Constrained Quantum Imaginary Time in Krylov Space

Abstract

We introduce constrained quantum imaginary time evolution (CQITE), extending quantum imaginary time evolution (QITE) to constrained-search dynamics for ground-state preparation with prescribed expectation-value constraints. A central challenge in practical QITE is constructing a compact unitary representation of the imaginary-time evolution while maintaining accuracy and controlling the approximation error. We resolve this challenge by deriving an exact characterization of the imaginary-time evolution from the Krylov space generated by repeated applications of the Hamiltonian to the initial state. This characterization induces a hierarchy of operator directions expressed through nested anti-commutators. Truncating the hierarchy produces a systematic static operator pool, with polynomial size in the system size at any fixed truncation order. The resulting construction provides a principled route to compact unitary approximations of imaginary-time dynamics. Numerical tests on spin models show close agreement with exact constrained imaginary-time evolution and small constraint deviations, while a fermionic example yields energies close to the analytic constrained-search reference.