Patent Publication Number: US-11663513-B2

Title: Quantum computer with exact compression of quantum states

Description:
BACKGROUND 
     Quantum computers promise to solve industry-critical problems which are otherwise unsolvable. Key application areas include chemistry and materials, bioscience and bioinformatics, and finance. Interest in quantum computing has recently surged, in part, due to a wave of advances in the performance of ready-to-use quantum computers. Although these machines are not yet able to solve useful industry problems, the precipice of utility seems to be rapidly approaching. 
     Broadly speaking, quantum autoencoders belong to an emerging genre of algorithms called variational quantum algorithms, where a parametrized quantum circuit is used for solving a certain task using a quantum computer and a classical computer is used for optimizing the circuit parameters with respect to some objective function that is efficiently measurable on the quantum computer. A major advantage of the variational quantum algorithm paradigm is that it takes advantage of both classical and quantum computers. This hybrid approach is particularly appealing in the current era of noisy intermediate scale quantum (NISQ) devices, where qubits are expensive and sensitive to error and noise. However, one potential challenge in implementing variational quantum algorithms effectively concerns optimizing the quantum circuit. As the problem size grows, the number of parameters that need to be optimized also grows, giving rise to high-dimensional black-box optimization problems that are difficult to tackle for a classical computer. Such difficulty is rigorously manifested in a recent study showing that the optimization landscape has vanishingly small gradient in most regions. To surmount the problem, one potential solution is to start from an educated initial guess of a good quantum circuit for the problem, a circuit ansatz so to speak, instead of a quantum circuit chosen in a random or ad hoc manner. As an example, for variational quantum eigensolvers (VQE) there are excellent quantum circuit ansatz constructions available for approximating the ground state of a physical system. However, for quantum autoencoders it remains unclear what circuit ansatz may be useful. 
     SUMMARY 
     A quantum computer includes an efficient and exact quantum circuit for performing quantum state compression. 
     Other features and advantages of various aspects and embodiments of the present invention will become apparent from the following description and from the claims. 
     In a first aspect, a hybrid quantum/classical computer includes a quantum computer component and a classical computer component having a processor, a non-transitory computer-readable medium, and computer program instructions stored on the non-transitory computer-readable medium. The computer program instructions are executable by the processor to cause the classical computer component to perform a first classical subroutine on a plurality of quantum states S related to a system of interest, wherein performing the first classical subroutine includes generating data representing a set S′ of matrix product state (MPS) approximations of the plurality of quantum states, and store the data representing the set S′ of MPS approximations in the non-transitory computer-readable medium. The computer program instructions are also executable by the processor to cause the classical computer component to perform a second classical subroutine on the data representing the set S′ of MPS approximations to generate data representing a quantum circuit U′, and storing the data representing the quantum circuit U′ in the non-transitory computer-readable medium for use in compressing the plurality of quantum states. The computer program instructions are also executable by the processor to cause the classical computer component to generate data representing a variational quantum circuit U(x) based on the data representing the quantum circuit U′, and store the data representing the variational quantum circuit U(x) in the non-transitory computer-readable medium, the data representing the variational quantum circuit U(x) for use in performing a quantum state compression on the plurality of quantum states S with the quantum computer component. The computer program instructions are also executable by the processor to cause the classical computer component and the quantum computer component to perform a quantum circuit training procedure on the data representing the variational quantum circuit U(x) to generate data representing an optimized circuit U* and to store the data representing the optimized circuit U* in the non-transitory computer-readable medium, the data representing the optimized circuit U* for use in compressing the plurality of the quantum states S into states of fewer qubits. 
     In certain embodiments of the first aspect, the computer program instructions to compress include computer program instructions executable by the processor to exactly compress the plurality of the quantum states S into the states of fewer qubits. 
     In certain embodiments of the first aspect, the computer program instructions stored on the non-transitory computer-readable medium include computer program instructions executable by the processor to cause the classical computer component to generate data representing a circuit ansatz for performing the exact compression on the plurality of quantum states S. In these embodiments, generating the data representing the circuit ansatz includes (i) selecting a parameter c that comprises an upper bound on a number of qubits on which an operation is performed or compiled on the quantum computer component, (ii) selecting a value of a parameter D such that D≤√{square root over (2 c-1 /M)}, D being an MPS bond dimension specifying an amount of quantum correlation between neighboring qubits captured in the MPS description of the plurality of quantum states S, and M being a selected number of quantum states in the plurality of quantum states S, (iii) causing the classical computer component to generate data representing a set S D  of MPS approximations, the set S D  comprising the set S′ of MPS approximations having the MPS bond dimension D, and to store the data representing the set S D  in the non-transitory computer-readable medium, (iv) causing the classical computer component to generate data representing a circuit U D  based on the data representing the set S D  of MPS approximations and causing the classical computer component to store the data representing the circuit U D  in the non-transitory computer-readable medium, (v) causing the classical computer component to generate the data representing the circuit ansatz using the data representing the circuit U D  as a template, and to store the data representing the circuit ansatz in the non-transitory computer-readable medium, for use in exactly compressing S D . 
     In certain embodiments of the first aspect, the computer program instructions stored in the non-transitory computer-readable medium include computer program instructions executable by the processor to cause the classical computer component to perform a new operations development subroutine to develop data representing new operations with variational parameters θ, using the data representing the circuit U D  as an initial template, and store the data representing the new operations in the non-transitory computer-readable medium. The computer program instructions are further executable to cause the classical computer component to perform a parametrized circuit development subroutine to add the data representing the new operations to data representing the circuit U D  to produce data representing a parametrized circuit U(θ), and store the data representing the parametrized circuit U(θ) in the non-transitory computer-readable medium for use in compressing the plurality of the quantum states. 
     In certain embodiments of the first aspect, the computer program instructions stored in the non-transitory computer-readable medium further include computer program instructions executable by the processor to cause the classical computer component to perform an entanglement capturing subroutine to generate data representing additional entanglement in the plurality of the quantum states S, the entanglement capturing subroutine having a circuit fine tuning subroutine to generate data representing a fine-tuned version of circuit U D  using additional parameterized operations. The computer program instructions are further executable to cause the classical computer component to store the data representing additional entanglement and the data representing the fine-tuned version of circuit U D  in the non-transitory computer-readable medium for use in generating the data representing the circuit ansatz. 
     In certain embodiments of the first aspect, the computer program instructions stored in the non-transitory computer-readable medium include computer program instructions executable by the processor to cause the classical computer component to generate data representing the additional parameterized operations with variational parameters θ using the circuit U D  as an initial template. The computer program instructions are further executable to cause the classical computer component to add the data representing the additional parameterized operations to data representing the circuit U D  to produce data representing a parametrized circuit U(θ), and store the data representing the parametrized circuit U(θ) in the non-transitory computer-readable medium for use in compressing the plurality of quantum states. 
     In certain embodiments of the first aspect, the plurality of quantum states includes a plurality of training states. The data representing a first training state in the plurality of training states is implicitly specified by data representing a Hamiltonian related to the system of interest. The computer program instructions to perform the quantum circuit training include computer program instructions executable by the processor to use the data representing the first training state to represent a ground state of the Hamiltonian in the performing of the quantum circuit training. 
