Solving quadratic optimization problems on orthogonal groups using quantum computers

Abstract

In one aspect, the method includes receiving data representing a quadratic optimization problem, where decision variables of the quadratic optimization problem take values ​​on an orthogonal group or a special orthogonal group; encoding the quadratic optimization problem as a quantum Hamiltonian, where the encoding includes mapping an orthogonal matrix or a special orthogonal matrix of the group to respective quantum states in Hilbert space using a Clifford algebra representation of the group; determining approximate eigenstates of the quantum Hamiltonian; and calculating expectation values ​​of Pauli operators for the approximate eigenstates, where the Pauli operators include operators obtained by mapping a multiplication operation of the Clifford algebra to Hilbert space; and rounding the expectation values ​​of the Pauli operators to elements of the orthogonal group to obtain a solution to the quadratic optimization problem.

Description

[Background technology]

[0001] The present disclosure relates to quantum computing.

[0002] Many optimization problems feature orthogonal matrices as decision variables. Such problems typically involve the joint alignment of points in Euclidean space via isometry, e.g., in the context of structural biology via cryo-electron microscopy and NMR spectroscopy, computer vision, robotics, and sensor network localization. There are two main difficulties in solving these problems. First, orthogonal groups are non-convex, which generally makes the optimization landscape difficult to navigate. Second, the objective functions for these types of problems are typically quadratic or higher-order polynomials in the optimization variables.

[0003] In the commutative setting, these problems are called quadratic combinatorial optimization problems,

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[0006] In the non-commutative setting, the decision variables x1,...,x m is the group of 1×1 orthogonal matrices

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[0013] Quadratic combinatorial optimization problems in the commutative setting can be transformed into a quantum information framework by equivalence with the classical Ising model, which naturally translates to quantum spin Hamiltonians. This correspondence is at the heart of much work investigating the potential of quantum computing for solving combinatorial optimization problems, including quantum annealing and a wide range of quantum approximate optimization algorithms.

[0014] This idea can also be generalized, for example, in the case of cyclic groups of degree k>2 (MAX-k-CUT).

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[0017] This disclosure describes techniques for solving quadratic optimization problems on orthogonal groups using quantum computing.

[0018] In general, one inventive aspect of the subject matter described herein includes receiving, by a classical computer, data representing a quadratic optimization problem, wherein decision variables of the quadratic optimization problem take values ​​in an orthogonal group or a special orthogonal group; The quadratic optimization problem can be implemented in a manner that includes encoding, by a classical computer, the quadratic optimization problem as a quantum Hamiltonian, where the encoding includes mapping an orthogonal matrix or a special orthogonal matrix of the group to respective quantum states in Hilbert space using a Clifford algebra representation of the group; determining, by a quantum computer, approximate eigenstates of the quantum Hamiltonian; computing, by a quantum computer, expectation values ​​of Pauli operators for the approximate eigenstates, where the Pauli operators include operators obtained by mapping a multiplication operation of the Clifford algebra to Hilbert space; and computing, by a classical computer, expectation values ​​of the Pauli operators to elements of the orthogonal group to obtain a solution to the quadratic optimization problem.

[0019] Other implementations of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. One or more classical or quantum computer systems can be configured to perform particular operations or actions by having installed thereon software, firmware, hardware, or a combination thereof that causes the system to perform the actions during operation. One or more computer programs can be configured to perform particular operations or actions by including instructions that, when executed by a data processing device, cause the device to perform the actions.

[0020] These and other embodiments may each optionally include one or more of the following features, alone or in combination: In some embodiments, the orthogonal group comprises an orthogonal group of dimension n≧1, or a special orthogonal group of dimension n≧1, and the group operation is given by matrix multiplication.

[0021] In some embodiments, using the Clifford algebraic representation of the group to map the orthogonal matrices of the group to respective quantum states in Hilbert space is nThe basis elements of the Clifford algebra in a 2-dimensional vector space are n to respective computational basis states in a n-dimensional Hilbert space, where n represents the dimension of the group; and mapping a Clifford algebra left multiplication operation and a Clifford algebra right multiplication operation to respective n-qubit operators in the Hilbert space.

[0022] In some implementations, the Clifford algebra left multiplication operation for the i-th basis element is

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[0025] In some embodiments, mapping an orthogonal matrix of the group to each quantum state in the Hilbert space using a Clifford algebra representation of the group comprises applying a quadratic mapping to elements of the Clifford algebra, the quadratic mapping being such that 2 n It maps inputs in an n-dimensional vector space to respective outputs in an n×n-dimensional vector space, where the matrix elements of each output contain the expectation values ​​of an n-qubit operator in Hilbert space.

[0026] In some implementations, the matrix elements of the output of the quadratic mapping comprise the expectation values ​​of the n-qubit operator given by:

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[0028] In some embodiments, the matrix elements of the output of the quadratic mapping are:

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[0031] In some implementations, the quadratic optimization problem defines a graph of vertices and edges, and rounding the computed expectations to elements of the orthogonal group includes implementing an edge bound rounding algorithm.

[0032] In some embodiments, implementing the edge bound rounding algorithm includes decomposing a square matrix containing expected values ​​of the edges of the graph as a product of a rectangular matrix and the transpose of the rectangular matrix, generating a normally distributed random matrix, multiplying the random matrix and the rectangular matrix to obtain a vector of matrices, and for each vertex of the graph, projecting each matrix in the vector of matrices to the nearest element of the orthogonal group.

[0033] In some implementations, projecting each matrix in the vector of matrices onto the nearest element of the orthogonal group includes computing a singular value decomposition of the vector of matrices.

