Classical phase unwrapping enables large-time-step unitary quantum Krylov subspace diagonalization

For computing molecular energies on early fault-tolerant hardware, unitary quantum Krylov subspace diagonalization (UQKSD) avoids both the deep circuits of quantum phase estimation and the Pauli-term sampling of variational methods by extracting the energies from a projected real-time propagator. At a small time step $τ$, nearly linearly dependent Krylov vectors make the overlap matrix ill-conditioned. Our tests at fixed total evolution time show that additional Krylov vectors then add no accuracy, while the thresholded energy becomes increasingly sensitive to the choice of cutoff. A larger time step improves the conditioning and reduces the Krylov dimension $M$ needed for a target accuracy, but the energies are then known only modulo $2π/τ$. We present Classical Unwrapping for Resolving Eigenvalues (CURE-QKSD), a classical post-processing method that leaves the UQKSD circuits and generalized eigenvalue problem unchanged and recovers the missing multiples of $2π/τ$ from phase-consistency landscapes of one or, when needed, two UQKSD runs. In noiseless tests on six molecules with 12--20 qubits, all 28 targeted energies are recovered in their aliased regimes with final errors of at most $1.1\times10^{-7}$~Ha. Under Gaussian noise with $σ=10^{-4}$, the median error of the ground or first excited state reaches chemical accuracy at large aliased time steps with $M=20$--$65$, whereas uncorrected principal-branch energies are off by $0.7$--$115$~Ha. CURE-QKSD therefore makes aliased time steps usable for UQKSD calculations with smaller, better-conditioned Krylov bases.