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 Art

[0001] This disclosure relates to quantum computing.

[0002] Many optimization problems are characterized by orthogonal matrices as decision variables. Such problems typically involve joint alignment of points within the context of structural biology via Euclidean spaces by isometric mappings, e.g., cryo-electron microscopy and NMR spectroscopy, computer vision, robotics, and position estimation in sensor networks. There are two main difficulties in solving these problems. First, since the orthogonal group is non-convex, it is generally difficult to navigate the optimization landscape. Second, the objective functions of these types of problems are usually quadratic or higher-order polynomials of the optimization variables.

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

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[0007] , , , , represent binary decision variables,

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

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[0013] Quadratic combinatorial optimization problems in commutative settings can be transformed into a quantum information framework through their equivalence to the classical Ising model, and naturally into a quantum spin Hamiltonian. This correspondence is central to a wide range of quantum annealing and quantum approximate optimization algorithms, and to numerous studies exploring the potential of quantum computing for solving combinatorial optimizations.

[0014] A generalization of this idea is, for example, a cyclic group of order k>2(MAX-k-CUT).

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

[0018] In general, one inventive aspect of the subject matter described herein is the reception of data representing a quadratic optimization problem by a classical computer, wherein the decision variables of the quadratic optimization problem take values ​​from an orthogonal group or a special orthogonal group. This can be implemented by a classical computer as encoding a quadratic optimization problem as a quantum Hamiltonian, the encoding including mapping an orthogonal or special orthogonal matrix of a group to each quantum state in a Hilbert space using a Clifford algebraic representation of the group; determining approximate eigenstates of the quantum Hamiltonian by a quantum computer; computing the expectation values ​​of Pauli operators for the approximate eigenstates by a quantum computer, the Pauli operators including operators obtained by mapping the multiplication operation of the Clifford algebra to a Hilbert space; and rounding the expectation values ​​of the Pauli operators to elements of an orthogonal group by a classical computer to obtain a solution to the quadratic optimization problem.

[0019] Other embodiments of this model include corresponding computer systems, devices, 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 specific operations or actions by having software, firmware, hardware, or a combination thereof installed on the system that causes the system to perform actions during operation. One or more computer programs, when executed by a data processing device, can be configured to perform specific operations or actions by including instructions that cause the device to perform actions.

[0020] Each of the above and other embodiments may optionally include one or more of the following features, either alone or in combination. In some embodiments, the orthogonal group includes an orthogonal group of n ≥ 1 dimension, or a special orthogonal group of n ≥ 1 dimension, and the group operation is given by matrix multiplication.

[0021] In some embodiments, using the Clifford algebraic representation of the group, mapping the orthogonal matrix of the group to each quantum state in the Hilbert space is 2 nThe basis elements of a Clifford algebra in a 2-dimensional vector space are given by 2 n This involves mapping each computational basis state in a Hilbert space of dimension n, where n represents the dimension of the group, and includes mapping the left multiplication and right multiplication operations of the Clifford algebra to the respective n-qubit operators of the Hilbert space.

[0022] In some embodiments, the left multiplication operation of the Clifford algebra on the i-th basis element is:

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[0025] In some embodiments, mapping an orthogonal matrix of a group to each quantum state in a Hilbert space using a Clifford algebraic representation of the group involves applying a quadratic mapping to the elements of the Clifford algebra, where the quadratic mapping is 2 n The inputs in a dimensional vector space are mapped to their respective outputs in an n×n dimensional vector space, and the matrix elements of each output contain the expectation values ​​of the n-qubit operator in the Hilbert space.

[0026] In some embodiments, the matrix elements of the output of the quadratic mapping include the expected values ​​of the n-qubit operator given by the following equation:

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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 embodiments, the quadratic optimization problem involves defining a graph of vertices and edges, and rounding the calculated expected values ​​to elements of an orthogonal group, which includes implementing an edge boundary rounding algorithm.

