Method and system for encoding intended matrix in quantum circuit

Abstract

To provide a computer-implemented method and system for encoding an intended matrix in a quantum circuit.SOLUTION: A computer-implemented method includes the steps of: acquiring a matrix product operator (MPO) representation of an intended matrix; determining an approximation rank of the intended matrix on the basis of the MPO representation; determining an initial guess for an orthogonal approximation of the intended matrix in the form of a tensor network with isometric sub-tensors of the approximation rank; and starting with the initial guess and iteratively optimizing the orthogonal approximation of the intended matrix on the basis of an optimization algorithm minimizing a cost function subject to an isometry constraint for the isometric sub-tensors. The cost function attributes a cost to the orthogonal approximation of the intended matrix on the basis of the quality of the orthogonal approximation with respect to the intended matrix. The method also includes the step of encoding the orthogonal approximation into a quantum circuit on the basis of the encoding of the isometric sub-tensors into quantum gates.SELECTED DRAWING: Figure 2

Description

[Technical field]

[0001] The present invention relates to the field of encoding computational operations into quantum circuits. More precisely, the invention relates to the optimization of the computer implementation of computational operations aimed at encoding in quantum circuits. [Background technology]

[0002] Quantum computers provide a platform for controllable quantum mechanical systems whose states and interactions can be controlled to perform computations. Computation is realized through the deterministic evolution of controllable quantum mechanical systems, e.g., qubits, which are the quantum analogues of classical bits, whose states can be measured to determine the outcome of the computation.

[0003] Control operations on these qubits are called quantum gates. Quantum gates can act coherently on qubits, for example to entangle the states of multiple qubits and any combination of them, both to cause a change in the state of a single qubit (so-called single-qubit gates) and to act on multiple qubits (so-called multi-qubit gates). For example, a single-qubit gate may cause a rotation of the spin state of an electron by a selectable value, for example π / 2. A multi-qubit gate may act coherently on more than one qubit, such as a coherent CNOT operation on the states of two qubits. Multiple quantum gates can be applied to qubits in a quantum computer in parallel or in sequence to perform a computation. Finally, the state of the qubits may be repeatedly measured after applying a sequence of quantum gates to determine the probability for each possible outcome of the computation.

[0004] In essence, the operation of a quantum computer can be thought of as relying on encoding a starting state into an internal quantum state of a qubit, followed by multiplying the internal quantum state of the qubit by a matrix that can be implemented by a combination of different quantum gates, and measuring the resulting result.

[0005] To compute solutions to problems that are considered intractable for classical computers, quantum computers can exploit the special properties of quantum mechanical states, in particular the superposition and entanglement of different quantum states, to find the solution in a relatively small number of computational steps. Moreover, because quantum operations can create complex superposition states of qubits, quantum computers may, in principle, have access to large internal memories for processing computational states.

[0006] However, as found by Shende et al. ("Minimal universal two-qubit controlled-not-based Circuits", Phys. Rev. A, 69:062321, June 2004), converting any operation into a combination of single- and two-qubit gates generally 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 computing. The outputs of quantum variational circuits are combined by a quantum nonlinear processor, and the Hamiltonian expressible as an MPS is compared with an MPS ansatz encoded as a quantum circuit according to the MPS quantum circuit encoding scheme. [Prior art documents] [Non-patent literature]

[0008] [Non-Patent Document 1] Shende et al., "Minimal universal two-qubit controlled-not-based Circuits," Phys. Rev. A, 69:062321, June 2004. [Non-Patent Document 2] Lubasch et al., “Variational quantum algorithms for nonlinear problems” Summary of the Invention [Problem to be solved by the invention]

[0009] However, current approaches are still limited by the significant complexity involved in implementing the steps of state encoding in qubits, matrix encoding via quantum gates, and measurement. Specifically, the exponential complexity of decomposing any multi-qubit gate with respect to the number of qubits, combined with the need for full connectivity between qubits, presents a significant obstacle. Although algorithms exist for converting large matrices into combinations of single quantum gates and two-qubit operations (e.g., CNOT gates), the complexity of the resulting circuits generally scales exponentially with the number of qubits. While the use of Hamiltonians expressible as MPSs constrains the problem cases applicable to the algorithm, variational approaches often rely on a large number of quantum circuit evaluations to optimize variational quantum circuits. [Means for solving the problem]

[0010] In view of this state of the art, it is an object of the present invention to provide a computer-implemented method, and corresponding system, for determining a matrix encoding of a desired computational operation, where the encoding operation can closely approximate the desired computational operation, but can also be efficiently implemented on quantum hardware.

[0011] This object is solved by a method and a system according to the independent claims. The dependent claims relate to preferred embodiments.

[0012] According to a first aspect, the present invention relates to a computer-implemented method for encoding a target matrix in a quantum circuit. The method includes obtaining a matrix product operator representation (MPO) of the target matrix and determining an approximation rank of the target matrix based on the MPO representation. The method further includes determining an initial guess for an orthogonal approximation of the target matrix in the form of a tensor network with isometric subtensors of approximate rank, and starting from the initial guess, iteratively optimizing the orthogonal approximation of the target matrix based on an optimization algorithm that minimizes a cost function subject to an isometry constraint of the isometric subtensors. The cost function attributes a cost to the orthogonal approximation of the target matrix based on a quality of the orthogonal approximation to the target matrix. The method further includes encoding the orthogonal approximation into a quantum circuit based on encoding the isometric subtensors into quantum gates.

[0013] The objective matrix codifies a desired multi-qubit operation, which may act on input states of computational qubits of a quantum processing system. It should be noted that although the objective matrix is ​​considered to be a multi-qubit operation, the method according to the first aspect does not require the objective matrix to be unitary. For example, matrices frequently used in the context of quantum computing algorithms include, for example, the Laplace matrix M in the solution of partial differential equations. L As another example, the diagonal matrix M D may be used as part of the optimization algorithm.

[0014] A common property of these two example object matrices is that, when considered as quantum operators, they have relatively small entanglement and / or that these matrices can be described by tensor networks with relatively small rank, i.e., ranks of 2 and 3 for diagonal matrices and Laplacian operators, respectively.

[0015] The method according to the first aspect exploits this property by first obtaining a tensor network representation of the target matrix, particularly in the form of a tensor sequence that may have a relatively small bond dimension. However, the matrix multiplication operator representation / approximation of the general target matrix does not directly correspond to a quantum circuit that can be determined without the difficulty of exponential calculations. Rather, based on the matrix multiplication operator representation of the target matrix, an approximation rank is determined that should be equal to or greater than the rank of the tensor sequence representation to approximate the target matrix with an orthogonal approximation.

