Products of unitaries as continuous evolutions at finite precision

arXiv:2610.06541v1 Announce Type: new Abstract: I present a short approach to discretizing continuous unitary evolution. It gives a correspondence in both directions between continuous unitary evolutions and products of unitaries in finite dimensions, and I use it to reduce discrete adiabatic theorems to continuous ones. At fine steps, holding truncated samples of a regulated Hamiltonian on a suitable finite partition gives an approximating product. Conversely, rounding the clock identifies a product of $T$ unit-step samples with a continuous evolution, up to two half-step phases. I bound its operator-norm distance from the original evolution by $O(1/T)$ through cancellation within each time step. This assumes bounded first and second derivatives, bounded spectral width, and eigenvalue differences bounded away from nonzero multiples of $2\pi$, uniformly in $T$. For a unitary walk with these derivative bounds, a fixed common free spectral arc supplies a Hermitian logarithm path satisfying the comparison hypotheses. Continuous simulation guarantees then transfer to the phase-corrected product, and continuous adiabatic bounds to the product itself, with an explicit additive $O(1/T)$ error and each adiabatic theorem's hypotheses retained. For positive definite quantum linear systems of condition number $\kappa$, this comparison and continuous adiabatic theory give a rescaled qubitized walk with $O(\kappa/\delta)$ steps for leakage $\delta$, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.

Abstract:I present a short approach to discretizing continuous unitary evolution. It gives a correspondence in both directions between continuous unitary evolutions and products of unitaries in finite dimensions, and I use it to reduce discrete adiabatic theorems to continuous ones. At fine steps, holding truncated samples of a regulated Hamiltonian on a suitable finite partition gives an approximating product. Conversely, rounding the clock identifies a product of $T$ unit-step samples with a continuous evolution, up to two half-step phases. I bound its operator-norm distance from the original evolution by $O(1/T)$ through cancellation within each time step. This assumes bounded first and second derivatives, bounded spectral width, and eigenvalue differences bounded away from nonzero multiples of $2\pi$, uniformly in $T$. For a unitary walk with these derivative bounds, a fixed common free spectral arc supplies a Hermitian logarithm path satisfying the comparison hypotheses. Continuous simulation guarantees then transfer to the phase-corrected product, and continuous adiabatic bounds to the product itself, with an explicit additive $O(1/T)$ error and each adiabatic theorem's hypotheses retained. For positive definite quantum linear systems of condition number $\kappa$, this comparison and continuous adiabatic theory give a rescaled qubitized walk with $O(\kappa/\delta)$ steps for leakage $\delta$, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.

arXiv:2610.06541 [quant-ph]

(or arXiv:2610.06541v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06541 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Products of unitaries as continuous evolutions at finite precision

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