[0032] In some embodiments, implementing an edge boundary rounding algorithm involves decomposing a square matrix containing the expected values ​​of the edges of the graph as the product of a rectangular matrix and its transpose; 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 onto the nearest element of the orthogonal group.

[0033] In some embodiments, projecting each matrix in a matrix-vector onto the nearest element of an orthogonal group involves computing the singular value decomposition of the matrix-vector.

[0034] In some embodiments, the quadratic optimization problem involves defining a graph of vertices and edges, and rounding the calculated expected values ​​to elements of an orthogonal group, which includes implementing a vertex boundary rounding algorithm.

[0035] In some embodiments, implementing a vertex boundary rounding algorithm involves projecting a matrix containing the expected values ​​of the vertices onto the nearest element of an orthogonal group for each vertex in the graph.

[0036] In some embodiments, projecting a matrix containing the expected values ​​of the vertices onto the nearest element of an orthogonal group involves computing a singular value decomposition of the matrix containing the expected values ​​of the vertices.

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

[0038] In some embodiments, determining the approximate eigenstates of a quantum Hamiltonian involves performing quantum phase estimation or variational algorithms, and the accuracy of the approximate eigenstates depends on the accuracy of the quantum phase estimation computation or variational algorithm.

[0039] In some embodiments, the received data includes the values ​​of the elements of a positive semi-definite matrix, and the solution to the quadratic optimization problem includes the rounded expected value and the values ​​of the elements of the positive semi-definite matrix.

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

[0041] In some embodiments, the quadratic optimization problem includes a small noncommutative Grothendieck problem on an orthogonal group or a special orthogonal group.

[0042] In some embodiments, the eigenstates of the quantum Hamiltonian include the maximum eigenstate, or the eigenstate that maximizes the energy relative to the initial state.

[0043] The subject matter described herein can be implemented in certain embodiments to achieve one or more of the following advantages:

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

[0045] The inventions described herein enable the transformation of such problems into a quantum information framework, thereby allowing them to be efficiently solved using a quantum computer. Therefore, the techniques described herein for solving quadratic optimization problems on orthogonal groups are particularly suited to specific technical implementations, namely 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 the corresponding qubit operator. Thus, a quantum computer can be used to obtain a solution to a quadratic optimization problem, since it can perform the operations necessary to solve the quantum formulation, e.g., phase estimation, variational algorithms, state preparation, and measurement of the qubit operator. In other words, the techniques described herein are motivated by technical considerations regarding the internal functionality of quantum computers.

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

[0047] Details of one or more embodiments of the subject matter of this specification are described in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from this specification, the drawings, and the claims. [Brief explanation of the drawing]

[0048] [Figure 1] This is a block diagram of an exemplary system for solving quadratic optimization problems on orthogonal groups using classical and quantum computing. [Figure 2] This is a flowchart illustrating an exemplary process for solving a quadratic optimization problem on an orthogonal group using classical and quantum computing. [Figure 3] This is a flowchart illustrating an exemplary process for edge boundary rounding of quantum states. [Figure 4] This is a flowchart illustrating an exemplary process for vertex boundary rounding of quantum states. [Figure 5] This illustrates an exemplary quantum computing system. [Figure 6] A schematic diagram of an exemplary classical processor system is shown. [Modes for carrying out the invention]

[0049] This specification describes a technique for encoding quadratic optimization problems on orthogonal groups into quantum Hamiltonians using Clifford algebraic representations of orthogonal groups. The solution to the quadratic optimization problem can then be obtained by computing the quantum states that optimize the energy of the quantum Hamiltonian and rounding the quantum states into an acceptable solution space.

[0050] Figure 1 is a conceptual block diagram of an exemplary system 100 that solves a quadratic optimization problem 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 can communicate electronically over one or more networks, or otherwise communicate and exchange information, for example, over one or more wired or wireless connections.