[0016] In a preferred embodiment, the approximation rank is greater than the rank of the matrix multiplication operator representation of the object matrix, which rank of the matrix multiplication operator representation is, in particular, the bond dimension of the matrix multiplication operator representation.

[0017] Since the orthogonal approximation relies on the constraints of the isometric subtensors, increasing the rank of the orthogonal approximation may increase the degrees of freedom to accurately approximate the objective matrix.

[0018] It should be noted that for some matrices, an exact matrix multiply operator representation may be determined, but in general, the matrix multiply operator representation may be obtained by an approximation algorithm, such as an algorithm based on cross approximation or singular value decomposition, to approximate the target matrix in MPO without significantly affecting the results of the method. Those skilled in the art will further appreciate that matrix multiply operator representations / approximations are already known and it may be sufficient to receive the matrix multiply operator representation from a suitable source.

[0019] The MPO representation may include non-isometric subtensors, e.g., less than 1% or 10, depending on the application. -6 The objective matrix is ​​approximated with an error that may be less than 1%, and may also be referred to as a non-orthogonal MPO approximation of the objective matrix. For example, to solve an optimization problem, an approximation of the objective matrix with an error of about 1% may be sufficient, while for solving a PDE problem, the objective matrix may be approximated with an error of less than 10 -5It may be approximated with the following error: Although no matrix can be represented by the MPO representation with arbitrarily small accuracy, those skilled in the art will understand that they may further choose to apply the present method taking into account increased inaccuracies of the MPO approximation, for example, when other ways of performing the calculations are not possible or when a relatively low accuracy is sufficient for the respective computational problem.

[0020] Based on the bond dimension (rank) of the matrix multiplication operator representation, the approximate rank is the next power of two, i.e. 2 N , where N may be selected as an integer or a power of two greater than the bond dimension of the matrix multiply operator representation. As an example, if the rank of the matrix multiply operator representation of the target matrix is ​​3, then the approximation rank may be 4 or 8. However, if the approximation rank is less than or equal to the rank of the matrix multiply operator representation of the target matrix, then although a suitable approximation may in principle be made, the orthogonal approximation is likely to be an inaccurate representation of the target matrix.

[0021] To determine the orthogonal approximation, the method includes determining an initial guess for the orthogonal approximation of the objective matrix, which may correspond to a tensor sequence of isometric subtensors of approximate rank. The initial guess may be constructed from random subtensors that satisfy the isometry condition.

[0022] In a preferred embodiment, the orthogonal approximation is a matrix multiplication operator, and the orthogonal approximation is mathematically equivalent to a tensor network, where each intermediate isometric subtensor has two exterior uncontracted indices, as well as two interior indices that are chain-contracted with adjacent isometric subtensors.

[0023] The uncontracted indices correspond to the dimensions of the input and output qubit states for each isometric subtensor, and the internal indices may be of approximate rank.

[0024] In a preferred embodiment, the approximate rank is the common bond dimension of all the subtensors of the tensor network.

[0025] The initial estimate is then optimized based on an iterative optimization algorithm such that the difference between the orthogonal approximation and the object matrix is ​​minimized. The difference may be based on a norm of the difference, for example the Frobenius norm or the cosine norm of the difference between the orthogonal approximation and the object matrix or a sub-tensor thereof.

[0026] In a preferred embodiment, the cost function is based on a difference function of the object matrix and a renormalized orthogonal approximation evaluated based on a tensor network calculation, where the renormalized orthogonal approximation is based on the orthogonal approximation multiplied by a renormalization constant, the cost function is based in particular on the Frobenius norm of the difference function evaluated based on the tensor network calculation, and / or the renormalization constant is selected such that the object matrix divided by the normalization constant does not increase the trace of the states on which the object matrix operates and / or such that the trace of the density matrix on which the renormalized orthogonal approximation operates is less than or equal to 1 regardless of the initial state.

[0027] The renormalization constant allows preserving the constraint that quantum operations cannot increase the trace of the density matrix of the quantum state, i.e., given an arbitrary input density matrix ρ, the orthogonal approximation A is given by Tr[AρA † ]≦1.

[0028] Given an arbitrary objective matrix M, we can find an orthogonal approximation A with a differential cost function of

number

[0029] In a preferred embodiment, a renormalization constant c is set to minimize the cost function:

number

[0030] For example, the renormalization constant may be updated using steepest descent on c to minimize a cost function.

[0031] Based on a cost function that quantifies the difference between the (renormalized) orthogonal approximation and the objective matrix, the elements of the orthogonal approximation may be updated to minimize the cost function.

[0032] In a preferred embodiment, the optimization algorithm comprises gradient descent, and in particular is based on Riemann gradient descent or an optimization algorithm derived therefrom.

[0033] In other words, the method may include a step of (probabilistically) determining the gradient of the difference between the orthogonal approximation and a matrix multiplication operator representation of the target matrix, and subsequently updating the orthogonal approximation according to the determined gradient.

[0034] Those skilled in the art will appreciate that in some embodiments, minimizing the difference between the orthogonal approximation and the matrix multiplication operator representation of the objective matrix may be preferred as this may reduce the computational complexity of the optimization. For example, the difference between the subtensors of the orthogonal approximation and the matrix multiplication operator representation of the objective matrix may be determined and each isometric subtensor updated based on the difference. The update of the isometric subtensors should preserve the isometry constraint, i.e., the updated isometric subtensors must be isometric.

[0035] In a preferred embodiment, the orthogonal approximation is a matrix multiplication operator, which can be expressed as:

[0036]

number

[0037] Here, V k is isometric, i.e., for any index k of the isometric subtensor

number

[0038] The alternating descent method may be performed as a stochastic gradient descent method on each of the subtensors, for example in a predetermined order or in a random order, such that each of the isometric subtensors of the orthogonal approximation approaches a subtensor of the matrix product operator representation of the target matrix.

[0039] In a preferred embodiment, the gradient of the cost function is determined based on multiple partial derivatives of the subtensors of the orthogonal approximation, e.g., by determining the partial derivatives of the cost function with respect to the values ​​of each subtensor, and by updating the orthogonal approximation based on multiple subtensors, e.g., the partial derivatives of all the subtensors. Those skilled in the art will appreciate that the optimization includes stochastic optimization, e.g., it is not necessary to determine partial derivatives for all parameters of the orthogonal approximation at each iteration, but may update a stochastically determined subset of the parameters, such as in stochastic gradient descent and related optimization algorithms.

