Quantum algorithms to simulate many-body physics of correlated fermions

TL;DR

This paper develops optimized quantum algorithms for simulating strongly correlated fermionic systems, including state preparation, Fourier transformation, and evolution simulation on 2D qubit arrays, advancing quantum simulation capabilities.

Contribution

It introduces new algorithms for fermionic state preparation, Fourier transform, and evolution simulation optimized for 2D qubit arrays, with minimal circuit depth and gate counts.

Findings

Algorithms for fermionic Gaussian state preparation are optimal in parameters.

Proposed 2D fermionic Fourier transform uses minimal circuit depth.

Simulation methods for 2D Fermi-Hubbard model are efficient and scalable.

Abstract

Simulating strongly correlated fermionic systems is notoriously hard on classical computers. An alternative approach, as proposed by Feynman, is to use a quantum computer. Here, we discuss quantum simulation of strongly correlated fermionic systems. We focus specifically on 2D and linear geometry with nearest neighbor qubit-qubit couplings, typical for superconducting transmon qubit arrays. We improve an existing algorithm to prepare an arbitrary Slater determinant by exploiting a unitary symmetry. We also present a quantum algorithm to prepare an arbitrary fermionic Gaussian state with $O(N^2)$ gates and $O(N)$ circuit depth. Both algorithms are optimal in the sense that the numbers of parameters in the quantum circuits are equal to those to describe the quantum states. Furthermore, we propose an algorithm to implement the 2-dimensional (2D) fermionic Fourier transformation on a 2D qubit array with only $O(N^{1.5})$ gates and $O(\sqrt N)$ circuit depth, which is the minimum depth required for quantum information to travel across the qubit array. We also present methods to simulate each time step in the evolution of the 2D Fermi-Hubbard model---again on a 2D qubit array---with $O(N)$ gates and $O(\sqrt N)$ circuit depth. Finally, we discuss how these algorithms can be used to determine the ground state properties and phase diagrams of strongly correlated quantum systems using the Hubbard model as an example.