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

[0052] An exemplary system 100 is configured to perform operations to solve quadratic optimization problems on an orthogonal group using classical and quantum computation. An n-dimensional orthogonal group, denoted as O(n), is the mathematical group of n × n orthogonal matrices, where an orthogonal matrix is ​​a real matrix whose inverse is equal to its transpose. Orthogonal groups have a group operation given by matrix multiplication. An orthogonal group containing matrices with determinant 1 is called a special orthogonal group and is a normal subgroup denoted as SO(n). A quadratic optimization problem on an orthogonal group G=O(n) or G=SO(n) can be defined as follows: (V,E) where 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 the quadratic optimization problem 108. For example, the classical processor 102 receives data that specifies the objective function given by 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 a quadratic optimization problem as a quantum Hamiltonian 110. Specifically, the classical processor 102 is configured to apply a quadratic mapping 106 to an objective function defined by the input data 108. The quadratic mapping 106 maps an orthogonal matrix of the orthogonal group to each quantum state of the Hilbert space. The quadratic mapping 106 is based on the Clifford algebra representation of the orthogonal group and will be described in more detail below with reference to FIG. 2. Under the quadratic mapping 106, the objective function is transformed from a classical form, for example, a form that can be accepted by a classical computing device, to a quantum form, for example, a form that can be accepted by 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(Ф)∈G is equivalent to the energy of the corresponding Hamiltonian with respect to the state |Ф〉∈Hd. Therefore, the task of maximizing the objective function given by 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 receives the data 112 and is configured to determine an approximate eigenstate of the quantum Hamiltonian, for example, an eigenstate that increases the energy more than the maximum 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. Then, the quantum processor 104 prepares the (approximate) eigenstate and the qubit operator P ij ,Γ ijThe quantum processor 104 can be configured to repeatedly prepare (approximate) eigenstates of a Hamiltonian using a ground state preparation method or an approximate group state preparation method (e.g., prepare multiple physical qubits with the (approximate) eigenstates of the Hamiltonian), apply a quantum circuit to evolve the (approximate) eigenstates according to a qubit operator (e.g., apply a quantum circuit to multiple physical qubits using a control system), and measure the evolved quantum state (e.g., measure multiple physical qubits using a measurement system).

[0059] The quantum processor 104 is configured to transmit data representing the result of a measurement operation 116 to the classical processor 102. The classical processor 102 is configured to receive the transmitted data 116 and, for example, calculate the expectation value of the qubit operator for the (approximate) eigenstate by averaging the measurement results 116 received from the quantum processor 104. The calculated expectation value defines the initial solution to the 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, since the space of quantum states is larger than the orthogonal group, not all quantum states are mapped to orthogonal matrices via the quadratic mapping 106. Thus, the Hamiltonian obtained by applying the quadratic mapping 106 corresponds to a relaxation of the original problem, and for example, its maximum eigenvalue is generally larger than the value of the objective function given by Equation 1. Therefore, the classical processor 102 is configured to apply a rounding algorithm 118 to the expectation value to recover an orthogonal matrix from the relaxed initial solution, for example, to inversely project the initial solution onto the acceptable solution space. For example, the classical processor 102 can be configured to perform one or both of the rounding algorithms described below with reference to Figures 3-4. The rounded expectation value is the matrix R that maximizes the objective function given by Equation 1 above. v The complexity is ∈ O(n) or SO(n).

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

[0062] Figure 2 is a flowchart 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 in one or more locations. For example, a appropriately programmed system 100 shown in Figure 1 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 the values ​​of an orthogonal group, for example, an orthogonal group of n ≥ 1 dimension, or a special orthogonal group of n ≥ 1 dimension. The group operation is matrix multiplication, which takes the quadratic form of the optimization problem. The received data is in the form of a matrix, for example, 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 noncommutative Grothendieck problem, as described above.

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

[0068] The Clifford algebra Cl(n) is 2 n It is a vector space of dimension, which can be specified as an n-qubit Hilbert space. To encode a quadratic optimization problem as a quantum Hamiltonian, the system is a basis element of the Clifford algebra e I The computational ground states of each Hilbert space

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

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[0073] Since the parity of a Clifford algebra can be converted to the parity of a qubit computation basis, a parity automorphism α is equivalent to an n-qubit parity operator.

