Solving quadratic optimization problems on orthogonal groups using quantum computers

Abstract

Methods, systems, and devices for solving quadratic optimization problems on orthogonal groups using quantum computing. In one aspect, a method includes receiving data representative of a quadratic optimization problem, where decision variables of the quadratic optimization problem take values in an orthogonal group or a special orthogonal group; encoding the quadratic optimization problem as a quantum Hamiltonian, the encoding including using a Crifford algebraic representation of the population to map orthogonal matrices or special orthogonal matrices in the population to corresponding quantum states in Hilbert space; determining an approximate eigenstate of the quantum Hamiltonian; calculating an expected value of a Fouli operator with respect to the approximate eigenstate, where the Fouli operator comprises an operator obtained by mapping a multiplication operation of a Crifford algebra into a Hilbert space; and rounding the expected value of the Fouli operator into the 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 their decision variables. Such problems typically involve the joint alignment of points in Euclidean space by isometric transformations, for example, in the context of structural biology, computer vision, robotics, and sensor network localization via cryo-EM microscopy and NMR spectroscopy. The main difficulties in solving these problems are twofold: First, the orthogonal group is non-convex, which generally makes it challenging to navigate the optimization landscape. Second, the objectives of these types of problems are typically quadratic or higher-order polynomials in the optimization variables.

[0003] In the commutative setting, these problems are known as quadratic combinatorial optimization problems and take the form where (V, E) represents an undirected graph with m = |V| vertices, x1,..., x m represents binary-valued decision variables, and is a positive semidefinite matrix. In general, quadratic combinatorial optimization problems are NP-hard.

[0004] In the non-commutative setting, the decision variables x1,..., x m can be interpreted as taking values in the group of 1×1 orthogonal matrices, where the group multiplication provides the quadratic form. This is a generalization from promoting the decision variables to take values in the group of n×n orthogonal matrices, where I n represents the n×n identity matrix. This leads to the small non-commutative Grothendieck (LNCG) problem on the orthogonal group, which is defined by the objective where R,..., R m represent the decision variables, and each forms the (u, v)-th n×n block of the positive semidefinite matrix . The LNCG problem can also be considered on the special orthogonal group where the objective is as given above. Although the determinant constraint may potentially further complicate the optimization process, SO(n) is encountered in many applications due to the importance of orientation preservation.

[0005] Quadratic combinatorial optimization problems in the context of swap settings can be transformed into the quantum information framework due to their equivalence with the classical Ising model, which naturally transforms into a quantum spin Hamiltonian. This correspondence is at the heart of a great deal of work studying the potential of quantum computing for solving combinatorial optimization under the umbrella of quantum annealing and quantum approximate optimization algorithms.

[0006] Generalizations of this idea have also been considered, such as to cyclic groups of order k > 2 (MAX-k-CUT) or to non-commuting spin operators (quantum MAXCUT). In both cases, the objective is mapped to a Hamiltonian whose energy is to be maximized, and the decision variables are mapped to quantum states. For example, in MAX-k-CUT, k-dimensional qubits are assigned to each vertex, and the group operation is represented by the action of a generalized Pauli X operator on those qubits. However, for the LNCG problem, no such formulation exists where the structure of the orthogonal group is manifested through the inherent properties of quantum mechanics. SUMMARY OF THE INVENTION

[0007] This disclosure describes techniques for using quantum computing to solve quadratic optimization problems over orthogonal groups.

[0008] Generally, one innovative aspect of the subject matter described in this specification can be implemented in a method that includes: receiving, by a classical computer, data representing a quadratic optimization problem, where the decision variables of the quadratic optimization problem take values in an orthogonal group or a special orthogonal group;

[0009] encoding, by the classical computer, the quadratic optimization problem into a quantum Hamiltonian, the encoding including mapping an orthogonal matrix or a special orthogonal matrix in the group to a corresponding quantum state in a Hilbert space using a Clifford algebra representation of the group; determining, by a quantum computer, an approximate eigenstate of the quantum Hamiltonian; computing, by the quantum computer, the expectation value of a Pauli operator with respect to the approximate eigenstate, where the Pauli operator includes an operator obtained by mapping the multiplication operation of the Clifford algebra to 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.

[0010] Other implementations of these aspects 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. Systems of one or more classical and quantum computers can be configured to perform particular operations or actions by means of software, firmware, hardware, or combinations thereof installed on the system, which in operation cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by means of instructions that, when executed by a data processing device, cause the device to perform the actions.

[0011] The foregoing and other implementations can each optionally include, individually or in combination, one or more of the following features. In some implementations, the orthogonal group includes an orthogonal group of dimension n≥1 or a special orthogonal group of dimension n≥1, where the group operation is given by matrix multiplication.

[0012] In some implementations, using the Clifford algebra representation of the group to map orthogonal matrices in the group to corresponding quantum states in a Hilbert space includes: mapping the basis elements of the Clifford algebra in a 2 n -dimensional vector space to the corresponding computational basis states in a Hilbert space of dimension 2 n , where n represents the dimension of the group; and mapping the left multiplication operation of the Clifford algebra and the right multiplication operation of the Clifford algebra to the corresponding n-qubit operators on the Hilbert space.

[0013] In some implementations, the left multiplication operation of the Clifford algebra for the i-th basis element is mapped to the n-qubit operator given by , and the right multiplication operation of the Clifford algebra for the i-th basis element is mapped to the n-qubit operator given by , where Z represents the Pauli Z operator and Y represents the Pauli Y operator.

[0014] In some implementations, using the Clifford algebra representation of the group to map orthogonal matrices in the group to corresponding quantum states in a Hilbert space includes applying a quadratic map to the elements of the Clifford algebra, where the quadratic map maps an input in a vector space of dimension 2 n to a corresponding output in a vector space of dimension n×n, where each matrix element of the output includes the expected value of an n-qubit operator on the Hilbert space.

[0015] In some implementations, the matrix elements of the output of the quadratic map include the expected values of the n-qubit operators given by the following formula

[0016]

[0017] where \(i, j\) represent Clifford algebra basis indices, \(X\) represents the Pauli \(X\) operator, \(Z\) represents the Pauli \(Z\) operator, and \(Y\) represents the Pauli \(Y\) operator.