[0040] However, the updated subtensors should preserve the isometry constraints, for example, by updating the isometric subtensors according to a Riemann gradient descent method, which may preserve the isometry constraints, for example, by projecting the update vectors and / or the update subtensors onto the Stiefel manifold of the isometric tensor.

[0041] For example, optimizing an orthogonal approximation under an isometry constraint is a projection of the partial derivatives of the cost function.

number

number

[0042] For example, subtensor V k is an optimization algorithm similar to steepest descent optimization,

number

[0043] In a preferred embodiment, optimizing the orthogonal approximation under the isometry constraint involves approximating the sub-tensor V based on partial derivatives of the cost function and, optionally, information from previous iteration steps of the optimization algorithm. k Search direction p k and determining point V k Stiefel manifold of m×p isometric matrices in

number

number

[0044] The partial derivatives and / or the search direction may then be used to determine an update direction, and based on the update direction, the isometric subtensor and / or the orthogonal approximation may be modified, for example, onto the Stiefel manifold to obtain an updated subtensor and / or an updated orthogonal approximation.

[0045] In some embodiments, optimizing the orthogonal approximation under the isometry constraint includes projecting the modified subtensor and / or the modified orthogonal approximation, where the modified subtensor and / or the modified orthogonal approximation are modified on a Stiefel manifold for the subtensor and / or the orthogonal approximation, respectively, based on a search direction and / or an update direction, and / or projecting the subtensor V k Retraction of updates to the manifold

number

number

[0046] Retraction, for example,

number

number

[0047] Those skilled in the art will appreciate that this technique can be extended to optimization techniques that may include, for example, momentum based on gradients determined in previous iterations, thereby providing an update vector

number

[0048] In a preferred embodiment, optimizing the orthogonal approximation under the isometry constraint comprises updating the orthogonal approximation by updating all the isometric subtensors, where for each isometric subtensor V k is the update direction point V k Stiefel manifold of m×p isometric matrices in

number

[0049] Those skilled in the art will appreciate that different retractions are used in embodiments such as Cayley retraction or singular value decomposition, to name just two examples, and that those skilled in the art may select an appropriate retraction from multiple known retractions.

[0050] The isometric subtensor can be reshaped into a matrix that is square with respect to the intermediate subtensor of the orthogonal approximation, and the reshaped matrix corresponding to the intermediate subtensor is unitary based on the isometry constraint of the subtensor during the iterative optimization of the orthogonal approximation.

[0051] In a preferred embodiment, the isometric subtensors of the orthogonal approximation at the ends of the orthogonal approximation are isometric and the intermediate tensors are unitary when reshaped into two-dimensional matrices before extending the orthogonal approximation, such that all isometric subtensors are unitary after reshaping the isometric subtensors into square matrices.

[0052] To obtain a set of unitary matrices that can be implemented in a quantum circuit, a sub-tensor at the end of the chain of tensors (sometimes called the boundary core) can be expanded using an ancilla qubit, for example by preparing a zero-state |0> ancilla qubit that is multiplied with the first isometric sub-tensor, and by implementing a post-selection at the opposite end of the chain of tensors.

[0053] Thus, the method may include extending the orthogonal approximations such that all isometric subtensors become unitary after reshaping the isometric subtensors into square matrices. The extending the orthogonal approximations may be part of determining an initial estimate of the orthogonal approximations or may be performed at the beginning or later when the orthogonal approximations are iteratively optimized. For example, the initial estimates may be constructed from subtensors that can be reshaped into unitary matrices and the determination of the cost function may take into account an implicit extension of the orthogonal approximations.

[0054] In some embodiments, the step of extending the orthogonal approximation is based on providing one or more additional ancilla qubits as input to the first end of the tensor network, and the method specifically provides at least logR additional ancilla qubits, where R is the approximation rank.

[0055] In some embodiments, the step of extending the orthogonal approximation includes measuring a qubit at a second end of the tensor network, and selecting an outcome having a predetermined measurement result of the qubit measured at the second end of the tensor network is part of the implementation of the objective matrix.

[0056] Based on the addition of ancilla qubits and post-selection, all subtensors may be square after appropriate reshaping, and thus form a set of unitary matrices that can be implemented in quantum circuits by decomposing the isometric subtensors into combinations of quantum gates of the respective quantum architecture, e.g., single-qubit rotations and CNOT gates.

[0057] Those skilled in the art will understand that the expansion can be performed before or after iteratively optimizing the orthogonal approximation. In other words, it is possible to define an initial guess based on an expanded matrix multiplication operator, or to expand the orthogonal approximation after it has been iteratively optimized, for example by introducing and filling additional elements of the boundary core, such that the resulting boundary core becomes unitary after appropriate reshaping.

[0058] In a preferred embodiment, the method further comprises implementing an orthogonal approximation of the object matrix as a quantum circuit on quantum hardware based on encoding the isometric subtensors into quantum gates.

[0059] The decompositions of each isometric subtensor are combined to perform an orthogonal approximation based on sequential application of each isometric subtensor, and the output state of the logR qubits resulting from a unitary operation corresponding to one of the subtensors may be the input for the unitary operation corresponding to the next isometric subtensor in the tensor network.

[0060] In a preferred embodiment, implementing an orthogonal approximation of the object matrix as a quantum circuit is based on sequentially applying isometric sub-operations to a set of qubits and rearranging unitary matrices in a quantum network.

[0061] As a result, the depth of a quantum circuit can only scale polynomially with the number of qubits, in contrast to the exponential scaling of typical decompositions of matrices into single- and two-qubit gates.

[0062] According to a second aspect, the invention relates to a processing system for encoding a target matrix in a quantum circuit. The system is configured to determine an approximation rank of a matrix product operator representation of the target matrix and to determine an initial guess for an orthogonal approximation of the target matrix in the form of a tensor network with an isometric subtensor of the approximation rank. Starting from the initial guess, the system is further configured to iteratively optimize the orthogonal approximation of the target matrix based on an optimization algorithm that minimizes a cost function subject to an isometry constraint of the isometric subtensor, the cost function attributing a cost to the orthogonal approximation of the target matrix based on a quality of the orthogonal approximation to the target matrix. The system is further configured to determine an implementation of the orthogonal approximation of the target matrix in a quantum circuit based on the encoding of the isometric subtensor into a quantum gate.

[0063] The system may implement the method according to the first aspect, or any combination of its embodiments. The system according to the second aspect may also benefit from any feature of the preferred embodiments of the first aspect.