[0034] In some implementations, the quadratic optimization problem defines a graph of vertices and edges, and rounding the computed expectations to elements of the orthogonal group includes implementing a vertex bound rounding algorithm.

[0035] In some embodiments, implementing the vertex bound rounding algorithm includes, for each vertex of the graph, projecting a matrix containing the expected value of the vertex onto the closest element of the orthogonal group.

[0036] In some implementations, projecting the matrix containing the vertex expectation values ​​onto the nearest element of the orthogonal group includes computing a singular value decomposition of the matrix containing the vertex expectation values.

[0037] In some embodiments, calculating the expectation value of the Pauli operator for the approximate eigenstate includes preparing, by a quantum computer, a copy of the approximate eigenstate using a ground state preparation method or an approximate ground state preparation method, and measuring, by the quantum computer, the copy of the approximate eigenstate, including measuring a matrix element of the 1-RDM or 2-RDM.

[0038] In some implementations, determining the approximate eigenstates of the quantum Hamiltonian includes performing a quantum phase estimation or variational algorithm, and the accuracy of the approximate eigenstates depends on the accuracy of the quantum phase estimation calculation or variational algorithm.

[0039] In some implementations, the received data includes values ​​of elements of a positive semidefinite matrix, and the solution to the quadratic optimization problem includes rounded expected values ​​and values ​​of elements of the positive semidefinite matrix.

[0040] In some embodiments, the quantum Hamiltonian includes two-body interactions of fermions.

[0041] In some embodiments, the quadratic optimization problem comprises a small non-commutative Grothendieck problem on an orthogonal group or a special orthogonal group.

[0042] In some embodiments, the eigenstates of the quantum Hamiltonian include maximal eigenstates, or eigenstates that maximize energy with respect to the initial state.

[0043] The subject matter described herein can be implemented in particular embodiments to realize one or more of the following advantages.

[0044] Quadratic optimization on orthogonal groups encompasses a wide class of optimization problems, such as swarm synchronization (with applications in structural biology, robotics, and wireless networking), point set registration, simultaneous localization and mapping, and generalized orthogonal Procrustes problems (with applications in areas such as shape and image recognition). These problems are typically computationally inefficient or difficult to solve accurately in all environments using classical computers.

[0045] The inventions described herein enable such problems to be transformed into a quantum information framework, thereby allowing them to be solved efficiently using quantum computers. Thus, the techniques described herein for solving quadratic optimization problems on orthogonal groups are particularly adapted for a specific technical implementation: quantum computing. In this disclosure, a quadratic optimization problem on an orthogonal group is first formulated for a classical computing device. This formulation is then mapped to a quantum formulation, e.g., a quantum Hamiltonian and corresponding qubit operators. Thus, a quantum computer can be used to obtain a solution to the quadratic optimization problem because it can perform the operations necessary to solve the quantum formulation, e.g., phase estimation, variational algorithms, state preparation, and measurement of the qubit operators. That is, the techniques described herein are motivated by technical considerations regarding the internal functionality of quantum computers.

[0046] Furthermore, quantum entanglement resources are predicted to provide higher quality solutions than classical approximation algorithms.

[0047] The details of one or more embodiments of the subject matter herein are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims. [Brief explanation of the drawings]

[0048] [Figure 1] FIG. 1 is a block diagram of an exemplary system for solving quadratic optimization problems on orthogonal groups using classical and quantum computing. [Figure 2] FIG. 1 is a flow diagram of an exemplary process for solving a quadratic optimization problem on an orthogonal group using classical and quantum computing. [Figure 3] FIG. 10 is a flow diagram of an exemplary process for edge boundary rounding of a quantum state. [Figure 4] FIG. 10 is a flow diagram of an exemplary process for vertex bound rounding of a quantum state. [Figure 5] 1 illustrates an exemplary quantum computing system. [Figure 6] 1 shows a schematic diagram of an exemplary classical processor system. DETAILED DESCRIPTION OF THE INVENTION

[0049] We describe techniques for encoding quadratic optimization problems on orthogonal groups into quantum Hamiltonians using the Clifford algebraic representation of the orthogonal groups. A solution to the quadratic optimization problem can then be obtained by computing a quantum state that optimizes the energy of the quantum Hamiltonian and rounding the quantum state to a space of feasible solutions.

[0050] 1 is a conceptual block diagram of an exemplary system 100 for solving quadratic optimization problems on an orthogonal group using classical and quantum computing. The exemplary system 100 includes a classical processor 102 and a quantum processor 104. The classical processor 102 and the quantum processor 104 may be in electronic communication over one or more networks or may be communicatively exchanged in other ways, such as via one or more wired or wireless connections.

[0051] Classical processor 102 is configured to perform classical computations. Quantum processor 104 is configured to perform quantum computations. For convenience, classical processor 102 and quantum processor 104 are shown as separate entities. For example, quantum processor 104 can be a quantum processor operated by an external party. However, in some implementations, classical processor 102 can be included in quantum processor 104. That is, quantum processor 104 can also include components for performing classical computational operations. In general, the classical computing component of a classical processor can be implemented as one or more classical computers having physical hardware such as that described with respect to FIG. 6, and the quantum computing component of quantum processor 104 can be implemented as a quantum computing device having physical hardware such as that described with respect to FIG. 5.