     In certain embodiments of the first aspect, the data representing the variational quantum circuit U(x) includes data representing quantum gates having an associated plurality of tuning parameters. 
     In certain embodiments of the first aspect, the system of interest includes optical switching. Furthermore, the data representing the variational quantum circuit U(x) comprises data representing quantum gates having associated tuning parameters corresponding to angles of individual optical elements. 
     In a second aspect, a method includes performing compression of quantum states with a hybrid quantum-classical computer system having a quantum computer component and a classical computer component with a processor, a non-transitory computer-readable medium, and computer program instructions stored in the non-transitory computer-readable medium. The computer program instructions are executable by the processor to perform the compression by causing the classical computer component to perform a first classical subroutine on a plurality of quantum states S related to a system of interest, wherein performing the first classical subroutine comprises generating data representing a set S′ of matrix product state (MPS) approximations of the plurality of quantum states, and storing the data representing the set S′ of MPS approximations in the non-transitory computer-readable medium. The computer program instructions are executable by the processor to perform the compression by causing the classical computer component to perform a second classical subroutine on the data representing the set S′ of MPS approximations to generate data representing a quantum circuit U′, and storing the data representing the quantum circuit U′ in the non-transitory computer-readable medium for use in compressing the plurality of quantum states. The computer program instructions are executable by the processor to perform the compression by causing the classical computer component to generate data representing a variational quantum circuit U(x) based on the data representing the quantum circuit U′, and storing the data representing the variational quantum circuit U(x) in the non-transitory computer-readable medium, the data representing the variational quantum circuit U(x) for use in performing a quantum state compression on the plurality of quantum states S with the quantum computer component. The computer program instructions are executable by the processor to perform the compression by causing the classical computer component and the quantum computer component to perform a quantum circuit training procedure on the data representing the variational quantum circuit U(x) to generate data representing an optimized circuit U*, and storing the data representing the optimized circuit U* in the non-transitory computer-readable medium for use in compressing the plurality of quantum states S into states of fewer qubits. 
     In certain embodiments of the second aspect, generating the data representing the variational quantum circuit U(x) includes adding data representing an additional quantum gate with at least one tuning parameter to data representing a gate sequence of the quantum circuit U′. 
     In certain embodiments of the second aspect, generating the data representing the variational quantum circuit U(x) includes causing the quantum computer component to combine data representing the quantum circuit U′ with data representing parameterized gates to generate data representing a parameterized variational quantum circuit, and storing the data representing the parameterized variational quantum circuit in the non-transitory computer-readable medium for use in further optimizing procedures. 
     In certain embodiments of the second aspect, the set S′ of matrix product state approximations of the plurality of quantum states comprises a set S′ of n-qubit matrix product states with n&gt;|S′|. The plurality of compressed quantum states S includes a set of [log 2 |S′|]-qubit states. Furthermore, compressing the plurality of quantum states S into states of fewer qubits further includes causing the quantum computer component to iteratively apply a (c,c−1)-compression subroutine to perform (c,c−1)-compressions on data representing the n-qubit matrix product states in parallel until a compression threshold is achieved. 
     In certain embodiments of the second aspect, the compression threshold is achieved when the n-qubit matrix product states act non-trivially on at most a selected number of qubits. The selected number includes a ceiling(log_2|S′|), with the (c,c−1)-compression procedure being a procedure for removing a qubit from the set S′, |S′| representing cardinality of the set S′, and the ceiling(log_2|S′|) being a smallest integer that is no smaller than log_2|S′|. 
     In certain embodiments of the second aspect, c is an upper bound on a number of qubits on which an operation is performable or compilable on the quantum computer component. 
     In a third aspect, a method for compressing a plurality of tensor networks, each formed from all of a plurality of tensors, includes identifying in all of the tensor networks a first subset of the tensors separated from a remaining second subset of the tensors by a boundary. A first total dimension of contracting indices between the first and second subsets across the boundary is identical for all of the tensor networks, and each of the tensors of the first subset is contractable with an external index such that a second total dimension of all the external indices is identical for all of the tensor networks. The method also includes identifying one tensor in the first subset having an external index with a dimension less than or equal to a ratio of (i) the second total dimension, and (ii) a product of the first total dimension and a number of the plurality of tensor networks. The method also includes recasting the first subset of the tensors as a matrix having a number of rows equal to a product of the first total dimension and the number of the plurality of tensor networks, and a number of columns equal to the second total dimension. The method also includes performing QR decomposition on the matrix to obtain a unitary matrix and an upper triangular matrix. The method also includes replacing, in the tensor networks, the first subset of the tensors with the upper triangular matrix to disentangle the identified one tensor from each of the tensor networks. 
     In certain embodiments of the third aspect, the method includes repeating (i) identifying the first subset, (ii) identifying the one tensor in the first subset, (iii) recasting the first subset, (iv) performing QR decomposition, and (v) replacing the first subset, until either the first subset cannot be identified, or the one tensor in the first subset cannot be identified. 
     In certain embodiments of the third aspect, the plurality of tensors represents a plurality of quantum states. 
     In certain embodiments of the third aspect, each of the plurality of quantum states is either a matrix product state or a projected entangled pair state. 
     In certain embodiments of the third aspect, the method further includes generating one of the plurality of quantum states on a quantum computer with a plurality of entangled qubits. The method further includes compressing said one of the plurality of quantum states by applying to the entangled qubits a unitary transformation implementing an inverse of the unitary matrix. 
     In certain embodiments of the third aspect, the unitary transformation is parameterized to implement a quantum autoencoder. 
     In certain embodiments of the third aspect, the method further includes training the quantum autoencoder to compress all of the plurality of quantum states. 
     In a fourth aspect, a computing system for compressing a plurality of tensor networks, each formed from all of a plurality of tensors, includes a processor and a memory communicably coupled with the processor and storing machine-readable instructions. When executed by the processor, the machine-readable instructions control the computing system to identify in all of the tensor networks a first subset of the tensors separated from a remaining second subset of the tensors by a boundary. A first total dimension of contracting indices between the first and second subsets across the boundary is identical for all of the tensor networks, and each of the tensors of the first subset is contractable with an external index such that a second total dimension of all the external indices is identical for all of the tensor networks. The machine-readable instructions also control the computing system to identify one tensor in the first subset having an external index with a dimension less than or equal to a ratio of (i) the second total dimension, and (ii) a product of the first total dimension and a number of the plurality of tensor networks. The machine-readable instructions also control the computing system to recast the first subset of the tensors as a matrix having a number of rows equal to a product of the first total dimension and the number of the plurality of tensor networks, and a number of columns equal to the second total dimension. The machine-readable instructions also control the computing system to perform QR decomposition on the matrix to obtain a unitary matrix and an upper triangular matrix. The machine-readable instructions also control the computing system to replace, in the tensor networks, the first subset of the tensors with the upper triangular matrix to disentangle the identified one tensor from each of the tensor networks. 
