We derive a geometric quantum speed limit (QSL) for imaginary-time evolution, where the dynamics is governed by a nonunitary Schrödinger equation. Based on the angular distance between the normalized evolving state and the initial state, we obtain a lower bound on the evolution time, expressed as the ratio between this geometric distance and the time-averaged energy dispersion. The present bound explicitly incorporates the norm-non-conserving nature of imaginary-time evolution and is applicable to time-dependent or time-independent Hamiltonians. It is saturable when the evolution direction is aligned with the geodesic on projective Hilbert space. We analytically evaluate this bound for several examples. For a two-level system, we obtain a closed-form expression for the minimal evolution time required to reach a target ground state. For the imaginary-time Grover's quantum search, our QSL reproduces the well-known logarithmic scaling $T=\mathcal{O}(lnN)$, in agreement with previous results by Okuyama and Ohzeki. We further show that the equality condition of the bound is exactly satisfied for imaginary-time evolutions generated by time-dependent but mutually commuting Hamiltonians, demonstrating that the derived QSL provides both a rigorous and achievable limit for nonunitary quantum dynamics.

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• Quantum computation