[0052] The exemplary system 100 is configured to perform operations for solving quadratic optimization problems on orthogonal groups using classical and quantum computation. An n-dimensional orthogonal group, denoted O(n), is a mathematical group of n-by-n orthogonal matrices, where an orthogonal matrix is ​​a real matrix whose inverse is equal to its transpose. An orthogonal group has a group operation given by matrix multiplication. An orthogonal group containing a matrix with determinant 1 is called a special orthogonal group and is a normal subgroup denoted SO(n). A quadratic optimization problem on an orthogonal group G = O(n) or G = SO(n) can be defined as follows: Let (V,E) be a set of n-by-n orthogonal matrices such that each edge (u,v)∈E has real-valued coefficients.

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[0055] The classical processor 102 is configured to receive input data that specifies a quadratic optimization problem 108. For example, the classical processor 102 receives data that specifies the objective function given in Equation 1 above, such as the size and coefficients of the orthogonal group.

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[0057] The classical processor 102 is configured to encode the quadratic optimization problem as a quantum Hamiltonian 110. Specifically, the classical processor 102 is configured to apply a quadratic mapping 106 to the objective function defined by the input data 108. The quadratic mapping 106 maps orthogonal matrices of the orthogonal group to respective quantum states in Hilbert space. The quadratic mapping 106 is based on a Clifford algebraic representation of the orthogonal group and is described in more detail below with reference to FIG. 2. Under the quadratic mapping 106, the objective function is transformed from a classical form, e.g., a form acceptable to a classical computing device, to a quantum form, e.g., a form acceptable to a quantum computing device. More specifically, the objective function is formulated as a qubit Hamiltonian. Thus, the value of the objective function for a given element of the orthogonal group Q(Φ)∈G is equivalent to the energy of the corresponding Hamiltonian with respect to the state |Φ〉∈Hd. Thus, the task of maximizing the objective function given in Equation 1 is mapped to the task of maximizing the energy of the corresponding Hamiltonian.

[0058] The classical processor 102 is configured to send data representing the quantum Hamiltonian to the quantum processor 104. The quantum processor 104 is configured to receive the data 112 and determine an approximate eigenstate of the quantum Hamiltonian, e.g., an eigenstate that increases in energy over the maximal eigenstate or the initial state. For example, the quantum processor 104 can be configured to implement a phase estimation algorithm or to train a variational quantum circuit to determine the eigenstate. The accuracy of the determined eigenstate depends on the technique used to determine the eigenstate. The quantum processor 104 then prepares the (approximate) eigenstate and calculates the qubit operator P, defined below with reference to FIG. 2, for the (approximate) eigenstate 114. ij ,Γ ijFor example, quantum processor 104 may be configured to iteratively prepare (approximate) eigenstates of a Hamiltonian using a ground state preparation method or an approximate group state preparation method (e.g., prepare a plurality of physical qubits in the (approximate) eigenstates of the Hamiltonian), apply a quantum circuit that evolves the (approximate) eigenstates according to a qubit operator (e.g., apply the quantum circuit to the plurality of physical qubits using a control system), and measure the evolved quantum states (e.g., measure the plurality of physical qubits using a measurement system).

[0059] Quantum processor 104 is configured to transmit data representing the results of measurement operations 116 to classical processor 102. Classical processor 102 is configured to receive the transmitted data 116 and calculate expectation values ​​of the qubit operators over the (approximate) eigenstates, for example, by averaging the measurement results 116 received from quantum processor 104. The calculated expectation values ​​define an initial solution to a quadratic optimization problem.

[0060] The Hamiltonian obtained by applying the quadratic mapping 106 is equivalent to the Hamiltonian of fermions with two-body interactions. Therefore, because the space of quantum states is larger than the orthogonal group, not all quantum states can be mapped to an orthogonal matrix via the quadratic mapping 106. Therefore, the Hamiltonian obtained by applying the quadratic mapping 106 corresponds to a relaxation of the original problem, and, for example, its largest eigenvalue is generally larger than the value of the objective function given in Equation 1. Therefore, the classical processor 102 is configured to apply a rounding algorithm 118 to the expectation to, for example, recover an orthogonal matrix from the relaxed initial solution in order to backproject the initial solution into the space of feasible solutions. For example, the classical processor 102 can be configured to perform one or both of the rounding algorithms described below with reference to FIGS. 3-4. The rounded expectation is the matrix R that maximizes the objective function given in Equation 1 above. v ∈O(n) or SO(n).

[0061] The classical processor 102 is configured to output data representing a solution to the quadratic optimization problem 120, for example, data specifying a rounded expected value.

[0062] 2 is a flow diagram of an exemplary process 200 for solving a quadratic optimization problem on an orthogonal group using classical and quantum computing. For convenience, process 200 is described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, system 100 shown in FIG. 1 , when appropriately programmed, can perform exemplary process 200.

[0063] The system receives data representing a quadratic optimization problem (step 202). As described above with reference to Equation 1, the decision variables of the quadratic optimization problem take values ​​in an orthogonal group, e.g., an orthogonal group of dimension n≧1, or a special orthogonal group of dimension n≧1. The group operation is matrix multiplication, resulting in a quadratic form of the optimization problem. The received data may be in the form of a matrix, e.g., a positive semidefinite matrix, as described above with reference to Equation 1.

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[0066] In some embodiments, the quadratic optimization problem is a small non-commutative Grothendieck problem, as described above.

[0067] The system encodes the quadratic optimization problem as a quantum Hamiltonian (step 204). The quantum Hamiltonian includes two-body interactions of fermions. To encode the quadratic optimization problem as a quantum Hamiltonian, the system uses a Clifford algebraic representation of the group to map an orthogonal matrix of the group to each quantum state in Hilbert space.

