Analytical Series Expansion for Efficient Gradient Evaluation in Multi-Qubit Optimal Control
Abstract
Gradient-based quantum optimal control and the theory of operator evolution are rarely discussed together. In this Letter, we bridge this gap by showing that the gradient of the time-evolved propagator involves the Heisenberg evolution of local operators. We present a unifying framework by deriving from first principles the formal solution for the gradient under an arbitrary pulse parameterization, and show that the same construction extends to derivatives of any order. We obtain a series of nested, time-independent commutators weighted by time-dependent scalar coefficients. The commutators are thus computed once, and each optimization step only updates the scalars. The method is particularly suited for simulating optimal control tasks in quantum systems with local interactions, which is a common situation in large multi-qubit platforms. In this setting, light-cone arguments and sparse-Pauli truncation heuristics may be used to keep the number of relevant terms in the series small. We compare the computational cost required for the series with the Gradient Optimization of Analytic conTrols (GOAT) method, and, focusing on the problem of preparation of a GHZ state, demonstrate more than an order of magnitude speedup for a qubit ladder and a chain geometry.
Related
- Analytical Time-Domain Waveforms for Black-Hole Scattering in the Post-Minkowskian Expansion
- Analytical Time-Domain Waveforms for Black-Hole Scattering in the Self-Force Expansion
- Heavy-quark expansion at leading power for local QCD operators in gradient flow
- Underdamped Stochastic Optimal Transport with Landauer boundary conditions: machine learning and analytical solution
- Breaking Behavioral Uniformity in Traffic Flow: Analytical Derivation and Numerical Verification of Optimal Cooperative Buffering
BibTeX
@misc{mishra2026analytical,
title = {{Analytical Series Expansion for Efficient Gradient Evaluation in Multi-Qubit Optimal Control}},
author = {Ashutosh Mishra and Elena Lupo and Frank K. Wilhelm and Alessandro Ciani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.26867},
eprint = {2607.26867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.26867},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analytical Series Expansion for Efficient Gradient Evaluation in Multi-Qubit Optimal Control
AU - Ashutosh Mishra
AU - Elena Lupo
AU - Frank K. Wilhelm
AU - Alessandro Ciani
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2607.26867
UR - https://arxiv.org/abs/2607.26867
AB - Gradient-based quantum optimal control and the theory of operator evolution are rarely discussed together. In this Letter, we bridge this gap by showing that the gradient of the time-evolved propagator involves the Heisenberg evolution of local operators. We present a unifying framework by deriving from first principles the formal solution for the gradient under an arbitrary pulse parameterization, and show that the same construction extends to derivatives of any order. We obtain a series of nested, time-independent commutators weighted by time-dependent scalar coefficients. The commutators are thus computed once, and each optimization step only updates the scalars. The method is particularly suited for simulating optimal control tasks in quantum systems with local interactions, which is a common situation in large multi-qubit platforms. In this setting, light-cone arguments and sparse-Pauli truncation heuristics may be used to keep the number of relevant terms in the series small. We compare the computational cost required for the series with the Gradient Optimization of Analytic conTrols (GOAT) method, and, focusing on the problem of preparation of a GHZ state, demonstrate more than an order of magnitude speedup for a qubit ladder and a chain geometry.
ER -