Abstract: Methods, systems, and apparatus for verified quantum phase estimation. In one aspect, a method includes repeatedly performing a experiment. Performing one repetition of the experiment includes: applying a second unitary to a system register of N qubits prepared in a target computational basis state; applying, conditioned on a state of a control qubit, a first unitary to the system register; applying an inverse of the second unitary to the system register and measuring each qubit to determine an output state of the system register; measuring the control qubit to obtain a corresponding measurement result m; and post-selecting on the target computational basis state by, in response to determining that the output state indicates that each qubit was in the target computational basis state prior to measurement, incrementing a first or second classical variable by (?1)m. Phases or expectation values of the first unitary are estimated based on the classical variables.

Abstract: Methods and apparatus for estimating the fidelity of quantum hardware. In one aspect, a method includes accessing a set of quantum gates; sampling a subset of quantum gates from the set of quantum gates, wherein the subset of quantum gates defines a quantum circuit; applying the quantum circuit to a quantum system and performing measurements on the quantum system to determine output information of the quantum system; calculating output information of the quantum system based on application of the quantum circuit to the quantum system; and estimating a fidelity of the quantum circuit based on the determined output information and the calculated output information of the quantum system.

Abstract: Methods, systems and apparatus for correcting a result of a quantum computation.

Abstract: Methods, systems and apparatus for error correction of fermionic quantum simulation. In one aspect, a method includes representing a fermionic system as a graph of vertices and edges, where each vertex represents a fermionic system fermionic mode and each edge represents an interaction between two respective fermionic modes; allocating a qubit to each edge in the graph to form a qubit system; determining qubit operators that satisfy a set of fermionic commutation and dependence relations, where the qubit operators are non-uniform with respect to the graph vertices; determining stabilizer operators corresponding to products of quadratic Majorana operators on respective loops in the graph, where a common eigenspace of the defined stabilizer operators defines a code subspace that encodes states of the fermionic system to be simulated; and simulating the fermionic system by evolving the qubit system under a qubit Hamiltonian that includes the determined qubit operators and stabilizer operators.

Abstract: Methods, systems and apparatus for simulating physical systems. In one aspect, a method includes the actions of selecting a first set of basis functions for the simulation, wherein the first set of basis functions comprises an active and a virtual set of orbitals; defining a set of expansion operators for the simulation, wherein expansion operators in the set of expansion operators approximate fermionic excitations in an active space spanned by the active set of orbitals and a virtual space spanned by the virtual set of orbitals; performing multiple quantum computations to determine a matrix representation of a Hamiltonian characterizing the system in a second set of basis functions, computing, using the determined matrix representation of the Hamiltonian, eigenvalues and eigenvectors of the Hamiltonian; and determining properties of the physical system using the computed eigenvalues and eigenvectors.

Abstract: Methods, systems and apparatus for approximating a target quantum state. In one aspect, a method for determining a target quantum state includes the actions of receiving data representing a target quantum state of a quantum system as a result of applying a quantum circuit to an initial quantum state of the quantum system; determining an approximate quantum circuit that approximates the specific quantum circuit by adaptively adjusting a number of T gates available to the specific quantum circuit; and applying the determined approximate quantum circuit to the initial quantum state to obtain an approximation of the target quantum state.

Abstract: Methods, systems, and apparatus for training quantum evolutions using sub-logical controls. In one aspect, a method includes the actions of accessing quantum hardware, wherein the quantum hardware includes a quantum system comprising one or more multi-level quantum subsystems; one or more control devices that operate on the one or more multi-level quantum subsystems according to one or more respective control parameters that relate to a parameter of a physical environment in which the multi-level quantum subsystems are located; initializing the quantum system in an initial quantum state, wherein an initial set of control parameters form a parameterization that defines the initial quantum state; obtaining one or more quantum system observables and one or more target quantum states; and iteratively training until an occurrence of a completion event.