[0068] The Clifford algebra Cl(n) is 2 n It is a dimensional vector space, which can be identified by a Hilbert space of n qubits. To encode a quadratic optimization problem as a quantum Hamiltonian, the system is constructed by using a basis element e of the Clifford algebra. I the respective computational basis states of the Hilbert space

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[0070] To represent the algebraic multiplication operation in this Hilbert space, the system maps the left multiplication operation of the Clifford algebra and the right multiplication operation of the Clifford algebra to respective n-qubit operators in Hilbert space. The left and right multiplications are linear automorphisms, with λ for elements x and y. x ,ρ x :Cl(n)→Cl(n). λ x (y)=xy,ρ x (y)=yx (2) Therefore, the algebraic action can be expressed as a linear operator H2n. Due to linearity, the generator e i It is sufficient to specify left and right multiplication by λ. The left multiplication operation of the Clifford algebra for the ith basis element is given by λ i and the right multiplication operation of the Clifford algebra for the ith basis element is given by p i is mapped to an n-qubit operator given by

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[0073] The parity of the Clifford algebra transforms into the parity of the qubit computational basis, so the parity automorphism α is equivalent to the n-qubit parity operator.

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[0075] The double covering of the group SO(n) is the even parity subalgebra Cl 0 (n) can be obtained from the basis vector e I Since only half of characterizes |I| mod 2 = 0, this subspace has only half the dimension of Cl(n). Therefore, we can project the projection operator from Cl(n) to Cl 0 It is useful to designate H2n to H2n-1, or equivalently H2n to H2n-1. This mapping is n-1 ×2 ncan be given by the matrix

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[0077] The above mappings connect objects in the Clifford algebra to objects in Hilbert space, e.g., quantum states and qubit operators. These quantities can be further linked to physical quantum systems and measurements by defining quadratic mappings that can be applied to the elements of the Clifford algebra. The quadratic mappings are n It maps inputs in an n-dimensional vector space (e.g., elements from the Clifford algebra) to respective outputs in an n×n-dimensional vector space (e.g., elements of a Hilbert space), where each output matrix element contains the expectation value of the combination of n-qubit operators described above with reference to Equation 3.

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[0080] The linear map λ as described above with reference to Equation 3 i , e i、 e j Left multiplication and right multiplication by ρ j, and the conjugate identity

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[0083] This formula can be converted into the quantum representation developed above: First, the Pauli operator for n qubits is defined.

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[0086] Equation 8 corresponds to optimization on an orthogonal group. The (n-1) qubit operator for optimization on a special orthogonal group is

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[0089] The Clifford algebraic formulation of the elements of the orthogonal group leads to the encoding of the quadratic optimization problem into a quantum Hamiltonian. The objective function f(R1,…R) (as defined in Eq. 1) m ) can be expanded explicitly in terms of matrix elements.

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[0091] Secondary Mapping

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[0097] Here, the quadratic optimization problem is reformulated as the optimization of the mn qubit Hamiltonian.

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[0102] Returning to Figure 2, the system uses a quantum computer to determine (approximate) eigenstates of the quantum Hamiltonian (step 206). For example, the system may perform known techniques such as quantum phase estimation or variational algorithms to determine the quantum state that maximizes the energy 〈ψ|H|ψ〉.

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[0104] The system uses a quantum computer to calculate the expectation value of the Pauli operator for the (approximate) maximal eigenstate, where the Pauli operator comprises an operator obtained by mapping the multiplication operation of the Clifford algebra onto Hilbert space as given in Equation 8 above (step 208). For example, the system prepares a copy of the (approximate) maximal eigenstate of the quantum Hamiltonian, e.g., using a ground state preparation method or an approximate ground state preparation method, and calculates the expectation value 〈ψ|P ij |ψ〉 and 〈ψ|Γ ij To calculate |ψ〉, we use the operator P for (approximate) the maximum eigenstate copy. ij and Γ ij For example, the system can measure the one-particle reduced density matrix (1-RDM) or the two-particle reduced density matrix (2-RDM) of fermions.

[0105] The system uses a classical computer to round the expectation value of the Pauli operator to an element of the orthogonal group to obtain a solution to the quadratic optimization problem (step 210). For example, the system can perform one (or both) of the algorithms described below with reference to Figures 3 and 4. In the first algorithm, for each pair of vertices (u,v) in the graph formulation of the quadratic optimization problem, the operator

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[0108] The system combines data representing rounded expected values ​​(which are the optimized decision variables, as explained in more detail below with reference to FIGS. 3 and 4) with coefficients C forming a positive semidefinite matrix. uv can be provided as output.

[0109] 3 is a flow diagram of an exemplary process 300 for edge bound rounding of a quantum state. For convenience, process 300 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, system 100 of FIG. 1 , when appropriately programmed, can perform exemplary process 300.

[0110] The quadratic optimization problem described above with reference to step 202 of the exemplary process 200 defines a graph of vertices and edges (V,E), where for each edge (u,v)∈E, the matrix

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[0113] The system calculates the expected value for each edge (u,v) of the graph.

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[0120] Therefore, the system inputs a square matrix M using the data obtained in step 302 (step 304). The system aggregates the data into a large nm×nm block matrix.

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[0124] The system decomposes the square matrix as the product of i) a rectangular matrix and ii) the transpose of the rectangular matrix (step 306). For square matrices M≧0, the system decomposes several

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[0129] The system generates a random matrix (step 308). For example, the system may sample values ​​from a normal distribution N(0,1 / n) and use the sampled values ​​to generate a random matrix

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[0131] The system multiplies the random matrix Z by the rectangular matrix L to obtain a vector of matrices LZ (step 310).

