Polylogarithmic-Depth Quantum Algorithm for Simulating the Extended Hubbard Model on a Two-Dimensional Lattice Using the Fast Multipole Method

TL;DR

This work develops Q2FMM, a quantum algorithm for simulating the time evolution of the 2D extended Hubbard model with long-range Coulomb interactions by recasting pairwise terms into inter-box interactions via a fast multipole framework. Leveraging a 0th-order FMM foundation and hierarchical box decompositions, the method achieves polylogarithmic circuit depth per Trotter step on 2D neutral-atom quantum hardware, aided by COPY, atom shuttling, and unbounded fan-out techniques. The paper further extends the framework to higher-order FMM, analyzes error contributions from Trotterization and FMM truncation, and provides resource estimates showing near-linear gate counts in system size with favorable depth under practical hardware assumptions. The approach generalizes to broader Hamiltonians, including ab initio molecular systems, and highlights the potential of hierarchical, multipole-expansion strategies for scalable quantum simulation on near-term to future quantum architectures.

Abstract

The extended Hubbard model on a two-dimensional lattice captures key physical phenomena, but is challenging to simulate due to the presence of long-range interactions. In this work, we present an efficient quantum algorithm for simulating the time evolution of this model. Our approach, inspired by the fast multipole method, approximates pairwise interactions by interactions between hierarchical levels of coarse-graining boxes. We discuss how to leverage recent advances in two-dimensional neutral atom quantum computing, supporting non-local operations such as long-range gates and shuttling. The resulting circuit depth for a single Trotter step scales polylogarithmically with system size.

Figures (13)

• Figure 1: The expression that originally depends on the two target points $a$ and $b$ is rewritten in terms of local contributions around the centers $A$ and $B$, together with the interaction between the two centers.
• Figure 2: Blue: interaction field of the target box $A$, handled at the current level. White: near field for box $A$. The distances to $A$ are too close to be considered at the current level and will be deferred to a finer level. Gray: far field, whose boxes have been considered at a coarser level, as shown on the right. Evaluating $K_C(\mathbf{r}_a,\mathbf{r}_b)$ is approximated by $K_C(\mathbf{r}_A,\mathbf{r}_B)$ within interaction field at each level in the $0^\text{th}$-order FMM.
• Figure 3: Illustration of the coarse-graining and time evolution algorithm. Each ball in the left diagram corresponds to a single qubit (lattice site) on the finest level. When reaching a coarser level (larger boxes), four child boxes are merged into one parent box. The time evolution phase is evaluated for all interacting pairs of boxes, displayed for one such pair in the center diagram. This procedure repeats until the coarsest level is reached. The ancilla qubits that record the occupation numbers are not drawn for visual clarity.
• Figure 4: Conceptual illustration of out-of-place quantum adders and multipliers. In contrast, in-place designs overwrite the input registers with the results. The explicit circuit structure varies across different implementations.
• Figure 5: Uncomputing the additions in Fig. \ref{['fig:2d_algo']} "splits" the boxes by applying the inverses of the adders. This operation proceeds from the coarsest to the finest level.