Stabilizer-based quantum simulation of fermion dynamics with local qubit encodings

TL;DR

This paper introduces a new framework for simulating fermion dynamics on quantum computers using local qubit encodings, classifying their structures, and designing low-depth circuits with stabilizer formalism, improving efficiency in fermionic simulations.

Contribution

It proposes a systematic classification of local fermion-to-qubit encodings via flow sets and develops low-depth quantum circuits for their time evolution, enhancing simulation efficiency.

Findings

Classified local fermionic encodings using flow sets.

Designed low-depth circuits for various encodings.

Found a space-time trade-off in encoding efficiency.

Abstract

Simulating the dynamical properties of large-scale many-fermion systems is a longstanding goal of quantum chemistry, material science and condensed matter. Local fermion-to-qubit encodings have opened a new path for practical fermionic simulations on digital quantum hardware where fermionic statistics are not enforced at the hardware level. In this paper, we explore these local encodings from the perspective of the corresponding time-evolution unitaries. Specifically, we propose a new framework for digital implementations of these qubit-encoded fermionic time-evolution unitaries based on \emph{flow sets}, which are one-dimensional subsets of the directed fermionic interaction graph. We find that any local fermionic encoding, when restricted to a given flow set, adopts a simple structure that we can classify systematically. For each categorized flow-set form, we propose a low-depth qubit quantum circuit that implements the time evolution unitary using the stabilizer formalism. As an application of our construction, we introduce novel flow-based decompositions for known two-dimensional encodings, leading to efficient circuit decompositions of time-evolution unitaries. We generally observe a space-time trade-off, where mappings with larger qubit-to-fermion ratios yield shallower time-evolution quantum circuits.