Nearly optimal quantum algorithm for generating the ground state of a free quantum field theory

TL;DR

This paper introduces a nearly optimal quantum algorithm for efficiently generating the ground state of a free quantum field theory, achieving a super-quadratic speedup over previous methods and applicable to both translationally invariant and broken symmetry cases.

Contribution

The authors develop two quantum algorithms, Fourier-based and wavelet-based, that generate the ground state of a free massive scalar bosonic QFT with quasilinear gate complexity, improving over prior approaches.

Findings

Fourier-based algorithm works for translationally invariant QFTs.

Wavelet-based algorithm handles broken translational invariance.

Cost of preparing particle excitations is independent of energy scale.

Abstract

We devise a quasilinear quantum algorithm for generating an approximation for the ground state of a quantum field theory (QFT). Our quantum algorithm delivers a super-quadratic speedup over the state-of-the-art quantum algorithm for ground-state generation, overcomes the ground-state-generation bottleneck of the prior approach and is optimal up to a polylogarithmic factor. Specifically, we establish two quantum algorithms -- Fourier-based and wavelet-based -- to generate the ground state of a free massive scalar bosonic QFT with gate complexity quasilinear in the number of discretized-QFT modes. The Fourier-based algorithm is limited to translationally invariant QFTs. Numerical simulations show that the wavelet-based algorithm successfully yields the ground state for a QFT with broken translational invariance. Furthermore, the cost of preparing particle excitations in the wavelet approach is independent of the energy scale. Our algorithms require a routine for generating one-dimensional Gaussian (1DG) states. We replace the standard method for 1DG-state generation, which requires the quantum computer to perform lots of costly arithmetic, with a novel method based on inequality testing that significantly reduces the need for arithmetic. Our method for 1DG-state generation is generic and could be extended to preparing states whose amplitudes can be computed on the fly by a quantum computer.