     In certain embodiments of the fourth aspect, the machine-readable instructions that control the computing system to (i) identify the first subset, (ii) identify the one tensor in the first subset, (iii) recast the first subset, (iv) perform QR decomposition, and (v) replace the first subset, are configured to repeat until either the first subset cannot be identified, or the one tensor in the first subset cannot be identified. 
     In certain embodiments of the fourth aspect, the plurality of tensors represents a plurality of quantum states. 
     In certain embodiments of the fourth aspect, each of the plurality of quantum states is either a matrix product state or a projected entangled pair state. 
     In certain embodiments of the fourth aspect, the computing system also includes a quantum computer communicably coupled with the processor and configured to generate one of the plurality of quantum states on a quantum computer with a plurality of entangled qubits. The quantum computer is also configured to compress the one of the plurality of quantum states by applying to the entangled qubits a unitary transformation implementing an inverse of the unitary matrix. 
     In certain embodiments of the fourth aspect, the unitary transformation is parameterized to implement a quantum autoencoder. 
     In certain embodiments of the fourth aspect, the machine-readable instructions are further configured to control the computing system to train the quantum autoencoder to compress all of the plurality of quantum states. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG.  1    is a diagram of a quantum computer according to one embodiment of the present invention; 
         FIG.  2 A  is a flowchart of a method performed by the quantum computer of  FIG.  1    according to one embodiment of the present invention; 
         FIG.  2 B  is a diagram of a hybrid quantum-classical computer which performs quantum annealing according to one embodiment of the present invention; 
         FIG.  3    is a diagram of a hybrid quantum-classical computer according to one embodiment of the present invention; 
         FIG.  4    is a diagram illustrating an example of two tensor networks formed from ten tensors according to an embodiment of the present invention; 
         FIG.  5    is a flowchart illustrating a method for compressing a plurality of tensor networks, each formed from all of a plurality of tensors according to an embodiment of the present invention; and 
         FIGS.  6 A and  6 B  are flowcharts of methods for compressing quantum states according to embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present invention are directed to a quantum computer which includes an efficient and exact quantum circuit for performing quantum state compression. Unlike quantum autoencoders, which need training to learn a compression operation, quantum computers implemented according to embodiments of the present invention are constructed explicitly to compress specific kinds of quantum states. As a result, quantum computers implemented according to embodiments of the present invention do not require training. Embodiments of the present invention may be scaled to handle many-qubit training states. 
     Furthermore, quantum computers implemented according to embodiments of the present invention may be used to compress states other than specific kinds of quantum states which they were designed to compress, in which case the resulting output may be used as an initial guess based on approximations of the training states. 
     Embodiments of the present invention are directed to quantum computers which implement exact and efficient techniques for compressing a given set of quantum states. The value of such quantum computers lies in the intersection between quantum computation and information compression. The latter is generally important for improving memory and algorithmic efficiency of computing devices, and the former is a revolutionary technology for handling data in the specific form of quantum states. Because of their quantum mechanical nature, quantum states are in general not known to admit efficient representations on a classical computer, making them hard to store and manipulate with conventional computing technologies. However, quantum computers are able to store and process quantum states efficiently in ways that are not possible with classical computation. A quantum algorithm that is able to compress quantum states therefore may improve the memory and algorithmic efficiency of quantum computers. This has motivated recent interest in quantum autoencoders, where a parametrized quantum circuit is trained to compress a given set of states into states of fewer qubits. 
     Quantum computers implemented according to embodiments of the present invention address the problem described above of identifying a useful circuit ansatz by starting from simple cases in which the quantum states to be compressed are so well structured that there is an efficient and explicit construction of quantum circuits which exactly compresses them. In these cases, since the circuit is explicitly known, there is no need to train a quantum circuit for the compression. Although in practice the quantum states that need to be compressed may not admit any of the simple structure assumed here, it is nonetheless useful to first compress instead well-structured approximations of the quantum states and to use the compression circuit as an educated initial guess, or an ansatz, for compressing the original set of quantum states. Specifically, one may introduce additional tuning parameters to the circuit ansatz and fine tune the parameters to account for the error incurred in the state approximations used for constructing the ansatz. A method performed by quantum computers implemented according to embodiments of the present invention is shown in  FIGS.  6 A and  6 B . 
     Because quantum computers implemented according to embodiments of the present invention are direct improvements to quantum autoencoders, their areas of application include, but are not limited to, those of quantum autoencoders, such as:
         1. Low dimensional latent space for variational quantum eigensolvers. This allows for fewer tuning parameters in the state preparation for VQE.   2. Quantum state compression for improving the bandwidth of quantum communication. More quantum information may be sent through a quantum communication channel with the same number of qubits if there is a method for compressing the messages.   3. Compression of classical data encoded as quantum states.       

     Referring to  FIG.  6 B , a quantum computer implemented according to an embodiment of the present invention may:
         Begin with a set S of training states. One or more of such training states may, for example, be implicitly specified by a Hamiltonian, where the ground state of the Hamiltonian is treated as the training state.   Perform a classical subroutine A, which receives the set of training states S as input, and produces as its output a set S′ of matrix product state approximations of the set S of training states. The classical subroutine A may, for example, be any of a variety of existing MPS approximation schemes for the specific setting of the problem.   Perform a classical subroutine B, which receives the set S′ of matrix product state approximations as its input, and which produces as its output a description of a quantum circuit U′ for compressing the set S′. The classical subroutine B may, for example, be implemented in any of the ways described in Section 2 of the attached document entitled, “Exact compression of quantum states and effective ansatz for quantum autoencoder.”   Construct a variational quantum circuit U(x) based on the quantum circuit U′. For example, U(x) may be constructed by taking U′ and adding to its gate sequence one or more additional quantum gates with parameters X which are tunable. Hence, U′ combined with the parameterized gates from a variational quantum circuit U(x).   Perform quantum circuit training on the variational quantum circuit U(x) such that U(x) compresses the training states S. The output of the quantum circuit training is a description of an optimized circuit U* for compressing the set S of training states.   Apply the optimized circuit U* to compress the set S of training states exactly.       

       FIG.  6 A  shows a method performed by a quantum computer implemented according to one embodiment of the present invention for compressing a set S′ of n-qubit matrix product states into [log 2 |S′|]-qubit states. Starting from the set S′ of n-qubit states with n&gt;|S′|, the method iteratively applies (c,c−1)-compression to the states in S′ in parallel until the states act non-trivially only on ceiling(log_2|S′|) qubits. Two notes:
         (1) the notation |A| represents the cardinality (number of elements) of a set A. Here |S|=|S′|.   (2) The notations in  FIG.  6 A  around “log_2|S|” represent a ceiling function, where ceiling(x) is the smallest integer that is no smaller than x.       

     It is to be understood that although the invention has been described above in terms of particular embodiments, the foregoing embodiments are provided as illustrative only, and do not limit or define the scope of the invention. Various other embodiments, including but not limited to the following, are also within the scope of the claims. For example, elements and components described herein may be further divided into additional components or joined together to form fewer components for performing the same functions. 