[0018] In some implementations, the matrix elements of the output of the quadratic mapping include the expected value of the \(n\)-qubit operator given by where \(i, j\) represent Clifford algebra basis indices, \(Z\) represents the Pauli \(Z\) operator, and \(\langle +|,\langle -|\) represent the plus state and the minus state, respectively. where \(i, j\) represent Clifford algebra basis indices, \(Z\) represents the Pauli \(Z\) operator, and \(\langle +|,\langle -|\) represent the plus state and the minus state, respectively.

[0019] In some implementations, the quadratic optimization problem defines a graph of vertices and edges, and rounding the computed expected value to an element of the orthogonal group includes implementing an edge-marginal rounding algorithm.

[0020] In some implementations, implementing the edge-marginal rounding algorithm includes: decomposing a square matrix including the expected values of the edges in the graph into a product of a rectangular matrix and the transpose of a rectangular matrix; generating a matrix of random normal distribution; multiplying the random matrix by the rectangular matrix to obtain a matrix vector; and for each vertex in the graph, projecting the corresponding matrix in the matrix vector to the nearest element in the orthogonal group.

[0021] In some implementations, projecting the corresponding matrix in the matrix vector to the nearest element in the orthogonal group includes computing the singular value decomposition of the matrix vector.

[0022] In some implementations, the quadratic optimization problem defines a graph of vertices and edges, and rounding the computed expected value to an element of the orthogonal group includes implementing a vertex-marginal rounding algorithm.

[0023] In some implementations, implementing the vertex-marginal rounding algorithm includes: for each vertex in the graph, projecting the matrix including the expected value of that vertex to the nearest element in the orthogonal group.

[0024] In some implementations, projecting the matrix including the expected value of the vertex to the nearest element in the orthogonal group includes computing the singular value decomposition of the matrix including the expected value of the vertex.

[0025] In some implementations, computing the expected value of a Pauli operator with respect to an approximate eigenstate includes: preparing a copy of the approximate eigenstate by a quantum computer using ground state preparation or approximate ground state preparation techniques; and measuring the copy of the approximate eigenstate by the quantum computer, including measuring the matrix elements of the 1-RDM or 2-RDM.

[0026] In some implementations, determining an approximate eigenstate of a quantum Hamiltonian includes performing quantum phase estimation or a variational algorithm, where the accuracy of the approximate eigenstate depends on the accuracy of the quantum phase estimation calculation or the variational algorithm.

[0027] In some implementations, the received data includes values of elements of a positive semi-definite matrix, and where the solution to the quadratic optimization problem includes the rounded expected value and the values of the elements of the positive semi-definite matrix.

[0028] In some implementations, the quantum Hamiltonian includes two-body fermionic interactions.

[0029] In some implementations, the quadratic optimization problem includes a small non-commutative Grothendieck problem on the orthogonal group or the special orthogonal group.

[0030] In some implementations, the eigenstates of the quantum Hamiltonian include the maximum eigenstate or the eigenstate that maximizes the energy relative to an initial state.

[0031] The subject matter described in this specification can be implemented in particular embodiments so as to achieve one or more of the following advantages.

[0032] Quadratic optimization on the orthogonal group encompasses a wide class of optimization problems such as group synchronization (which has applications in structural biology, robotics, and wireless networking), point-set registration, simultaneous localization and mapping, and the generalized orthogonal Procrustes problem (which has applications in fields such as shape and image recognition). These problems are generally computationally inefficient or intractable when solved with high accuracy using classical computers in all settings.

[0033] The presently described invention enables such problems to be transformed into a quantum information framework such that such problems can be efficiently solved using a quantum computer. Thus, the presently described techniques for solving quadratic optimization problems on the orthogonal group are particularly well-suited for a particular computational implementation - quantum computing. In the present disclosure, the quadratic optimization problem on the orthogonal group was initially formulated for classical computing devices. This formulation was 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 a quantum computer is capable of performing the operations required to solve the quantum formulation, e.g., phase estimation, variational algorithms, state preparation, and measurement of qubit operators. That is, the presently described techniques are motivated by the technical considerations of the internal functionality of a quantum computer.

[0034] Additionally, it is expected that quantum entanglement resources provide a higher quality solution than classical approximation algorithms.

[0035] Details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the specification, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is a block diagram of an example system for solving quadratic optimization problems on the orthogonal group using classical and quantum computing.

[0037] Figure 2 is a flowchart of an example process for solving quadratic optimization problems on the orthogonal group using classical and quantum computing.

[0038] Figure 3 is a flowchart of an example process for edge-marginal rounding of a quantum state.

[0039] Figure 4 is a flowchart of an example process for vertex-marginal rounding of a quantum state.

[0040] Figure 5 depicts an example quantum computing system.

[0041] Figure 6 shows a schematic diagram of an exemplary classical processor system. DETAILED DESCRIPTION

[0042] This specification describes techniques for encoding quadratic optimization problems on the orthogonal group onto a quantum Hamiltonian using a Clifford algebra representation of the orthogonal group. The 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 the feasible solution space.

[0043] Figure 1 shows a conceptual block diagram of an example system 100 for solving quadratic optimization problems on the orthogonal group using classical and quantum computing. The example system 100 includes a classical processor 102 and a quantum processor 104. The classical processor 102 and the quantum processor 104 can communicate electronically via one or more networks, or can exchange communications in another manner - such as via one or more wired or wireless connections.

[0044] The classical processor 102 is configured to perform classical computing. The quantum processor 104 is configured to perform quantum computing. 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 implementations, 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 computing operations. Generally speaking, the classical computing components of the classical processor may 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 may be implemented as a quantum computing device having physical hardware as described with respect to Figure 5 .

[0045] The example system 100 is configured to perform operations for solving a quadratic optimization problem on the orthogonal group using classical and quantum computing. The orthogonal group of dimension n, denoted as O(n), is a mathematical group of n×n orthogonal matrices, where an orthogonal matrix is a real matrix whose inverse is equal to its transpose. The orthogonal group has a group operation given by matrix multiplication. The orthogonal group containing matrices with determinant 1 is a normal subgroup called the special orthogonal group and denoted as S0(n). The quadratic optimization problem on the orthogonal group G = 0(n) or G = SO(n) can be defined as follows: Let (V, E) be a graph, where each edge (u, v) ∈ E is associated with a matrix with real-valued coefficients, and the vertices are labeled by V = [m]. The goal of the quadratic optimization problem is to assign an element to the matrix R v ∈ 0(n) for each v ∈ V such that the objective function

[0046]

[0047] is maximized, where tr denotes the trace operation. When G = 0(1) = {±1}, the difficulty of this problem is already obvious, which includes the NP-hard MAXCUT problem as a special case. For larger n, the above formulation can thus be considered a generalization of MAXCUT to continuous non-commuting variables.

