qubits, allowing more complex computations on near-term quantum hardware.

[0085] Figure 3 Schematically illustrates a method for calculating the expected value based on a given expected matrix 24 (shown as a square matrix M) according to an example. NxN And the flowchart of the matrix product operator representation 22 is determined in Penrose graphical representation 24a. The expected matrix 24 may be the expected computational operation to be performed on quantum hardware, for example, given the order N=2 n The diagonal operator M D or the Laplace operator M L , which may correspond to the computational ground state of a qubit register 12 having n qubits. As an example, the diagonal operator M D and the Laplace operator M L It can be expressed in matrix form as:

[0086]

[0087] These matrices can be involved in algorithms related to quantum computing, such as solving partial differential equations and optimization algorithms, but are not unitary.

[0088] The expected matrix 24 can be represented by a circle with two unmerged N-dimensional indices in a Penrose diagram representation 24a. The expected matrix 24 can be approximated or represented by a matrix product operator representation 22 consisting of a plurality of sub-tensors 26, where summing based on the merged indices restores the individual elements of the expected matrix 24, for example, based on:

[0089]

[0090] Among them, the expected matrix 24A l mThe MPO representation 22 of can be composed of log(N) sub-tensors 26, where the unmerged index is l k and m k , and the merge index is j k 、j k-1 .exist Figure 3 In FIG. 2 , each subtensor 26 indicated by a circle has four indices, except for the boundary core at the opposite end of the MPO representation 22 (which has three indices), where the merged indices (extending horizontally) are the MPO bond indices with a bond dimension of r. In the example, the unmerged indices have the dimension of the quantum state of one qubit, i.e., 2.

[0091] Such an MPO representation 22 can be found for the expected matrix 24 or its approximation using a variety of known algorithms. In principle, any expected matrix 24 can be represented in a matrix product operator (MPO) representation 22 by decomposing the expected matrix 24 into a set of matrix products whose characteristic is that their matrix product is equal to a particular element of the expected matrix 24.

[0092] For a large class of operators that includes all Hamiltonians with short-range interactions in one dimension (such as the diagonal operator M D or the Laplace operator M L ), the required MPO bond dimension r is both small (e.g., ≈5) and constant with the system size, so that the expected matrix 24 can be stored using multiple relatively small subtensors 26. However, while MPS states can be efficiently encoded into quantum circuits (e.g., as shown by Lubasch et al.), no corresponding strategy exists for general MPO.

[0093] despite this, Figure 2 The method shown in the example of can be used to provide a quantum circuit for approximating the expected matrix 24.

[0094] Figure 4 Schematically illustrated is a strategy according to an example for determining a quantum circuit 28 encoding an approximation of an expected matrix 24. Based on a suitable decomposition and / or approximation algorithm, the expected matrix 24 is initially represented by an MPO representation 22 having log(N) subtensors 26 and rank r.

[0095] Based on the rank r of the MPO representation 22, an approximate rank R is selected, which should be equal to or greater than the rank r of the MPO representation 22 and can be a power of two, i.e., 2 k, where k is an integer. Based on the selected approximation rank R, the method can generate an initial guess for the orthogonal approximation 30 using the isometric subtensors 32, 32a, 32b so that all isometric subtensors 32, 32a, 32b are isometric when reshaped into a two-dimensional matrix, where the bond dimension of the orthogonal approximation 30 is equal to the approximation rank R. The isometric subtensors 32, 32a, 32b used for the initial guess of the orthogonal approximation 30 can be randomly generated, for example, selected from random isometric subtensors 32, 32a, 32b.

[0096] Starting from an initial guess of the orthogonal approximation 30, the method can proceed by iteratively optimizing the orthogonal approximation 30 based on an optimization algorithm (whose goal is to minimize the distance measure with the expected matrix 24), wherein the intermediate step size of the optimization algorithm can maintain the isometry of the sub-tensors 32, 32a, 32b.