[0064] The processing system may include a single processing unit or 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 processing devices that operate 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 interfaces for connecting to sensors, devices, instruments, integrated logic circuits, other controllers, etc., and the interfaces may be configured to receive or transmit signals, such as electrical signals, optical signals, radio signals, acoustic signals, etc.

[0065] The system may be configured to obtain the MPO representation of the objective matrix, for example by executing a decomposition algorithm on the objective matrix, or by receiving the MPO representation from a suitable source, such as a database.

[0066] In some embodiments, the system is further configured to extend the orthogonal approximation such that all isometric subtensors are unitary after reshaping the isometric subtensors into square matrices.

[0067] In a preferred embodiment, the system is further configured to communicate the implementation to a quantum computing system for performing the orthogonal approximation on quantum hardware.

[0068] According to a third aspect, the present invention relates to a hybrid quantum-classical computing system comprising a system according to the second aspect and quantum computing hardware, the hybrid quantum-classical computing system being configured to receive an implementation from a processing system, to implement the implementation in the quantum computing hardware and to receive a computation result from the quantum computing hardware.

[0069] According to a fourth aspect, the present invention relates to a non-transitory medium comprising machine-readable instructions which, when executed by a processing system, implements the method according to the first aspect and / or the system according to the second aspect and / or the third aspect. [Brief description of the drawings]

[0070] The features and many advantages of the method and system according to the present invention will be best understood from the following detailed description of the preferred embodiment, when read in conjunction with the accompanying drawings. [Figure 1] FIG. 1 is a schematic diagram of an example quantum computing system. [Diagram 2] FIG. 1 illustrates an example of a computer-implemented method for encoding an objective matrix in a quantum circuit. [Diagram 3] FIG. 13 illustrates generally a flow chart for determining a matrix multiplication operator representation based on a given target matrix. [Figure 4]FIG. 13 illustrates a schematic diagram of a strategy for determining a quantum circuit encoding for an approximation of an objective matrix, according to an example. [Diagram 5] FIG. 5 illustrates the results obtained from applying the strategy described in relation to FIG. 4 to two different objective matrices. [Figure 6] FIG. 5 illustrates the results obtained from applying the strategy described in relation to FIG. 4 to two different objective matrices. [Figure 7A] FIG. 13 illustrates experimental runtime results by implementing a method for determining a quantum circuit when approximating a diagonal matrix according to one example. [Figure 7B] FIG. 13 illustrates experimental runtime results by implementing a method for determining a quantum circuit when approximating a diagonal matrix according to one example. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0071] 1 illustrates a schematic diagram of an example quantum computing system 10. System 10 includes a qubit register 12 with computation qubits, a state preparation module 14 for preparing the quantum states of the computation qubits in 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 results of the quantum computation, e.g., by projective measurement of the states of some or all of the qubits. A processing system 20 can initialize the computation qubits in qubit register 12 and define the operation of state preparation module 14 and / or quantum matrix multiplication module 16, and may retrieve the measurement results from measurement module 18, e.g., by determining and sending control actions to hardware of quantum computing system 10.

[0072] State preparation module 14 and quantum matrix multiplication module 16 may be similar in configuration and may implement a combination of single and multi-qubit quantum gates that affect the state of the qubits in qubit register 12. Note that in some embodiments, state preparation module 14 may not be required and the combination of quantum gates determined by quantum matrix multiplication module 16 may directly act on the quantum states of the qubits initialized in qubit register 12 at some initialized quantum states. The types of quantum gates implemented in the quantum circuit by state preparation module 14 and / or quantum matrix multiplication module 16 may depend on the architecture of the particular quantum hardware utilized, for example, based on the native gates of the quantum hardware. State preparation module 14 and quantum matrix multiplication module 16 may calculate quantum gate configurations and / or send control operations to control the quantum hardware so that the quantum hardware realizes the desired computational operations for the particular quantum states.

[0073] However, decomposing any computational operation into quantum circuits is generally a hard computational problem whose complexity usually scales exponentially with the number of quantum bits involved. Furthermore, for common operations, the depth of the circuit, i.e., the number of successive quantum gates to realize a quantum operation, may scale exponentially with the number of quantum bits. While efficient encoding of the problem into combinations of quantum gates is known for certain types of problems, the encoding of many desirable problem types into quantum circuits may still be a significant obstacle to realizing quantum computing.

[0074] 2 illustrates an example of a computer-implemented method for encoding a target matrix in a quantum circuit. The method includes obtaining a matrix product operator representation of the target matrix (S10) and determining an approximation rank for the target matrix based on the MPO representation (S12). The method further includes determining an initial guess of an orthogonal approximation of the target matrix in the form of a tensor network with isometric subtensors of approximate rank (S14) and, starting from the initial guess, iteratively optimizing the orthogonal approximation of the target matrix based on an optimization algorithm that minimizes a cost function subject to an isometry constraint of the isometric subtensors (S16). The method further includes encoding the orthogonal approximation into a quantum circuit based on the encoding of the isometric subtensors into quantum gates (S18).

[0075] The cost function attributes a cost to the orthogonal approximation of the object matrix based on the quality of the orthogonal approximation to the object matrix. Thus, instead of directly implementing the object matrix in a quantum circuit, the method includes approximating the object matrix with an orthogonal approximation, constructed in the form of a tensor network with an isometric subtensor (also referred to below as an isometric core).

[0076] The isometry constraint of the subtensors of the orthogonal approximation can be enforced at each step of the iterative optimization, for example based on a modified gradient descent method and / or a retraction of any updated subtensors into isometric subtensors, so that the optimization algorithm converges towards an approximation of the objective matrix with the isometric subtensor. In general, the approximation rank should be at least equal to the rank of the matrix multiplication operator representation of the objective matrix, so that the isometric subtensor constraint can be at least partially compensated for by the large degrees of freedom of the reduced index of the orthogonal approximation.

[0077] The isometric subtensors are reshaped into square matrices for any subtensors between the ends of the tensor network (so-called boundary cores), which may be unitary after appropriate reshaping. These subtensors that may be reshaped into unitary matrices can be directly implemented as quantum circuits with known decomposition algorithms, which may be computed efficiently since the size of the reshaped matrices / isometric subtensors is small. Those skilled in the art will understand that the subtensors may be implemented without explicitly reshaping the isometric subtensors into matrices.

