# US 12,009,063 B2 · Quantum information processing method and device for finding a differential of energy

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- Number: US 12,009,063 B2
- Document ID: US12009063B2
- Status: Granted
- Kind: B2
- Application number: 17/159,954
- Filed: 2021-01-27
- Published: 2021-06-17
- Granted: 2024-06-11
- Earliest claimed: 2019-05-13
- Assignees: Qunasys Inc.
- Applicants: Qunasys Inc.
- Inventors: Kosuke Mitarai; Yuya Nakagawa
- Web page: https://patentgrep.com/patent/US12009063B2

## CPC

- G16C 10/00: Computational theoretical chemistry, i.e. ICT specially adapted for theoretical aspects of quantum chemistry, molecular mechanics, molecular dynamics or the like
- G06F 17/16: Matrix or vector computation {, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization (matrix transposition G06F7/78)}
- G06N 10/20: Models of quantum computing, e.g. quantum circuits or universal quantum computers
- G06N 10/60: Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms
- G16C 20/30: Prediction of properties of chemical compounds, compositions or mixtures
- G16C 20/90: Programming languages; Computing architectures; Database systems; Data warehousing
- G16C: COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS

## Abstract

A classical computer outputs a Hamiltonian and initial information of a parameter expressing a quantum circuit. The classical computer, according to a parameter expressing a first quantum circuit that was output from a quantum computer and was generated by quantum computation employing a Variational Quantum Eigensolver (VQE) based on the Hamiltonian and the initial information, generates a parameter expressing a second quantum circuit including a rotation gate and outputs the parameter expressing the second quantum circuit. The classical computer, based on measurement results of quantum computation that were output from the quantum computer and computed according to the parameter expressing the second quantum circuit, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generates a derivative function of energy corresponding to the Hamiltonian and outputs the derivative function of energy.

## Claims

### Claim 1

A quantum information processing method for finding a differential of energy by processing executed on a hybrid system including a classical computer and a quantum computer, the quantum information processing method comprising:
the classical computer outputting a Hamiltonian and initial information of a parameter of a quantum circuit, wherein the Hamiltonian is a Hamiltonian of a substance;
the quantum computer, based on the Hamiltonian and the initial information output from the classical computer, executing quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit to generate a parameter of a first quantum circuit for computing energy, and outputting the parameter of the first quantum circuit;
the classical computer, based on the parameter of the first quantum circuit output from the quantum computer, generating a parameter of a second quantum circuit including a rotation gate by inserting the rotation gate between a plurality of unitary gates in the first quantum circuit, and outputting the parameter of the second quantum circuit;
the quantum computer, based on the parameter of the second quantum circuit output from the classical computer, executing quantum computation and outputting measurement results of this quantum computation; and
the classical computer, based on the measurement results output from the quantum computer, the Hamiltonian, and a derivative function of the Hamiltonian, computing a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 2 (depends on 1)

The quantum information processing method for finding a differential of energy of claim 1, wherein:
the classical computer and the quantum computer are connected over a computer network; and
the classical computer and the quantum computer exchange information with each other over the computer network.

### Claim 3

A quantum information processing method for finding a differential of energy by processing executed by a classical computer, the processing comprising:
outputting a Hamiltonian and initial information of a parameter expressing of a quantum circuit;
according to a parameter of a first quantum circuit that was output from a quantum computer and was generated by quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit based on the Hamiltonian and the initial information, generating a parameter of a second quantum circuit including a rotation gate and outputting the parameter of the second quantum circuit; and
based on measurement results of quantum computation that were output from the quantum computer and computed according to the parameter of the second quantum circuit, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 4

A quantum information processing method for finding a differential of energy by processing executed by a quantum computer, the processing comprising:
based on a Hamiltonian and initial information of a parameter of a quantum circuit that were output from a classical computer, executing quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit to generate a parameter of a first quantum circuit and outputting the parameter of the first quantum circuit;
based on a parameter that expresses a second quantum circuit including a rotation gate, that was output from a classical computer, and that accords with the parameter of the first quantum circuit, executing quantum computation and outputting measurement results of the quantum computation; and
based on measurement results of quantum computation, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 5

A classical computer configured to execute processing, the processing comprising:
outputting a Hamiltonian and initial information of a parameter of a quantum circuit;
according to a parameter of a first quantum circuit that was output from a quantum computer and was generated by quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit based on the Hamiltonian and the initial information, generating a parameter of a second quantum circuit including a rotation gate and outputting the parameter of the second quantum circuit; and
based on measurement results of quantum computation that were output from the quantum computer and computed according to the parameter of the second quantum circuit, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 6

A quantum computer configured to execute processing, the processing comprising:
based on a Hamiltonian and initial information of a parameter of a quantum circuit that were output from a classical computer, executing quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit to generate a parameter of a first quantum circuit for computing energy and outputting the parameter of the first quantum circuit;
based on a parameter that expresses a second quantum circuit including a rotation gate, that was output from a classical computer, and that accords with the parameter of the first quantum circuit, executing quantum computation and outputting measurement results of the quantum computation; and
based on measurement results of quantum computation, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 7