[0097] Since the expected matrix 24 may not be a unitary matrix that can be directly implemented on a quantum computer, the quantum operation A will have to preserve the trace

[0098]

[0099] Therefore, it would be preferable to minimize the difference metric with respect to the renormalized orthogonal approximation 30, for example according to

[0100] C=|cA-M|| 2 , (4)

[0101] So that the inequality of equation (3) is satisfied. Therefore, the renormalized expected matrix 24 (e.g., c -1 M), rather than the expected matrix 24, M. From a practical point of view, some computations may require remembering this normalization factor c and taking it into account at the end of the computation, for example to interpret the results when they depend on complex output quantum states.

[0102] For the optimization algorithm, the task of the processing system (such as a computer) may be to perform the multiplication of all sub-tensors 32, 32a, 32b (abbreviated as V k ) finds the minimum value of the cost function under the isometry constraint, that is,

[0103]

[0104] Wherein, ||*|| represents a suitable norm, such as the Frobenius norm, and wherein both c and A take optimal values. The optimization can be implemented as an alternating descent by determining the partial derivatives of the cost function with respect to any isometric subtensors 32, 32a, 32b under an isometry constraint, and by optimizing the orthogonal approximation 30 based on the partial derivatives, such as by a gradient descent-based algorithm, such as stochastic gradient descent or a momentum-based algorithm (such as adaptive moment estimation).

[0105] The isometry constraint may be obeyed by ensuring the isometry of the sub-tensors 32, 32a, 32b in each update step of the optimization algorithm and / or by considering the isometry constraint as part of the gradient direction.

[0106] In general, any m-by-p isometric matrix belongs to the Stiefel manifold

[0107]

[0108] This is a Riemannian manifold. Therefore, the cost function C to be minimized can be defined on the Cartesian product of n Stiefel manifolds. In order to find a set of isometric matrices V that minimize this cost function k , we can use gradient-based optimization methods, such as the Riemann gradient descent method, which can solve the optimization problem under constraints in this case.

[0109] In conventional gradient-based methods (such as gradient descent), the variable being optimized is updated by moving in the opposite direction of the gradient of the cost function:

[0110]

[0111] where α is the step size. In the Riemann generalization of these methods, to determine the direction of this movement, one can use its position at point V k The projection δ k R C, without using δ k C. Further, in order to update the matrix V based on the partial derivatives and / or their projections k The value of can be used to shrink on the manifold:

[0112]

[0113] In optimization methods that use information from previous optimization steps (such as gradient descent with momentum), the concept of vector transfer is also used, which is a generalization of the concept of parallel transfer and can therefore be implemented directly based on the above-mentioned scheme.

[0114] The Riemann formula for gradient descent can be implemented as an iterative procedure that computes at each step the Riemann gradient of the optimization function C at the current point on the manifold, and can then find the next point using a retraction chosen in the direction of the negative gradient.

[0115] Then, the algorithm can be implemented as follows:

[0116] Step 0: Get an initial guess of the orthogonal approximation 30;

[0117] Step 1: Find a renormalization constant c that minimizes the cost function of, for example, equation (4), where the orthogonal approximation 30 is fixed (ie, A is constant in this step);

[0118] Step 2: For the problem of equation (5), find a new approximation A via a step of Riemann gradient descent, where the renormalization constant c can be constant during this step, and then return to step 1.

[0119] The iterative algorithm may terminate at any point, such as after a predetermined number of iterations, or when the cost function has reached a predetermined cost threshold.

[0120] Once the optimized orthogonal approximation 30 is obtained (which may be a renormalized orthogonal approximation 30 of the expected matrix 24), c -1 M), all internal sub-tensors 32 of the orthogonal approximation 30 should already be unitary matrices after corresponding reshaping into two-dimensional matrices.

