Optimal measurement-based feedback control for a single qubit: a candidate protocol

TL;DR

This paper proposes a candidate optimal measurement-based feedback protocol for a single qubit under continuous measurement, valid across various feedback strengths, and suggests it is optimal in the regime of good control.

Contribution

The authors introduce a specific form for the optimal feedback protocol for a broad class of noise sources, supported by numerical searches and conjectured to be optimal to leading order.

Findings

Protocol is valid for all feedback strengths in the good control regime.

Contains a discontinuity as a function of feedback strength.

Conjectured to be optimal to leading order in small parameters.

Abstract

Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system close to the desired state. With the assumption of this form the remaining parameters can be determined via a numerical search. The result is a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. We conjecture that this protocol is optimal to leading order in the small parameters that define this regime. The protocol can be described relatively simply, and as a notable feature contains a discontinuity as a function of the feedback strength.