[0048] The classical processor 102 is configured to receive input data 108 specifying the quadratic optimization problem. For example, the classical processor 102 may receive data specifying the objective function given by Equation 1 above, e.g., data specifying the size and coefficients of the orthogonal group.

[0049] The classical processor 102 is configured to encode a quadratic optimization problem into 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 applies an orthogonal matrix in the orthogonal group to the corresponding quantum state in the Hilbert space. The quadratic mapping 106 is based on the Clifford algebra representation of the orthogonal group and is described in more detail below with reference to Figure 2 Under the quadratic mapping 106, the objective function is transformed from a classical format (e.g., a format acceptable for a classical computing device) to a quantum format (e.g., a format acceptable for 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 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.

[0050] 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., the maximum eigenstate or an eigenstate with increased energy compared to the initial state. For example, the quantum processor 104 can be configured to implement a phase estimation algorithm or 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 can then prepare (approximate) the eigenstate and measure the qubit operator P Figure 2 defined with respect to the (approximate) eigenstate 114 ij , Γ ij . For example, the quantum processor 104 can be configured to repeatedly: prepare an (approximate) eigenstate of the Hamiltonian using a ground state preparation or approximate group state preparation technique (e.g., prepare multiple physical qubits in the (approximate) eigenstate of the Hamiltonian), apply a quantum circuit that evolves the (approximate) eigenstate according to the qubit operator (e.g., apply the 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).

[0051] The quantum processor 104 is configured to send data 116 representing the result of the measurement operation to the classical processor 102. The classical processor 102 is configured to receive the sent data 116 and calculate the expected value of the qubit operator with respect to the (approximate) eigenstate, e.g., by averaging the measurement results 116 received from the quantum processor 104. The calculated expected value defines an initial solution to the quadratic optimization problem.

[0052] The Hamiltonian obtained by applying the quadratic mapping 106 is equivalent to a fermionic Hamiltonian with two-body interactions. Thus, the space of quantum states is larger than the orthogonal group, and thus not every quantum state is mapped to an orthogonal matrix via the quadratic mapping 106. Thus, the Hamiltonian obtained by applying the quadratic mapping 106 corresponds to a relaxation of the original problem, e.g., its largest eigenvalue is generally greater than the value of the objective function given by Equation 1. Thus, the classical processor 102 is configured to apply a rounding algorithm 118 to the expected value to project the initial solution back into the feasible solution space, e.g., to recover an orthogonal matrix from the relaxed initial solution. For example, the classical processor 102 may be configured to perform one or both of the rounding algorithms described below with reference to Figure 3 and Figure 4 The rounded expected value is the matrix R that maximizes the objective function given by Equation 1 above v ∈0(n) or SO(n).

[0053] The classical processor 102 is configured to output data 120 representing the solution to the quadratic optimization problem, e.g., data specifying the rounded expected value.

[0054] Figure 2 is a flowchart of an example process 200 for solving a quadratic optimization problem on an orthogonal group using classical and quantum computing. For convenience, process 200 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, the Figure 1 system 100 shown can perform example process 200.

[0055] The system receives data representing the quadratic optimization problem (step 202). As described above with reference to Equation 1, the decision variables of the quadratic optimization problem take values in the orthogonal group (e.g., the orthogonal group of dimension n≥1 or the special orthogonal group of dimension n≥1). The group operation is matrix multiplication that provides the quadratic form of the optimization problem. The received data may include data in matrix form, e.g., a set of matrices forming the (u, v)-th n×n block of the positive semi-definite matrix as described above with reference to Equation 1. In some implementations, the matrix C uv may represent, for example, an image or measurement of a 2D electron microscopy projection.

[0056] In some implementations, the quadratic optimization problem is a small non-commutative Grothendieck problem, as described above.

[0057] The system encodes the quadratic optimization problem as a quantum Hamiltonian (step 204). The quantum Hamiltonian includes two-body fermion interactions. To encode the quadratic optimization problem as a quantum Hamiltonian, the system uses the Clifford algebra representation of the group to map the orthogonal matrices in the group to the corresponding quantum states in the Hilbert space.

[0058] The Clifford algebra Cl(n) is a 2 n -dimensional vector space and can be considered equivalent to the Hilbert space of n qubits. To encode the quadratic optimization problem as a quantum Hamiltonian, the system maps the basis elements e I of the Clifford algebra to the corresponding computational basis states in the Hilbert space where b i = 1 if i ∈ I, and b i = 0 otherwise. The inner products on the two spaces agree because this associates one orthonormal basis to another. This correspondence also naturally equates the grade (number of generators in the product) of the Clifford algebra with the Hamming weight of the qubit states in their respective standard bases. The concept of parity is then also preserved between these isomorphisms because parity is just the result of the grade / Hamming weight modulo 2.

[0059] To represent the multiplication operation of the algebra 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 the corresponding n-qubit operators on the Hilbert space. The left multiplication and right multiplication are linear automorphisms represented by λ x , ρ x : Cl(n) → Cl(n) for elements x, y:

[0060] λ x (y) = xy, ρ x (y) = yx. (2)

[0061] Thus, the action of the algebra can be represented as linear operators on . Due to linearity, it is sufficient to specify the left multiplication and right multiplication of the generators e i . The left multiplication operation of the Clifford algebra for the i-th basis element is mapped to the n-qubit operator given by λ i below, and the right multiplication operation of the Clifford algebra for the i-th basis element is mapped to the n-qubit operator given by ρ i below

[0062]

[0063] where \(Z\) represents the Pauli \(Z\) operator, \(Y\) represents the Pauli \(Y\) operator, \(I\) represents the identity operator, and represents the tensor product operator.