[0132] For each vertex of the graph, the system projects each matrix in the vector of matrices onto the closest element of the orthogonal group (step 312). That is, the system finds the closest (special) orthogonal matrix

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[0134] An exemplary process 300 can be manifested in pseudo-code as follows:

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[0136] 4 is a flow diagram of an exemplary process 400 for vertex bound rounding of a quantum state. For convenience, process 400 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, system 100 of FIG. 1 , when appropriately programmed, can perform exemplary process 400.

[0137] The quadratic optimization problem described above with reference to step 202 of the exemplary process 200 defines a graph of vertices and edges (V,E), where for each edge (u,v)∈E, the matrix

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[0140] The system calculates the expected value for each vertex v in the graph.

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[0142] A single vertex boundary is defined by tracing the qudits associated with all but one vertex.

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[0145] Thus, for each vertex in the graph, the system v ) onto the closest element of the orthogonal group (step 404). That is, the system projects the closest (special) orthogonal matrix

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[0149] By performing this rounding procedure at each vertex, we return a feasible solution. Then, for each v∈V, v The set of ,is provided as a solution to a quadratic optimization problem. In principle, this means that each

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[0153] An exemplary process 400 can be manifested in pseudo-code as follows:

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[0155] 5 illustrates an exemplary quantum computer 500 for performing the quantum operations described herein. The exemplary quantum computer 500 includes an exemplary quantum computing device 502. The quantum computing device 502 is intended to represent various forms of quantum computing devices. The components shown here, their connections and relationships, and their functions are for illustrative purposes only and do not limit the embodiments of the invention described and / or claimed herein.

[0156] Exemplary quantum computing device 502 includes a qubit assembly 552 and a control and measurement system 504. The qubit assembly includes a plurality of qubits, e.g., qubit 506, that are used to perform algorithmic operations or quantum computations. While the qubits shown in FIG. 5 are arranged in a rectangular array, this is schematic and not intended to be limiting. Qubit assembly 552 also includes adjustable coupling elements, e.g., couplers 508, that enable interactions between coupled qubits. In the schematic diagram of FIG. 5, each qubit is adjustably coupled to each of its four neighboring qubits by respective coupling elements. However, this is an exemplary arrangement of qubits and couplers; other arrangements are possible, including non-rectangular arrangements, arrangements that allow coupling between non-adjacent qubits, and arrangements that include adjustable coupling between three or more qubits.

[0157] Each qubit can be a physical two-level quantum system or device with levels representing logical values ​​0 and 1. The specific physical implementation of the qubits and how the qubits interact with each other depends on various factors, such as the type of quantum computing device 502 included in the exemplary computer 500 or the type of quantum computation the quantum computing device is performing. For example, in an atomic quantum computer, the qubits may be implemented via atomic quantum systems, molecular quantum systems, or solid-state quantum systems, e.g., hyperfine atomic states. As another example, in a superconducting quantum computer, the qubits may be implemented via superconducting qubits or semiconductor qubits, e.g., superconducting transmon states. As another example, in an NMR quantum computer, the qubits may be implemented via nuclear spin states.

[0158] In some implementations, quantum computation can proceed, for example, by loading qubits from a quantum memory and applying a sequence of unitary operators to the qubits. Applying the unitary operators to the qubits can include applying a corresponding sequence of quantum logic gates to the qubits, for example, to implement the qubit operators described herein. Examples of quantum logic gates include single-qubit gates, such as Pauli X, Pauli Y, and Pauli Z (also referred to as X, Y, and Z), Hadamard gates, S gates, and rotations; two-qubit gates, such as controlled X, controlled Y, and controlled Z (also referred to as CX, CY, and CZ), controlled NOT gates (also referred to as CNOT), controlled swap gates (also referred to as CSWAP), and iSWAP gates; and gates involving three or more qubits, such as Toffoli gates. Quantum logic gates can be implemented by applying control signals 510 generated by the control and measurement system 504 to the qubits and couplers.

[0159] For example, in some implementations, the qubits in qubit assembly 552 can be frequency tunable. In these examples, each qubit can have an associated operating frequency that can be adjusted by applying voltage pulses through one or more drive lines coupled to the qubit. Exemplary operating frequencies include a qubit idle frequency, a qubit interaction frequency, and a qubit readout frequency. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to a corresponding idle frequency can place the qubit in a state where it does not strongly interact with other qubits and can be used to perform single-qubit gates. As another example, when qubits interact through fixed-coupling couplers, the qubits can be configured to interact with each other by setting their respective operating frequencies at some gate-dependent frequency that detunes from their common interaction frequency. In other cases, for example, when qubits interact through tunable couplers, the qubits can be configured to interact with each other by setting the parameters of each coupler to enable interaction between the qubits and then setting the respective operating frequencies of the qubits at some gate-dependent frequency that detunes from their common interaction frequency. Such interactions can be performed to perform multi-qubit gates.

[0160] The type of control signal 510 used depends on the physical implementation of the qubit. For example, the control signal may include an RF or microwave pulse in an NMR or superconducting quantum computer system, or an optical pulse in an atomic quantum computer system.