     Various physical embodiments of a quantum computer are suitable for use according to the present disclosure. In general, the fundamental data storage unit in quantum computing is the quantum bit, or qubit. The qubit is a quantum-computing analog of a classical digital computer system bit. A classical bit is considered to occupy, at any given point in time, one of two possible states corresponding to the binary digits (bits) 0 or 1. By contrast, a qubit is implemented in hardware by a physical medium with quantum-mechanical characteristics. Such a medium, which physically instantiates a qubit, may be referred to herein as a “physical instantiation of a qubit,” a “physical embodiment of a qubit,” a “medium embodying a qubit,” or similar terms, or simply as a “qubit,” for ease of explanation. It should be understood, therefore, that references herein to “qubits” within descriptions of embodiments of the present invention refer to physical media which embody qubits. 
     Each qubit has an infinite number of different potential quantum-mechanical states. When the state of a qubit is physically measured, the measurement produces one of two different basis states resolved from the state of the qubit. Thus, a single qubit can represent a one, a zero, or any quantum superposition of those two qubit states; a pair of qubits can be in any quantum superposition of 4 orthogonal basis states; and three qubits can be in any superposition of 8 orthogonal basis states. The function that defines the quantum-mechanical states of a qubit is known as its wavefunction. The wavefunction also specifies the probability distribution of outcomes for a given measurement. A qubit, which has a quantum state of dimension two (i.e., has two orthogonal basis states), may be generalized to a d-dimensional “qudit,” where d may be any integral value, such as 2, 3, 4, or higher. In the general case of a qudit, measurement of the qudit produces one of d different basis states resolved from the state of the qudit. Any reference herein to a qubit should be understood to refer more generally to a d-dimensional qudit with any value of d. 
     Although certain descriptions of qubits herein may describe such qubits in terms of their mathematical properties, each such qubit may be implemented in a physical medium in any of a variety of different ways. Examples of such physical media include superconducting material, trapped ions, photons, optical cavities, individual electrons trapped within quantum dots, point defects in solids (e.g., phosphorus donors in silicon or nitrogen-vacancy centers in diamond), molecules (e.g., alanine, vanadium complexes), or aggregations of any of the foregoing that exhibit qubit behavior, that is, comprising quantum states and transitions therebetween that can be controllably induced or detected. 
     For any given medium that implements a qubit, any of a variety of properties of that medium may be chosen to implement the qubit. For example, if electrons are chosen to implement qubits, then the x component of its spin degree of freedom may be chosen as the property of such electrons to represent the states of such qubits. Alternatively, the y component, or the z component of the spin degree of freedom may be chosen as the property of such electrons to represent the state of such qubits. This is merely a specific example of the general feature that for any physical medium that is chosen to implement qubits, there may be multiple physical degrees of freedom (e.g., the x, y, and z components in the electron spin example) that may be chosen to represent 0 and 1. For any particular degree of freedom, the physical medium may controllably be put in a state of superposition, and measurements may then be taken in the chosen degree of freedom to obtain readouts of qubit values. 
     Certain implementations of quantum computers, referred as gate model quantum computers, comprise quantum gates. In contrast to classical gates, there is an infinite number of possible single-qubit quantum gates that change the state vector of a qubit. Changing the state of a qubit state vector typically is referred to as a single-qubit rotation, and may also be referred to herein as a state change or a single-qubit quantum-gate operation. A rotation, state change, or single-qubit quantum-gate operation may be represented mathematically by a unitary 2×2 matrix with complex elements. A rotation corresponds to a rotation of a qubit state within its Hilbert space, which may be conceptualized as a rotation of the Bloch sphere. (As is well-known to those having ordinary skill in the art, the Bloch sphere is a geometrical representation of the space of pure states of a qubit.) Multi-qubit gates alter the quantum state of a set of qubits. For example, two-qubit gates rotate the state of two qubits as a rotation in the four-dimensional Hilbert space of the two qubits. (As is well-known to those having ordinary skill in the art, a Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.) 
     A quantum circuit may be specified as a sequence of quantum gates. As described in more detail below, the term “quantum gate,” as used herein, refers to the application of a gate control signal (defined below) to one or more qubits to cause those qubits to undergo certain physical transformations and thereby to implement a logical gate operation. To conceptualize a quantum circuit, the matrices corresponding to the component quantum gates may be multiplied together in the order specified by the gate sequence to produce a 2n×2n complex matrix representing the same overall state change on n qubits. A quantum circuit may thus be expressed as a single resultant operator. However, designing a quantum circuit in terms of constituent gates allows the design to conform to a standard set of gates, and thus enable greater ease of deployment. A quantum circuit thus corresponds to a design for actions taken upon the physical components of a quantum computer. 
     A given variational quantum circuit may be parameterized in a suitable device-specific manner. More generally, the quantum gates making up a quantum circuit may have an associated plurality of tuning parameters. For example, in embodiments based on optical switching, tuning parameters may correspond to the angles of individual optical elements. 
     In certain embodiments of quantum circuits, the quantum circuit includes both one or more gates and one or more measurement operations. Quantum computers implemented using such quantum circuits are referred to herein as implementing “measurement feedback.” For example, a quantum computer implementing measurement feedback may execute the gates in a quantum circuit and then measure only a subset (i.e., fewer than all) of the qubits in the quantum computer, and then decide which gate(s) to execute next based on the outcome(s) of the measurement(s). In particular, the measurement(s) may indicate a degree of error in the gate operation(s), and the quantum computer may decide which gate(s) to execute next based on the degree of error. The quantum computer may then execute the gate(s) indicated by the decision. This process of executing gates, measuring a subset of the qubits, and then deciding which gate(s) to execute next may be repeated any number of times. Measurement feedback may be useful for performing quantum error correction, but is not limited to use in performing quantum error correction. For every quantum circuit, there is an error-corrected implementation of the circuit with or without measurement feedback. 
     Some embodiments described herein generate, measure, or utilize quantum states that approximate a target quantum state (e.g., a ground state of a Hamiltonian). As will be appreciated by those trained in the art, there are many ways to quantify how well a first quantum state “approximates” a second quantum state. In the following description, any concept or definition of approximation known in the art may be used without departing from the scope hereof. For example, when the first and second quantum states are represented as first and second vectors, respectively, the first quantum state approximates the second quantum state when an inner product between the first and second vectors (called the “fidelity” between the two quantum states) is greater than a predefined amount (typically labeled ∈). In this example, the fidelity quantifies how “close” or “similar” the first and second quantum states are to each other. The fidelity represents a probability that a measurement of the first quantum state will give the same result as if the measurement were performed on the second quantum state. Proximity between quantum states can also be quantified with a distance measure, such as a Euclidean norm, a Hamming distance, or another type of norm known in the art. Proximity between quantum states can also be defined in computational terms. For example, the first quantum state approximates the second quantum state when a polynomial time-sampling of the first quantum state gives some desired information or property that it shares with the second quantum state. 
