Probing quantum advantage for solving the Fermi-Hubbard model with entropy benchmarking

TL;DR

The paper introduces a practical, hardware-agnostic entropy-based benchmarking framework that uses Gibbs-state boundaries in energy-entropy space to assess when quantum advantage is achievable for optimization problems, with a focus on the 2D Fermi-Hubbard model. It derives tractable, lower-bound relaxations to the Gibbs boundary by decomposing the Hamiltonian into tractable parts and applying Gibbs states to each part, enabling estimation of an entropy threshold beyond which quantum devices cannot outperform classical solvers. Applying the method to the 2D FHM on square lattices up to $N_{sites}=144$ and several partitionings reveals scale-invariant bounds for Plaquettes and a nuanced dependence on $U/t$, guiding the evaluation of quantum circuits (LDCA, HVA) under depolarizing noise. The results indicate no-go results for large instances on current near-term quantum hardware, while outlining a clear, hardware-application separation framework and avenues for extension to fault-tolerant regimes and embedding approaches.

Abstract

We developed a practical quantum advantage benchmarking framework that connects the accumulation of entropy in a quantum processing unit and the degradation of the solution to a target optimization problem. The benchmark is based on approximating from below the Gibbs states boundary in the energy-entropy space for the application of interest. We believe the proposed benchmarking technique creates a powerful bridge between hardware benchmarking and application benchmarking, while remaining hardware-agnostic. It can be extended to fault-tolerant scenarios and relies on computationally tractable numerics. We demonstrate its applicability on the problem of finding the ground state of the two-dimensional Fermi-Hubbard.

Figures (7)

• Figure 1: Gibbs states entropy benchmarking framework. Gibbs states (blue line) separate the energy-entropy parameter space $(E,S)$ into a region of physically achievable parameters from unachievable ones (light red-shaded area). A classical solver providing a candidate solution $E_{\mathrm{class}}$ leads to an entropy threshold $S_{\mathrm{th}}$. Any quantum algorithm running on a realistic QPU surpassing $S_{\mathrm{th}}$ (bright red area) cannot provide quantum advantage. A lower-bound to the Gibbs state boundary (purple) can only yield a more conservative benchmark ($S'_{\mathrm{th}} \geq S_{\mathrm{th}}$).
• Figure 2: Partitioning techniques. The three different decompositions of the periodic 2D FHM used in this work to compute lower bounds to the Gibbs state energies. Example of a size $L=4$ model. PBC are materialized with colored dots representing the same site. Phenomenological partitioning (Phenom.) between the kinetic and the atomic terms, and geometric partitionings into respectively 1D Fermi-Hubbard systems (One-dim.) and Fermi-Hubbard plaquettes (Plaq., valid for even values of $L\geq 4$).
• Figure 3: Onset of scale-invariance for the Gibbs state energy lower-bounds. Results are presented in challenging regimes of correlation $U/t=5, 10$ for the Phenom. and One-dim. lower-bounds, as well as the combination obtained upon taking locally the best lower-bound among the three investigated Phenom., One-dim. and Plaq.. The latter lower bound only applies to cases of even $L\geq4$ and is not shown here, as scale invariance is inherent in this case.
• Figure 4: Behaviour of the combination of lower-bounds. Best lower bound to the Gibbs states energy density as a function of the entropy density for $N_{\mathrm{sites}}=144$, for different values of the correlation $U/t$. The dashed lines materialize the continuation of each lower-bound.
• Figure 5: Maximum number of layers allowed for the LDCA and HVA circuit ansatze in order to remain below the entropy density threshold above which classical superiority is certified. The considered application is the ground state preparation of the 8 $\times$ 8 half-filled Hubbard model in 2D in the challenging intermediate regime of correlation $U/t = 4$. The dotted red line materializes the expected required depth for the HVA circuit to explore a suitable portion of the Hilbert space.

Theorems & Definitions (10)

• Theorem 1
• proof
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• proof
• Lemma 3
• proof