[0121] However, the sub-tensors 32a, 32b (i.e., V1 and V2) at the boundary n , also called boundary cores) can only be isometric. However, the boundary cores 32a, 32b can be expanded to unitary matrices by expanding the subtensors under an isometry condition with an addition of a |0> state 36 at one end of the tensor network and a projection onto a |0> state 38 at the other end of the tensor network to maintain the correspondence with the expected matrix 24. By expanding the orthogonal approximation 30, an expanded orthogonal approximation 34 can be obtained, which is characterized only by having a subtensor 32 that can be reshaped into a unitary matrix.

[0122] Since all sub-tensors 32 can be reshaped into unitary matrices, they can be obtained by performing quantum operations 40 (implemented as unitary operations U1 to U n ) in a cascade of isometric subtensors 32 to implement the conversion to quantum circuits, such as Figure 4As shown in the quantum circuit 28 in the example. Here, the unmerged indices of the isometric subtensor 32 may correspond to the input states and output states of the corresponding quantum operation 40, while the merged indices of the isometric subtensor 32 correspond to the quantum state that is the input of the next unitary operation 40 in the serial chain of unitary operations 40. Therefore, each unitary operation 40 has 1+log(R) qubit inputs and qubit outputs, and log(R) qubits are processed by the subsequent unitary operation 40 in the serial chain of unitary operations 40. The decomposition of these subtensors 32 into quantum gates can be performed based on known decomposition techniques for the corresponding quantum hardware architecture.

[0123] The upper |0> state 36 as the input of the upper boundary core 32a of the extended orthogonal approximation 34 may correspond to preparing the additional auxiliary qubit 42 to the "zero" state. At the same time, the lower |0> state 38 as the input of the upper boundary core 32a of the extended orthogonal approximation 34 may correspond to a projection onto the zero state, i.e., a measurement with post-selection.

[0124] The skilled person will recognize that the actual quantum states prepared or measured for these operations can be changed, as long as the correspondence with the expected matrix 24 is maintained, such as by preparing and post-selecting based on the |1> state or a different state, based on an appropriate basis change. However, for the sake of simplicity, the preparation of the "|0>" state and the corresponding post-selection will be considered below.

[0125] Therefore, if Figure 4 As shown in the example of FIG. 2 , the quantum circuit 28 can be implemented by the following steps of the quantum processing system:

[0126] Step 1: Prepare auxiliary qubit 42 to zero state The remaining qubits remain "branch open" (solid horizontal line on the left side of quantum circuit 28), ie the qubits will be encoded according to some initial state based on the input to the computation.

[0127] Step 2: Apply gates U1,...,U sequentially n , these gates should be decomposed into quantum gate implementations based on the quantum hardware employed, such as a combination of single-qubit operations and two-qubit operations.

[0128] Step 3: Measure the auxiliary quantum bit 44 on the computational basis, and if all the measurement results in step 3 are equal to "0", record the results, otherwise return to step 1.

[0129] Note that for post-selection, the qubits originally introduced as auxiliary qubits 42 into the initial boundary core 32a should not be measured, but rather the qubits from the other end of the orthogonal approximation 30 (i.e., corresponding to the opposite boundary core 32b (see Figure 4 ))'s quantum bit.

[0130] Only when all auxiliary qubit measurements 44 are equal to |0> (e.g., a string of zero bits for the corresponding qubits is obtained), the algorithm implements the quantum circuit 28 corresponding to the expected matrix 24. The state of the quantum system before the measurement determines the probability of this "successful" outcome. This means that the initial state of the system to which the quantum scheme is applied affects this probability.

[0131]

[0132] The initial state of the system is in It can be determined by the specific computing task. The probability values ​​of all possible initial states of the system can be averaged. <Pr suc > to ignore this dependence. To this end, ρ in can be viewed as a maximally mixed state, i.e., a uniform mixture of states from an orthogonal basis. In this case, ρ in =1 / 2 n I, using <0| a2 U|0> a1 =A, and the average probability can be obtained as

[0133]

[0134] That is, the success probability can be mainly determined by the norm of the orthogonal approximation 30. At the same time, since A is approximately equal to c -1 M, we can see how the success probability is related to the normalization coefficient c, i.e. the higher the normalization coefficient, the lower the success probability. In other words, there may be a trade-off between the accuracy of the approximation and the success probability. It is found that the optimal theoretical value of the normalization constant c is c = λ max (M), which is equal to the maximum eigenvalue of the expected matrix 24. As long as the approximation rank R of the orthogonal approximation 30 is large enough, the expected matrix 24 can be approximated accurately enough.