     Not all quantum computers are gate model quantum computers. Embodiments of the present invention are not limited to being implemented using gate model quantum computers. As an alternative example, embodiments of the present invention may be implemented, in whole or in part, using a quantum computer that is implemented using a quantum annealing architecture, which is an alternative to the gate model quantum computing architecture. More specifically, quantum annealing (QA) is a metaheuristic for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations. 
       FIG.  2 B  shows a diagram illustrating operations typically performed by a computer system  250  which implements quantum annealing. The system  250  includes both a quantum computer  252  and a classical computer  254 . Operations shown on the left of the dashed vertical line  256  typically are performed by the quantum computer  252 , while operations shown on the right of the dashed vertical line  256  typically are performed by the classical computer  254 . 
     Quantum annealing starts with the classical computer  254  generating an initial Hamiltonian  260  and a final Hamiltonian  262  based on a computational problem  258  to be solved, and providing the initial Hamiltonian  260 , the final Hamiltonian  262  and an annealing schedule  270  as input to the quantum computer  252 . The quantum computer  252  prepares a well-known initial state  266  ( FIG.  2 B , operation  264 ), such as a quantum-mechanical superposition of all possible states (candidate states) with equal weights, based on the initial Hamiltonian  260 . The classical computer  254  provides the initial Hamiltonian  260 , a final Hamiltonian  262 , and an annealing schedule  270  to the quantum computer  252 . The quantum computer  252  starts in the initial state  266 , and evolves its state according to the annealing schedule  270  following the time-dependent Schrödinger equation, a natural quantum-mechanical evolution of physical systems ( FIG.  2 B , operation  268 ). More specifically, the state of the quantum computer  252  undergoes time evolution under a time-dependent Hamiltonian, which starts from the initial Hamiltonian  260  and terminates at the final Hamiltonian  262 . If the rate of change of the system Hamiltonian is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian. If the rate of change of the system Hamiltonian is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem Hamiltonian, i.e., diabatic quantum computation. At the end of the time evolution, the set of qubits on the quantum annealer is in a final state  272 , which is expected to be close to the ground state of the classical Ising model that corresponds to the solution to the original computational problem  258 . An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal. 
     The final state  272  of the quantum computer  252  is measured, thereby producing results  276  (i.e., measurements) ( FIG.  2 B , operation  274 ). The measurement operation  274  may be performed, for example, in any of the ways disclosed herein, such as in any of the ways disclosed herein in connection with the measurement unit  110  in  FIG.  1   . The classical computer  254  performs postprocessing on the measurement results  276  to produce output  280  representing a solution to the original computational problem  258  ( FIG.  2 B , operation  278 ). 
     As yet another alternative example, embodiments of the present invention may be implemented, in whole or in part, using a quantum computer that is implemented using a one-way quantum computing architecture, also referred to as a measurement-based quantum computing architecture, which is another alternative to the gate model quantum computing architecture. More specifically, the one-way or measurement based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is “one-way” because the resource state is destroyed by the measurements. 
     The outcome of each individual measurement is random, but they are related in such a way that the computation always succeeds. In general, the choices of basis for later measurements need to depend on the results of earlier measurements, and hence the measurements cannot all be performed at the same time. 
     Any of the functions disclosed herein may be implemented using means for performing those functions. Such means include, but are not limited to, any of the components disclosed herein, such as the computer-related components described below. 
     Referring to  FIG.  1   , a diagram is shown of a system  100  implemented according to one embodiment of the present invention. Referring to  FIG.  2 A , a flowchart is shown of a method  200  performed by the system  100  of  FIG.  1    according to one embodiment of the present invention. The system  100  includes a quantum computer  102 . The quantum computer  102  includes a plurality of qubits  104 , which may be implemented in any of the ways disclosed herein. There may be any number of qubits  104  in the quantum computer  102 . For example, the qubits  104  may include or consist of no more than 2 qubits, no more than 4 qubits, no more than 8 qubits, no more than 16 qubits, no more than 32 qubits, no more than 64 qubits, no more than 128 qubits, no more than 256 qubits, no more than 512 qubits, no more than 1024 qubits, no more than 2048 qubits, no more than 4096 qubits, or no more than 8192 qubits. These are merely examples, in practice there may be any number of qubits  104  in the quantum computer  102 . 
     There may be any number of gates in a quantum circuit. However, in some embodiments the number of gates may be at least proportional to the number of qubits  104  in the quantum computer  102 . In some embodiments the gate depth may be no greater than the number of qubits  104  in the quantum computer  102 , or no greater than some linear multiple of the number of qubits  104  in the quantum computer  102  (e.g., 2, 3, 4, 5, 6, or 7). 
     The qubits  104  may be interconnected in any graph pattern. For example, they be connected in a linear chain, a two-dimensional grid, an all-to-all connection, any combination thereof, or any subgraph of any of the preceding. 
     As will become clear from the description below, although element  102  is referred to herein as a “quantum computer,” this does not imply that all components of the quantum computer  102  leverage quantum phenomena. One or more components of the quantum computer  102  may, for example, be classical (i.e., non-quantum components) components which do not leverage quantum phenomena. 
     The quantum computer  102  includes a control unit  106 , which may include any of a variety of circuitry and/or other machinery for performing the functions disclosed herein. The control unit  106  may, for example, consist entirely of classical components. The control unit  106  generates and provides as output one or more control signals  108  to the qubits  104 . The control signals  108  may take any of a variety of forms, such as any kind of electromagnetic signals, such as electrical signals, magnetic signals, optical signals (e.g., laser pulses), or any combination thereof. 
