Preparing Fermions via Classical Sampling and Linear Combinations of Unitaries

Abstract

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

Figures (5)

• Figure 1: Schematic implementation of the state preparation portion of proposed algorithm. It can take a classically-obtained fermionic initial state obtained from many sources, and compute time-evolved expectation values. The quantum circuit is for Eq. \ref{['eq:classic']} with $\theta_1=0.57081,\theta_2=2.0663,\theta_3=0.62978$. The Prep circuit prepares the dense state while the Select circuit is comprised of multicontrolled gates which select the appropriate bit strings.
• Figure 2: Analysis of real part of the recurrence probability $R(t)$ of the ground state for of $N_s=16$, $g=0.1$, $m_0=1.0$ for different $M$. Top: $R(t)$ and difference between the simulated and exact value. Bottom: Fourier transformation of $R(t)$.
• Figure 3: Values of $M$ needed to ensure $\epsilon<10^{-2}$ for $N_s=16$.
• Figure 4: Analysis of first excited state with (top) $R(t)$ and (bottom) Fourier spectrum (Bottom) for the first excited state of $N_s=16$, $g=0.1$, $m_0=1.0$ for different $M$.
• Figure 5: Two-point correlation function $C^{0,0}(6, t)$ vs. $t$ and $\bar{\varepsilon}$ vs. $M$ for $m_0=0.6$, $g=0.4$, and $N=16$. Couplings are different from other examples to show case broader applicability and in different regime.