From QMCGroup
Quantum Feedback and Optimal Control
Overview
Since its modern conception in the 20th century, control theory has had great success as an abstract means for analyzing dynamical systems. From stabilizing aircraft to hedging financial risk, the mathematical framework is broadly applicable. Within the last decade, the ability to interact with distinctly quantum mechanical systems has provided a new regime for using control theory techniques. However, due to the inherent uncertainty in quantum mechanics, we cannot simply apply classical control theory techniques without modification.
In open-loop quantum control, one seeks to engineer robust controls that satisfy given design constraints and optimize a target objective. For example, controls can be designed to create a desired quantum state or to maximize the expected value of an observable. These controls are designed given a known set of initial states and known dynamics of the quantum system. In the QMC Group , we focus more on closed-loop or feedback quantum control, which generalizes the open-loop problem to include continuous measurements of quantum systems and controls which depend on those measurements. The obvious advantage is that controls can now depend on real-time observations of the underlying system. However, this additional control advantage makes the optimal control problem difficult to solve.
Closed-Loop Quantum Error Correction[CQEC 1]
Our group has investigated the use of closed-loop feedback in quantum error correction, which plays an important role in quantum computing. In a quantum computer, quantum information is stored in the state of qubits, which most physicists call spin-1/2 particles. However, the stored quantum state is easily destroyed when the qubits interact with the environment in a process called decoherence. This noise results in errors in the quantum computation. Quantum error correction attempts to dull the impact of decoherence by encoding the logical state of the qubits across several physical qubits. A judicious choice of encoding enables one to diagnose certain errors and recover from them.
While the standard approach of performing error correction between stages of computation works, our work considers the case when the speed measurement and error correction are not significantly faster than the rate of errors. In this case, one would expect a continuous application of error correction to perform better. As a first step, we studied this in the context of a quantum memory, in which we protect a state without doing any computation. Previous work of Ahn, Doherty and Landahl [CQEC 1]devised a control law for performing closed-loop error correction using continuous measurements. However, the controller required computational resources which scaled exponentially in the number of physical qubits in the quantum code. Generalizing results of Mabuchi and van Handel [CQEC 1], we were able to construct a filter that only scaled as the square of the number of physical qubits, but whose performance was identical to the previous controller.
Future work involves searching for optimal solutions which minimize the control cost, as opposed to the heuristic, but useful, control law we studied. More importantly, the results need to be extended to allow for computational stages. This will likely involve developing novel encodings, so that the error detection/correction processes can occur simultaneously with computation.
Optical Phase Estimation
Given our success in parameter estimation at the quantum limit, see Atomic Magnetometry, we are interested in devising feedback controls which make the estimation faster, more accurate and more robust to non-systematic error. In particular, we are interested in the problem of adaptive phase estimation, in which we seek to estimate the optical phase of a coherent state. Berry and Wiseman [AP 1] have been able to devise an ad-hoc method which reaches the quantum limit. We seek to recover similar results by solving an optimal control problem, with the hope that the feedback and filtering laws are more amenable to experimental implementation.