     For example:
         In embodiments in which some or all of the qubits  104  are implemented as photons (also referred to as a “quantum optical” implementation) that travel along waveguides, the control unit  106  may be a beam splitter (e.g., a heater or a mirror), the control signals  108  may be signals that control the heater or the rotation of the mirror, the measurement unit  110  may be a photodetector, and the measurement signals  112  may be photons.   In embodiments in which some or all of the qubits  104  are implemented as charge type qubits (e.g., transmon, X-mon, G-mon) or flux-type qubits (e.g., flux qubits, capacitively shunted flux qubits) (also referred to as a “circuit quantum electrodynamic” (circuit QED) implementation), the control unit  106  may be a bus resonator activated by a drive, the control signals  108  may be cavity modes, the measurement unit  110  may be a second resonator (e.g., a low-Q resonator), and the measurement signals  112  may be voltages measured from the second resonator using dispersive readout techniques.   In embodiments in which some or all of the qubits  104  are implemented as superconducting circuits, the control unit  106  may be a circuit QED-assisted control unit or a direct capacitive coupling control unit or an inductive capacitive coupling control unit, the control signals  108  may be cavity modes, the measurement unit  110  may be a second resonator (e.g., a low-Q resonator), and the measurement signals  112  may be voltages measured from the second resonator using dispersive readout techniques.   In embodiments in which some or all of the qubits  104  are implemented as trapped ions (e.g., electronic states of, e.g., magnesium ions), the control unit  106  may be a laser, the control signals  108  may be laser pulses, the measurement unit  110  may be a laser and either a CCD or a photodetector (e.g., a photomultiplier tube), and the measurement signals  112  may be photons.   In embodiments in which some or all of the qubits  104  are implemented using nuclear magnetic resonance (NMR) (in which case the qubits may be molecules, e.g., in liquid or solid form), the control unit  106  may be a radio frequency (RF) antenna, the control signals  108  may be RF fields emitted by the RF antenna, the measurement unit  110  may be another RF antenna, and the measurement signals  112  may be RF fields measured by the second RF antenna.   In embodiments in which some or all of the qubits  104  are implemented as nitrogen-vacancy centers (NV centers), the control unit  106  may, for example, be a laser, a microwave antenna, or a coil, the control signals  108  may be visible light, a microwave signal, or a constant electromagnetic field, the measurement unit  110  may be a photodetector, and the measurement signals  112  may be photons.   In embodiments in which some or all of the qubits  104  are implemented as two-dimensional quasiparticles called “anyons” (also referred to as a “topological quantum computer” implementation), the control unit  106  may be nanowires, the control signals  108  may be local electrical fields or microwave pulses, the measurement unit  110  may be superconducting circuits, and the measurement signals  112  may be voltages.   In embodiments in which some or all of the qubits  104  are implemented as semiconducting material (e.g., nanowires), the control unit  106  may be microfabricated gates, the control signals  108  may be RF or microwave signals, the measurement unit  110  may be microfabricated gates, and the measurement signals  112  may be RF or microwave signals.       

     Although not shown explicitly in  FIG.  1    and not required, the measurement unit  110  may provide one or more feedback signals  114  to the control unit  106  based on the measurement signals  112 . For example, quantum computers referred to as “one-way quantum computers” or “measurement-based quantum computers” utilize such feedback signal  114  from the measurement unit  110  to the control unit  106 . Such feedback signal  114  is also necessary for the operation of fault-tolerant quantum computing and error correction. 
     The control signals  108  may, for example, include one or more state preparation signals which, when received by the qubits  104 , cause some or all of the qubits  104  to change their states. Such state preparation signals constitute a quantum circuit also referred to as an “ansatz circuit.” The resulting state of the qubits  104  is referred to herein as an “initial state” or an “ansatz state.” The process of outputting the state preparation signal(s) to cause the qubits  104  to be in their initial state is referred to herein as “state preparation” ( FIG.  2 A , section  206 ). A special case of state preparation is “initialization,” also referred to as a “reset operation,” in which the initial state is one in which some or all of the qubits  104  are in the “zero” state i.e. the default single-qubit state. More generally, state preparation may involve using the state preparation signals to cause some or all of the qubits  104  to be in any distribution of desired states. In some embodiments, the control unit  106  may first perform initialization on the qubits  104  and then perform preparation on the qubits  104 , by first outputting a first set of state preparation signals to initialize the qubits  104 , and by then outputting a second set of state preparation signals to put the qubits  104  partially or entirely into non-zero states. 
     Another example of control signals  108  that may be output by the control unit  106  and received by the qubits  104  are gate control signals. The control unit  106  may output such gate control signals, thereby applying one or more gates to the qubits  104 . Applying a gate to one or more qubits causes the set of qubits to undergo a physical state change which embodies a corresponding logical gate operation (e.g., single-qubit rotation, two-qubit entangling gate or multi-qubit operation) specified by the received gate control signal. As this implies, in response to receiving the gate control signals, the qubits  104  undergo physical transformations which cause the qubits  104  to change state in such a way that the states of the qubits  104 , when measured (see below), represent the results of performing logical gate operations specified by the gate control signals. The term “quantum gate,” as used herein, refers to the application of a gate control signal to one or more qubits to cause those qubits to undergo the physical transformations described above and thereby to implement a logical gate operation. 
     It should be understood that the dividing line between state preparation (and the corresponding state preparation signals) and the application of gates (and the corresponding gate control signals) may be chosen arbitrarily. For example, some or all the components and operations that are illustrated in  FIGS.  1  and  2 A- 2 B  as elements of “state preparation” may instead be characterized as elements of gate application. Conversely, for example, some or all of the components and operations that are illustrated in  FIGS.  1  and  2 A- 2 B  as elements of “gate application” may instead be characterized as elements of state preparation. As one particular example, the system and method of  FIGS.  1  and  2 A- 2 B  may be characterized as solely performing state preparation followed by measurement, without any gate application, where the elements that are described herein as being part of gate application are instead considered to be part of state preparation. Conversely, for example, the system and method of  FIGS.  1  and  2 A- 2 B  may be characterized as solely performing gate application followed by measurement, without any state preparation, and where the elements that are described herein as being part of state preparation are instead considered to be part of gate application. 
     The quantum computer  102  also includes a measurement unit  110 , which performs one or more measurement operations on the qubits  104  to read out measurement signals  112  (also referred to herein as “measurement results”) from the qubits  104 , where the measurement results  112  are signals representing the states of some or all of the qubits  104 . In practice, the control unit  106  and the measurement unit  110  may be entirely distinct from each other, or contain some components in common with each other, or be implemented using a single unit (i.e., a single unit may implement both the control unit  106  and the measurement unit  110 ). For example, a laser unit may be used both to generate the control signals  108  and to provide stimulus (e.g., one or more laser beams) to the qubits  104  to cause the measurement signals  112  to be generated. 
     In general, the quantum computer  102  may perform various operations described above any number of times. For example, the control unit  106  may generate one or more control signals  108 , thereby causing the qubits  104  to perform one or more quantum gate operations. The measurement unit  110  may then perform one or more measurement operations on the qubits  104  to read out a set of one or more measurement signals  112 . The measurement unit  110  may repeat such measurement operations on the qubits  104  before the control unit  106  generates additional control signals  108 , thereby causing the measurement unit  110  to read out additional measurement signals  112  resulting from the same gate operations that were performed before reading out the previous measurement signals  112 . The measurement unit  110  may repeat this process any number of times to generate any number of measurement signals  112  corresponding to the same gate operations. The quantum computer  102  may then aggregate such multiple measurements of the same gate operations in any of a variety of ways. 
     After the measurement unit  110  has performed one or more measurement operations on the qubits  104  after they have performed one set of gate operations, the control unit  106  may generate one or more additional control signals  108 , which may differ from the previous control signals  108 , thereby causing the qubits  104  to perform one or more additional quantum gate operations, which may differ from the previous set of quantum gate operations. The process described above may then be repeated, with the measurement unit  110  performing one or more measurement operations on the qubits  104  in their new states (resulting from the most recently-performed gate operations). 
     In general, the system  100  may implement a plurality of quantum circuits as follows. For each quantum circuit C in the plurality of quantum circuits ( FIG.  2 A , operation  202 ), the system  100  performs a plurality of “shots” on the qubits  104 . The meaning of a shot will become clear from the description that follows. For each shot S in the plurality of shots ( FIG.  2 A , operation  204 ), the system  100  prepares the state of the qubits  104  ( FIG.  2 A , section  206 ). More specifically, for each quantum gate G in quantum circuit C ( FIG.  2 A , operation  210 ), the system  100  applies quantum gate G to the qubits  104  ( FIG.  2 A , operations  212  and  214 ). 
