# US 10,311,370 B2 · Efficient reduction of resources for the simulation of Fermionic Hamiltonians on quantum hardware

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- Number: US 10,311,370 B2
- Document ID: US10311370B2
- Status: Granted
- Kind: B2
- Application number: 15/437,895
- Filed: 2017-02-21
- Published: 2018-02-22
- Granted: 2019-06-04
- Earliest claimed: 2016-08-17
- Assignees: International Business Machines Corporation
- Applicants: International Business Machines Corporation
- Inventors: Sergey Bravyi; Jay M. Gambetta; Antonio Mezzacapo; Paul Kristan Temme
- Web page: https://patentgrep.com/patent/US10311370B2

## CPC

- G06N 10/40: Physical realisations or architectures of quantum processors or components for manipulating qubits, e.g. qubit coupling or qubit control
- G06F 17/16: Matrix or vector computation {, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization (matrix transposition G06F7/78)}
- G06F 30/392: Floor-planning or layout, e.g. partitioning or placement
- G06N 10/60: Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms
- G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

A technique relates to reducing qubits required on a quantum computer. A Fermionic system is characterized in terms of a Hamiltonian. The Fermionic system includes Fermions and Fermionic modes with a total number of 2M Fermionic modes. The Hamiltonian has a parity symmetry encoded by spin up and spin down parity operators. Fermionic modes are sorted such that the first half of 2M modes corresponds to spin up and the second half of 2M modes corresponds to spin down. The Hamiltonian and the parity operators are transformed utilizing a Fermion to qubit mapping that transforms parity operators to a first single qubit Pauli operator on a qubit M and a second single qubit Pauli operator on a qubit 2M. The qubit M having been operated on by the first single qubit Pauli operator and the qubit 2M having been operated on by the second single qubit Pauli operator are removed.

## Claims

### Claim 1

A computer-implemented method of reducing a number of hardware qubits required on a quantum computer, the method comprising:
characterizing a Fermionic system in terms of a Hamiltonian, the Fermionic system including Fermions and Fermionic modes with a total number of 2M Fermionic modes, wherein the Hamiltonian has a parity symmetry encoded by spin up and spin down parity operators;
sorting the Fermionic modes on the Hamiltonian, such that the first half of the 2M modes corresponds to spin up and the second half of the 2M modes corresponds to spin down;
transforming the Hamiltonian and the parity operators utilizing a Fermion to qubit mapping, wherein the Fermion to qubit mapping transforms the parity operators to a first single qubit Pauli operator on a qubit M and a second single qubit Pauli operator on a qubit 2M; and
removing the qubit M having been operated on by the first single qubit Pauli operator and the qubit 2M having been operated on by the second single qubit Pauli operator, the removing causing a reduced number of hardware qubits to be utilized in the quantum cormputer.

### Claim 2 (depends on 1)

The method of claim 1, wherein the Fermion to qubit mapping is a generalized Jordan-Wigner transformation.

### Claim 3 (depends on 1)

The method of claim 1, wherein the first single qubit Pauli operator on the qubit M is a Pauli Z-matrix on the qubit M.

### Claim 4 (depends on 3)

The method of claim 3, wherein the second single qubit Pauli operator on the qubit 2M is a product of two Pauli Z-matrices at sites of the qubit M and the qubit 2M.

### Claim 5 (depends on 1)

The method of claim 1, wherein removing the qubit M having been operated on by the first single qubit Pauli operator comprises replacing the qubit M with an eigenvalues of +1 or −1 according to a parity of the qubit M.

### Claim 6 (depends on 5)

The method of claim 5, wherein removing the qubit 2M having been operated on by the second single qubit Pauli operator comprises replacing the qubit 2M with an eigenvalues of +1 or −1 according to a parity of the qubit 2M.

### Claim 7 (depends on 6)

The method of claim 6, wherein the qubit M encodes the parity of qubit M and the qubit 2M encode the parity of qubit 2M.

### Claim 8 (depends on 7)

The method of claim 7, wherein the parities of the qubit M and the qubit 2M are known in advance.

### Claim 9 (depends on 1)

The method claim 1 further comprising executing the Hamiltonian on the quantum computer.

### Claim 10

A computer-implemented method of reducing a number of hardware qubits required on a quantum computer, the method comprising:
characterizing a Fermionic system in terms of a Hamiltonian, the Fermionic system including Fermions and Fermionic modes;
transforming the Hamiltonian utilizing a Fermion to qubit mapping;
finding Pauli symmetry operators of the Hamiltonian;
transforming the Pauli symmetry operators into single qubit Pauli operators; and
removing every qubit that the single qubit Pauli operators are acting upon, the removing causing a reduced number of hardware qubits to be utilized in the quantum computer.

### Claim 11 (depends on 10)

The method of claim 10, wherein the Fermion to qubit mapping is a generalized Jordan-Wigner transformation or a standard Jordan-Wigner transformation.

### Claim 12 (depends on 10)

The method of claim 10, wherein finding Pauli symmetry operators of the Hamiltonian comprises:
performing a parity check matrix on the Hamiltonian to determine the Pauli symmetry operators;
if no Pauli symmetry operators are found, determining that no further reduction can be performed; and
if the Pauli symmetry operators are found, determining a commuting set of the Pauli symmetry operators.