[0064] The parity of the Clifford algebra is transformed into the parity of the qubit computational basis, such that the parity automorphism \(\alpha\) is equivalent to the \(n\)-qubit parity operator,

[0065]

[0066] where \(Z\) represents the Pauli \(Z\) operator.

[0067] The double cover of the group \(SO(n)\) can be obtained from the even-parity subalgebra \(Cl\) 0 (n). This subspace has only half the dimension of \(Cl(n)\) because only half of the basis vectors \(e\) I are characterized by \(|I|\bmod 2 = 0\). Thus, it is useful to specify the projection operator from \(Cl(n)\) to \(Cl\) 0 (n) or equivalently from to . Such a mapping can be given by a \(2\) n-1 × \(2\) n matrix

[0068]

[0069] where \(\langle +|\) represents the plus state and \(\langle -|\) represents the minus state.

[0070] The mapping described above connects objects in the Clifford algebra to objects in the Hilbert space, e.g., quantum states and qubit operators. These quantities can be further related to physical quantum systems and measurements by defining a quadratic mapping that can be applied to elements of the Clifford algebra. The quadratic mapping maps an input in a vector space of dimension \(2\) n (e.g., an element from the Clifford algebra) to a corresponding output in a vector space of dimension \(n\times n\) (e.g., an element in the Hilbert space), where the matrix elements of each output include the expected values of combinations of \(n\)-qubit operators described above with reference to Equation 3.

[0071] In the standard basis, the linear mapping has matrix elements

[0072] [Q(x)] ij =\(\langle e\) i ,Q(x)(e\) j )\(=\langle e\) i ,\(\alpha(x)e\) j x\(.\) (6)

[0073] where the notation \(\langle A,B\rangle=\text{tr}(A\)T B) represents the Frobenius inner product on the matrix.

[0074] Use e as described above with reference to Equation 3 i , e j The linear mappings λ i , ρ j , and the conjugate identity These matrix elements of Q(x) can be rearranged as

[0075]

[0076] This expression can be transferred to the quantum representation expanded above. First, the n-qubit Pauli operators

[0077]

[0078] are defined, where i, j denote the Clifford algebra basis indices, X denotes the Pauli X operator, Z denotes the Pauli Z operator, Y denotes the Pauli Y operator, I denotes the identity operator, and the explicit expressions are obtained from Equations 4 and 5. Then it follows that

[0079] [Q(x)] ij =(x|P ij ||x>, (9)

[0080] where is considered equivalent to the quantum state x ∈ Cl(n). Thus, the matrix elements of the output of the mapping can be obtained by calculating the expected values of a set of n 2 Pauli observables. In addition, this matrix Q(x) is orthogonal if and only if x ∈ Pin(n), and lies in SO(n) if and only if x ∈ Spin(n). Generally speaking, these double covers are only subsets of the unit sphere in

[0081] Equation 8 corresponds to the optimization on the orthogonal group. The (n - 1)-qubit operator for the optimization on the special orthogonal group is defined as where i, j denote the Clifford algebra basis indices, Z denotes the Pauli Z operator, and <+|, <-| denote the plus state and the minus state respectively.

[0082] The Clifford algebra formulation of the elements of the orthogonal group provides that the quadratic optimization problem becomes the encoding of a quantum Hamiltonian. The objective function f(R1,..., R m )(as defined in Equation 1) can be explicitly expanded and expressed in terms of matrix elements:

[0083]

[0084] From the quadratic mapping it is known that for each R ∈ G, there exists some such that R ij = <x|P ij |x>. Thus, the matrix product can be expressed as

[0085]

[0086] It is now the expectation value of the 2 n qubit Pauli operators with respect to the product state (separable into two subsystems each with n qubits). To extend this to all m vertices, define the Hilbert space of mn qubits, which is naturally partitioned into m registers of n qubits. For each edge (u, v), introduce the following Hamiltonian terms:

[0087]

[0088] where

[0089]

[0090] To simplify the notation, the identity operators in the tensor product are omitted when the context is clear.

[0091] The quadratic optimization problem is now reformulated as optimizing the mn qubit Hamiltonian

[0092]

[0093] The quantum states that optimize the classical objective function are product states on local qubits: where moreover, each or equivalently each |x v > is a Slater determinant of the appropriate parity. Since such states only form a subset of all quantum states, it follows that the largest eigenvalue of H provides an upper bound on the optimal expected objective value:

[0094]

[0095] In other words, the quantum formulation described currently yields an outer approximation of the original problem. The approximation can be rounded to recover a feasible quantum solution using classical post - processing, as discussed in reference Figure 3 and Figure 4as described. (Note that in the physics literature, the minimum value of the Hamiltonian energy, which is referred to as the ground state energy, is of interest due to the physical relevance of the ground state. On the other hand, computer science convention typically formulates problems in a maximization manner. The computer science convention is adopted here. In this context, the two perspectives are equivalent.)

[0096] Return to Figure 2 , the system uses a quantum computer to determine the (approximate) eigenstates of the quantum Hamiltonian (step 206). For example, the system can perform known techniques such as quantum phase estimation or variational algorithms to determine the quantum state that maximizes the energy <ψ|H|ψ> of the approximation. The quantum state |ψ> can be a pure or mixed state.

[0097] The system uses a quantum computer to calculate the expectation values of Pauli operators with respect to the (approximate) maximum eigenstate, where the Pauli operators include the operators obtained by mapping the multiplication operation of the Clifford algebra to the Hilbert space as given by Equation 8 above (step 208). For example, the system can prepare a copy of the (approximate) maximum eigenstate of the quantum Hamiltonian using, for example, ground state preparation or approximate ground state preparation techniques, and measure the operators P ij and Γ ij to calculate the expectation values <ψ|P ij |ψ> and <ψ|Γ ij |ψ>. For example, the system can measure the single-particle reduced density matrix (1-RDM) or two-particle reduced density matrix (2-RDM) of fermions.

[0098] The system uses a classical computer to round the expectation values of the Pauli operators to elements 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 Figure 3 and Figure 4 . The first algorithm requires measuring the expectation value of the operator for each pair of vertices (u, v) in the graph formulation of the quadratic optimization problem. Since this information content involves the quantum marginals of |ψ> across edges, this procedure is referred to herein as edge-marginal rounding. The second algorithm directly rounds the expectation value of for each vertex v in the graph. In this case, only the information generated by the quantum marginals of the individual vertices is required, and thus, this approach is referred to herein as vertex-marginal rounding. In both algorithms, techniques such as partial state tomography can be used to efficiently estimate the relevant expectation values. Then, the rounding algorithm operates as classical post-processing after obtaining the necessary matrix elements.

