arXiv:2610.21596

Time-optimal robust quantum control for partially known errors

Llorenç Balada Gaggioli

quant-phSubmitted Oct 15, 2026• Announced Oct 16, 2026• 0 citations

Abstract

Robust quantum control usually assumes one of two extremes: the error is of a known form and only its strength is unknown, or nothing at all is known about it. In practice one often sits in between. The size of a static error is unknown, but its class is known, for example, it only shifts energy levels, or it acts on each subsystem separately. We ask how fast a control can cancel, to first order, every error in such a class. A time-independent Hamiltonian can do this if and only if all errors in the class have zero diagonal in a common basis. In that case an equally spaced energy ladder in this basis does it after one period. A lower bound that holds for any time-dependent control with bounded drive power shows that this simple constant control is optimal for several natural classes, including unknown level shifts and unknown local errors on two subsystems. Completely unknown errors need a time-dependent control, and there, knowing the error class shortens the minimum time by a factor $\sqrt{d^2+1}$. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.

Comments: 11 pages total, 7 pages main text, 2 figures

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arXiv:2610.21596

Time-optimal robust quantum control for partially known errors

Llorenç Balada Gaggioli

quant-phSubmitted Oct 15, 2026• Announced Oct 16, 2026• 0 citations

arXiv HTML PDF

Abstract

\sqrt{d^2+1}

. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.

Comments: 11 pages total, 7 pages main text, 2 figures

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\sqrt{d^2+1}

. So partial knowledge of the error changes not only how long an optimal robust control takes, but also its form.

We study Hamiltonian learning for

k

-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown

s

-sparse Hamiltonian with fixed locality

k

and

\|H\|_{\mathrm{op}}\leqΛ

, our algorithm recovers its Pauli coefficients to

\ell_2

error

\varepsilon

in total evolution time

\widetilde{O}(s/\varepsilon)

, matching the best known scaling while allowing a near-maximal step size

\widetildeΘ(1/Λ)

. As an independent contribution, we strengthen the evolution-time lower bound to

Ω_k(s^{1-1/(2k)}/\varepsilon)

, placing our algorithm within a factor

\widetilde{O}(s^{1/(2k)})

of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a

\sqrt n

overhead in

\ell_2

learning time for

n

qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.

2N

for a field robust to the

N\mathrm{th}

order, which leads to an estimate of its complexity.

We derive a universal performance limit for coherent quantum control in the presence of modeled and unmodeled uncertainties. For any target unitary

W

that is implementable in the absence of error, we prove that the worst-case (and hence the average) gate fidelity obeys the lower bound $F \ge \Flb\bigl(\tf \Omeff\bigr)$, where $\tf$ is the gate duration and $\Omeff$ is a single frequency-like measure that aggregates \emph{all} bounded uncertainty sources, e.g., coherent control imperfections, unknown couplings, and residual environment interactions, without assuming an initially factorizable system-bath state or a completely positive map. The bound is obtained by combining an interaction-picture averaging method with a Bellman-Gronwall inequality and holds for any finite-norm Hamiltonian decomposition. Hence it applies equally to qubits, multi-level qudits, and ancilla-assisted operations. Because $\Flb$ depends only on the dimensionless product $\tf\Omeff$, it yields a device-independent metric that certifies whether a given hardware platform can, in principle, reach a specified fault-tolerance threshold, and also sets a quantitative target for robust-control synthesis and system identification.