Google Inc.

Ryan Babbush

Jarrod Ryan McClean

Hartmut Neven

Zhang Jiang

Nan Ding

Craig Gidney

Ian David Kivlichan

John Martinis

Austin Greig Fowler

Vadim Smelyanskiy

Sergio Boixo Castrillo

Nicholas Charles Rubin

Sergei V Isakov

William Huggins

Thomas Eugene O'Brien

Jarrod Ryan Mcclean

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.

Increasing representation accuracy of quantum simulations without additional quantum resources

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). Phases or expectation values of the first unitary are estimated based on the classical variables.

Verified quantum phase estimation

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.

Variational quantum state preparation

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.

Preparing superpositions of computational basis states on a quantum computer

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.

Techniques for obtaining accurate diagonal electronic structure Hamiltonians

Methods, systems and apparatus for correcting a result of a quantum computation. In one aspect, a method includes selecting a quantum error correcting code for the quantum computation, wherein the quantum error correcting code is defined by multiple stabilizer generators; determining a set of symmetry operators, comprising: selecting a subset of the stabilizer generators, determining, for each selected stabilizer generator, a sum between an identity operator and the stabilizer generator, and multiplying the determined sums together to form a summation of terms, wherein each term in the summation is equal to a respective symmetry operator; measuring a projective correction of a physical observable over an output quantum state of the quantum computation using the determined set of symmetry operators, wherein the physical observable corresponds to the result of the quantum computation; and determining a corrected result of the quantum computation using the measured projective correction of the physical observable.

Decoding errors using quantum subspace expansion

Methods, systems and apparatus for measuring the energy of a quantum chemical system. In one aspect, a method includes obtaining a Hamiltonian describing the chemical system, where the Hamiltonian is expressed in an orthonormal basis; decomposing the Hamiltonian into a sum of terms where each term comprises a respective operator that effects a respective single particle basis rotation, and one or more particle density operators; repeatedly, for each group comprising terms with a same operator that effects a respective single particle basis rotation, measuring expectation values of the terms included in the group, comprising: performing the respective single particle basis rotation on a qubit system encoding a state of the chemical system; and measuring Jordan-Wigner transformations of the one or more particle density operators in the group to obtain a respective measurement result for the group; and determining the energy of the chemical system using the obtained measurement results.

Efficient and noise resilient measurements for quantum chemistry

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.

Fidelity estimation for quantum computing systems

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.

Performing unitary iteration and indexed operations

Methods, systems and apparatus for approximating a target quantum state that is defined as a result of applying a specific rotation operation to an initial quantum state. A method includes determining multiple configurations of T-gates. Each configuration of T-gates includes a number of T-gates that is less than or equal to a predefined total number of T-gates and represents a rotation operation that, when applied to the initial quantum state, produces an evolved quantum state that is an approximation of the target quantum state. A configuration of T-gates that represents a rotation operation with a rotation angle that is closest to a rotation angle of the specific rotation operation is selected from the multiple configurations of T-gates. A rotation operation represented by the selected configuration of T-gates is applied to the initial quantum state to obtain the approximation of the target quantum state.

Error corrected variational algorithms

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.

Efficient fault-tolerant trotter simulation of molecular hamiltonians

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.

Training quantum evolutions using sublogical controls

Methods, systems and apparatus for simulating quantum systems. In one aspect, a method includes the actions of obtaining a first Hamiltonian describing the quantum system, wherein the Hamiltonian is written in a plane wave basis comprising N plane wave basis vectors; applying a discrete Fourier transform to the first Hamiltonian to generate a second Hamiltonian written in a plane wave dual basis, wherein the second Hamiltonian comprises a number of terms that scales at most quadratically with N; and simulating the quantum system using the second Hamiltonian.

Plane wave dual basis for quantum simulation

Methods, systems and apparatus for estimating an expectation value of a quantum mechanical observable. In one aspect, a method includes identifying a first operator associated with the observable, wherein the first operator comprises a linear combination of terms. One or more constraints on expectation values of one or more of the terms in the linear combination are determined. A second operator is defined, wherein the second operator comprises a combination of the first operator and one or more of the determined constraints. The expectation value of the quantum mechanical observable is estimated using the second operator.

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Venice, CA, United States of America

Ryan Babbush