[0161] A quantum computation can be completed by measuring the state of the qubit using a quantum observable, such as X or Z, using a respective control signal 510. The measurement causes a readout signal 512 representing the measurement result to be communicated back to the measurement and control system 504. The readout signal 512 may include an RF signal, a microwave signal, or an optical signal, depending on the physical form of the quantum computing device and / or qubit. For convenience, the control signals 510 and readout signals 512 shown in FIG. 5 are shown addressing only selected elements of the qubit assembly (i.e., the top and bottom rows), but in operation, the control signals 510 and readout signals 512 can address each element in the qubit assembly 552.

[0162] Control and measurement system 504 is an example of a classical computer system that can be used to perform various operations on qubit assembly 552, as described above, and other classical subroutines or calculations. Control and measurement system 504 includes one or more classical processors, e.g., classical processor 514, one or more memories, e.g., memory 516, and one or more I / O units, e.g., I / O unit 518, connected by one or more data buses. Control and measurement system 504 can be programmed to send sequences of control signals 510 to the qubit assembly, e.g., to perform a selected series of quantum gate operations, and to receive sequences of readout signals 512 from the qubit assembly, e.g., as part of performing a measurement operation.

[0163] Processor 514 is configured to process instructions for execution within control and measurement system 504. In some embodiments, processor 514 is a single-threaded processor. In other embodiments, processor 514 is a multi-threaded processor. Processor 514 can process instructions stored in memory 516.

[0164] The memory 516 stores information within the control and measurement system 504. In some implementations, the memory 516 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the memory 516 may include a storage device capable of providing mass storage to the system 504, such as a hard disk device, an optical disk device, a storage device shared over a network by multiple computing devices (e.g., a cloud storage device), and / or some other mass storage device.

[0165] Input / output devices 518 provide input and output operations for control and measurement system 504. Input / output devices 518 may include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers to send control signals 510 to and receive readout signals 512 from the qubit assemblies, as appropriate for the physics of the quantum computer. In some implementations, input / output devices 518 may also include one or more network interface devices, such as Ethernet cards, serial communication devices, such as RS-232 ports, and / or wireless interface devices, such as 802.11 cards. In other implementations, input / output devices 518 may include driver devices configured to receive input data and send output data to other external devices, such as keyboards, printers, and display devices.

[0166] Although an exemplary control and measurement system 504 is shown in FIG. 5, embodiments of the subject matter and functional operations described herein can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware, or in one or more combinations thereof, including the structures disclosed herein and structural equivalents thereof.

[0167] 6 illustrates a schematic diagram of an exemplary classical processor system 600. System 600 can be used for classical computing described herein, according to some embodiments. System 600 is intended to represent various forms of digital computers, such as laptops, desktops, workstations, personal digital assistants, servers, blade servers, mainframes, mobile devices, and other suitable computers. The components shown here, their connections and relationships, and their functions are for illustrative purposes only and do not limit the embodiments of the invention described and / or claimed herein.

[0168] System 600 includes a processor 610, a memory 620, a storage device 630, and an input / output device 640. Each of components 610, 620, 630, and 620 are interconnected using a system bus 650. Processor 610 may be capable of processing instructions for execution within system 600. In one embodiment, processor 610 is a single-threaded processor. In other embodiments, processor 610 is a multi-threaded processor. Processor 610 may be capable of processing instructions stored in memory 620 or storage device 630 to display graphical information for a user interface on input / output device 640.

[0169] Memory 620 stores information within system 600. In one embodiment, memory 620 is a computer-readable medium. In one embodiment, memory 620 is a volatile memory unit. In other embodiments, memory 620 is a non-volatile memory unit.

[0170] The storage device 630 may enable the system 600 to provide mass storage. In one embodiment, the storage device 630 is a computer-readable medium. In various different embodiments, the storage device 630 may be a floppy disk device, a hard disk device, an optical disk device, or a tape device.

[0171] The input / output devices 640 provide input and output operations for the system 600. In one embodiment, the input / output devices 640 include a keyboard and / or a pointing device. In another embodiment, the input / output devices 640 include a display unit for displaying a graphical user interface.

[0172] Embodiments of the subject matter and operations described herein may be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry, or more generally, a quantum computing system, including the structures disclosed herein and structural equivalents thereof, in tangibly embodied software or firmware, in hardware, or in one or more combinations thereof. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0173] Implementations of the subject matter described herein may be implemented as one or more computer programs, i.e., as one or more modules of computer program instructions encoded on a tangible, non-transitory storage medium for execution by or to control the operation of a data processing apparatus. The computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more thereof. Alternatively or additionally, the program instructions may be encoded in an artificially generated propagated signal capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal generated to encode digital and / or quantum information for transmission to a receiving device suitable for execution by a data processing apparatus.

[0174] The terms quantum information and quantum data refer to information or data carried by, held, or stored in a quantum system, with the smallest nontrivial system being a qubit, i.e., a system defining a unit of quantum information. It will be understood that the term "qubit" encompasses all quantum systems that can be suitably approximated as a two-level system in a corresponding context. Such quantum systems may include, for example, multi-level systems having two or more levels. By way of example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, a computational basis state is specified using a ground state and a first excited state, although it will be understood that other setups are possible in which a computational state is specified using a higher-level excited state.

[0175] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all types of apparatus, devices, and machines for processing digital and / or quantum data, including, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. An apparatus can also be or include special-purpose logic circuitry, e.g., an FPGA (field-programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus designed to simulate or generate information about a particular quantum system. Specifically, a quantum simulator is a special-purpose quantum computer that does not have the capability to perform universal quantum computation. In addition to hardware, an apparatus can also optionally include code that creates an execution environment for digital and / or quantum computer programs, e.g., code constituting processor firmware, a protocol stack, a database management system, an operating system, or one or more combinations thereof.

