Products of unitaries as continuous evolutions at finite precision
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 2π, 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 κ, this comparison and continuous adiabatic theory give a rescaled qubitized walk with O(κ/δ) steps for leakage δ, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.
Related
- On Separately Continuous Functions of the First Baire Class on Squares and Mixed Products
- Scott products and sobriety of countable meet-continuous dcpos
- Representation zeta functions of wreath products with finite groups
- Finite Orders of Products of Two Class Transpositions Divide 840
- Development of approximate scheduling-adaptive controllers for multi-products continuous chemical processes using deep learning techniques and model predictive control
BibTeX
@misc{jarret2026products,
title = {{Products of unitaries as continuous evolutions at finite precision}},
author = {Michael Jarret},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06541},
eprint = {2610.06541},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06541},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Products of unitaries as continuous evolutions at finite precision
AU - Michael Jarret
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06541
UR - https://arxiv.org/abs/2610.06541
AB - 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 2π, 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 κ, this comparison and continuous adiabatic theory give a rescaled qubitized walk with O(κ/δ) steps for leakage δ, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.
ER -