     Then, for each of the qubits Q  104  ( FIG.  2 A , operation  216 ), the system  100  measures the qubit Q to produce measurement output representing a current state of qubit Q ( FIG.  2 A , operations  218  and  220 ). 
     The operations described above are repeated for each shot S ( FIG.  2 A , operation  222 ), and circuit C ( FIG.  2 A , operation  224 ). As the description above implies, a single “shot” involves preparing the state of the qubits  104  and applying all of the quantum gates in a circuit to the qubits  104  and then measuring the states of the qubits  104 ; and the system  100  may perform multiple shots for one or more circuits. 
     Referring to  FIG.  3   , a diagram is shown of a hybrid quantum-classical (HQC) computer  300  implemented according to one embodiment of the present invention. The HQC  300  includes a quantum computer component  102  (which may, for example, be implemented in the manner shown and described in connection with  FIG.  1   ) and a classical computer component  306 . The classical computer component may be a machine implemented according to the general computing model established by John Von Neumann, in which programs are written in the form of ordered lists of instructions and stored within a classical (e.g., digital) memory  310  and executed by a classical (e.g., digital) processor  308  of the classical computer. The memory  310  is classical in the sense that it stores data in a storage medium in the form of bits, which have a single definite binary state at any point in time. The bits stored in the memory  310  may, for example, represent a computer program. The classical computer component  304  typically includes a bus  314 . The processor  308  may read bits from and write bits to the memory  310  over the bus  314 . For example, the processor  308  may read instructions from the computer program in the memory  310 , and may optionally receive input data  316  from a source external to the computer  302 , such as from a user input device such as a mouse, keyboard, or any other input device. The processor  308  may use instructions that have been read from the memory  310  to perform computations on data read from the memory  310  and/or the input  316 , and generate output from those instructions. The processor  308  may store that output back into the memory  310  and/or provide the output externally as output data  318  via an output device, such as a monitor, speaker, or network device. 
     The quantum computer component  102  may include a plurality of qubits  104 , as described above in connection with  FIG.  1   . A single qubit may represent a one, a zero, or any quantum superposition of those two qubit states. The classical computer component  304  may provide classical state preparation signals  332  to the quantum computer  102 , in response to which the quantum computer  102  may prepare the states of the qubits  104  in any of the ways disclosed herein, such as in any of the ways disclosed in connection with  FIGS.  1  and  2 A- 2 B . 
     Once the qubits  104  have been prepared, the classical processor  308  may provide classical control signals  334  to the quantum computer  102 , in response to which the quantum computer  102  may apply the gate operations specified by the control signals  332  to the qubits  104 , as a result of which the qubits  104  arrive at a final state. The measurement unit  110  in the quantum computer  102  (which may be implemented as described above in connection with  FIGS.  1  and  2 A- 2 B ) may measure the states of the qubits  104  and produce measurement output  338  representing the collapse of the states of the qubits  104  into one of their eigenstates. As a result, the measurement output  338  includes or consists of bits and therefore represents a classical state. The quantum computer  102  provides the measurement output  338  to the classical processor  308 . The classical processor  308  may store data representing the measurement output  338  and/or data derived therefrom in the classical memory  310 . 
     The steps described above may be repeated any number of times, with what is described above as the final state of the qubits  104  serving as the initial state of the next iteration. In this way, the classical computer  304  and the quantum computer  102  may cooperate as co-processors to perform joint computations as a single computer system. 
     The technique described thus far is not restricted to compressing 1D tensor network as is the case for matrix product states. In fact it can be generalized to compressing a general set of M tensor networks T 1 , . . . , T M  each of which can be described as an undirected graph G j (V,E j ) with a subset of nodes P ⊂ V such that for all j, the total dimension D P  of the edges K across the cut between P and V\P is the same. (Commonly tensor networks that do not contract to a scalar will have edges that correspond to open indices. Here we ignore these open indices and only consider indices that connect tensors as edges.) See  FIG.  4    for an example where M=2 and K=9. For G j , let D u  be the dimension of the index i u  coming out of the tensor node u∈E j . Let N max  be the largest dimensional QR factorization that can be handled classically in the setting that the user considers. The criterion for compressibility which in this most general case would be 
     
       
         
           
             
               
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     In one example, N max=2     c   , c=3, M is the number of tensor networks, 
             P   =         {       A     (   k   )       ,     A     (     k   +   1     )       ,     A     (     k   +   2     )         }     ⁢           ⁢   and   ⁢           ⁢       min     w   ∈   P       ⁢     D   w         =     2   .             
The general workflow may be summarized as the following:
         1. Given a set of tensor networks S={T 1 , . . . , T M }, find a partition P such that the total dimension across the cut is identically D P , and also that the total dimension of edges pointing outwards Π w∈P D w ≤N max . If no such partition can be found, terminate the algorithm.   2. Find a node u∈P such that D P M≤Π w∈P\{u} D w  and perform compression by QR factorization. This step disentangles u for all of the tensor networks in S. If no such partition can be found, terminate the algorithm.   3. Repeat the procedure until one of the previous two steps terminate.       

     The above algorithm may also parallelized by considering partitions P, P′, P″, that are disjoint to each other at the same step. 
       FIG.  5    is a flow chart illustrating a method  500  for compressing a plurality of tensor networks, each formed from all of a plurality of tensors. Method  500  may be performed on either a classical computer, or a classical computing component of a hybrid quantum-classical computer. Method  500  starts at a block  502 . In a block  502 , a first subset of the tensors is identified in all of the tensor networks. The first subset of the tensors is separated from a remaining second subset of the tensors by a boundary. A first total dimension of contracting indices between the first and second subsets across the boundary is identical for all of the tensor networks. Each of the tensors of the first subset is contractable with an external index such that a second total dimension of all the external indices is identical for all of the tensor networks. 
       FIG.  4    shows an example of two tensor networks formed from ten tensors. In  FIG.  4   , the first subset includes four tensors (labeled u 1  through u 4 ) that are separated from the second subset of remaining tensors (labeled v 1  through v 6 ) by a boundary that is indicated as a dashed line. In  FIG.  4   , each contracting index is shown as an edge connecting two tensors, and each edge is labeled with a dimension of the contracting index (i.e., the number of values the contracting index may have). The first total dimension equals a product of the dimensions of the contracting indices crossing the boundary (i.e., 32,768 for the example of  FIG.  4   ). Note that the first total dimension is the same for all of the tensor networks. 
     In  FIG.  4   , each of the four tensors of the first subset is shown with a vertical line indicated an external index (labeled u u     1    through i u     4   ) that may be contracted when connected to a tensor. Although not shown in  FIG.  4   , each external index has a dimension equal to the number of values that the external index may have. The second total dimension equals a product of the dimensions of the external indices. 