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[0075] The double cover of a group SO(n) is an even parity subalgebra Cl 0 It can be obtained from (n). Basis vector e I Since only half of it characterizes |I|mod2=0, this subspace has only half the dimension of Cl(n). Therefore, the projection operator can be converted from Cl(n) to Cl 0 It is useful to specify (n), or equivalently, from H2n to H2n-1. This mapping is 2 n-1 ×2 nThis can be given by a matrix.

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[0077] The above mapping connects objects in Clifford algebras to objects in Hilbert spaces, such as 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 elements of Clifford algebras. The quadratic mapping is 2 n An input in a dimensional vector space (e.g., an element from a Clifford algebra) is mapped to its respective output in an n × n dimensional vector space (e.g., an element from a Hilbert space), where the matrix elements of each output contain the expected values ​​of the combinations of the n qubit operators described above, referencing Equation 3.

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[0080] Refer to Equation 3 to see the linear map λ described above. i , e i、 e j The ρ of left multiplication and right multiplication by j, and conjugate identities

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[0083] This equation can be transformed into the quantum representation developed above. First, an n-qubit Pauli operator 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 algebra formulation of elements of an orthogonal group leads to the encoding of a quadratic optimization problem into a quantum Hamiltonian. The objective function f(R1,…R) (as defined in Equation 1) m ) can be explicitly extended with respect to 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 an mn-qubit Hamiltonian.

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[0102] Returning to Figure 2, the system uses a quantum computer to determine the (approximate) eigenstates of the quantum Hamiltonian (step 206). For example, the system uses 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 compute the expectation values ​​of the Pauli operators for the (approximate) maximum eigenstate, the Pauli operators including operators obtained by mapping the multiplication operations of the Clifford algebra to Hilbert space, as given in equation 8 above (step 208). For example, the system prepares a copy of the (approximate) maximum eigenstate of the quantum Hamiltonian, for example, 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 The operator P on the (approximate) maximum eigenstate copy is used to calculate |ψ〉. ij and Γ ij The system can measure the following: 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 of the Pauli operators to the elements of the orthogonal group in order 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, in the graph formulation of the quadratic optimization problem, for each pair of vertices (u,v), the operator

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[0108] The system uses data representing rounded expectations (which are optimized decision variables, as will be explained in more detail below with reference to Figures 3 and 4) and coefficients C that form a positive semi-definite matrix. uv This can be provided as output.

[0109] Figure 3 is a flowchart of an exemplary process 300 for edge boundary rounding of a quantum state. For convenience, the process 300 is described as being performed by a system of one or more classical and quantum computing devices located in one or more locations. For example, a appropriately programmed system 100 in Figure 1 can perform the exemplary process 300.

[0110] Referring to step 202 of the exemplary process 200, the quadratic optimization problem described above defines a graph of vertices and edges (V,E), where each edge (u,v)∈E, and a 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 a square matrix as the product of i) a rectangular matrix and ii) the transpose of the rectangular matrix (step 306). Since the square matrix M≧0, the system can, for example, use Cholesky decomposition to obtain several

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[0129] The system generates a random matrix (step 308). For example, the system samples values ​​from a normal distribution N(0,1 / n) and uses 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 matrix LZ (step 310).

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

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[0134] An exemplary process 300 can be explicitly shown in pseudocode as follows:

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[0136] Figure 4 is a flowchart of an exemplary process 400 for vertex boundary rounding of a quantum state. For convenience, process 400 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, a well-programmed system 100 in Figure 1 can perform exemplary process 400.

[0137] Referring to step 202 of the exemplary process 200, the quadratic optimization problem described above defines a graph of vertices and edges (V,E), where each edge (u,v)∈E, and a 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 queudites associated with all vertices except one.

