Preparing Fermions via Classical Sampling and Linear Combinations of Unitaries

We present an extension of the Evolving density matrices on Qubits (EρOQ) 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}\(M²\) 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).