[0099] The system can provide data representing the rounded expected values (which are the optimized decision variables, as described in more detail below with reference to Figure 3 and Figure 4 ), and the coefficients C that form the positive semi - definite matrix uv as output.

[0100] Figure 3 FIG. 300 is a flow chart of an example process 300 for edge - marginal rounding of quantum states. 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, a system 100 that is appropriately programmed Figure 1 can perform example process 300.

[0101] The quadratic optimization problem described above with reference to step 202 of example process 200 defines a graph of vertices and edges (V, E), where for each edge (u, v) ∈ E, a matrix and the vertices are labeled by V = [m]. Then, the task of solving the quadratic optimization problem is to assign an R v ∈ G to each v ∈ V such that the objective function is maximized.

[0102] The system obtains data representing the expected value of the corresponding edge (u, v) in the graph (step 302). For each pair of vertices u, v ∈ V, u ≠ v, for all i, j ∈ [n], the operator is given by

[0103]

[0104] The system can receive this data from a quantum computer or a classical computer after performing step 208 of example process 200. The operator is constructed such that if the state is a tensor product of elements, then its expected value returns the matrix element of the product of two orthogonal matrices:

[0105]

[0106] However, for a general state, these expected values will not form an orthogonal matrix.

[0107] Accordingly, the system uses the data obtained at step 302 to fill the square matrix M (step 304). The system collects the data into a large nm×nm block matrix

[0108]

[0109] where for u < v, is defined as

[0110]

[0111] To ensure that M is positive semi - definite, object X vu is defined as X vu = [X uv T , and the diagonal blocks are set to nL because

[0112] the system decomposes the square matrix into the product of i) a rectangular matrix and ii) the transpose of a rectangular matrix (step 306). Since the square matrix it is thus decomposed, for example, via Cholesky decomposition into the form M = LL for some T . This provides the form of the square matrix M as a Gram matrix, where each n×n block of M is the product of the rectangular matrix such that

[0113] The system generates a random matrix (step 308). For example, the system can sample values from a normal distribution and use the sampled values to generate a random matrix

[0114] The system multiplies the random matrix Z by the rectangular matrix L to obtain the matrix - vector LZ (step 310).

[0115] For each vertex in the graph, the system projects the corresponding matrix in the matrix - vector onto the nearest element in the orthogonal group (step 312). That is, for each v ∈ V, the system identifies the nearest (special) orthogonal matrix as

[0116]

[0117] In some implementations, the system can compute R v by computing the singular - value decomposition of the matrix - vector L v Z. Then the set of R v for each v ∈ V is provided as the solution to a quadratic optimization problem.

[0118] The example process 300 can be expressed in pseudocode as:[[]]

[0119]

[0120] where the input data is the quantum state |ψ> on a graph of m vertices, and if G = O(n), then each vertex has a local Hilbert space of dimension d = 2 n , or if G = SO(n), then each vertex has a dimension d = 2​n-1 a local Hilbert space, and the output data is each vertex R1, ..., R m ∈G is an orthogonal matrix on

[0121] Figure 4 is a flowchart of an example process 400 for vertex - marginal rounding of quantum states. 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, a Figure 1 system 100 that is appropriately programmed can perform the example process 400.

[0122] The quadratic optimization problem described above with reference to step 202 of example process 200 defines a graph of vertices and edges (V, E), where for each edge (u, v) ∈ E, the matrix and the vertices are labeled by V = [m]. Then, the task of solving the quadratic optimization problem is to assign R v ∈G for each v ∈ V such that the objective function is maximized.

[0123] The system obtains data representing the expected value of the corresponding vertex v in the graph (step 402). The operator P ij for vertex v was defined above with reference to equation 8. The system can receive this data from a quantum computer or a classical computer after performing step 208 of example process 200.

[0124] The single - vertex marginals are defined by tracing out the qubits associated with all but one vertex,

[0125]

[0126] Then, the matrix R v is the image of σ v under the convex extension of the quadratic mapping Q(. To see this, consider the spectral decomposition σ v = ∑ α p α |φ α ><φ α |. The corresponding convex combination of Q(φ α ) is defined as

[0127]

[0128] This matrix does not necessarily lie in G, but in its convex hull conv(G).

[0129] Thus, for each vertex in the graph, the system will Q(σ v) Nearest element projected into the orthogonal group (step 404). That is, for each v ∈ V, the system identifies the nearest (special) orthogonal matrix as

[0130]

[0131] where Q(σ is defined in Equation 9 v ). In some implementations, the system can calculate R by computing the singular value decomposition of Q(σ v ), for example, writing the special singular decomposition of v to obtain Performing this rounding procedure for each vertex returns a feasible solution. Then, the set of R

[0132] for each v ∈ V is provided as the solution to the quadratic optimization problem. It should be noted that, in principle, this corresponds to rounding |ψ> to a product state v , where each Since each |x > is equivalent to a Slater determinant, its fermionic one-body reduced density matrix completely specifies it, and thus the n v one-body observables of the fermionic are sufficient to completely represent it. 2

[0133] Example process 400 can be expressed in pseudocode as:

[0134]

[0135] where the input data is the quantum state |ψ> on a graph of m vertices, and if G = O(n), each vertex has a local Hilbert space of dimension d = 2 n , or if G = SO(n), each vertex has a local Hilbert space of dimension d = 2 n-1 , and the output data is the orthogonal matrices R1,..., R m ∈ G for each vertex.

[0136] Figure 5 FIG. depicts an example quantum computer 500 for performing the quantum operations described in this specification. Example quantum computer 500 includes an example 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 merely exemplary and do not limit the implementations of the invention described and / or claimed in this document.

