Speaker
Description
Imaginary-time evolution (ITE) is a widely used quantum computing primitive for ground-state preparation, with asymptotic convergence guaranteed when the initial state has nonzero ground-state overlap. In practice, however, its runtime can be dominated by long finite-time stagnation plateaus, during which the energy decreases negligibly while the state remains far from the ground state. We formalize this bottleneck by defining a stagnation time, derive spectral-overlap bounds on its onset and duration, and show that it creates an intrinsic \emph{false-convergence} problem: gradient-based stopping criteria can signal success even when the ground-state fidelity remains below $1/2.$ Under an \emph{edge-bulk isolation} (EBI) spectral condition, the total stagnation time is asymptotically governed by the first excited-state plateau, yielding a direct mapping from the ground-state gap distribution to the stagnation-time tail distribution $P(T)$. For random-matrix ensembles with Dyson index $\beta$, this gives $P_\beta(T) \sim T^{-(\beta+2)}$, showing that level repulsion suppresses rare long stagnation events. We further show that spectral edge isolation is essential for this gap-controlled heavy-tail mechanism: despite its rapidly closing algebraic gap, the Sherrington-Kirkpatrick spin glass can exhibit \emph{weaker} stagnation than the random transverse-field Ising model because its low-energy spectrum is too dense to isolate the spectral edge.
These results show that the practical hardness of ITE is governed not by asymptotic convergence alone, but by how low-energy spectral structure controls finite-time stagnation.