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[0145] Therefore, for each vertex of the graph, the system is Q(σ v Project ) onto the nearest element of the orthogonal group (step 404). That is, the system is the nearest (special) orthogonal matrix

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[0149] Performing this rounding procedure at each vertex returns an acceptable solution. Then, for each v ∈ V, R v The set is provided as a solution to a quadratic optimization problem. In principle, this is because each of them is

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[0153] The exemplary process 400 can be explicitly expressed in pseudocode as follows.

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[0155] FIG. 5 shows 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] The 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, such as qubit 506, used to perform algorithm operations or quantum calculations. The qubits shown in FIG. 5 are arranged in a rectangular array, which is a schematic diagram and is not intended to be limiting. The qubit assembly 552 also includes adjustable coupling elements, such as coupler 508, that enable interaction between the coupled qubits. In the schematic diagram of FIG. 5, each qubit is adjustably coupled to each of its four adjacent qubits by respective coupling elements. However, this is an exemplary arrangement of qubits and couplers, and other arrangements are possible, such as non-rectangular arrangements, arrangements that enable 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 that has levels representing logical values 0 and 1. The specific physical realization of the plurality of qubits and how the plurality of qubits interact with each other depend on various factors, such as the type of quantum computing device 502 included in the exemplary computer 500 or the type of quantum calculation being performed by the quantum computing device. For example, in an atomic quantum computer, the qubits can be realized via an atomic quantum system, a molecular quantum system, or a solid-state quantum system, such as a hyperfine atomic state. As another example, in a superconducting quantum computer, the qubits can be realized via a superconducting qubit or a semiconductor qubit, such as a superconducting transmon state. As another example, in an NMR quantum computer, the qubits can be realized via a nuclear spin state.

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

[0159] For example, in some embodiments, the qubits in the qubit assembly 552 may be frequency tunable. In these examples, each qubit may 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 idling frequency, a qubit interaction frequency, and a qubit readout frequency. Different frequencies correspond to different operations that the qubits can perform. For example, setting the operating frequency to the corresponding idling frequency may put the qubit into a state where it does not strongly interact with other qubits and can be used to perform a single qubit gate. In another example, if the qubits interact via a coupler whose coupling is fixed, the qubits may be configured to interact with each other by setting their respective operating frequencies at a gate-dependent frequency that detunes from their common interaction frequency. In other cases, for example, if the qubits interact via a tunable coupler, the qubits may be configured to interact with each other by setting the parameters of their respective couplers to enable interaction between the qubits, and then setting the respective operating frequencies of the qubits at a gate-dependent frequency that detunes from their common interaction frequency. Such interactions can be performed to create multi-qubit gates.

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

[0161] Quantum computation can be completed by measuring the state of a qubit using the respective control signals 510, for example, using a quantum observable quantity such as X or Z. The measurement is communicated so that a readout signal 512 representing the measurement result is returned 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 scheme of the quantum computing device and / or the qubit. For convenience, the control signals 510 and readout signals 512 shown in Figure 5 are shown to deal with only selected elements of the qubit assembly (i.e., the top row and bottom row), but during operation, the control signals 510 and readout signals 512 can deal with each element in the qubit assembly 552.

[0162] The control and measurement system 504 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 552 and other classical subroutines or calculations, as described above. The 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. The control and measurement system 504 can be programmed, for example, to send a sequence of control signals 510 to the qubit assembly to perform a selected set of quantum gate operations, and to receive a sequence of read signals 512 from the qubit assembly, for example, as part of performing a measurement operation.

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

[0164] Memory 516 stores information within the control and measurement system 504. In some embodiments, memory 516 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, memory 516 may include a storage device that can provide 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] The input / output device 518 provides input / output operations for the control and measurement system 504. The input / output device 518 includes a D / A converter, an A / D converter, and an RF / microwave / optical signal generator, transmitter, and receiver, which can transmit control signals 510 to a qubit assembly and receive read signals 512 from the qubit assembly, in a manner appropriate to the physical scheme of a quantum computer. In some embodiments, the input / output device 518 may also include one or more network interface devices, e.g., an Ethernet card, a serial communication device, e.g., an RS-232 port, and / or a wireless interface device, e.g., an 802.11 card. In other embodiments, the input / output device 518 may include a driver device configured to receive input data and transmit output data to other external devices such as a keyboard, printer, and display device.