[0135] Note that the depth of quantum circuit 28 can grow polynomially with the approximation rank R, and the size of the approximation rank R can further increase the complexity of decomposing the isometric subtensor 32 into a combination of quantum gates, since each unitary operation U k Both act on log2 R+1 qubits, where R is the tensor sequence (TT)-rank of the orthogonal approximation A.

[0136] Note also that the numerical strategy of approximating the decomposition of general multi-qubit unitary operations using a combination of single-qubit and CNOT gates can achieve a number of CNOT gates close to the theoretical lower limit proposed by Shende et al. (Minimal universal two-qubit controlled-not-based circuits, Physical Review A, 2004), i.e., 4 for any decomposition of m qubits. m-1 CNOT gates, so in this case, a total of n*4 log R+1 CNOT gates, which is about n*R 2 Therefore, in general, the lower the approximation rank R selected for the orthogonal approximation 30, the simpler the quantum circuit 28 is, but the less accurate the orthogonal approximation 30 is.

[0137] To demonstrate the power of this approach, we consider commonly used matrices and encode them into multi-qubit quantum circuits28. Specifically, we consider D and the Laplace operator M L The matrix with discrete linear functions (corresponding to the discretized second-order derivative matrix) on , neither of which is unitary.

[0138] Figure 5 and Figure 6 The diagram shows the combination Figure 2 and Figure 4 The discussed algorithms are applied to the diagonal matrix M of equation (1) D and the Laplace matrix M L The result obtained. Laplace operator M L can be used to solve partial differential equations (PDEs), and the diagonal operator M D It can be used for optimization and PDE solving. For any n>1, M D and M L The MPO form of the matrix has TT-rank with r equal to 2 and 3 respectively.

[0139] The upper diagram illustrates the relationship between the diagonal matrix ( Figure 5 ) and the Laplacian matrix ( Figure 6 ) as a function of the approximation rank R (in terms of the number of auxiliary qubits 42 used being equal to log2 R) (n being the number of qubits). The lower graph illustrates the corresponding average probability of success based on the auxiliary qubit measurement 44. The dashed line represents ||M|| 2 λ max -2 (M)2 -n(See equation (10)), for both matrices, this value is independent of n when the number of quantum bits n is large.

[0140] For these two matrices, the method can achieve reasonable accuracy (M D and M L The errors of about 0.01% and 0.5% for qubits and 50 qubits, respectively, and a sufficient success probability (over 5%) up to 50 qubits are not a limitation of the method. The important property found in the experiments is that the errors and success probability are almost independent of the number of qubits, making the method very likely to be applicable to more complex tasks.

[0141] To assess computational efficiency, the algorithm can be considered to consist of three main steps, namely, finding an orthogonal approximation 30, expanding the boundary core 32a, 32b to allow reshaping into a unitary matrix, and decomposing the resulting 2R*2R unitary matrix into a series of single-qubit and two-qubit gates.

[0142] The time required for the last step is determined by the decomposition method used and will not be discussed here. Furthermore, in the example, the expansion step is skipped and the values ​​corresponding to U1 and U are directly optimized in an iterative optimization algorithm based on an initial guess of the orthogonal approximation 30. n The expanded sub-tensor 32 instead of optimizing kernels V1 and V n itself.

