Piecewise Constant Control

Rahel Heule, Christoph Bruder, Daniel Klaus Burgarth +1 more

Motivated by some recent results of quantum control theory, we discuss the feasibility of local operator control in arrays of interacting qubits modeled as isotropic Heisenberg spin chains. Acting on one of the end spins, we aim at finding piecewise-constant control pulses that lead to optimal fidelities for a chosen set of quantum gates. We analyze the robustness of the obtained results for the gate fidelities to random errors in the control fields, finding that with faster switching between piecewise-constant controls the system is less susceptible to these errors. The observed behavior falls into a generic class of physical phenomena that are related to a competition between resonance- and relaxation-type behavior, exemplified by motional narrowing in NMR experiments. Finally, we discuss how the obtained optimal gate fidelities are altered when the corresponding rapidly varying piecewise-constant control fields are smoothened through spectral filtering.

Andreas H. Trabesinger

Mikhail A. Noginov

E Dionis, Dominique Sugny

We apply an extension of the Pontryagin maximum principle to derive time-optimal controls of two-level quantum systems by means of piecewise constant pulses. Global optimal solutions are obtained for state-to-state transfer in the cases with one and two controls. Exact quantum speed limits are established as a function of the sampling period. We observe numerically an exponential convergence towards the minimum time in the continuous limit when this period goes to zero. We show that this convergence is only polynomial for a linearized quantum system. We discuss the experimental impact of this result.

S. Reichert

Ilan Degani, Antonella Zanna, L. Sælen +1 more

One of the long-standing problems in the field of computational quantum control is to devise algorithms that converge to high yield controls with a desired structure; particularly, controls that can be reproduced by laboratory laser pulse shapers. In this paper we propose an approach that is a considerable step forward toward the solution of this problem. Our treatment employs piecewise constant controls. For these, the concept of “desired structure” is addressed by including appropriate quadratic forms in the performance functionals. Thereafter, correct extremal control equations are derived, whereas the equations appearing in the literature are correct only to first or second order. We illustrate numerically that our approach can give extremely high yield controls with a desired structure. Our results are valid not only in the context of quantum control, but also for similiar bilinear systems arising in different areas. In particular, they are suitable for controlling linear flows with drift on unitary matrix groups.