Revealing symmetries in quantum computing for many-body systems

TL;DR

This paper introduces a method to uncover hidden symmetries in many-body Hamiltonians when represented in Pauli matrices for quantum computing, enabling more efficient simulations by reducing qubits.

Contribution

It provides a general theorem to identify symmetry transformations in Pauli representations and derives an affine qubit encoding scheme to eliminate redundant qubits.

Findings

Theorem for symmetry transformation of Pauli strings.

Group representation of symmetries within Clifford group.

Reduced qubit count via Boolean symmetries.

Abstract

We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan-Wigner form for evaluation on quantum computers. Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian. They are, however, masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits. To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations. They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices. We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces computational effort in quantum computing applications.