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Evaluating Sample-Based Krylov Quantum Diagonalization for Heisenberg Models with Applications to Materials Science

Roman Firt, Neel Misciasci, Jonathan E. Mueller, Triet Friedhoff, Chinonso Onah, Aaron Schulze, Sarah Mostame

TL;DR

The paper evaluates Sample-based Krylov Quantum Diagonalization (SKQD) for spin-1/2 Heisenberg models in 1D and 2D, including dense ground-state regimes. It combines Krylov subspace diagonalization with problem-informed initial states and magnetization-sector sweeps, supported by both classical benchmarks (DMRG, Bethe ansatz) and hardware demonstrations on IBM quantum devices. SKQD reliably estimates ground-state energies and magnetization across regimes, with performance improving as anisotropy or field increases and extending to 2D systems. The results indicate SKQD's practicality for strongly correlated spin systems and its potential relevance to materials science applications.

Abstract

We evaluate the Sample-based Krylov Quantum Diagonalization (SKQD) algorithm on one- and two-dimensional Heisenberg models, including strongly correlated regimes in which the ground state is dense. Using problem-informed initial states and magnetization-sector sweeps, SKQD accurately reproduces ground-state energies and field-dependent magnetization across a range of anisotropies. Benchmarks against DMRG and exact diagonalization show consistent qualitative agreement, with accuracy improving systematically in more anisotropic regimes. We further demonstrate SKQD on quantum hardware by implementing 18- and 30-qubit Heisenberg chains, obtaining magnetization curves that match theoretical expectations. Simulations on small 2D square-lattice systems further demonstrate that the method applies effectively beyond 1D geometries.

Evaluating Sample-Based Krylov Quantum Diagonalization for Heisenberg Models with Applications to Materials Science

TL;DR

The paper evaluates Sample-based Krylov Quantum Diagonalization (SKQD) for spin-1/2 Heisenberg models in 1D and 2D, including dense ground-state regimes. It combines Krylov subspace diagonalization with problem-informed initial states and magnetization-sector sweeps, supported by both classical benchmarks (DMRG, Bethe ansatz) and hardware demonstrations on IBM quantum devices. SKQD reliably estimates ground-state energies and magnetization across regimes, with performance improving as anisotropy or field increases and extending to 2D systems. The results indicate SKQD's practicality for strongly correlated spin systems and its potential relevance to materials science applications.

Abstract

We evaluate the Sample-based Krylov Quantum Diagonalization (SKQD) algorithm on one- and two-dimensional Heisenberg models, including strongly correlated regimes in which the ground state is dense. Using problem-informed initial states and magnetization-sector sweeps, SKQD accurately reproduces ground-state energies and field-dependent magnetization across a range of anisotropies. Benchmarks against DMRG and exact diagonalization show consistent qualitative agreement, with accuracy improving systematically in more anisotropic regimes. We further demonstrate SKQD on quantum hardware by implementing 18- and 30-qubit Heisenberg chains, obtaining magnetization curves that match theoretical expectations. Simulations on small 2D square-lattice systems further demonstrate that the method applies effectively beyond 1D geometries.

Paper Structure

This paper contains 12 sections, 17 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction and background
  2. method
  3. Benchmark Setup
  4. Model Hamiltonians and Lattice Geometries
  5. Initial State Preparation
  6. Singlet State
  7. Single Particle Sector Sweep
  8. Results
  9. Conclusions and Outlook
  10. Exploring Magnetization Sectors with W States
  11. Postprocessing: Magnetization Sector Filtering
  12. Setup and Parametrization of Calculations

Figures (5)

  • Figure 1: Ground state sparsity for 20 spin system with $J=1$ in $(\Delta, h_z)$-space plotted as the logarithm of the inverse participation ratio.
  • Figure 2: Relative Magnetization (A, B) and ground state energy (C, D) of CuPzN approximated with SKQD for 18 (A, C) and 30 (B, D) spin systems. Compared with theoretical values derived from Bethe ansatz PhysRevB.59.1008 and state-of-the art DMRG calculations.
  • Figure 3: Comparison between SKQD energy estimates and exact energies on a 6x4 square lattice (left) and snake-like initial state (green) on a 2D square lattice layout used for SKQD simulation (right). The red lines show the state's singlet couples.
  • Figure 4: Magnetization Sector in the presence of external field.
  • Figure 5: Histogram of sampled unique bitstrings on 18 spin system. Green bar marks the initialization particle sector and red line shows the maximum number of unique bitstrings, explaining the sampling distribution mean value shift towards the half-filling sector in the middle.

Theorems & Definitions (1)

  • Definition 1: ($\alpha_L$, $\beta_L$)-sparsity