[0078] To implement boundary cores, i.e., isometric subtensors placed at the ends of the tensor network in the form of tensor strings, the orthogonal approximation may be extended such that all isometric subtensors can be reshaped into unitary matrices, for example, by providing an ancilla qubit at one end of the tensor network and introducing a post-selection measurement at the opposite end, thereby reshaping the boundary core equally to a square matrix. Such an extension may be pre-configured as part of the initial guess of the orthogonal approximation, or may be configured as the orthogonal approximation is iteratively optimized, for example, by completing the boundary core with additional indices such that it is a square matrix, and filling in missing matrix elements such that when the appropriate reshaping is applied, the resulting matrix is ​​unitary.

[0079] As a result, the orthogonal approximation can be implemented in a quantum circuit based on a decomposition of isometric subtensors that are smaller in memory than the object matrix. As a result, the circuit depth of the quantum circuit for the entire object matrix need only scale polynomially with the number of qubits, making complex computations possible on near-term quantum hardware.

[0080] FIG. 3 shows an example of a square matrix M NxNFIG. 2 shows a schematic flow diagram for determining a matrix multiplication operator representation 22 based on a given target matrix 24, shown as n The diagonal operator M D or the Laplace operator M L which may correspond to the computational basis states of the qubit register 12 having n qubits. As an example, the diagonal operator M D and the Laplace operator M L may be expressed in matrix form as:

number

[0081] These matrices can be involved in algorithms related to quantum computing, such as solving partial differential equations and optimization algorithms, but are not unitary.

[0082] The objective matrix 24 may be represented in a Penrose diagram representation 24a by a circle with two uncontracted indices of dimension N. This objective matrix 24 may be approximated or represented by a matrix multiplication operator representation 22 composed of multiple sub-tensors 26, where summation by the contracted indices recovers each element of the objective matrix 24, for example according to:

number

[0083] Objective matrix 24, A l m The MPO representation of 22 is the uncontracted index l k , m k , and the contracted index j k , j k-1 26, V kEach subtensor 26, shown as a circle in Figure 3, has four indices, except for the boundary cores at either end of the MPO representation 22, which have three, and the contracted indices (projecting horizontally) are the MPO bond indices of bond dimension r. The uncontracted indices have the dimension of the quantum state by one qubit, in this example two.

[0084] Such an MPO representation 22 may use a number of known algorithms to find the objective matrix 24 or an approximation thereof. In principle, any objective matrix 24 may be represented in a matrix multiplication operator (MPO) representation 22 by decomposing the objective matrix 24 into a set of matrix products that have the property that those matrix products are equal to certain elements of the objective matrix 24.

[0085] The diagonal operator M, which contains all Hamiltonians with short-range interactions in one dimension D or the Laplace operator M L For a large class of operators such as , the required MPO bond dimension r is small (e.g., around 5) and constant in the size of the system, so that the objective matrix 24 may be stored with multiple relatively small sub-tensors 26. However, although it is possible to efficiently encode MPS states in quantum circuits, as shown, for example, by Lubasch et al., no general MPO-compatible strategy exists.

[0086] Nevertheless, the method illustrated in the example of FIG. 2 may be used to provide a quantum circuit for approximation of the objective matrix 24 .

[0087] 4 illustrates a schematic of a strategy for determining quantum circuits 28 to encode for an approximation of a target matrix 24 according to one example. The target matrix 24 is first represented by an MPO representation 22 with log(N) sub-tensors 26 of rank r, based on a suitable decomposition and / or approximation algorithm.

[0088] Based on the rank r of the MPO representation 22, an approximate rank R is selected, which should be greater than or equal to the rank r of the MPO representation 22 and be a power of two, i.e., 2 k where k is an integer. Based on the selected approximation rank R, the method may generate an initial estimate of the orthogonal approximation 30 having isometric subtensors 32, 32a, 32b such that all isometric subtensors 32, 32a, 32b are isometric when reshaped into a two-dimensional matrix and the bond dimension of the orthogonal approximation 30 is equal to the approximation rank R. The isometric subtensors 32, 32a, 32b for the initial estimate of the orthogonal approximation 30 may be generated randomly, e.g., selected from random isometric subtensors 32, 32a, 32b.

[0089] Starting with an initial estimate of the orthogonal approximation 30, the method may proceed by iteratively optimizing the orthogonal approximation 30 based on an optimization algorithm with the objective of minimizing a distance metric to the objective matrix 24, wherein intermediate steps of the optimization algorithm may maintain the isometry of the sub-tensors 32, 32a, 32b.

[0090] Since the object matrix 24 may not be a unitary matrix that can be directly implemented on a quantum computer, the quantum operation A must preserve the trace.

[0091] Tr[AρA † ]≦1 (3)

[0092] For example, it is preferable to minimize a difference metric for the renormalized orthogonal approximation 30 according to:

[0093]

number

[0094] This satisfies the inequality in equation (3). As a result, the renormalized objective matrix 24, e.g., c -1M may be realized in a quantum circuit instead of the objective matrix 24, i.e. M. From a practical point of view, in some computations it may be necessary to store this normalization factor c and take it into account at the end of the computation, for example to interpret the result when it depends on a complex output quantum state.

[0095] In the case of optimization algorithms, a processing system such as a computer k One may be asked to find the minimum of the cost function under an isometry constraint on all sub-tensors 32, 32a, 32b, abbreviated as

number

[0096] where ||*|| represents an appropriate norm, such as the Frobenius norm, and the optimum is taken over both c and A. This optimization may be performed as an alternating descent method by determining partial derivatives of the cost function with respect to any isometric subtensors 32, 32a, 32b under the isometry constraint, and optimizing an orthogonal approximation 30 based on the partial derivatives, such as by a gradient descent-based algorithm, e.g., stochastic gradient descent, or a momentum-based algorithm such as adaptive moment estimation.

[0097] The isometry constraint may be observed by ensuring isometry of the sub-tensors 32, 32a, 32b at each update step of the optimization algorithm and / or by considering the isometry constraint as part of the gradient direction.

[0098] In general, any m×p isometric matrix belongs to a Stiefel manifold.

[0099]

number

[0100] This is a Riemannian manifold. Therefore, the cost function C to be minimized can be defined on a Cartesian product of n Stiefel manifolds. The isometric matrix V that minimizes this cost function is k To find the set of , one can use a gradient-based optimization method such as Riemann gradient descent, which works with optimization problems under constraints, as in this case.

[0101] In normal gradient-based methods, such as gradient descent, the variables to be optimized are updated by moving in the opposite direction of the cost function gradient.

