Quantifying fault tolerant simulation of strongly correlated systems using the Fermi-Hubbard model

TL;DR

The paper argues that strongly correlated materials can be meaningfully explored with fault-tolerant quantum computing by using the Fermi-Hubbard model as a scalable benchmark. It develops explicit quantum-resource estimates for computing static and dynamic correlation functions, including a full resource breakdown for time evolution on a two-fridge MBQC architecture and realistic error-correction overheads, up to 7×7 lattices. It further translates these resource needs into a probabilistic utility framework, highlighting first-order savings (HPC costs, experimental guidance, knowledge generation) and second-order effects (spillovers, human capital), with stage-2/3 and stage-4/5 utility projections spanning billions of dollars. The results show that while close to utility-scale problems, significant improvements in quantum algorithms, hardware, and compiler tools are required to make these FH-based computations practical in the near term, establishing FH as a rigorous benchmark for progress in fault-tolerant quantum simulation of correlated systems.

Abstract

Understanding the physics of strongly correlated materials is one of the grand challenge problems for physics today. A large class of scientifically interesting materials, from high-$T_c$ superconductors to spin liquids, involve medium to strong correlations, and building a holistic understanding of these materials is critical. Doing so is hindered by the competition between the kinetic energy and Coulomb repulsion, which renders both analytic and numerical methods unsatisfactory for describing interacting materials. Fault-tolerant quantum computers have been proposed as a path forward to overcome these difficulties, but this potential capability has not yet been fully assessed. Here, using the multi-orbital Fermi-Hubbard model as a representative model and a source of scalable problem specifications, we estimate the resource costs needed to use fault-tolerant quantum computers for obtaining experimentally relevant quantities such as correlation function estimation. We find that advances in quantum algorithms and hardware will be needed in order to reduce quantum resources and feasibly address utility-scale problem instances.

Figures (11)

• Figure 1: The Fermi-Hubbard model. The base case (left) is a single orbital model on a two-dimensional square lattice. By increasing complexity through adding more degrees of freedom (orbital, lattice, spin, charge), a microscopic description of a realistic material can be achieved.
• Figure 2: Resolution test for a set of given frequencies $\omega_g = \{-1.5,-1.4,-0.05,0.5,1.5,1.8\}$ with $T_{\mathrm{max}} = \{100,700\}$ and $\epsilon = \{0.01,0.5\}$. The blue lines in frequency plots correspond to Fourier transformed signal generated using specified $T_{\mathrm{max}}$ with $\epsilon=0$, and Violet dashed lines are for the signal generated with non-zero specified $\epsilon$. Using Python's peak finder, frequency values are identified (marked with 'x') to reconstruct the signal shown. Lower $T_{\mathrm{max}} = 100$ fails to resolve frequencies close to each other, (-1.5 and -1.4), and, higher $\epsilon$ makes it worse. Increasing $T_{\mathrm{max}}$ resolves the values better even with high $\epsilon$.
• Figure 3: Resolution test for combinations of $\epsilon$ and $T_{\mathrm{max}}$, to compare the given frequencies and amplitudes with reconstructed values. (a) Difference in the frequencies given ($\omega_g$) and reconstructed values ($\omega_r$) and (b) The difference in frequencies and amplitudes.
• Figure 4: The core quantum circuit used in the dynamic correlation function estimation. This circuit prepares a state for which the overlap with $\ket{0}$ is a known function of $g(t)$. Depending on whether the real or imaginary part of $g(t)$ is being estimated, either a Hadamard gate $H$ or a Hadamard and phase gate $HS$ are applied.
• Figure 5: Physical resource scaling relationship for the time evolution portion of calculating dynamic correlation functions. The Hamiltonian considered is the Fermi-Hubbard model \ref{['eq:fh_hamiltonian']} with $U=2, V_{nn}=1, \mu=1$ and various lattice sizes. The bus size represents all physical qubits that are not involved in T state distillation. Run time denotes total run time of the quantum computer given in seconds. This run time accounts for all circuit invocations and shots.