Exact and Tunable Quantum Krylov Subspaces via Unitary Decomposition

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 $\sin(εH)/ε$. QKUD reduces to the exact Hamiltonian-power Krylov recursion as $ε\rightarrow0$, while finite $ε$ 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.