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arXiv:1911.03505

Site-by-site quantum state preparation algorithm for preparing vacua of fermionic lattice field theories

Ali Hamed Moosavian, James R. Garrison, Stephen P. Jordan

Published Nov 8, 2019• 10 citations
arXivHTMLPDF

Abstract

Answering whether quantum computers can efficiently simulate quantum field theories has both theoretical and practical motivation. From the theoretical point of view, it answers the question of whether a hypothetical computer that utilizes quantum field theory would be more powerful than other quantum computers. From the practical point of view, when reliable quantum computers are eventually built, these algorithms can help us better understand the underlying physics that govern our world. In the best known quantum algorithms for simulating quantum field theories, the time scaling is dominated by initial state preparation. In this paper, we exclusively focus on state preparation and present a heuristic algorithm that can prepare the vacuum of fermionic systems in more general cases and more efficiently than previous methods. With our method, state preparation is no longer the bottleneck, as its runtime has the same asymptotic scaling with the desired precision as the remainder of the simulation algorithm. We numerically demonstrate the effectiveness of our proposed method for the 1+1 dimensional Gross-Neveu model.

DOI
10.48550/arxiv.1911.03505

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Answering whether quantum computers can efficiently simulate quantum field theories has both theoretical and practical motivation. From the theoretical point of view, it answers the question of whether a hypothetical computer that utilizes quantum field theory would be more powerful than other quantum computers. From the practical point of view, when reliable quantum computers are eventually built, these algorithms can help us better understand the underlying physics that govern our world. In the best known quantum algorithms for simulating quantum field theories, the time scaling is dominated by initial state preparation. In this paper, we exclusively focus on state preparation and present a heuristic algorithm that can prepare the vacuum of fermionic systems in more general cases and more efficiently than previous methods. With our method, state preparation is no longer the bottleneck, as its runtime has the same asymptotic scaling with the desired precision as the remainder of the simulation algorithm. We numerically demonstrate the effectiveness of our proposed method for the 1+1 dimensional Gross-Neveu model.
In this thesis, we begin by reviewing some of the most important Hamiltonian simulation algorithms that are applied in simulation of quantum field theories. Then we focus on state preparation which has been the slowest subroutine in previously known algorithms. We present two distinct methods that improve upon prior results. The first method utilizes classical computational tools such as Density Matrix Renormalization Group to produce an efficient quantum algorithm for simulating fermionic quantum field theories in 1+1 dimensions. The second method presented is a heuristic algorithm that can prepare the vacuum of fermionic systems in more general cases and more efficiently than previous methods. With our last method, state preparation is no longer the bottleneck, as its runtime has the same asymptotic scaling with the desired precision as the remainder of the simulation algorithm. We then numerically demonstrate the effectiveness of this last method for the 1+1 dimensional Gross-Neveu model.
In quantum algorithms discovered so far for simulating scattering processes in quantum field theories, state preparation is the slowest step. We present an algorithm for preparing particle states to use in simulation of fermionic quantum field theory (QFT) on a quantum computer, which is based on the matrix product state ansatz. We apply this to the massive Gross-Neveu model in one spatial dimension to illustrate the algorithm, but we believe the same algorithm with slight modifications can be used to simulate any one-dimensional massive fermionic QFT. In the case where the number of particle species is 1, our algorithm can prepare particle states using $O\left({\ensuremath{\epsilon}}^{\ensuremath{-}3.23...}\right)$ gates, which is much faster than previous known results, namely, $O\left({\ensuremath{\epsilon}}^{\ensuremath{-}8\ensuremath{-}o\left(1\right)}\right)$. Furthermore, unlike previous methods which were based on adiabatic state preparation, the method given here should be able to simulate quantum phases unconnected to the free theory.
An important aspect of quantum simulation is the preparation of physically interesting states on a quantum computer, and this task can often be costly or challenging to implement. A digital, ``site-by-site'' scheme of state preparation was introduced in arXiv:1911.03505 as a way to prepare the vacuum state of certain fermionic field theory Hamiltonians with a mass gap. More generally, this algorithm may be used to prepare ground states of Hamiltonians by adding one site at a time as long as successive intermediate ground states share a non-zero overlap and the Hamiltonian has a non-vanishing spectral gap at finite lattice size. In this paper, we study the ground state overlap as a function of the number of sites for a range of quadratic fermionic Hamiltonians. Using analytical formulas known for free fermions, we are able to explore the large-NNN behavior and draw conclusions about the state overlap. For all models studied, we find that the overlap remains large (e.g. >0.1> 0.1>0.1) up to large lattice sizes (N=64,72N=64,72N=64,72) except near quantum phase transitions or in the presence of gapless edge modes. For one-dimensional systems, we further find that two N/2N/2N/2-site ground states also share a large overlap with the NNN-site ground state everywhere except a region near the phase boundary. Based on these numerical results, we additionally propose a recursive alternative to the site-by-site state preparation algorithm.
Extending previous work on scalar field theories, we develop a quantum algorithm to compute relativistic scattering amplitudes in fermionic field theories, exemplified by the massive Gross-Neveu model, a theory in two spacetime dimensions with quartic interactions. The algorithm introduces new techniques to meet the additional challenges posed by the characteristics of fermionic fields, and its run time is polynomial in the desired precision and the energy. Thus, it constitutes further progress towards an efficient quantum algorithm for simulating the Standard Model of particle physics.
Preparing Gibbs states on a quantum computer offers a way to computationally probe how microscopic interactions give rise to macroscopic phenomena at finite temperature. However, an uncomfortable tension has set in: our strongest general guarantees for Gibbs state preparation hold in regimes where the resulting states can also be simulated classically. We propose investigating physical models with competing energy scales. The Fermi--Hubbard model is a natural target: the competition between electron hopping, on-site repulsion and chemical potential encapsulates many mysteries of condensed matter physics, including Mott insulators, striping and superconductivity. Our main result identifies a simple guiding principle: the temperature scale for efficient preparation is set by the couplings between sites, while the on-site energy scale is unrestricted. For the Fermi--Hubbard model on an nnn-site nearest-neighbor lattice of fixed dimension, we construct a quantum Gibbs sampler that mixes in O(log⁡(n/ε))\mathcal{O}(\log(n/ε))O(log(n/ε)) time whenever the temperature is at least a constant factor larger than the hopping strength, but is allowed to be independent of the repulsion and chemical potential. We obtain analogous guarantees for local qubit systems with bounded intersite interactions and arbitrary external fields. As a consequence, our algorithms can efficiently prepare Fermi--Hubbard Gibbs states that are provably outside the convex hull of fermionic Gaussian states and qubit Gibbs states for locally interacting Hamiltonians outside the set of product states.
While quantum computers are capable of simulating many quantum systems efficiently, the simulation algorithms must begin with the preparation of an appropriate initial state. We present a method for generating physically relevant quantum states on a lattice in real space. In particular, the present algorithm is able to prepare general pure and mixed many-particle states of any number of particles. It relies on a procedure for converting from a second-quantized state to its first-quantized counterpart. The algorithm is efficient in that it operates in time that is polynomial in all the essential descriptors of the system, the number of particles, the resolution of the lattice, and the inverse of the maximum final error. This scaling holds under the assumption that the wave function to be prepared is bounded or its indefinite integral is known and that the Fock operator of the system is efficiently simulatable.