Preparing Fermions via Classical Sampling and Linear Combinations of Unitaries

TL;DR

This paper introduces an improved method for preparing fermionic quantum states on quantum computers by combining classical sampling with a linear combination of unitaries, reducing circuit complexity and enabling practical simulations.

Contribution

It extends the E$ ho$OQ framework to efficiently prepare fermionic states using a hybrid classical-quantum approach that mitigates the sign problem and reduces circuit scaling.

Findings

Requires $ ext{O}(M^2)$ $R_Z$ rotations for state preparation.

Validated on the Thirring model for ground and excited states.

Empirical scaling of $M$ with system parameters.

Abstract

We present an extension of the Evolving density matrices on Qubits (E$\rho$OQ) framework that enables efficient fault-tolerant preparation of fermionic quantum states. The original method circumvents state preparation by stochastic sampling, but faces a sign problem in fermionic systems leading to a large number of circuits necessary. We resolve this by combining classical stochastic sampling with a linear combination of unitaries method that avoids the exponential circuit scaling that plagued na\"{i}ve implementations. The resulting algorithm requires $\mathcal{O}(M^2)$ $R_Z$ rotations for circuit preparation, where $M$ is the number of retained basis states. We validate the method for ground and excited states in the Thirring model, including by computing two-point correlation functions relevant to scattering. In this model for fixed accuracy $\varepsilon$, $M$ is found to scale empirically as $M \propto \frac{1}{mg}\log(1/g)\log(1/m)$.