Exact and Tunable Quantum Krylov Subspaces via Unitary Decomposition

TL;DR

This paper introduces QKUD, a novel quantum Krylov subspace method that avoids time evolution, improves conditioning, and enhances the accuracy of quantum simulations for complex many-body systems.

Contribution

QKUD provides a time-evolution-free, tunable approach to quantum Krylov subspaces, addressing conditioning issues and improving convergence in quantum simulations.

Findings

QKUD reproduces exact Krylov convergence in well-conditioned regimes.

QKUD restores variational improvements when traditional methods stagnate.

Overlap conditioning is identified as key for robust quantum Krylov simulation.

Abstract

Quantum Krylov subspace methods can extract ground and excited states by diagonalizing the Hamiltonian in a compact variational space. In practice, these spaces are almost always generated by real or imaginary time evolution, forcing a timestep trade-off between dynamical accuracy and basis collapse and often producing ill-conditioned overlap matrices that stall convergence. Here we introduce Quantum Krylov using Unitary Decomposition (QKUD), a time-evolution-free construction that maps Hamiltonian powers to implementable unitaries via the Hermitian transform $\sin(\epsilon H)/\epsilon$. QKUD reduces to the exact Hamiltonian-power Krylov recursion as $\epsilon\rightarrow0$, while finite $\epsilon$ provides a controllable deformation that tunes subspace geometry and improves conditioning. Across molecular active-space benchmarks and a frustrated 2D J1-J2 Heisenberg model, QKUD reproduces exact-Krylov convergence in well-conditioned regimes and systematically restores variational improvement when both exact Krylov and time-evolution Krylov stagnate. These results identify overlap conditioning, instead of time-evolution fidelity, is the key resource for robust quantum Krylov simulation and provide a resilient way forward for accurate quantum simulation of challenging quantum many-body problems.