Constructing Fermionic Hamiltonians with Non-Gaussianic low-energy states

TL;DR

This paper constructs fermionic Hamiltonians where Gaussian states, a subset of fermionic states, have energies bounded below by a constant, providing insights into quantum complexity conjectures related to ground state energies.

Contribution

It introduces a new class of fermionic Hamiltonians with energy bounds for Gaussian states, extending techniques from previous work on Clifford states.

Findings

Gaussian states have a constant lower energy bound on the constructed Hamiltonians.

The construction adapts techniques from Clifford state analysis to fermionic systems.

Provides evidence supporting the complexity of approximating ground state energies in fermionic systems.

Abstract

Quantum PCP conjecture is one of the most influential open problems in quantum complexity theory, which states that approximating the ground state energy for a sparse local Hamiltonian upto a constant is QMA-complete. However, even though the problem remains unsolved, weaker versions of it-such as the NLTS [FH13, ABN22] and NLSS [GG22] conjectures-have surfaced in the hope of providing evidence for QPCP. While the NLTS hamiltonians were first constructed in[ABN22], NLSS conjecture still remains unsolved. Weaker versions of the NLSS conjecture were addressed in [CCNN23, CCNN24], demonstrating that Clifford and almost-Clifford states-a subclass of sampleable states-have a lower energy bound on Hamiltonians prepared by conjugating the NLTS Hamiltonians from [ABN22]. In similar spirit, we construct a class of fermionic Hamilltonians for which energy of Gaussian states, a subclass of sampleable fermionic states, is bounded below by a constant. We adapt the technique used in [CCNN23] to our context.