Abstract: Methods, systems and apparatus for determining properties of a physical system described by an electronic structure Hamiltonian. In one aspect, a Hamiltonian describing the physical system is transformed into a qubit Hamiltonian describing a corresponding system of qubits. The qubit Hamiltonian comprises multiple two-qubit interaction terms, each comprising a respective translation invariant coefficient. The system of qubits is evolved under a unitary operator generated by the multiple two-qubit interaction terms. The evolution includes applying layers of quantum logic gates to the system of qubits, wherein each application of a layer evolves the system of qubits under a unitary operator generated by a respective subset of the multiple two-qubit interaction terms and wherein the value of the coefficients of the subset of the multiple two-qubit interaction terms that generate the unitary operator is constant. The evolved system of qubits is measured and properties of the physical system is determined.

Abstract: Methods, systems and apparatus for preparing arbitrary superposition quantum states of a quantum register on a quantum computer, the quantum state comprising a superposition of L computational basis states. In one aspect, a register of log L qubits is prepared in a weighted sum of register basis states, where each register basis state indexes a corresponding quantum state computational basis state, and the amplitude of each register basis state in the weighted sum of register basis states is equal to the amplitude of the corresponding computational basis state in the superposition of L computational basis states. A unitary transformation that maps the register basis states to the corresponding L computational basis states is then implemented, including, for each index 1 to L, controlling, by the register of log L qubits, transformation of the quantum system register state for the index to the corresponding computational basis state for the index.

Abstract: Methods, systems, and apparatus for training quantum evolutions using sub-logical controls. In one aspect, a method includes the actions of accessing quantum hardware, wherein the quantum hardware includes a quantum system comprising one or more multi-level quantum subsystems; one or more control devices that operate on the one or more multi-level quantum subsystems according to one or more respective control parameters that relate to a parameter of a physical environment in which the multi-level quantum subsystems are located; initializing the quantum system in an initial quantum state, wherein an initial set of control parameters form a parameterization that defines the initial quantum state; obtaining one or more quantum system observables and one or more target quantum states; and iteratively training until an occurrence of a completion event.

Abstract: Methods, systems and apparatus for performing quantum state preparation. In one aspect, a method includes the actions of defining a target quantum state of a quantum system, wherein time evolution of the quantum system is governed by a target Hamiltonian, and defining a total Hamiltonian that interpolates between an initial Hamiltonian and the target Hamiltonian, wherein the total Hamiltonian is equal to the initial Hamiltonian at an initial time and is equal to the target Hamiltonian at a final time; approximating the time evolution of the total Hamiltonian using a truncated linear combination of unitary simulations to generate a truncated time evolution operator; evolving a ground state of the initial Hamiltonian according to the truncated time evolution operator for a truncated number of time steps to generate an intermediate state; and variationally adjusting the intermediate state to determine a wavefunction that approximates the target quantum state of the quantum system.

Abstract: Methods, systems and apparatus for performing indexed operations using a unary iteration quantum circuit. In one aspect, a method includes encoding an index value in an index register comprising index qubits; encoding the index value in a control register comprising multiple control qubits; and repeatedly computing and uncomputing the control qubits to perform, conditioned on the state of the control qubits, the operation on one or more target qubits corresponding to the index value, wherein during the encoding, computing and uncomputing: the multiple control qubits are made available in sequence, and the multiple control qubits correspond to a one-hot encoding of the encoded index value.

Abstract: Solution of a problem of determining values of a set of N problem variables xi makes use of a quantum processor that has a limited number of hardware elements for representing quantum bits and/or limitations on coupling between quantum bits. A method includes accepting a specification of the problem that includes a specification of a set of terms where each term corresponds to a product of at least three variables and is associated with a non-zero coefficient. A set of ancilla variables, each ancilla variable corresponding to a pair of problem variables, is determined by applying an optimization procedure to the specification of the set of the terms. The accepted problem specification is then transformed according to the determined ancilla variables to form a modified problem specification for use in configuring the quantum processor and solution of problem.

Abstract: Generating a computing specification to be executed by a quantum processor includes: accepting a problem specification that corresponds to a second-quantized representation of a fermionic Hamiltonian, and transforming the fermionic Hamiltonian into a first qubit Hamiltonian including a first set of qubits that encode a fermionic state specified by occupancy of spin orbitals. An occupancy of any spin orbital is encoded in a number of qubits that is logarithmic in the number of spin orbitals, and a parity for a transition between any two spin orbitals is encoded in a number of qubits that is logarithmic in the number of spin orbitals. An eigenspectrum of a second qubit Hamiltonian, including the first set of qubits and a second set of qubit, includes a low-energy subspace and a high-energy subspace, and an eigenspectrum of the first qubit Hamiltonian is approximated by a set of low-energy eigenvalues of the low-energy subspace.