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

[0167] Figure 6 shows a schematic diagram of an exemplary classical processor system 600. System 600 can be used for the classical arithmetic described herein in several 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 herein, their connections and relationships, and their functions are for illustrative purposes only and do not limit the embodiments of the invention described herein and / or claimed herein.

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

[0169] Memory 620 stores information within the 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 have high-capacity 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 device 640 provides input / output operations for the system 600. In one embodiment, the input / output device 640 includes a keyboard and / or a pointing device. In other embodiments, the input / output device 640 includes a display unit for displaying a graphical user interface.

[0172] The subjects and embodiments of operation described herein can be implemented in digital electronic circuits, analog electronic circuits, suitable quantum circuits, or more generally, quantum computing systems, including structures disclosed herein and their structural equivalents, in tangible 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 technology system, or a quantum simulator.

[0173] Embodiments of the subject matter described herein can be implemented as one or more computer programs, i.e., as one or more modules of computer program instructions encoded in a tangible non-temporary storage medium for execution by a data processing device or for controlling the operation of a data processing device. The computer storage medium can be a machine-readable storage device, a machine-readable storage board, a random or serial access memory device, one or more qubits, or one or more combinations thereof. Alternatively or additionally, the program instructions can be encoded in artificially generated propagating signals capable of encoding digital and / or quantum information, for example, machine-generated electrical, optical, or electromagnetic signals generated to encode digital and / or quantum information for transmission to a receiving device suitable for execution by a data processing device.

[0174] The terms quantum information and quantum data refer to information or data that is carried by, or held or stored in, a quantum system, with the smallest non-trivial system being a qubit, i.e., a system that defines the unit of quantum information. The term “qubit” will be understood to encompass all quantum systems that can be suitably approximated as two-level systems in the corresponding context. Such quantum systems may include, for example, multi-level systems having two or more levels. As an example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational ground state is identified using the ground state and a first excited state, but it will be understood that other setups are possible in which the computational state is identified using excited states of higher levels.

[0175] The term “data processing device” refers to digital and / or quantum data processing hardware and encompasses all types of devices, machines, and equipment for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. A device may also be, or further include, a dedicated logic circuit, e.g., an FPGA (Field Programmable Gate Array), an ASIC (Application-Specific Integrated Circuit), or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a particular quantum system. Specifically, a quantum simulator is a dedicated quantum computer that does not have the capability to perform universal quantum computing. In addition to hardware, a device may also optionally include code that creates an execution environment for digital and / or quantum computer programs, e.g., processor firmware, protocol stacks, database management systems, operating systems, or code that constitutes one or more combinations thereof.

[0176] A digital computer program, also 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 compiled languages ​​or interpreted languages, or declarative languages ​​or procedural languages, and can be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, also 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 compiled languages ​​or interpreted languages, or declarative languages ​​or procedural languages, and can be translated into a suitable quantum programming language, or can be written in a quantum programming language, such as QCL or Quipper.

[0177] A computer program may correspond to a file in a file system, but it does not necessarily have to. The program may be held in another program or data, for example, a part of a file that holds 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, for example, files that store one or more modules, subprograms, or parts of code. A computer program can be deployed to be executed on one computer, or located at one site, or distributed over multiple sites and interconnected by digital and / or quantum data communication networks. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system, such as a qubit. In general, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0178] The processes and logical flows described herein can be performed by one or more programmable computers operating using one or more processors, appropriately operating on input data to perform functions by generating output, by executing one or more computer programs. The processes and logical flows can also be performed as dedicated logic circuits, such as FPGAs or ASICs, or as quantum simulators, or by a combination of dedicated logic circuits or quantum simulators and one or more programmed digital and / or quantum computers, and the apparatus can be implemented as such.