[0102]

number

[0103] where α is the step size. In the Riemann generalization of these methods, δ is used to determine the direction of this movement. k Instead of C, point V k Its projection onto the space tangent to the Stiefel manifold in δ k R C can be used. Furthermore, the matrix V can be calculated based on the partial derivatives and / or their projections. k Retraction onto the manifold can be used to update the value of

[0104]

number

[0105] Optimization methods that use information from previous optimization steps, such as gradient descent with momentum, also use the concept of vector translation, which is a generalization of the concept of translation translation and may therefore be implemented directly based on the scheme described above.

[0106] The Riemann formulation of gradient descent may be implemented as an iterative procedure, where at every step one computes the Riemann gradient of the optimization function C at the current point on the manifold, and then finds the next point using a selected retraction in the direction of the negative of the gradient.

[0107] The algorithm can then be implemented as follows: Step 0: Obtain an initial guess for the orthogonal approximation 30. Step 1: Find a renormalization constant c that minimizes the cost function in equation (4), for example using a fixed orthogonal approximation 30 (i.e., A is constant during this step). Step 2: For the problem in equation (5), find a new approximation A via one step of Riemann gradient descent, where the renormalization constant c may be constant during this step, then return to step 1.

[0108] The iterative algorithm may be terminated at any point, for example, after a predetermined number of iterations or when a predetermined cost threshold is achieved by the cost function.

[0109] Objective matrix 24, c -1 Once an optimized orthogonal approximation 30 is obtained, which may be a renormalized orthogonal approximation 30 of M, all inner sub-tensors 32 of the orthogonal approximation 30 should already be unitary matrices after reshaping into corresponding two-dimensional matrices.

[0110] However, the subtensors 32a, 32b at the boundaries (i.e., V1 and V2, also called boundary cores) n) may be isometric only. However, the boundary cores 32a, 32b may be expanded to unitary matrices by expanding the subtensors under the isometric condition with the addition of |0> states 36 at one end of the tensor network and a projection to |0> states 38 at the opposite end of the tensor network to maintain the correspondence with the objective matrix 24. By expanding the orthogonal approximation 30, an expanded orthogonal approximation 34 can be obtained, which features only subtensors 32 that can be reshaped into unitary matrices.

[0111] Since all subtensors 32 can be reshaped into unitary matrices, the transition to the quantum circuit can be made by unitary operations U1 to U n The quantum operation 40 may be implemented by concatenating isometric subtensors 32 in a cascade of quantum operations 40 implemented as: where the uncontracted indices of the isometric subtensors 32 may correspond to the input and output states of the respective quantum operations 40, while the contracted indices of the isometric subtensors 32 correspond to the input quantum states for the next unitary operation 40 in the concatenated chain of unitary operations 40. Thus, each unitary operation 40 has a number of 1+log(R) qubits input and qubits output, where log(R) qubits are processed by the subsequent unitary operation 40 in the concatenated chain of unitary operations 40. The decomposition of these subtensors 32 into quantum gates may be implemented based on known decomposition techniques of the respective quantum hardware architecture.

[0112] The upper |0> state 36 as input to the upper boundary core 32a of the extended orthogonal approximation 34 may correspond to preparing an additional ancilla qubit 42 in the “zero” state. At the same time, the lower |0> state 38 as input to the upper boundary core 32a of the extended orthogonal approximation 34 may correspond to a projection onto the zero state, i.e., a post-selection measurement.

[0113] Those skilled in the art will appreciate that the actual quantum states prepared or measured for these operations may be modified, for example by preparing based on a |1> state and post-selecting, or by a different state based on an appropriate basis change, as long as the correspondence with the target matrix 24 is preserved. However, for simplicity, in the following the preparation of a "|0>" state and the corresponding post-selection will be considered.

[0114] Thus, as shown in the example of FIG. 4, quantum circuit 28 may be implemented by the following steps of a quantum processing system: Step 1: Set ancilla qubit 42 to the zero state

number

[0115] Note that for post-selection, one should measure the quantum bit from the other end of the orthogonal approximation 30, i.e., the quantum bit corresponding to the opposite boundary core 32b (see FIG. 4), rather than the quantum bit initially introduced into the initial boundary core 32a as the ancilla quantum bit 42.

[0116] The algorithm implements the quantum circuit 28 corresponding to the objective matrix 24 only if all ancilla qubit measurements 44 are equal to |0>, e.g., resulting in a string of zeros for the corresponding qubits. The state of the quantum system immediately before the measurement determines the likelihood of this "successful" outcome. This means that the initial state of the system to which the quantum method is applied influences this probability.

[0117]

number

[0118] The initial state of the system, ρ in may be determined by the specific computational task. To ignore this dependency, <Pr suc To do this, we can average the probability values ​​over all possible initial states of ρ in can be thought of as a maximally mixed state that is a uniform mixture of states from an orthonormal basis. In this case, <0| a2 U|0> a1 Using =A, ρ in =1 / 2 n I, and the average probability is:

[0119]

number

[0120] That is, the success probability can be determined mainly by the norm of the orthogonal approximation 30. On the other hand, if A is c -1 Since M is approximately equal to λ, we can see how the success probability is related to the normalization factor c, i.e., the higher the normalization factor, the lower the success probability. In other words, there may be a trade-off between the accuracy of the approximation and the success probability. The best theoretical value for the normalization constant c is c = λ. max (M), i.e., the largest eigenvalue of the objective matrix 24. If the approximation rank R of the orthogonal approximation 30 is sufficiently large, the objective matrix 24 can be approximated with sufficient accuracy.

[0121] The depth of the quantum circuit 28 scales polynomially with the approximation rank R, and the size of the approximation rank R is proportional to the unitary operation U k Note that this may further increase the complexity of decomposing the isometric subtensor 32 into a combination of quantum gates, since each of these operates on log2R+1 qubits, where R is the tensor train (TT) rank of the orthogonal approximation A.

[0122] Moreover, a numerical strategy that approximately decomposes general multi-qubit unitary operations40 using a combination of single qubits and CNOT gates can achieve CNOT gate counts close to the theoretical lower bound presented by Shende et al. (Minimal universal two-qubit controlled-not-based circuits, Phys, Rev. A, 2004), i.e., any decomposed m-qubit gate can be calculated as follows: m-1 CNOT gates are needed, so in this case, a total of n*4 logR+1 , i.e., approximately n*R 2 Note that this requires CNOT gates. Thus, in general, the lower the approximation rank R selected for the orthogonal approximation 30, the simpler the quantum circuit 28 will be, but the less accurate the orthogonal approximation 30 will be.

