Fermionic Hamiltonians without trivial low-energy states

TL;DR

This paper constructs local fermionic Hamiltonians that lack low-energy trivial states, extending the NLTS theorem to fermionic systems and introducing a fermionic quantum PCP analogue.

Contribution

It introduces a fermionic NLTS construction, defining trivial states via fermionic circuits and relating it to a fermionic quantum PCP class.

Findings

Fermionic Hamiltonians without low-energy trivial states are constructed.

The fermionic NLTS property is established using qubit NLTS Hamiltonians.

Discussion of a fermionic quantum PCP class and its relation to the qubit version.

Abstract

We construct local fermionic Hamiltonians with no low-energy trivial states (NLTS), providing a fermionic counterpart to the NLTS theorem. Distinctly from the qubit case, we define trivial states via finite-depth $\textit{fermionic}$ quantum circuits. We furthermore allow free access to Gaussian fermionic operations, provided they involve at most $O(n)$ ancillary fermions. The desired fermionic Hamiltonian can be constructed using any qubit Hamiltonian which itself has the NLTS property via well-spread distributions over bitstrings, such as the construction in [Anshu, Breuckmann, Nirkhe, STOC 2023]. We define a fermionic analogue of the class quantum PCP and discuss its relation with the qubit version.