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

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

[0181] The elements of a computer include a central processing unit for issuing 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 into dedicated logic circuits or quantum simulators. Generally, a computer will also include one or more mass storage devices for storing data, such as magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information, or be operablely coupled to them for receiving data from them, transferring data to them, or both. However, a computer is not required to 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 data operations in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent and operate information in multiple states simultaneously. Examples of superconducting quantum circuit elements include, among others, 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).

[0183] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively execute instructions in a computer program by performing basic mathematical, logical, and / or input / output operations on data, and 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 circuit configurations, fast single-flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices, and ERSFQ devices, which are energy-efficient versions of RSFQ that do not use bias resistors.

[0184] In certain cases, some or all of quantum circuit elements and / or classical circuit elements may be implemented using, for example, superconducting quantum and / or classical circuit elements. The fabrication of superconducting circuit elements may 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., vapor deposition or sputtering), or epitaxial techniques, among several other deposition processes. The processes for fabricating the circuit elements described herein may involve the removal of one or more materials from the device during fabrication. Depending on the materials to be removed, the removal process may include, for example, wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithography techniques (e.g., photolithography or electron beam lithography).

[0185] During the operation of quantum computing systems using superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled in a cryostat to a temperature at which the superconducting material can exhibit superconducting properties. A superconducting (or 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). Therefore, superconducting structures such as superconducting traces and superconducting ground surfaces 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 electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form.

[0187] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memory, media, and memory devices, including, for example, semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. Quantum memory will be understood as a device capable of storing quantum data for long periods with high fidelity and efficiency, such as an optical-material interface where light is used for transmission, and a material for storing and preserving the quantum features of quantum data, such as superposition or quantum coherence.

[0188] The control of the various systems or parts thereof described herein can be implemented in a computer program product that includes instructions stored in one or more non-temporary machine-readable storage media and executable on one or more processing devices. Each of the systems or parts thereof described herein can be implemented as an apparatus, method, or system that may include one or more processing devices and a memory for storing executable instructions for performing the operations described herein.

[0189] While this specification includes details of many specific embodiments, these should not be construed as limiting the scope of claims, but rather as descriptions of features that may be specific to a particular embodiment. Certain features described herein in the context of a separate embodiment may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented individually in multiple embodiments or in any preferred partial combination. Furthermore, features may be described above as acting in a particular combination, and may even be initially claimed as such, but one or more features from a claimed combination may, in some cases, be removed from that combination, and the claimed combination may cover a partial combination or a variation of a partial combination.

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

[0191] Specific embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions described in the claims may be performed in a different order, and the desired results may still be achieved. As an example, the process shown in the accompanying drawings does not necessarily require the specific order or sequence shown to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. The process involves receiving data representing a quadratic optimization problem using a classical computer, wherein the decision variables of the quadratic optimization problem take values ​​from an orthogonal group or a special orthogonal group. Encoding the quadratic optimization problem as a quantum Hamiltonian using a classical computer, wherein encoding includes mapping the orthogonal or special orthogonal matrix of the group to the respective quantum states of the Hilbert space using the Clifford algebraic representation of the group. The approximate eigenstates of the quantum Hamiltonian are determined by a quantum computer, The quantum computer is used to calculate the expectation value of the Pauli operator for the approximate eigenstate, wherein the Pauli operator includes an operator obtained by mapping the multiplication operation of the Clifford algebra to the Hilbert space. A method implemented on a computer, comprising rounding the expected value of the Pauli operator to the elements of the orthogonal group in order to obtain a solution to the quadratic optimization problem using the classical computer.

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

3. Using the Clifford algebraic representation of the group, mapping the orthogonal matrix of the group to the respective quantum states of the Hilbert space is: 2 ｎ The basis elements of the Clifford algebra in the 3D vector space are, ｎ The mapping involves mapping each computational ground state in a Hilbert space of a certain dimension, where n represents the dimension of the group. The method according to claim 1, comprising mapping the left multiplication operation of the Clifford algebra and the right multiplication operation of the Clifford algebra to the respective n-qubit operators of the Hilbert space.