[0143] Therefore, the complexity of the algorithm used to determine the quantum circuit 28 can be mainly determined by the complexity of the iterative optimization process. The execution time of the optimization program is generally proportional to the number of iterations and the execution time of each iteration. The number of iterations required for the optimization algorithm to converge may depend on multiple factors, including the required accuracy, the number of optimization parameters, the selected learning rate and the optimization starting point. Therefore, it may be difficult to determine a sufficient number of iterations in advance, and the following will focus on the complexity of one iteration of the optimization program consisting of the following elements:

[0144] Element 1: The calculation of the normalization constant c can have O(nR 3 ), which is based on the computational complexity of multiplying two MPOs of rank r1 and r2 while taking their traces (whose computational complexity is O(nr1r2max(r1,r2))$).

[0145] Element 2: The computation of the cost function can have O(n(r+R) 3 ). After summing the two MPOs, their ranks are added, and then the resulting r1+r2 rank MPO is combined with its complex conjugate, which has a computational complexity of O(n(r1+r2) 3 ).

[0146] Element 3: Use automatic differentiation to evaluate the derivative of the cost function. When automatic differentiation is used, its algorithmic complexity is theoretically guaranteed to be no greater than the algorithmic complexity of the original program, which is equivalent to the previous element (2) and is equal to O(n R 3 ).

[0147] Element 4: The optimization process includes a Riemann optimization step, which may involve for each V k Compute the Riemann gradient, retraction, and vector transfer. For a 2R*2R matrix, the first and third steps of the Riemann optimization procedure have O(R 3 ) is calculated with the same complexity. The complexity of the second step depends on the selected shrinking method. In the example implementation, SVD shrinking can be used, and for a 2R*2R matrix, its complexity is also O(R 3 ). Therefore, the Riemann optimization step per iteration can have O(n R 3 ) of the image.

[0148] Fig. 7A and Figure 7B The diagonal matrix M according to the approximation equation (1) is shown in FIG. D When implementing the combination Figure 2 and Figure 4 Experimental runtime results of the discussed method for determining quantum circuit 28.

[0149] Fig. 7A The execution time of one iteration of the iterative optimization algorithm for different approximation ranks R of the orthogonal approximation 30 is illustrated as a function of the number of qubits n (which depends on the size of the expected matrix 24 ).

[0150] Figure 7B The experimentally determined execution time of each iteration and its elements (1)-(4) (as discussed in the previous section) is plotted as a function of the number of qubits n.

[0151] In full agreement with theoretical estimates of the algorithm’s complexity, these running times are polynomial in the number of qubits n and therefore in line with the target matrix size N=2 of the expected matrix 24. n A multi-logarithmic relationship.

[0152] Therefore, the method for determining quantum circuit 28 is expected to provide an efficient algorithm for determining the quantum gate decomposition of matrices in quantum computing applications. It is noted that the depth of quantum circuit 28 grows only linearly with the number of qubits n, and therefore grows logarithmically with the matrix size of the expected matrix 24.

[0153] The algorithm can exploit the relative strengths of quantum and classical hardware to generate an optimal algorithm for a given computation. Specifically, an orthogonal approximation 30 can be efficiently optimized on classical hardware with respect to a tensor network representation / approximation 22 of the expected matrix 24, even if the expected matrix 24 is large. At the same time, the resulting orthogonal approximation, which consists of compact sub-operations, can be efficiently implemented on quantum hardware where the circuit depth can grow only linearly with the number of qubits.

[0154] The description of the preferred embodiments and the accompanying drawings is only for illustrating the present invention and the beneficial effects associated therewith, and should not be understood as implying any limitation. The scope of the present invention will be determined only by the appended claims.

