Rapid ground state energy estimation with a Sparse Pauli Dynamics-enabled Variational Double Bracket Flow

TL;DR

This paper tackles the problem of efficiently estimating ground-state energies in strongly correlated quantum systems by fusing Sparse Pauli Dynamics with a variational double bracket flow (vDBF). The method uses a greedily chosen, SPD-based set of Pauli rotations with coefficient truncation and energy-variance extrapolation to achieve high accuracy, often surpassing classical DMRG performance in wall time for 2D lattices. Key results show sub-1% error relative to DMRG benchmarks for 1×100 and 10×10 Heisenberg lattices and 1×64, 4×4, and 8×8 Hubbard models, with notable speedups on large systems (e.g., 10×10 Heisenberg in minutes on a single CPU). The work demonstrates that circuit-simulation techniques developed for quantum advantage benchmarking can be repurposed into practical classical tools for many-body physics, while outlining avenues to improve convergence, symmetry handling, and observable accuracy.

Abstract

Ground state energy estimation for strongly correlated quantum systems remains a central challenge in computational physics and chemistry. While tensor network methods like DMRG provide efficient solutions for one-dimensional systems, higher-dimensional problems remain difficult. Here we present a variational double bracket flow (vDBF) algorithm that leverages Sparse Pauli Dynamics, a technique originally developed for classical simulation of quantum circuits, to efficiently approximate ground state energies. By combining greedy operator selection with coefficient truncation and energy-variance extrapolation, the method achieves less than 1% error relative to DMRG benchmarks for both Heisenberg and Hubbard models in one and two dimensions. For a 10x10 Heisenberg lattice (100 qubits), vDBF obtains accurate results in approximately 10 minutes on a single CPU thread, compared to over 50 hours on 64 threads for DMRG. For an 8x8 Hubbard model (128 qubits), the speedup is even more pronounced. These results demonstrate that classical simulation techniques developed in the context of quantum advantage benchmarking can provide practical tools for many-body physics.

Figures (6)

• Figure 1: Schematic depiction of the evolution of a single Pauli string (top) following 3 sequential small-angle Pauli rotations. Background colors indicate the additional set of Paulis that are included under tighter levels of coefficient truncation.
• Figure 2: Energy vs Variance extrapolations for Heisenberg lattices using a series of vDBF thresholds: $\epsilon=$1e-2 (blue), 1e-3 (orange), 1e-4 (green), 1e-5 (red). (a) 1$\times$100 lattice, (b) 6$\times$6 lattice, and (c)10$\times$10 lattice. Extrapolations shown as solid lines. Uncertainties (difference between linear and quadratic fits) shown as shaded regions. DMRG best estimates (dashed grey line). 1% error region w.r.t. DMRG best estimate (grey shaded area). For (c), DMRG extrapolation is shown, with DMRG discarded weight at the top axis.
• Figure 3: Spin-spin correlation functions for 1x100 site Heisenberg lattice. $C(r) = \expval{S_1S_r} - \expval{S_1}\expval{S_r}.$ Blue are for converged DMRG results. Black line is vDBF( 1e-4)
• Figure 4: Energy vs Variance extrapolations for Hubbard lattice using a series of vDBF thresholds: $\epsilon=$1e-2 (blue), 1e-3 (orange), 5e-4 (green), 2e-4 (red). Two qubits per fermionic site require 128, 32, and 128 qubits, respectively. Extrapolations shown as solid lines. Uncertainties (difference between linear and quadratic fits) are shown as shaded regions. DMRG best estimates (dashed grey line). 1% error region w.r.t. DMRG best estimate (grey shaded area).
• Figure 5: Relationship between number of Pauli strings vs truncation threshold for both Heisenberg models (squares) and Hubbard models (circles).