Precision-preserving qubit reduction based on spatial symmetries in fermionic systems

Patent metadata

Field	Value
Publication number	US-11894107-B2
Application number	US-201916660059-A
Country	US
Kind code	B2
Filing date	Oct 22, 2019
Priority date	Oct 22, 2019
Publication date	Feb 6, 2024
Grant date	Feb 6, 2024

Abstract

Systems and techniques that facilitate precision-preserving qubit reduction based on spatial symmetries in fermionic systems are provided. In one or more embodiments, a symmetry component can generate a diagonalized second quantization representation of a spatial point group symmetry operation. The spatial point group symmetry operation can be associated with a molecule (e.g., a geometrical rotation, reflection, and/or inversion of a physical molecule that results in a new molecular orientation that is substantially the same as the original molecular orientation). In one or more embodiments, a transformation component can convert the diagonalized second quantization representation into a single Pauli string. In one or more embodiments, a tapering component can taper off qubits in a computational quantum algorithm that models properties of the molecule, based on the single Pauli string. Various embodiments can thus leverage geometrical spatial symmetries of a molecule to reduce a number of qubits needed to simulate the molecule.

First claim

What is claimed is:

1. A system, comprising:
a memory that stores computer-executable components; and
a processor, operably coupled to the memory, that executes the computer-executable components stored in the memory, wherein the processor;
obtains a computational quantum algorithm that models properties of the molecule, wherein the computational quantum algorithm is employable for simulating the molecule at a defined precision using a first number of qubits of a quantum device;
identifies a spatial point group symmetry operation associated with the molecule;
generates a unitary matrix that corresponds to the spatial point group symmetry operation;
generates a Hermitian matrix based on the unitary matrix;
generates a diagonalized second quantization representation of the spatial point group symmetry operation based on the Hermitian matrix converts the diagonalized second quantization representation into a single Pauli string;
tapers the computational quantum algorithm based on the single Pauli string, wherein the tapered computational quantum algorithm is employable for simulating the molecule at the defined precision using a second number of qubits of the quantum device that is less than the first number of qubits; and
simulates, using the tapered computational quantum algorithm with the second number of qubits of the quantum device, the molecule.

2. The system of claim 1 , wherein the processor converts the diagonalized second quantization representation by one from a group consisting of a Jordan-Wigner transformation, a Parity transformation, a Bravyi-Kitaev transformation, a Superfast Encoding transformation, and a Generalized Superfast Encoding transformation.

3. The system of claim 1 , wherein:
the molecule is one from a group consisting of hydrogen (H 2 ), hydrogen chloride (HCl), beryllium hydride (BeH 2 ), ammonia (NH 3 ), water (H 2 O), ethylene (C 2 H 4 ), boron trifluoride (BF 3 ), lithium hydride (LiH), carbon dioxide (CO 2 ), and acetylene (C 2 H 2 ).

4. The system of claim 1 , wherein:
the spatial point group symmetry operation is one from a group consisting of a proper rotation about a principal axis, a reflection through a plane of symmetry, an inversion about a center of symmetry, and an improper rotation about the principal axis.

5. The system of claim 1 , wherein:
the generating the unitary matrix comprises generating a unitary n-by-n permutation matrix, R, that corresponds to the spatial point group symmetry operation, such that there exists a first matrix, {circumflex over (R)}, wherein {circumflex over (R)}a p {circumflex over (R)} † =Σ q=1 n p|R|q a q for all 1≤p≤n, wherein n is a number of fermionic modes of the molecule, and wherein a p and a q are fermionic annihilation operators;
the generating the Hermitian matrix comprises generating the Hermitian matrix, G, wherein R=exp (iG) such that there exists a second matrix, Ĝ, wherein R=exp(iG), such that there exists a second matrix, Ĝ, wherein Ĝ=Σ p,q=1 n p|G|q a p † a q ; and
the generating the diagonalized second quantization representation comprises:
generating a unitary n-by-n matrix, V, that diagonalizes G such that V † GV=Σ p=1 n λ p |p) (p|, wherein λ p is a real eigenvalue, such that there exists a third matrix, {circumflex over (V)}, wherein {circumflex over (V)}, wherein {circumflex over (V)}a p {circumflex over (V)} † =Σ q=1 n p|V|q a q for all 1≤p≤n; and
generating as the diagonalized second quantization representation an operator S such that S={circumflex over (V)}{circumflex over (R)}{circumflex over (V)} † =exp({circumflex over (V)}Ĝ{circumflex over (V)} † )=Π p=1 n exp (iλ p a p † a p ).