[0155] List of Reference Numerals

[0156] 10 System

[0157] 12 qubit register

[0158] 14 State Preparation Module

[0159] 16 Quantum Matrix Multiplication Module

[0160] 18 Measurement Module

[0161] 20 Processing System

[0162] 22 MPO approximation

[0163] 24 Expectation Matrix

[0164] 24 Penrose diagram representation of the expectation matrix

[0165] 26 Subtensors

[0166] 28 Quantum Circuits

[0167] 30 Orthogonal approximation

[0168] 32, 32a, 32b Isometric subtensors

[0169] 34 Extended Orthogonal Approximation

[0170] 36,38 |0> state of the boundary core

[0171] 40 Quantum Operations

[0172] 42 auxiliary qubits

[0173] 44 Auxiliary qubit measurement

Claims

1. A computer-implemented method for encoding an expectation matrix (24) into a quantum circuit (28), the method comprising the steps of: Obtaining a matrix product operator MPO representation (22) of the expected matrix (24); determining an approximate rank (R) of the expectation matrix (24) based on the MPO representation (22); determining an initial guess of an orthogonal approximation (30, 34) of the expected matrix (24) in the form of a tensor network of isometric sub-tensors (32, 32a, 32b) having the approximation rank (R); starting from the initial guess, iteratively optimizing the orthogonal approximation (30, 34) of the expected matrix (24) based on an optimization algorithm, the optimization algorithm minimizing a cost function under the isometry constraint of the isometric subtensors (32, 32a, 32b), wherein the cost function assigns a cost to the orthogonal approximation (30, 34) of the expected matrix (24) based on the quality of the orthogonal approximation (30, 34) with respect to the expected matrix (24), and The orthogonal approximation (30, 34) is encoded into the quantum circuit (28) based on encoding the isometric subtensors (32, 32a, 32b) as quantum gates.

2. The method of claim 1, wherein: The approximate rank (R) is greater than the rank (r) of the matrix product operator representation (22) of the expected matrix (24), wherein the rank (r) of the matrix product operator representation (22) is in particular the key dimension of the matrix product operator representation (22); and / or Wherein, the approximation rank (R) is the common bond dimension of all isometric subtensors (32, 32a, 32b) of the orthogonal approximation (30, 34).

3. A method as claimed in any one of the preceding claims, wherein: The orthogonal approximation (30, 34) is a matrix product operator, and wherein the orthogonal approximation (30, 34) is mathematically equivalent to a tensor network in which each intermediate isometric sub-tensor (32) has two outer unmerged indices and two inner indices merged with adjacent isometric sub-tensors (32, 32a, 32b) in a chained manner.

4. A method as claimed in any one of the preceding claims, wherein: The cost function is based on a difference function of the expected matrix (24) and a renormalized orthogonal approximation (30, 34) evaluated based on tensor network calculations, wherein the renormalized orthogonal approximation (30, 34) is based on the orthogonal approximation (30, 34) multiplied by a renormalization constant, wherein the cost function is particularly based on the Frobenius norm of the difference function evaluated based on tensor network calculations, and / or wherein the renormalization constant is selected such that the division of the expected matrix (24) by the normalization constant does not increase the trace of the state (36, 38) on which the expected matrix acts, and / or such that the trace of the density matrix on which the renormalized orthogonal approximation (30, 34) acts is equal to or less than 1, regardless of the initial state, The renormalization constant c is specifically for Or obtained by optimizing the orthogonal approximation (30, 34) A of the expected matrix M (24) according to a gradient-based optimization algorithm so as to minimize each iterative step of the cost function.

5. A method as claimed in any one of the preceding claims, wherein: The optimization algorithm comprises gradient descent and in particular is based on Riemannian gradient descent or an optimization algorithm derived therefrom.

6. A method as claimed in any one of the preceding claims, wherein: The orthogonal approximation (30, 34) is a matrix product operator and can be expressed as Among them, V k is isometric, i.e., for any index k of the isometric subtensor (32, 32a, 32b), And where each isometric subtensor V k In particular, an iterative optimization is performed by alternating descent to optimize the orthogonal approximation (30, 34).

7. A method as claimed in any one of the preceding claims, wherein: Optimizing the orthogonal approximation (30, 34) under the isometry constraint comprises: Based on the partial derivatives of the cost function and optionally on information from the previous iteration step of the optimization algorithm, a subtensor V is determined k The search direction p k ;as well as Determine the search direction p k At point V k Stiefel manifolds with m×p isometric matrices Projection on tangent space Among them, when reshaped into a two-dimensional matrix, the isometric subtensor V k The dimension is m×p.