Abstract: Methods, systems and apparatus for simulating a physical system described by an electronic structure Hamiltonian expressed in an orthonormal basis. In one aspect, a method includes decomposing the electronic structure Hamiltonian into a sum of sub-Hamiltonians, wherein each sub-Hamiltonian in the sum of sub-Hamiltonians is expressed in one of multiple bases; simulating evolution of the physical system using the decomposed electronic structure Hamiltonian; and using the simulated evolution of the physical system using the decomposed electronic structure Hamiltonian to determine properties of the physical system.

Abstract: Methods, systems and apparatus for targeting many-body states on a quantum computer. In one aspect, a method includes an adaptive phase shift method that includes preparing the quantum system in an initial state, wherein the initial state has non-zero overlap with the target eigenstate; preparing an ancilla qubit in a zero computational basis state; and iteratively applying a quantum eigenstate locking circuit to the quantum system and ancilla qubit until the state of the quantum system approximates the target eigenstate, wherein the quantum eigenstate locking circuit comprises a phase gate that, at each n-th iteration, is updated using a current average energy estimate of the quantum system.

Abstract: Methods, systems, and apparatus for simulating a physical system. In one aspect, a method includes transforming a Hamiltonian describing the physical system into a qubit Hamiltonian describing a corresponding system of qubits, the qubit Hamiltonian comprising a transformed kinetic energy operator; simulating evolution of the system of qubits under the qubit Hamiltonian, comprising simulating the evolution of the system of qubits under the transformed kinetic energy operator by applying a fermionic swap network to the system of qubits; and using the simulated evolution of the system of qubits under the qubit Hamiltonian to determine properties of the physical system.

Abstract: Generating a computing specification to be executed by a quantum processor includes: accepting a problem specification that corresponds to a second-quantized representation of a fermionic Hamiltonian, and transforming the fermionic Hamiltonian into a first qubit Hamiltonian including a first set of qubits that encode a fermionic state specified by occupancy of spin orbitals. An occupancy of any spin orbital is encoded in a number of qubits that is logarithmic in the number of spin orbitals, and a parity for a transition between any two spin orbitals is encoded in a number of qubits that is logarithmic in the number of spin orbitals. An eigenspectrum of a second qubit Hamiltonian, including the first set of qubits and a second set of qubit, includes a low-energy subspace and a high-energy subspace, and an eigenspectrum of the first qubit Hamiltonian is approximated by a set of low-energy eigenvalues of the low-energy subspace.

Abstract: Methods, systems, and apparatus for quantum phase estimation. In one aspect, an apparatus includes a quantum circuit comprising a first quantum register comprising at least one ancilla qubit, quantum gates, comprising at least (i) two Hadamard gates, (ii) a phase gate, (iii) a unitary operator, and (iv) a measurement operator, a second quantum register comprising one or more qubits, wherein the second quantum register is prepared in an arbitrary quantum state that is not an eigenstate of the unitary operator; and a phase learning system, configured to perform phase estimation experiments on the quantum circuit, comprising repeatedly measuring the state of an ancilla qubit for each phase estimation experiment to determine an expectation value of the state of the ancilla qubit and learn phases of the eigenvalues of the unitary operator.

Abstract: Solution of a problem of determining values of a set of N problem variables xi makes use of a quantum processor that has a limited number of hardware elements for representing quantum bits and/or limitations on coupling between quantum bits. A method includes accepting a specification of the problem that includes a specification of a set of terms where each term corresponds to a product of at least three variables and is associated with a non-zero coefficient. A set of ancilla variables, each ancilla variable corresponding to a pair of problem variables, is determined by applying an optimization procedure to the specification of the set of the terms. The accepted problem specification is then transformed according to the determined ancilla variables to form a modified problem specification for use in configuring the quantum processor and solution of problem.