[0123] To demonstrate the power of the method, we consider a frequently used matrix and encode it into a multi-qubit quantum circuit 28. Specifically, the diagonal M D The matrix with the discretized linear function above and the Laplace operator M corresponding to the second derivative matrix after discretization L , both of which are not unitary.

[0124] 5 and 6 show the algorithm described in relation to FIG. 2 and FIG. 4 for the diagonal matrix M D and the Laplace matrix M L The results obtained by applying the Laplace operator M Lcan be used to solve partial differential equations (PDEs), while the diagonal operator M D can be used for optimization and PDE solving. For any n>1, M D and M L The MPO forms of the matrices have TT ranks r equal to 2 and 3, respectively.

[0125] The top graph shows the minimum achievable error in approximating diagonal (FIG. 5) and Laplace (FIG. 6) matrices by the orthogonal approximation 30 as a function of the approximation rank R (in terms of the number of used ancilla qubits 42, which is equal to log2R) for four different sizes of the objective matrix 24 (where n is the number of qubits). The bottom graph shows the corresponding average probability of success based on the ancilla qubit measurements 44. The dotted lines indicate the mean probability of success for ||M|| 2 λ max -2 (M)2 -n (see equation (10)) and for both matrices is independent of the number n of qubits.

[0126] For both matrices, the method achieves reasonable accuracy (M D and M L It is possible to achieve error rates of about 0.01% and 0.5%, respectively, for qubits and a satisfactory probability of success (above 5%) for up to 50 qubits, which is not a limitation of the method. An important property found in the experiments is that the error and probability of success are nearly independent of the number of qubits, making the method likely to scale well for complex tasks.

[0127] To evaluate the efficiency of the computation, the algorithm can be considered to consist essentially of three steps: finding the orthogonal approximation 30; expanding the boundary cores 32a, 32b so that they can be reshaped into a unitary matrix; and decomposing the resulting 2R*2R unitary matrix into a series of 1- and 2-qubit gates.

[0128] The time required for the last step is determined by the decomposition method used and will not be described here. Furthermore, in this example, the expansion step is skipped and the kernels V1 and V2 are calculated based on the initial estimate of the orthogonal approximation 30. n Instead of optimizing itself, U1 and U n The expanded sub-tensor 32 corresponding to is directly optimized in an iterative optimization algorithm.

[0129] Therefore, the complexity of the algorithm for determining the quantum circuit 28 may be determined primarily by the complexity of the iterative optimization process. The running time of the optimization procedure is generally proportional to the number of iterations and the running time of each iteration. The number of iterations required for the optimization algorithm to converge may depend on several 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 in advance the number of sufficient iterations, and in the following we focus on the complexity of one iteration of the optimization procedure, which consists of the following elements: Element 1: The calculation of the normalization constant c is based on the computational complexity of multiplication while simultaneously taking the traces of two MPOs with ranks r1 and r2, which has computational complexity of O(nr1r2max(r1,r2))$. 3 ) computational complexity scaling. Element 2: The calculation of the cost function is O(n(r+R) 3 After the sum of the two MPOs, their ranks are added and the resulting MPO of rank r1+r2 is reduced by its complex conjugate, which can have a computational complexity of O(n(r1+r2) 3 ) computational complexity. Element 3: Evaluation of partial derivatives of the cost function using automatic differentiation. When using automatic differentiation, its algorithmic complexity is theoretically guaranteed to be less than or equal to that of the original program, which is equivalent to the preceding element (2) and takes O(nR 3 ). Element 4: The optimization process includes a Riemannian optimization step, which optimizes each V kFor a 2R*2R matrix, the first and third steps of the Riemann optimization step are O(R 3 ) The complexity of the second step depends on the retraction method chosen. In the exemplary implementation, it is O(R 3 ) complexity. Thus, the Riemannian optimization step for each iteration is O(nR 3 ) scaling.

[0130] 7A and 7B show an example of the diagonal matrix M D 2 and 4 to determine a quantum circuit 28 when approximating

[0131] FIG. 7A shows the running time of one iteration of the iterative optimization algorithm for different approximation ranks R of the orthogonal approximation 30 as a function of the number n of qubits, which depends on the size of the objective matrix 24 .

[0132] FIG. 7B shows the experimentally determined execution time of each iteration and its components (1)-(4) (as explained in the sections above) as a function of the number of qubits n.

[0133] In full agreement with theoretical estimates of the complexity of the algorithm, these execution times depend polynomially on the number of qubits, n, and hence on the target matrix size of the objective matrix 24, N=2 n It depends polynomially on

[0134] Thus, the present method for determining quantum circuits 28 promises to provide an efficient algorithm for determining quantum gate decompositions of matrices in quantum computing applications. Note that the depth of quantum circuits 28 scales only linearly with the number of qubits n, and therefore scales logarithmically with the matrix size of object matrix 24.

[0135] The algorithm may exploit the relative strengths of quantum and classical hardware to generate an optimal algorithm for a given computation. Specifically, the orthogonal approximation 30 may be efficiently optimized on classical hardware for a tensor network representation / approximation 22 of the objective matrix 24, even when the objective matrix 24 is large. At the same time, the resulting orthogonal approximation, composed of compact sub-operations, may be efficiently implemented on quantum hardware with a circuit depth that may scale only linearly with the number of qubits.

[0136] The description of the preferred embodiment and the drawings merely serve to illustrate the invention and its associated beneficial effects, and should not be understood as implying any limitation, the scope of the invention being to be determined solely by the appended claims. [Explanation of symbols]

[0137] 10. System 12 qubit register 14 State Preparation Module 16 Quantum Matrix Multiplication Module 18 Measurement Module 20 Processing System 22 MPO approximation 24 Objective matrix 24 Penrose diagram representation of the objective matrix 26 Subtensor 28 Quantum circuit 30 Orthogonal Approximation 32, 32a, 32b Isometric Subtensors 34 Extended Orthogonal Approximation 36,38 Boundary core |0> state 40 Quantum Operations 42 Ancila Qubit 44 Ancila qubit measurement

Claims

1. 1. A computer-implemented method for encoding an objective matrix (24) in a quantum circuit (28), the method comprising: obtaining a matrix multiplication operator representation (MPO representation) (22) of the objective matrix (24); determining an approximate rank (R) of the objective matrix (24) based on the MPO representation (22); determining an initial guess for an orthogonal approximation (30, 34) of said objective matrix (24) in the form of a tensor network having isometric subtensors (32, 32a, 32b) of said approximation rank (R); starting from the initial estimate, iteratively optimizing the orthogonal approximation (30, 34) of the objective matrix (24) based on an optimization algorithm that minimizes a cost function subject to an isometry constraint of the isometric subtensors (32, 32a, 32b), the cost function attributing a cost to the orthogonal approximation (30, 34) of the objective matrix (24) based on a quality of the orthogonal approximation (30, 34) to the objective matrix (24); encoding the orthogonal approximation (30, 34) into a quantum circuit (28) based on encoding the isometric subtensors (32, 32a, 32b) into quantum gates. method.