[0137] The example quantum computing device 502 includes a qubit assembly 552 and a control and measurement system 504. The qubit assembly includes a plurality of qubits for performing algorithm operations or quantum computations, e.g., qubit 506. Although Figure 5 the qubits shown are arranged in a rectangular array, this is a schematic depiction and not intended to be limiting. The qubit assembly 552 also includes adjustable coupling elements that allow interactions between the coupled qubits, e.g., coupler 508. In Figure 5 the schematic depiction, each qubit is adjustably coupled to each of its four neighboring qubits via a respective coupling element. However, this is an example arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that allow coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits.

[0138] Each qubit can be a physical two-level quantum system or device having energy levels representing the logical values 0 and 1. The specific physical implementation of the plurality of qubits and how they interact with each other depends on various factors, including the type of quantum computing device 502 included in the example computer 500 or the type of quantum computation that the quantum computing device is performing. For example, in an atomic quantum computer, the qubits can be implemented via atoms, molecules, or solid-state quantum systems (e.g., hyperfine atomic states). As another example, in a superconducting quantum computer, the qubits can be implemented via superconducting qubits or semiconductor qubits (e.g., superconducting transmon states). As another example, in an NMR quantum computer, the qubits can be implemented via nuclear spin states.

[0139] In some implementations, the quantum computation can proceed, e.g., by loading qubits from a quantum memory and applying a sequence of unitary operators to the qubits. Applying a unitary operator to a qubit can include applying a corresponding sequence of quantum logic gates to the qubit, e.g., to implement the qubit operators described in this specification. Example quantum logic gates include: single-qubit gates, such as Pauli X, Pauli Y, Pauli Z (also known as X, Y, Z), Hadamard gate, S gate, rotations; two-qubit gates, such as controlled-X gate, controlled-Y gate, controlled-Z gate (also known as CX, CY, CZ), controlled-not gate (also known as CNOT), controlled-swap gate (also known as CSWAP), iSWAP gate, and gates involving three or more qubits, such as Toffoli gate. The quantum logic gates can be implemented by applying control signals 510 generated by the control and measurement system 504 to the qubits and couplers.

[0140] 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 via one or more drive lines coupled to the qubit. Example operating frequencies include qubit idle frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubits can perform. For example, setting the operating frequency to the corresponding idle frequency can place the qubit in a state in which it does not strongly interact with other qubits and it can be used to perform single-qubit gates. As another example, in the case where qubits interact via a coupler with a fixed coupling, the qubits can be configured to interact with each other by setting their respective operating frequencies to some gate-related frequency that is detuned from their common interaction frequency. In other cases, for example, when qubits interact via a tunable coupler, the qubits can be configured to interact with each other by setting the parameters of their respective couplers such that interaction between the qubits is enabled and then by setting the respective operating frequencies of the qubits to some gate-related frequency that is detuned from their common interaction frequency. Such interactions can be performed in order to perform multi-qubit gates.

[0141] The type of control signal 510 used depends on the physical implementation of the qubits. For example, the control signal can include RF or microwave pulses in NMR or superconducting quantum computer systems, or optical pulses in atomic quantum computer systems.

[0142] Quantum computation can be done by measuring the state of the qubits (e.g., using quantum observables such as X or Z) using the respective control signal 510. The measurement causes a readout signal 512 representing the measurement result to be transmitted back to the measurement and control system 504. Depending on the quantum computing device and / or the physical scheme of the qubits, the readout signal 512 can include RF, microwave, or optical signals. For convenience, Figure 5 the control signal 510 and the readout signal 512 shown are depicted as only addressing selected elements of the qubit assembly (i.e., the top and bottom rows), but during operation, the control signal 510 and the readout signal 512 can address each element in qubit assembly 552.

[0143] 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 as described above, as well as other classical subroutines or computations. 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 to issue a sequence of control signals 510 to the qubit assembly, e.g., to perform a selected sequence of quantum gate operations, and to receive a sequence of readout signals 512 from the qubit assembly, e.g., as part of performing a measurement operation.

[0144] The processor 514 is configured to process instructions for execution within the control and measurement system 504. In some implementations, the processor 514 is a single-threaded processor. In other implementations, the processor 514 is a multi-threaded processor. The processor 514 is capable of processing instructions stored in the memory 516.

[0145] The memory 516 stores information within the control and measurement system 504. In some implementations, the memory 516 includes computer-readable media, volatile memory units, and / or non-volatile memory units. In some cases, the memory 516 can include storage devices capable of providing mass storage for the system 504, such as hard disk devices, optical disk devices, storage devices shared by multiple computing devices over a network (e.g., cloud storage devices), and / or some other mass storage device.

[0146] The input / output device 518 provides input / output operations for the control and measurement system 504. The input / output device 518 can include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers, whereby control signals 510 are issued to the qubit assembly and readout signals 512 are received from the qubit assembly, as appropriate for the physical scheme used in a quantum computer. In some implementations, the input / output device 518 can also include one or more network interface devices (e.g., Ethernet cards), serial communication devices (e.g., RS-232 ports), and / or wireless interface devices (e.g., 802.11 cards). In some implementations, the input / output device 518 can include driver devices configured to receive input data and issue output data to other external devices (e.g., keyboards, printers, and display devices).

[0147] Although it has been in Figure 5An example control and measurement system 504 is depicted, but the implementation of the subject matter and functional operations described in this specification can be implemented in other types of digital electronic circuitry or in computer software, firmware, or hardware (including the structures disclosed in this specification and their structural equivalents) or in a combination of one or more of them.

[0148] Figure 6 A schematic diagram of an exemplary classical processor system 600 is shown. According to some implementations, system 600 can be used for the classical operations described in this specification. System 600 is intended to represent various forms of digital computers, such as laptop computers, desktop computers, 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 merely exemplary and do not limit the implementations of the inventions described and / or claimed in this document.

[0149] System 600 includes a processor 610, a memory 620, a storage device 630, and an input / output device 640. Each of the components 610, 620, 630, and 620 is interconnected using a system bus 650. Processor 610 can be enabled to process instructions for execution within system 600. In one implementation, processor 610 is a single-threaded processor. In another implementation, processor 610 is a multi-threaded processor. Processor 610 can be enabled to process instructions stored in memory 620 or on storage device 630 to display image information for a user interface on input / output device 640.

[0150] Memory 620 stores information within system 600. In one implementation, memory 620 is a computer-readable medium. In one implementation, memory 620 is a volatile memory unit. In another implementation, memory 620 is a non-volatile memory unit.