[0176] A digital computer program, which may also be called or described as a program, software, software application, module, software module, script, or code, can be written in any form of programming language, including a compiled or interpreted language, or a declarative or procedural language, and can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be called or described as a program, software, software application, module, software module, script, or code, can be written in any form of programming language, including a compiled or interpreted language, or a declarative or procedural language, and converted to a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper.

[0177] A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file holding other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files storing one or more modules, subprograms, or code portions. A computer program can be deployed to run on one computer or on multiple computers located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using quantum systems, e.g., qubits. Generally, digital data communication networks cannot transmit quantum data, but quantum data communication networks can transmit both quantum data and digital data.

[0178] The processes and logic flows described herein may be performed by one or more programmable computers, operating with one or more processors, executing one or more computer programs to perform functions by operating on input data and generating output, as appropriate. The processes and logic flows may also be performed by, and apparatus may be implemented as, special purpose logic circuitry, e.g., an FPGA or ASIC, or a quantum simulator, or a combination of special purpose logic circuitry or a quantum simulator with one or more programmed digital and / or quantum computers.

[0179] When a system of one or more computers is "configured to" perform a particular operation or action, it means that the system has installed thereon software, firmware, hardware, or a combination thereof that, when in operation, causes the system to perform the operation or action. When one or more computer programs are configured to perform a particular operation or action, it means that the one or more programs contain instructions that, when executed by a data processing device, cause the device to perform the operation or action. For example, a quantum computer may receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform an operation or action.

[0180] A computer suitable for executing a computer program can be based on a general-purpose or special-purpose processor, or any other type of central processing unit. Typically, the central processing unit will receive instructions and data from a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data, e.g., photons, or a combination thereof.

[0181] Elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a computer will also include, or be operatively coupled to receive data from, or transfer data to, one or more mass storage devices for storing data, e.g., magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a computer need not have such devices.

[0182] Quantum circuit elements (also called quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, quantum circuit elements are configured to utilize quantum mechanical phenomena such as superposition and quantum entanglement to perform operations on data in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent information and operate in multiple states simultaneously. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUIDs or DC-SQUIDs), among others.

[0183] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively execute the instructions of a computer program by performing basic mathematical, logical, and / or input / output operations on data, where the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and / or receive data from quantum circuit elements via electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices, and ERSFQ devices, which are an energy-efficient version of RSFQ that do not use bias resistors.

[0184] In certain cases, some or all of the quantum and / or classical circuit elements may be implemented using, for example, superconducting quantum and / or classical circuit elements. Fabrication of superconducting circuit elements can involve the deposition of one or more materials, such as superconductors, dielectrics, and / or metals. Depending on the materials selected, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among others. Processes for fabricating circuit elements described herein can involve removing one or more materials from the device during fabrication. Depending on the materials removed, the removal process can include, for example, wet etching techniques, dry etching techniques, or lift-off processes. Materials forming the circuit elements described herein can be patterned using known lithography techniques (e.g., photolithography or electron beam lithography).

[0185] During operation of a quantum computing system using superconducting quantum and / or classical circuit elements, such as those described herein, the superconducting circuit elements are cooled in a cryostat to a temperature at which the superconducting material can exhibit superconducting properties. A superconductor (alternatively, superconducting) material can be understood as a material that exhibits superconducting properties below its superconducting critical temperature. Examples of superconducting materials include aluminum (superconducting critical temperature 1.2 Kelvin) and niobium (superconducting critical temperature 9.3 Kelvin). Thus, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from materials that exhibit superconducting properties below their superconducting critical temperature.

[0186] In certain embodiments, control signals for quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form.

[0187] Suitable computer-readable media for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memories, media, and memory devices, including, by way of example, semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices, magnetic disks, e.g., internal hard disks or removable disks, magneto-optical disks, CD-ROM and DVD-ROM disks, and quantum systems, e.g., trapped atoms or electrons. It will be understood that a quantum memory is a device that can store quantum data with high fidelity and efficiency for long periods of time, e.g., a light-matter interface where light is used for transmission and material for storing and preserving the quantum characteristics of quantum data, such as superposition or quantum coherence.

[0188] Control of the various systems described herein, or portions thereof, may be implemented in a computer program product including instructions stored on one or more non-transitory machine-readable storage media and executable on one or more processing devices. The systems described herein, or portions thereof, may each be implemented as an apparatus, method, or system, which may include one or more processing devices and a memory for storing executable instructions for performing the operations described herein.

[0189] While this specification contains details of many specific embodiments, these should not be construed as limiting the scope of what may be claimed, but rather as descriptions of features that may be inherent to particular embodiments. Certain features described herein in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Furthermore, while features may be described above as acting in a particular combination, and may even be initially claimed as such, one or more features from a claimed combination may, in some cases, be deleted from the combination, and the claimed combination may be directed to a subcombination or variations of the subcombination.

[0190] Similarly, while operations are illustrated in a particular order, this should not be understood as requiring such operations to be performed in the particular order or sequence illustrated, or that all of the illustrated operations be performed, to achieve desirable results. In certain situations, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the above-described embodiments should not be understood as requiring such separation in all embodiments, and it should be understood that the program components and systems described above may generally be integrated together in a single software product or packaged in multiple software products.