A non-transitory recording medium storing a quantum information processing program to cause a classical computer to execute processing, the processing comprising:
outputting a Hamiltonian and initial information of a parameter of a quantum circuit;
according to a parameter of a first quantum circuit for computing energy that was output from a quantum computer and was generated by quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit based on the Hamiltonian and the initial information, generating a parameter of a second quantum circuit including a rotation gate and outputting the parameter of the second quantum circuit; and
based on measurement results of quantum computation that were output from the quantum computer and computed according to the parameter of the second quantum circuit, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 8

A non-transitory recording medium storing a quantum information processing program to cause a quantum computer to execute processing, the processing comprising:
based on a Hamiltonian and initial information of a parameter of a quantum circuit that were output from a classical computer, executing quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit to generate a parameter of a first quantum circuit for computing energy and outputting the parameter of the first quantum circuit;
based on a parameter that expresses a second quantum circuit including a rotation gate, that was output from a classical computer, and that accords with the parameter of the first quantum circuit, executing quantum computation and outputting measurement results of the quantum computation; and
based on measurement results of quantum computation, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 9 (depends on 5)

A hybrid system comprising:
the classical computer of claim 5; and
the quantum computer configured to execute processing, the processing comprising:
based on a Hamiltonian and initial information of a parameter of a quantum circuit that were output from a classical computer, executing quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit to generate a parameter of a first quantum circuit for computing energy and outputting the parameter of the first quantum circuit;
based on a parameter that expresses a second quantum circuit including a rotation gate, that was output from a classical computer, and that accords with the parameter of the first quantum circuit, executing quantum computation and outputting measurement results of the quantum computation.

### Claim 10

A quantum information processing method for finding a differential of energy by processing executed by a classical computer, the processing comprising:
according to a parameter of a first quantum circuit that was output from a quantum computer and was generated by quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit based on a Hamiltonian and initial information, generating a parameter of a second quantum circuit including a rotation gate and outputting the parameter of the second quantum circuit; and
based on measurement results of quantum computation that were output from the quantum computer and computed according to the parameter of the second quantum circuit, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 11

A classical computer configured to execute processing, the processing comprising:
according to a parameter of a first quantum circuit that was output from a quantum computer and was generated by quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit on a Hamiltonian and initial information, generating a parameter of a second quantum circuit including a rotation gate and outputting the parameter of the second quantum circuit; and
based on measurement results of quantum computation that were output from the quantum computer and computed according to the parameter of the second quantum circuit, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 12

A non-transitory recording medium storing a quantum information processing program to cause a classical computer to execute processing, the processing comprising:
according to a parameter of a first quantum circuit for computing energy that was output from a quantum computer and was generated by quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit based on a Hamiltonian and initial information, generating a parameter of a second quantum circuit including a rotation gate and outputting the parameter of the second quantum circuit; and
based on measurement results of quantum computation that were output from the quantum computer and computed according to the parameter of the second quantum circuit, based on the Hamiltonian, and based on a derivative function of the Hamiltonian, generating a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.

### Claim 13

A quantum information processing method for finding a differential of energy by processing executed on a hybrid system including at least one or more classical computers and at least one or more quantum computers, the quantum information processing method comprising:
a classical computer outputting a Hamiltonian and initial information of a parameter of a quantum circuit;
a quantum computer, based on the Hamiltonian and the initial information output from a classical computer, executing quantum computation employing a method for computing an approximation of an eigenvalue for minimum of a Hamiltonian by variational updating of a parameter of a quantum circuit to generate a parameter of a first quantum circuit for computing energy, and outputting the parameter of the first quantum circuit;
a classical computer, based on the parameter of the first quantum circuit output from a quantum computer, generating a parameter of a second quantum circuit including a rotation gate by inserting the rotation gate between a plurality of unitary gates in the first quantum circuit, and outputting the parameter of the second quantum circuit;
a quantum computer, based on the parameter of the second quantum circuit output from a classical computer, executing quantum computation and outputting measurement results of this quantum computation; and
a classical computer, based on the measurement results output from a quantum computer, the Hamiltonian, and a derivative function of the Hamiltonian, computing a derivative function of energy corresponding to the Hamiltonian and outputting the derivative function of energy,
wherein the classical computer inserts a rotation gate R±a,μ expressed by Equation (1) below between a plurality of unitary gates in the first quantum circuit in order to compute the derivative function of energy for a ground state, or the classical computer inserts a rotation gate R(s),+a,μ expressed by Equation (2) below between a plurality of unitary gates in the first quantum circuit to compute the derivative function of energy for an excited state
Ra,μ±=exp(±iπPa,μ/4)
Pa,μ∈
={I,X,Y,Z}⊗n (1)
Ra,μ(s),+=exp(iπPa,μ(s)/4)
Pa,μ(s)∈
={I,X,Y,Z}⊗n (2)
wherein a is an index from among indices a, b, c, . . . used to discriminate between elements of a parameter vector of a quantum circuit, μ is an index used to discriminate between Pauli operators, s is an index to represent excited states, is a Pauli operator collection, I, X, Y, and Z are Pauli operators, ⊗ is the Kronecker product symbol, and n is a natural number.