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

3. the orthogonal approximation (30, 34) is a matrix multiplication operator, the orthogonal approximation (30, 34) is mathematically equivalent to a tensor network, each intermediate isometric subtensor (32) having two external uncontracted indices and two internal indices that are chain-contracted with adjacent isometric subtensors (32, 32a, 32b); The method according to claim 1 or 2.

4. the cost function is based on a difference function of the objective matrix (24) and a renormalized orthogonal approximation (30, 34) evaluated on the basis of a tensor network calculation, the renormalized orthogonal approximation (30, 34) being based on the orthogonal approximation (30, 34) multiplied by a renormalization constant, the cost function being based in particular on the Frobenius norm of the difference function evaluated on the basis of a tensor network calculation, and / or the renormalization constant is selected such that the objective matrix (24) divided by the renormalization constant does not increase the trace of the states (36, 38) on which the objective matrix acts and / or such that the trace of the density matrix on which the renormalized orthogonal approximation (30, 34) acts is less than or equal to 1 regardless of the initial state, The renormalization constant c is determined to minimize the cost function by: [0010] or according to a gradient-based optimization algorithm, in particular to obtain The method according to claim 1 or 2.

5. The optimization algorithm includes a gradient descent method, and in particular is based on the Riemann gradient descent method or an optimization algorithm derived from the Riemann gradient descent method. The method according to claim 1 or 2.

6. The orthogonal approximation (30, 34) is a matrix multiplication operator, which can be expressed as: [0025] Here, V ｋ is isometric, i.e. for any index k of the isometric subtensors (32, 32a, 32b), [0030] and each isometric subtensor V ｋ is iteratively optimized by an alternating descent method to optimize said orthogonal approximation (30, 34), The method according to claim 1 or 2.

7. optimizing the orthogonal approximation (30, 34) under the isometry constraint, Based on partial derivatives of the cost function and, optionally, information from previous iteration steps of the optimization algorithm, a sub-tensor V ｋ Search direction p ｋ determining Point V ｋ Stiefel manifold of m×p isometric matrices in [0045] Search direction p on the space tangent to ｋ Projection of [0050] determining the isometric subtensor V ｋ has dimensions m×p when reshaped into a two-dimensional matrix; The method according to claim 1 or 2.

8. optimizing the orthogonal approximation (30, 34) under the isometry constraint, Updating the orthogonal approximation (30, 34) by updating all isometric subtensors (32, 32a, 32b), where each isometric subtensor Vk is updated to a point V in the update direction. ｋ Stiefel manifold of m×p isometric matrices in [006] The isometric subtensor V is updated by performing retraction on Ｋ has dimensions m×p when reshaped into a two-dimensional matrix, the update direction is a direction that minimizes the cost function based on a partial derivative of the cost function, and in particular, the projection p ｋ，ｔａｎｇｅｎｔ The isometric subtensor V ｋ using the update direction for updating the The method according to claim 1 or 2.

9. isometric sub-tensors (32a, 32b) of the orthogonal approximation (30, 34) at the ends of the orthogonal approximation (30, 34) are isometric and the intermediate tensor (32) is unitary when reshaped into a two-dimensional matrix before expanding the orthogonal approximation (30, 34), whereby all isometric sub-tensors (32, 32a, 32b) are unitary after reshaping the isometric sub-tensors (32, 32a, 32b) into square matrices; The method according to claim 1 or 2.

10. The method includes providing one or more additional ancilla qubits (42) as input to a first end of the tensor network, the method particularly providing at least logR additional ancilla qubits (42), where R is an approximate rank (R); the method in particular comprises a step of measuring a qubit (44) at a second end of the tensor network, and a selection of a result having a predetermined measurement result of the qubit (44) measured at the second end of the tensor network is part of the implementation of the objective matrix (24), The method according to claim 1 or 2.

11. the method further comprising implementing the orthogonal approximation (30, 34) of the target matrix (24) as a quantum circuit (28) on quantum hardware based on the encoding of the isometric subtensors (32, 32a, 32b) into quantum gates; implementing the orthogonal approximation (30, 34) of the target matrix (24) as a quantum circuit (28) is based in particular on applying isometric sub-operations to a set of qubits in sequence and rearranging unitary matrices in a quantum network, The method according to claim 1 or 2.

12. A processing system (20) for encoding a target matrix (24) in a quantum circuit (28), the system (10) comprising: determining an approximate rank (R) of a matrix multiplication operator (MPO) representation (22) of said target matrix (24); determining an initial guess for an orthogonal approximation (30, 34) of said objective matrix (24) in the form of a tensor network having isometric subtensors (32, 32a, 32b) of said approximation rank (R); starting from the initial estimate, iteratively optimizing the orthogonal approximation (30, 34) of the objective matrix (24) based on an optimization algorithm that minimizes a cost function subject to an isometry constraint of the isometric subtensors (32, 32a, 32b), the cost function attributing a cost to the orthogonal approximation (30, 34) of the objective matrix (24) based on a quality of the orthogonal approximation (30, 34) with respect to the objective matrix (24); determining an implementation of the orthogonal approximation (30, 34) of the target matrix (24) in a quantum circuit (28) based on encoding the isometric subtensors (32, 32a, 32b) into quantum gates; configured to implement A processing system (20).

13. The system (10) of claim 12, further configured to communicate the implementation to a quantum computing system for executing the orthogonal approximation (30, 34) on quantum hardware.

14. A hybrid quantum-classical computing system comprising the system (10) of claim 12 or 13 and quantum computing hardware, the hybrid quantum-classical computing system comprising: receiving said embodiment from said processing system (20); Implementing the embodiment on the quantum computing hardware; and receiving a computation result from the quantum computing hardware; configured to implement A hybrid quantum-classical computing system.

15. A computer program comprising machine-readable instructions which, when executed by a processing unit, causes the processing unit to perform a method according to claim 1 or 2 or to perform and / or control a system according to claim 12 or 13.

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