Direct Estimation of the Density of States for Fermionic Systems

Simulating time evolution is one of the most natural applications of quantum computers and is thus one of the most promising prospects for achieving practical quantum advantage. Here we develop quantum algorithms to extract thermodynamic properties by estimating the density of states (DOS), a central object in quantum statistical mechanics. We introduce key innovations that significantly improve the practicality and extend the generality of previous techniques. First, our approach allows one to estimate the DOS for a specific subspace of the full Hilbert space. This is crucial for fermionic systems, since fermion-to-qubit mappings partition the full Hilbert space into subspaces of fixed number, on which both canonical and grand canonical ensemble properties depend. Second, in our approach, by time evolving very simple, random initial states (e.g. random computational basis states), we can exactly recover the DOS on average. Third, due to circuit-depth limitations, we only reconstruct the DOS up to a convolution with a Gaussian window - thus all imperfections that shift the energy levels by less than the width of the convolution window will not significantly affect the estimated DOS. For these reasons we find the approach is a promising candidate for early quantum advantage as even short-time, noisy dynamics yield a semi-quantitative reconstruction of the DOS (convolution with a broad Gaussian window), while early fault tolerant devices will likely enable higher resolution DOS reconstruction through longer time evolution. We demonstrate the practicality of our approach in representative Fermi-Hubbard and spin models and find that our approach is highly robust to algorithmic errors in the time evolution and to gate noise. We show that our approach is compatible with NISQ-friendly variational methods, introducing a new technique for variational time evolution in noisy DOS computations.