# A Quantum Solution for Efficient Use of Symmetries in the Simulation of Many-Body Systems

## TL;DR

This paper introduces a quantum algorithm leveraging Grover's search to efficiently identify symmetry-adapted basis states in many-body Hamiltonian simulations, offering exponential memory savings and quadratic speedup over classical methods.

## Contribution

The paper presents a novel quantum approach using Grover's minimization to find symmetry group representatives, reducing memory and computational complexity in many-body system simulations.

## Key findings

- Quantum method achieves exponential memory reduction.
- Quadratic speedup over classical algorithms.
- Error mitigation scheme improves robustness without extra qubits.

## Abstract

A many-body Hamiltonian can be block-diagonalized by expressing it in terms of symmetry-adapted basis states. Finding the group orbit representatives of these basis states and their corresponding symmetries is currently a memory/computational bottleneck on classical computers during exact diagonalization. We apply Grover's search in the form of a minimization procedure to solve this problem. Our quantum solution provides an exponential reduction in memory, and a quadratic speedup in time over classical methods. We discuss explicitly the full circuit implementation of Grover minimization as applied to this problem, finding that the oracle only scales as polylog in the size of the group, which acts as the search space. Further, we design an error mitigation scheme that, with no additional qubits, reduces the impact of bit-flip errors on the computation, with the magnitude of mitigation directly correlated with the error rate, improving the utility of the algorithm in the Noisy Intermediate Scale Quantum era.