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. Intuitive candidates do not even 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.
Highlights
Tremendous experimental progress has been made in the last few years in the real-time measurement of mesoscopic systems
While progress has been made in understanding the dynamics induced by continuous measurements, and its implications for feedback control [16, 28, 24, 29, 30, 31, 32], except in certain special cases [33, 34, 35] it is still unknown how to use feedback to best control a single qubit
Our method is to show that by focussing on the regime of good control, and analyzing the dynamics under measurement, one can make a number of well-motivated conjectures about the form the optimal protocol should take
Summary
Tremendous experimental progress has been made in the last few years in the real-time measurement of mesoscopic systems. The problem of finding a superior feedback protocol is not in choosing the Hamiltonian at each time: as yet unproven, so long as there is no restriction on what observable can be measured, and the noise is not unusually asymmetric, it is obvious that the optimal Hamiltonian is the one that moves the state closest to the desired state at each timestep. Our method is to show that by focussing on the regime of good control, and analyzing the dynamics under measurement, one can make a number of well-motivated conjectures about the form the optimal protocol should take. Assuming these conjectures to be true leaves only a single parameter of the measurement basis to be chosen as a function of a single parameter of the state.
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