     Method  500  also includes a block  508  in which one tensor of the first subset is identified for disentangling (see block  516 ). To qualify for disentangling, the one tensor must have an external index with a dimension less than or equal to a ratio of (i) the second total dimension, and (ii) a product of the first total dimension and a number of the plurality of tensor networks. Method  500  also includes a block  512  in which the first subset of the tensors is recast as a matrix having a number of rows equal to a product of the first total dimension and the number of the plurality of tensor networks, and a number of columns equal to the second total dimension. Method  500  also includes a block  514  in which QR decomposition is performed on the matrix to obtain a unitary matrix and an upper triangular matrix. Method  500  also includes a block  516  in which the first subset tensors is replaced, in the tensor networks, with the upper triangular matrix to disentangle the identified one tensor from each of the tensor networks. 
     In some embodiments, blocks  504 ,  508 ,  512 ,  514 , and  516  repeat until either (i) the first subset cannot be identified, or (ii) the one tensor in the first subset cannot be identified. With each iteration, another tensor is disentangled from the first subset such that number of tensors in the first subset is reduced by one. In some of these embodiments, method  500  includes a decision block  506  in which it is determined if the first subset was identified. If not, method  500  continues to a block  518  and ends. If the first subset was identified, then method  500  continues to block  508 . The first subset may not be identified when, for example, no subset of the tensors can be found in which the first total dimension is the same for all of the tensor networks. 
     In some of the embodiments in which blocks  504 ,  508 ,  512 ,  514 , and  516  repeat, method  500  also includes a decision block  510  in which it is determined if one tensor in the first subset is identified. If not, method  500  continues to a block  518  and ends. If the one tensor was identified, then method  500  continues to block  512 . The one tensor may not be identified when, for example, every tensor in the first subset has, for its external index, a dimension that is greater than the ratio of (i) the second total dimension, and (ii) the product of the first total dimension and the number of tensor networks. 
     In some embodiments of method  500 , the plurality of tensors represents a plurality of quantum states. For example, each of the plurality of quantum states may be a matrix product state or a projected entangled pair state. In some of these embodiments, method  500  further includes generating one of the plurality of quantum states on a quantum computer with a plurality of entangled qubits, and compressing said one of the plurality of quantum states by applying to the entangled qubits a unitary transformation implementing an inverse of the unitary matrix. The unitary transformation may be parameterized to implement a quantum autoencoder. In embodiments in which the quantum autoencoder is implemented, method  500  may also include training the quantum autoencoder to compress all of the plurality of quantum states. 
     Although certain functions may be described herein as being performed by a classical computer and other functions may be described herein as being performed by a quantum computer, these are merely examples and do not constitute limitations of the present invention. A subset of the functions which are disclosed herein as being performed by a quantum computer may instead be performed by a classical computer. For example, a classical computer may execute functionality for emulating a quantum computer and provide a subset of the functionality described herein, albeit with functionality limited by the exponential scaling of the simulation. Functions which are disclosed herein as being performed by a classical computer may instead be performed by a quantum computer. 
     The techniques described above may be implemented, for example, in hardware, in one or more computer programs tangibly stored on one or more computer-readable media, firmware, or any combination thereof, such as solely on a quantum computer, solely on a classical computer, or on a hybrid quantum-classical (HQC) computer. The techniques disclosed herein may, for example, be implemented solely on a classical computer, in which the classical computer emulates the quantum computer functions disclosed herein. 
     The techniques described above may be implemented in one or more computer programs executing on (or executable by) a programmable computer (such as a classical computer, a quantum computer, or an HQC) including any combination of any number of the following: a processor, a storage medium readable and/or writable by the processor (including, for example, volatile and non-volatile memory and/or storage elements), an input device, and an output device. Program code may be applied to input entered using the input device to perform the functions described and to generate output using the output device. 
     Embodiments of the present invention include features which are only possible and/or feasible to implement with the use of one or more computers, computer processors, and/or other elements of a computer system. Such features are either impossible or impractical to implement mentally and/or manually. For example, a quantum computer is required to describe the training states S for even a moderate number of qubits (e.g., 50 or more qubits). Even a classical computer cannot be used to describe the training states S for such a number of qubits, due to the infeasible amount of storage space that would be required. This is merely one example of an aspect of the present invention that is inherently rooted in quantum computing technology and which is either impossible or impractical to implement mentally and/or manually. 
     Any claims herein which affirmatively require a computer, a processor, a memory, or similar computer-related elements, are intended to require such elements, and should not be interpreted as if such elements are not present in or required by such claims. Such claims are not intended, and should not be interpreted, to cover methods and/or systems which lack the recited computer-related elements. For example, any method claim herein which recites that the claimed method is performed by a computer, a processor, a memory, and/or similar computer-related element, is intended to, and should only be interpreted to, encompass methods which are performed by the recited computer-related element(s). Such a method claim should not be interpreted, for example, to encompass a method that is performed mentally or by hand (e.g., using pencil and paper). Similarly, any product claim herein which recites that the claimed product includes a computer, a processor, a memory, and/or similar computer-related element, is intended to, and should only be interpreted to, encompass products which include the recited computer-related element(s). Such a product claim should not be interpreted, for example, to encompass a product that does not include the recited computer-related element(s). 
     In embodiments in which a classical computing component executes a computer program providing any subset of the functionality within the scope of the claims below, the computer program may be implemented in any programming language, such as assembly language, machine language, a high-level procedural programming language, or an object-oriented programming language. The programming language may, for example, be a compiled or interpreted programming language. 
     Each such computer program may be implemented in a computer program product tangibly embodied in a machine-readable storage device for execution by a computer processor, which may be either a classical processor or a quantum processor. Method steps of the invention may be performed by one or more computer processors executing a program tangibly embodied on a computer-readable medium to perform functions of the invention by operating on input and generating output. Suitable processors include, by way of example, both general and special purpose microprocessors. Generally, the processor receives (reads) instructions and data from a memory (such as a read-only memory and/or a random access memory) and writes (stores) instructions and data to the memory. Storage devices suitable for tangibly embodying computer program instructions and data include, for example, all forms of non-volatile memory, such as semiconductor memory devices, including EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROMs. Any of the foregoing may be supplemented by, or incorporated in, specially-designed ASICs (application-specific integrated circuits) or FPGAs (Field-Programmable Gate Arrays). A classical computer can generally also receive (read) programs and data from, and write (store) programs and data to, a non-transitory computer-readable storage medium such as an internal disk (not shown) or a removable disk. These elements will also be found in a conventional desktop or workstation computer as well as other computers suitable for executing computer programs implementing the methods described herein, which may be used in conjunction with any digital print engine or marking engine, display monitor, or other raster output device capable of producing color or gray scale pixels on paper, film, display screen, or other output medium. 
     Any data disclosed herein may be implemented, for example, in one or more data structures tangibly stored on a non-transitory computer-readable medium (such as a classical computer-readable medium, a quantum computer-readable medium, or an HQC computer-readable medium). Embodiments of the invention may store such data in such data structure(s) and read such data from such data structure(s).