[0191] Specific implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As an example, the processes depicted in the accompanying figures do not necessarily require the particular order or sequential order shown to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. receiving, by a classical computer, data representing a quadratic optimization problem, wherein decision variables of the quadratic optimization problem take values ​​in an orthogonal group or a special orthogonal group; encoding, by a classical computer, the quadratic optimization problem as a quantum Hamiltonian, the encoding comprising mapping an orthogonal matrix or a special orthogonal matrix of the group to respective quantum states in Hilbert space using a Clifford algebra representation of the group; determining, by a quantum computer, approximate eigenstates of the quantum Hamiltonian; calculating, by the quantum computer, expectation values ​​of Pauli operators for the approximate eigenstates, the Pauli operators comprising operators obtained by mapping multiplication operations of the Clifford algebra onto the Hilbert space; and rounding, by the classical computer, the expectation value of the Pauli operator to an element of the orthogonal group to obtain a solution to the quadratic optimization problem.

2. The method of claim 1 , wherein the orthogonal group comprises an orthogonal group of dimension n≧1 or a special orthogonal group of dimension n≧1, and the group operation is given by matrix multiplication.

3. Mapping an orthogonal matrix of the group to each quantum state in the Hilbert space using the Clifford algebra representation of the group comprises: 2 ｎ Let the basis elements of the Clifford algebra in the 2-dimensional vector space be ｎ mapping the n-dimensional Hilbert space to respective computational basis states, where n represents the dimension of the group; and mapping a left multiplication operation of the Clifford algebra and a right multiplication operation of the Clifford algebra to respective n-qubit operators in the Hilbert space.

4. The left multiplication operation of the Clifford algebra for the i-th basis element is [Equation 1] and the right multiplication operation of the Clifford algebra on the i-th basis element is given by [Equation 2] 4. The method of claim 3, wherein Z represents the Pauli Z operator and Y represents the Pauli Y operator.

5. Mapping the orthogonal matrices of the group to respective quantum states in the Hilbert space using the Clifford algebra representation of the group comprises applying a quadratic mapping to elements of the Clifford algebra, the quadratic mapping being ｎ 4. The method of claim 3, wherein inputs in an n-dimensional vector space are mapped to respective outputs in an n×n-dimensional vector space, the matrix elements of each output comprising expectation values ​​of the n-qubit operator in the Hilbert space.

6. The matrix elements of the output of the quadratic mapping comprise the expectation values ​​of the n-qubit operators given by: [Equation 3] 6. The method of claim 5, wherein i, j represent Clifford algebra basis indices, X represents the Pauli X operator, Z represents the Pauli Z operator, and Y represents the Pauli Y operator.

7. The matrix elements of the output of the quadratic mapping are: [Equation 4] , including the expectation value of the n-qubit operator given by: [Equation 5] 6. The method of claim 5, wherein i, j represent Clifford algebra basis indices, Z represents the Pauli Z operator, and <+|, <-| represent plus and minus states, respectively.

8. 8. The method of claim 1, wherein the quadratic optimization problem defines a graph of vertices and edges, and wherein rounding the calculated expected value to an element of the orthogonal group comprises implementing an edge bound rounding algorithm.

9. Implementing the edge boundary rounding algorithm comprises: decomposing a square matrix containing the expected values ​​of the edges of the graph as a product of a rectangular matrix and a transpose of the rectangular matrix; generating a normally distributed random matrix; multiplying the random matrix by the rectangular matrix to obtain a vector of matrices; and for each vertex of the graph, projecting each matrix in the vector of matrices onto the nearest element of the orthogonal group.

10. 10. The method of claim 9, wherein projecting each matrix in the vector of matrices onto a nearest element of the orthogonal group comprises computing a singular value decomposition of the vector of matrices.

11. 11. The method of claim 1, wherein the quadratic optimization problem defines a graph of vertices and edges, and wherein rounding the calculated expected value to an element of the orthogonal group comprises implementing a vertex bound rounding algorithm.

12. 12. The method of claim 11, wherein implementing the vertex bound rounding algorithm comprises: for each vertex of the graph, projecting a matrix containing the expected value of the vertex onto a nearest element of the orthogonal group.

13. 13. The method of claim 12, wherein projecting the matrix containing the vertex expectation values ​​onto the nearest element of the orthogonal group comprises computing a singular value decomposition of the matrix containing the vertex expectation values.

14. Calculating the expectation value of the Pauli operator for the approximate eigenstate preparing, by the quantum computer, a copy of the approximate eigenstate using a ground state preparation method or an approximate ground state preparation method; and measuring, by the quantum computer, the copy of the approximate eigenstate, which comprises measuring a matrix element of a 1-RDM or a 2-RDM.

15. 15. The method of claim 1, wherein determining the approximate eigenstates of the quantum Hamiltonian comprises performing a quantum phase estimation or variational algorithm, and the accuracy of the approximate eigenstates depends on the accuracy of the quantum phase estimation calculation or the variational algorithm.

16. 16. The method of claim 1, wherein the received data comprises values ​​of elements of a positive semidefinite matrix, and the solution to the quadratic optimization problem comprises the rounded expected value and the values ​​of the elements of the positive semidefinite matrix.

17. The method of any one of claims 1 to 16, wherein the quantum Hamiltonian comprises a two-body interaction of fermions.

18. 18. The method of any one of claims 1 to 17, wherein the quadratic optimization problem comprises a small non-commutative Grothendieck problem on the orthogonal group or the special orthogonal group.

19. The method of any one of claims 1 to 18, wherein the eigenstates of the quantum Hamiltonian comprise maximal eigenstates, or eigenstates that maximize energy with respect to an initial state.

20. one or more classical processors; one or more quantum computing devices in data communication with the one or more classical processors; An apparatus configured to perform the method of any one of claims 1 to 19.

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