### Claim 13 (depends on 12)

The method of claim 12, wherein transforming the Pauli symmetry operators into the single qubit Pauli operators comprises constructing a Clifford transformation for the commuting set of the Pauli symmetry operators to map to the commuting set of the Pauli symmetry operators to the single qubit Pauli operators.

### Claim 14

A computer-implemented method of reducing a number of hardware qubits required on a quantum computer, the method comprising:
characterizing a Fermionic system in terms of a Hamiltonian, the Fermionic system including Fermions and Fermionic modes with a total number of M Fermionic modes, wherein the Hamiltonian has particle number symmetry and N particles;
transforming the Hamiltonian utilizing a Fermion to qubit mapping that transforms from M Fermionic modes to M qubits, wherein the M qubits are represented by M-bit strings in a computational basis; and
applying a compression map to the Hamiltonian such that the Hamiltonian having the M qubits is mapped to a transformed Hamiltonian with Q qubits where Q<M, wherein the compression map maps the M-bit strings labeling the M qubits in the computational basis with Hamming weight N to Q-bit strings, the transformed Hamiltonian causing a reduced number of the hardware qubits to be utilized in the quantum computer.

### Claim 15 (depends on 14)

The method of claim 14, wherein the computational basis is a 0 and 1 for each of the M qubits.

### Claim 16 (depends on 14)

The method of claim 14 further comprising:
in response to the quantum computer measuring an energy of the transformed Hamiltonian with the Q qubits, receiving measured energy for each compressed term of the transformed Hamiltonian, the quantum computer including a quantum measurement circuit on Q+1 qubits;
performing a decoding on the measured energy to obtain measured results of each uncompressed term in the Hamiltonian; and
combining the measured results to obtain the energy of the Hamiltonian prior to applying the compression map.

### Claim 17 (depends on 16)

The method of claim 16, wherein:
the transformed Hamiltonian comprises compressed terms; and
the Hamiltonian prior to applying the compression map comprises uncompressed terms.

### Claim 18

A system comprising:
memory including computer-executable instructions for reducing a number of hardware qubits required on a quantum computer; and
a processor executing the computer-executable instructions, the computer-executable instructions causing the processor to perform operations comprising:
characterizing a Fermionic system in terms of a Hamiltonian, the Fermionic system including Fermions and Fermionic modes with a total number of 2M Fermionic modes, wherein the Hamiltonian has a parity symmetry encoded by spin up and spin down parity operators;
sorting the Fermionic modes on the Hamiltonian, such that the first half of the 2M modes corresponds to spin up and the second half of the 2M modes corresponds to spin down;
transforming the Hamiltonian and the parity operators utilizing a Fermion to qubit mapping, wherein the Fermion to qubit mapping transforms the parity operators to a first single qubit Pauli operator on a qubit M and a second single qubit Pauli operator on a qubit 2M; and
removing the qubit M having been operated on by the first single qubit Pauli operator and the qubit 2M having been operated on by the second single qubit Pauli operator, the removing causing a reduced number of hardware qubits to be utilized in the quantum computer.

### Claim 19 (depends on 18)

The system of claim 18, wherein the Fermion to qubit mapping is a generalized Jordan-Wigner transformation.

### Claim 20 (depends on 19)

The system of claim 19, wherein the first single qubit Pauli operator on the qubit M is a Pauli Z-matrix on the qubit M.

### Claim 21 (depends on 20)

The system of claim 20, wherein the second single qubit Pauli operator on the qubit 2M is a product of two Pauli Z-matrices at sites of the qubit M and the qubit 2M.

### Claim 22 (depends on 18)

The system of claim 18, wherein removing the qubit M having been operated on by the first single qubit Pauli operator comprises replacing the qubit M with an eigenvalues of +1 or −1 according to a parity the qubit M.

### Claim 23 (depends on 22)

The system of claim 22, wherein removing the qubit 2M having been operated on by the second single qubit Pauli operator comprises replacing the qubit 2M with an eigenvalues of +1 or −1 according to a parity the qubit 2M.

### Claim 24 (depends on 23)

The system of claim 23, wherein:
the qubit M encodes the parity of the qubit M and the qubit 2M encodes the parity of qubit 2M; and
the parities of the qubit M and the qubit 2M are known in advance.

### Claim 25

A system comprising:
memory including computer-executable instructions for reducing a number of hardware qubits required on a quantum computer; and
a processor executing the computer-executable instructions, the computer-executable instructions causing the processor to perform operations comprising:
characterizing a Fermionic system in terms of a Hamiltonian, the Fermionic system including Fermions and Fermionic modes;
transforming the Hamiltonian utilizing a Fermion to qubit mapping;
finding Pauli symmetry operators of the Hamiltonian;
transforming the Pauli symmetry operators into single qubit Pauli operators; and
removing every qubit that the single qubit Pauli operators are acting upon, the removing causing a reduced number of hardware qubits to be utilized in the quantum computer.