Quantum supremacy regime for compressed fermionic models

Compressible models extend the domain of simulable systems in quantum computers, but little is known about their precise limits of applicability. Using the theory of compressible matchgate circuits, we identify a class of quadratic fermionic Hamiltonians that can be simulated in compressed space. In particular, for systems of $n$ orbitals encoded to 2-local qubit models with nearest neighbour interactions, the ground state energy can be evaluated with $O\left(\log n\right)$ sets of measurements, independently of the number of dimensions in which the $n$ sites are arranged. We also provide an expressible circuit ansatz in a logarithmic number of qubits for finding the compressed ground state with a variational quantum eigensolver. From the complexity analysis of the compressed circuits, we find a regime of quantum supremacy for sampling compressed Gaussian fermionic models.