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 superquadratic 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.
27 More- Received 30 December 2021
- Accepted 26 May 2022
DOI:https://doi.org/10.1103/PRXQuantum.3.020364
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Preparing the ground state of free quantum field theory (QFT) is the most expensive part of current approaches for simulating a bosonic QFT on a quantum computer. We develop two quantum algorithms for preparing the ground state using resources that scale almost linearly with the discretized QFT number of modes. Our algorithms are near optimal, provide more-than-quadratic speedup, and remove ground-state preparation as the most expensive part. For ground-state preparation, we must discretize the QFT and digitize the discretized QFT, both of which introduce errors. We account for these errors in a controlled way and take special care in translating physical inputs for QFT simulation into constructing quantum circuits for state preparation and thoroughly analyzing resource costs.
Our first algorithm is based on the wavelet approach and involves matrix operations. The involved matrices have many zero elements that we exploit to reduce the state-preparation cost. Our second algorithm employs translational invariance of the QFT to eliminate costly matrix operations of prior approaches. The wavelet approach is advantageous for two cases: simulating theories with inhomogeneous mass and preparing excitations above the ground state at variable length scales, which is useful for scattering simulations.
Moreover, we construct two quantum algorithms for preparing one-dimensional Gaussian states, which have applications beyond QFT simulation. Our first algorithm uses a standard state-preparation method, which requires costly arithmetic. We employ novel techniques in our second algorithms to significantly reduce arithmetic operations. Our state-preparation techniques could be used broadly in simulating quantum systems and improving current quantum algorithms for differential equations.