[0151] Storage device 630 can be enabled to provide mass storage for system 600. In one implementation, storage device 630 is a computer-readable medium. In various different implementations, storage device 630 can be a floppy disk device, a hard disk device, an optical disk device, or a tape device.

[0152] Input / output device 640 provides input / output operations for system 600. In one implementation, input / output device 640 includes a keyboard and / or a pointing device. In another implementation, input / output device 640 includes a display unit for displaying a graphical user interface.

[0153] The subject matter and the implementation of the operations described in this specification can be implemented in digital electronic circuitry, in analog electronic circuitry, in suitable quantum circuitry or more generally quantum computing systems, in tangibly embodied software or firmware, in computer hardware (including the structures disclosed in this specification and their structural equivalents), or in combinations of one or more of them. The term "quantum computing system" can include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptographic systems, or quantum simulators.

[0154] The implementation of the subject matter described in this specification can be realized as one or more computer programs, i.e., 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 can 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 of them. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, which is generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing apparatus.

[0155] The terms quantum information and quantum data refer to information or data carried, held, or stored by a quantum system, where the smallest non-trivial system is a qubit, i.e., the system that defines the unit of quantum information. It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as two-level systems in the corresponding context. Such quantum systems can include multi-level systems, e.g., having two or more energy levels. By way of example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational basis states are considered equivalent to the ground state and the first excited state, however, it should be understood that other settings where the computational states are considered equivalent to higher energy level excited states are also possible.

[0156] The term "data processing device" refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, apparatuses, and machines 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. The device can also be or further include dedicated logic circuitry, such as FPGAs (field-programmable gate arrays), ASICs (application-specific integrated circuits), or quantum simulators, i.e., quantum data processing devices 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 ability to perform universal quantum computing. In addition to the hardware, the device can optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code that constitutes processor firmware, protocol stacks, database management systems, operating systems, or combinations of one or more of them.

[0157] A digital computer program, which can also be referred to as 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 or interpreted languages, or declarative or procedural languages), and it 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 can also be referred to as 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 or interpreted languages, or declarative or procedural languages) and translated into a suitable quantum programming language, or can be written in a quantum programming language such as QCL or Quipper.

[0158] A computer program can, but need not, correspond to a file in a file system. The program can be stored as part of a file that holds other programs or data (such as 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 (such as files that store one or more modules, subroutines, or portions of code). A computer program can be deployed to execute 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 send quantum data using quantum systems (such as qubits). In general, a digital data communication network cannot send quantum data; however, a quantum data communication network can send both quantum data and digital data.

[0159] The processes and logical flows described in this specification can be performed by one or more programmable computers that operate, optionally, using one or more processors to execute one or more computer programs to perform functions by operating on input data and generating output. The processes and logical flows can also be performed by special-purpose logic circuitry, and the apparatus can also be implemented as special-purpose logic circuitry, such as an FPGA or an ASIC or a quantum simulator, or by a combination of special-purpose logic circuitry or a quantum simulator with one or more programmed digital and / or quantum computers.

[0160] For a system of one or more computers “configured to” perform a particular operation or action, it means that the system has installed thereon software, firmware, hardware, or a combination thereof that in operation causes the system to perform the operation or action. For one or more computer programs configured to perform a particular operation or action, it means that the one or more programs include instructions that, when executed by a data processing apparatus, cause the apparatus to perform the operation or action. For example, a quantum computer can receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operation or action.

[0161] A computer suitable for executing a computer program can be based on a general-purpose or special-purpose processor or any other kind of central processing unit. In general, 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.

[0162] The elements of a computer include a central processing unit for executing or performing instructions and one or more memory devices for storing the instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special-purpose logic circuitry or a quantum simulator. In general, 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 operatively coupled to receive data from, or transfer data to, or both, the one or more mass storage devices. However, a computer need not have such devices.

[0163] A quantum circuit element (also referred to as a quantum computing circuit element) includes circuit elements for performing quantum processing operations. That is, a quantum circuit element is configured to perform operations on data in a non-deterministic manner by leveraging quantum mechanical phenomena such as superposition and entanglement. Some quantum circuit elements, such as qubits, can be configured to represent information in more than one state simultaneously and operate on that information. 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.

[0164] In contrast, classical circuit elements typically process data in a deterministic manner. Classical circuit elements can be configured to execute the instructions of a computer program by performing basic arithmetic, 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 send 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, where the ERSFQ device is an energy-efficient version of the RSFQ that does not use a bias resistor.

[0165] In some cases, some or all of the quantum and / or classical circuit elements can be implemented using, for example, superconducting quantum and / or classical circuit elements. The fabrication of superconducting circuit elements may require depositing one or more materials such as superconductors, dielectrics, and / or metals. Depending on the materials selected, deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes, can be used to deposit these materials. The processes described herein for fabricating circuit elements may require removing one or more materials from the device during fabrication. Depending on the materials to be removed, the removal processes can 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).

[0166] During operation of a quantum computing system that uses superconducting quantum circuit elements and / or superconducting classical circuit elements (such as the circuit elements described herein), the superconducting circuit elements are cooled within a cryostat to a temperature that allows the superconducting material to exhibit superconducting properties. A superconductor (or superconducting) material can be understood as a material that exhibits superconducting properties at or below the superconducting critical temperature. Examples of superconducting materials include aluminum (superconducting critical temperature of 1.2 Kelvin) and niobium (superconducting critical temperature of 9.3 Kelvin). Thus, superconducting structures such as superconducting traces and superconducting ground planes are formed from materials that exhibit superconducting properties at or below the superconducting critical temperature.

[0167] In some implementations, classical circuit elements that are electrically and / or electromagnetically coupled to quantum circuit elements (e.g., qubits and qubit couplers) can be used to provide control signals for these quantum circuit elements. The control signals can be provided in digital and / or analog form.

[0168] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memories, 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. It should be understood that a quantum memory is a device capable of storing quantum data with high fidelity and high efficiency over a long period of time, such as an optical-matter interface that uses light for transmission and matter for storing and preserving quantum characteristics (such as superposition or quantum coherence) of the quantum data.