8. A method as claimed in any one of the preceding claims, wherein: Optimizing the orthogonal approximation (30, 34) under the isometry constraint further comprises: The orthogonal approximation (30, 34) is updated by updating all isometric subtensors (32, 32a, 32b), wherein each isometric subtensor V k All pass through point V k Stiefel manifold of m×p isometric matrices along the update direction The isometric subtensor V is updated when reshaped into a two-dimensional matrix. k The dimension is m×p, and wherein the update direction is the direction that minimizes the cost function based on the partial derivative of the cost function, and in particular uses the projection p k,tangent As used to update the isometric subtensor V k Update direction.

9. A method as claimed in any one of the preceding claims, wherein: Before the orthogonal approximation (30, 34) is expanded, when the isometric sub-tensors (32a, 32b) and the intermediate tensor (32) of the orthogonal approximation (30, 34) at the ends of the orthogonal approximation (30, 34) are reshaped into a two-dimensional matrix, the isometric sub-tensors (32a, 32b) are isometric and the intermediate tensor (32) is unitary, so that after the isometric sub-tensors (32, 32a, 32b) are reshaped into a square matrix, all the isometric sub-tensors (32, 32a, 32b) are unitary.

10. A method as claimed in any one of the preceding claims, wherein: The method comprises providing one or more additional auxiliary qubits (42) as input to the first end of the tensor network, wherein the method in particular provides at least logR additional auxiliary qubits (42), wherein R is the approximate rank (R), Therein, the method in particular comprises measuring a quantum bit (44) at a second end of the tensor network, wherein the selection of a result having a predetermined measurement result of the quantum bit (44) measured at the second end of the tensor network is part of implementing the expectation matrix (24).

11. A method as claimed in any one of the preceding claims, wherein: The method further comprises implementing the orthogonal approximation (30, 34) of the expected matrix (24) as a quantum circuit (28) on quantum hardware based on encoding the isometric subtensors (32, 32a, 32b) as quantum gates, Therein, the orthogonal approximation (30, 34) of the expected matrix (24) is implemented as a quantum circuit (28) based in particular on sequentially applying isometries to a set of quantum bits and rearranging unitary matrices into a quantum network.

12. A processing system (20) for encoding an expected matrix (24) into a quantum circuit (28), wherein: The system (10) is configured as follows: Determining an approximate rank (R) of a matrix product operator MPO representation (22) of the expected matrix (24); determining an initial guess of an orthogonal approximation (30, 34) of the expected matrix (24) in the form of a tensor network of isometric sub-tensors (32, 32a, 32b) having the approximation rank (R); starting from the initial guess, iteratively optimizing the orthogonal approximation (30, 34) of the expected matrix (24) based on an optimization algorithm, the optimization algorithm minimizing a cost function under the isometry constraint of the isometric subtensors (32, 32a, 32b), wherein the cost function assigns a cost to the orthogonal approximation (30, 34) of the expected matrix (24) based on the quality of the orthogonal approximation (30, 34) with respect to the expected matrix (24), and Based on encoding the isometric subtensors (32, 32a, 32b) as quantum gates, an implementation of the orthogonal approximation (30, 34) of the expected matrix (24) in a quantum circuit (28) is determined.

13. The system (10) of claim 12, wherein: The system (10) is further configured to transmit the implementation to a quantum computing system to execute the orthogonal approximation (30, 34) on quantum hardware.

14. A hybrid quantum-classical computing system, comprising a system (10) as claimed in claim 12 or 13 and quantum computing hardware, wherein: The hybrid quantum-classical computing system is configured to receive the implementation from the processing system (20), implementing said embodiments in said quantum computing hardware, and A computation result is received from the quantum computing hardware.

15. A computer program comprising machine-readable instructions which, when executed by a processing unit, cause the processing unit to implement the method according to any one of claims 1 to 11 and / or to implement and / or control the system according to any one of claims 12 to 14.

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