On the stability to noise of fermion-to-qubit mappings

TL;DR

This paper analyzes how local fermionic encodings in quantum simulations provide stability against noise, especially in states with decaying correlations, unlike non-local encodings like Jordan-Wigner.

Contribution

It demonstrates that local fermionic encodings ensure stability to noise in quadratic observables for states with decaying correlations, unlike non-local encodings.

Findings

Local fermionic encodings provide noise stability for quadratic observables.

Stability depends on the decay rate of correlations in the state.

Non-local encodings like Jordan-Wigner lack this stability in 2D.

Abstract

Quantum simulations before fault tolerance suffer from the intrinsic noise present in quantum computers. In this regime, extracting meaningful results greatly benefits from stability against that noise. This stability, defined as an error in observables that is independent of the system's size, is expected in local systems under local noise. In fermionic systems, the encoding of the fermionic degrees of freedom into qubits can introduce non-locality, making stability more delicate. Here, we investigate the stability to noise of fermion-to-qubit mappings. We consider noisy quantum circuits in $D$ dimensions modeled by alternating layers of local unitaries and general, single-qubit Pauli noise. We show that, when using local fermionic encodings, expectation values of quadratic fermionic observables are stable to noise in states with spatially decaying correlations: a power-law decay with exponent $\mu>D$ is sufficient for stability. By contrast, we show that this stability cannot be achieved by non-local encodings such as Jordan-Wigner in $2D$, or quasi-local ones such as the Bravyi-Kitaev transform. Our findings formalize the intuition that decaying correlations of the physical systems under study provide protection against noise for local fermionic encodings, and help inform design principles in near-term quantum simulations.