[0169] The control of various systems or portions thereof described in this specification can be implemented with a computer program product that includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more processing devices. The systems or portions thereof described in this specification can each be implemented as a device, method, or system that can include one or more processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0170] While this specification contains many specific implementation details, these details should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations may also be implemented in combination within a single implementation. Conversely, the various features that are described in the context of a single implementation may also be implemented separately or in any suitable sub-combination in multiple implementations. Additionally, while the features are described above as acting in certain combinations and even initially claimed as such, one or more features from the claimed combination may in some cases be removed from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0171] Similarly, although operations are depicted in the drawings in a particular order, this should not be construed as requiring that such operations be performed in the particular order shown or in sequential order, or that all of the illustrated operations be performed, to achieve the desired result. In certain circumstances, multitasking and parallel processing may be advantageous. Additionally, the separation of the various system modules and components in the implementations described above should not be construed as requiring such separation in all implementations, and it should be understood that the described program components and systems may generally be integrated together in a single software product or packaged in multiple software products.

[0172] Specific implementations of the subject matter have been described. Other implementations are within the scope of the appended claims. For example, the acts recited in the claims may be performed in a different order and still achieve the desired result. As one example, the processes depicted in the figures do not necessarily require the particular order or sequential order shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A computer-implemented method, comprising: 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 the classical computer, the quadratic optimization problem into a quantum Hamiltonian, the encoding including mapping an orthogonal matrix or a special orthogonal matrix in the group to a corresponding quantum state in a Hilbert space using a Clifford algebra representation of the group; determining, by a quantum computer, an approximate eigenstate of the quantum Hamiltonian; computing, by the quantum computer, an expectation value of a Pauli operator with respect to the approximate eigenstate, wherein the Pauli operator includes an operator obtained by mapping a multiplication operation of the Clifford algebra to 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 according to claim 1, wherein The orthogonal group includes an orthogonal group of dimension n≥1 or a special orthogonal group of dimension n≥1, wherein the group operation is given by matrix multiplication.

3. The method according to claim 1 or claim 2, wherein Using the Clifford algebra representation of the group to map an orthogonal matrix in the group to a corresponding quantum state in the Hilbert space includes: Map 2 n the basis elements of the Clifford algebra in the n-dimensional vector space to the corresponding computational basis states in a Hilbert space of dimension 2 n , 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 corresponding n-qubit operators on the Hilbert space.

4. The method according to claim 3, wherein The left multiplication operation of the Clifford algebra for the i-th basis element is mapped to the n-qubit operator given by and the right multiplication operation of the Clifford algebra for the i-th basis element is mapped to the n-qubit operator given by where Z represents the Pauli Z operator and Y represents the Pauli Y operator.

5. The method according to claim 3, wherein Mapping an orthogonal matrix in the group to a corresponding quantum state in the Hilbert space using the Clifford algebra representation of the group involves applying a quadratic map to elements of the Clifford algebra, where the quadratic map maps an input in a vector space of dimension 2 n to a corresponding output in a vector space of dimension n×n, where the matrix elements of each output include the expected value of the n-qubit operator on the Hilbert space.

6. The method according to claim 5, wherein, The matrix elements of the output of the quadratic mapping include the expectation value of the n-qubit operator given by the following formula 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 method according to claim 5, wherein, The matrix elements of the output of the quadratic mapping include the expected value of the n-qubit operator given by , where i, j denote Clifford algebra basis indices, Z denotes the Pauli Z operator, and <+|, <-| denote the plus state and the minus state, respectively.

8. The method according to any one of claims 1 to 7, wherein, The quadratic optimization problem defines a graph of vertices and edges, and wherein rounding the computed expectation value to an element of the orthogonal group includes implementing an edge-marginal rounding algorithm.

9. The method according to claim 8, wherein, Implementing the edge-marginal rounding algorithm includes: decomposing a square matrix including the expectation values of the edges in the graph into a product of a rectangular matrix and the transpose of the rectangular matrix; generating a matrix with a random normal distribution; multiplying the random matrix by the rectangular matrix to obtain a matrix vector; and for each vertex in the graph, projecting the corresponding matrix in the matrix vector to the nearest element in the orthogonal group.

10. The method according to claim 9, wherein, Projecting the corresponding matrix in the matrix vector to the nearest element in the orthogonal group includes computing a singular value decomposition of the matrix vector.

11. The method according to any one of claims 1 to 10, wherein, The quadratic optimization problem defines a graph of vertices and edges, and wherein rounding the computed expectation value to an element of the orthogonal group includes implementing a vertex-marginal rounding algorithm.

12. The method according to claim 11, wherein, Implementing the vertex-marginal rounding algorithm includes: for each vertex in the graph, projecting a matrix including the expectation value of the vertex to the nearest element in the orthogonal group.

13. The method according to claim 12, wherein, Projecting a matrix including the expectation value of the vertex to the nearest element in the orthogonal group includes computing a singular value decomposition of the matrix including the expectation value of the vertex.

14. The method according to any one of claims 1 to 13, wherein Computing the expectation value of the Pauli operator with respect to the approximate eigenstate includes: preparing, by the quantum computer, a copy of the approximate eigenstate using a ground state preparation or approximate ground state preparation technique; and The copy of the approximate eigenstate is measured by the quantum computer, including measuring matrix elements of a 1-RDM or 2-RDM.

15. The method according to any one of claims 1 to 14, wherein Determining the approximate eigenstate of the quantum Hamiltonian includes performing quantum phase estimation or a variational algorithm, wherein the accuracy of the approximate eigenstate depends on the accuracy of the quantum phase estimation calculation or the variational algorithm.

16. The method according to any one of claims 1 to 15, wherein, The received data includes values of elements of a positive semi-definite matrix, and wherein the solution to the quadratic optimization problem includes the rounded expectation value and the values of the elements of the positive semi-definite matrix.

17. The method according to any one of claims 1 to 16, wherein The quantum Hamiltonian includes two-body fermion interactions.

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

19. The method according to any one of claims 1 to 18, wherein The eigenstates of the quantum Hamiltonian include the maximum eigenstate or an eigenstate that maximizes the energy relative to an initial state.

20. A device, comprising: one or more classical processors; and one or more quantum computing devices that communicate data with the one or more classical processors; wherein the device is configured to perform the method according to any one of claims 1 to 19.

Images