6. The system of claim 5 , wherein:
R 2 =I, wherein I is an identity matrix, such that λ p ∈{0, π}, and such that S=Π p∈M (−1) a p † a p wherein M is a subset of fermionic modes having λ p =π.

7. The system of claim 6 , wherein the processor converts the operator S into a single Pauli-Z string via a Jordan-Wigner transformation, such that S=Π p∈M σ z p , wherein σ z is a Pauli-Z matrix.

8. A computer-implemented method, comprising:
obtaining, by a system operatively coupled to a processor, a computational quantum algorithm that models properties of the molecule, wherein the computational quantum algorithm is employable for simulating the molecule at a defined precision using a first number of qubits of a quantum device;
identifying, by the system, a spatial point group symmetry operation associated with the molecule;
generating, by the system, a unitary matrix that corresponds to the spatial point group symmetry operation;
generating, by the system, a Hermitian matrix based on the unitary matrix;
generating, by a system operatively coupled to a processor, a diagonalized second quantization representation of the spatial point group symmetry operation based on the Hermitian matrix;
converting, by the system, the diagonalized second quantization representation into a single Pauli string; and
tapering off, by the system, the computational quantum algorithm based on the single Pauli string, wherein the tapered computational quantum algorithm is employable for simulating the molecule at the defined precision using a second number of qubits of the quantum device that is less than the first number of qubits; and
simulating, by the system, using the tapered computational quantum algorithm with the second number of qubits of the quantum device, the molecule.

9. The computer-implemented method of claim 8 , wherein:
the converting the diagonalized second quantization representation includes comprises employing one from a group consisting of a Jordan-Wigner transformation, a Parity transformation, a Bravyi-Kitaev transformation, a Superfast Encoding transformation, and a Generalized Superfast Encoding transformation.

10. The computer-implemented method of claim 8 , wherein:
the molecule is one from a group consisting of hydrogen (H 2 ), hydrogen chloride (HCl), beryllium hydride (BeH 2 ), ammonia (NH 3 ), water (H 2 O), ethylene (C 2 H 4 ), boron trifluoride (BF 3 ), lithium hydride (LiH), carbon dioxide (CO 2 ), and acetylene (C 2 H 2 ).

11. The computer-implemented method of claim 8 , wherein:
the spatial point group symmetry operation is one from a group consisting of a proper rotation about a principal axis, a reflection through a plane of symmetry, an inversion about a center of symmetry, and an improper rotation about the principal axis.

12. The computer-implemented method of claim 8 , wherein:
the generating the unitary matrix comprises generating a unitary n-by-n permutation matrix, R, that corresponds to the spatial point group symmetry operation, such that there exists a first matrix, {circumflex over (R)}, wherein {circumflex over (R)}a p {circumflex over (R)} † =Σ q=1 n p|R|q a q for all 1≤p≤n, wherein n is a number of fermionic modes of the molecule, and wherein a p and a q are fermionic annihilation operators;
the generating the Hermitian matrix comprises generating the Hermitian matrix, G, wherein R=exp (iG) such that there exists a second matrix, Ĝ, wherein R=exp(iG), such that there exists a second matrix, Ĝ, wherein Ĝ=Σ p,q=1 n p|G|q a p † a q ; and
the generating the diagonalized second quantization representation comprises:
generating a unitary n-by-n matrix, V, that diagonalizes G such that V † GV=Σ p=1 n λ p | p|, wherein λ p is a real eigenvalue, such that there exists a third matrix, {circumflex over (V)}, wherein {circumflex over (V)}, wherein {circumflex over (V)}a p {circumflex over (V)} † =Σ q=1 n p|V|q a q for all 1≤p≤n; and
generating as the diagonalized second quantizatio

Assignees

• IBM

Inventors

• Setia Kanav
• Bravyi Sergey
• Mezzacapo Antonio
• Chen Richard
• Pistoia Marco
• Rice Julia Elizabeth

Classifications

• G06N10/60 Primary

  Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms · CPC title

• G16C10/00 Primary

  Computational theoretical chemistry, i.e. ICT specially adapted for theoretical aspects of quantum chemistry, molecular mechanics, molecular dynamics or the like · CPC title

• G06F17/16

  Matrix or vector computation {, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization (matrix transposition G06F7/78)} · CPC title

• G06N10/00

  Quantum computing, i.e. information processing based on quantum-mechanical phenomena · CPC title