Preparing Fermions via Classical Sampling and Linear Combinations of Unitaries

Abstract

We present an extension of the Evolving density matrices on Qubits (E$\ddot{I}$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 naïve 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 $\epsilon$, $M$ is found to scale empirically as $M\propto \frac{1}{mg}\log (1/g)\log (1/m)$.

9 pages, 5 figures