4. The left multiplication operation of the Clifford algebra with respect to the i-th basis element is: [Math 1] The n-qubit operator given by is mapped to the right multiplication operation of the Clifford algebra with respect to the i-th basis element, [Math 2] The method according to claim 3, which is mapped to an n-qubit operator given by, where Z represents the Pauli Z operator and Y represents the Pauli Y operator.

5. Mapping the orthogonal matrices of the group to the respective quantum states of the Hilbert space using the Clifford algebraic representation of the group involves applying a quadratic mapping to the elements of the Clifford algebra, wherein the quadratic mapping is 2 ｎ The method according to claim 3, wherein inputs in a dimensional vector space are mapped to outputs in an n × n dimensional vector space, and the matrix elements of each output include the expectation value of the n qubit operator in the Hilbert space.

6. The matrix elements of the output of the quadratic mapping include the expected value of the n-qubit operator given by the following equation: [Math 3] The method according to claim 5, wherein i and j represent Clifford algebraic basis exponents, 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, [Math 4] The formula includes the expected value of the n-qubit operator given by, [Math 5] The method according to claim 5, wherein i and j represent Clifford algebraic basis exponents, Z represents the Pauli Z operator, and <+| and <-| represent the positive and negative states, respectively.

8. The method according to claim 1, wherein the quadratic optimization problem includes defining a graph of vertices and edges, and rounding the calculated expected value to the elements of the orthogonal group, which includes implementing an edge boundary rounding algorithm.

9. Implementing the aforementioned edge boundary rounding algorithm is Decomposing a square matrix containing the expected values ​​of the edges of the graph by multiplying a rectangular matrix by the transpose of the rectangular matrix, Generating a normally distributed random matrix, To obtain a matrix vector, the random matrix and the rectangular matrix are multiplied, The method according to claim 8, further comprising projecting each vertex of the graph onto the nearest element of the orthogonal group.

10. The method according to claim 9, wherein projecting each matrix in the vector of the matrix onto the nearest element of the orthogonal group includes calculating the singular value decomposition of the vector of the matrix.

11. The method according to claim 1, wherein the quadratic optimization problem includes defining a graph of vertices and edges, and rounding the calculated expected value to the elements of the orthogonal group, which includes implementing a vertex boundary rounding algorithm.

12. The method according to claim 11, wherein implementing the vertex boundary rounding algorithm includes projecting a matrix containing the expected value of each vertex of the graph onto the nearest element of the orthogonal group.

13. The method according to claim 12, wherein projecting a matrix containing the expected values ​​of the vertices onto the nearest element of the orthogonal group includes calculating a singular value decomposition of the matrix containing the expected values ​​of the vertices.

14. Calculating the expectation value of the Pauli operator for the aforementioned approximate eigenstate is: The quantum computer prepares copies of the approximate eigenstates using a ground state preparation method or an approximate ground state preparation method, The method according to claim 1, comprising measuring a copy of the approximate eigenstate, which includes measuring the matrix elements of a 1-RDM or a 2-RDM using the quantum computer.

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

16. The method according to claim 1, wherein the received data includes the values ​​of the elements of a positive semi-definite matrix, and the solution to the quadratic optimization problem includes the rounded expected value and the values ​​of the elements of the positive semi-definite matrix.

17. The method according to claim 1, wherein the quantum Hamiltonian includes a two-body interaction of fermions.

18. The method according to claim 1, wherein the quadratic optimization problem includes a small noncommutative Grothendieck problem on the orthogonal group or the special orthogonal group.

19. The method according to claim 1, wherein the eigenstates of the quantum Hamiltonian include the maximum eigenstate or an eigenstate that maximizes the energy relative to the initial state.

20. One or more classical processors, The system includes one or more classical processors and one or more quantum computing devices that communicate data with them. An apparatus configured to perform the method described in any one of claims 1 to 19.

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