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Chinese Control Conference (CCC)

Controlling dephasing decoherence by periodically performing tracking control and generalized measurement

Ming Zhang

Published Jul 1, 2008• 1 citation
Publisher

Abstract

We propose a method to overcome phase damping decoherence by periodically performing open loop tracking control and generalized measurement. Although it is impossible to stabilize a qubit subject to Markovian dynamics only by open loop coherent control, one can attain a dasiasoftenedpsila control goal with the help of periodical generalized measurement. The dasiasoftenedpsila control objective in our scheme is to keep the state of the controlled qubit to stay near a reference pure state all the time. Two sub-optimal control problems are given in the sense of trace distance and fidelity, respectively, and they are eventually reduced to the choice of a period T.

DOI
10.1109/chicc.2008.4605361

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We propose a method to overcome phase damping decoherence by periodically performing open loop tracking control and generalized measurement. Although it is impossible to stabilize a qubit subject to Markovian dynamics only by open loop coherent control, one can attain a dasiasoftenedpsila control goal with the help of periodical generalized measurement. The dasiasoftenedpsila control objective in our scheme is to keep the state of the controlled qubit to stay near a reference pure state all the time. Two sub-optimal control problems are given in the sense of trace distance and fidelity, respectively, and they are eventually reduced to the choice of a period T.
We propose a scheme to overcome phase damping decoherence by periodically performing open loop tracking control and projective measurement. Although it is impossible to stabilize a qubit subject to Markovian dynamics only by open loop coherent control, one can attain a ``softened'' control goal with the help of periodical projective measurement. The ``softened'' control objective in our scheme is to keep the state of the controlled qubit to stay near a reference pure state with a high probability for a sufficiently long time. Two suboptimal control problems are given in the sense of trace distance and fidelity, respectively, and they are eventually reduced to the design of a period TTT. In our scheme, one can choose the period TTT as long as possible if the ``softened'' control goal is attained. This is in contrast to the observation that quantum Zeno effect takes place only if measurements are performed in a very frequent manner, i.e., the period T\mathit{T}T must be extremely small.
We consider an open quantum system subjected to a noise channel under measurement-based feedback control and two prototypical classes of decoherence channels are considered: phase damping and generalized amplitude damping. Based on quantum trajectory theory, we obtain an extended master equation for the dynamics of the reduced system in the presence of feedback control. For a qubit system we analytically solve this master equation and obtain the solution of the state vector dynamics. Then we propose an effective feedback control scheme for preparing an arbitrary quantum pure state. We also study how to protect two nonorthogonal states effectively, and find that projective measurement with unbiased basis is not optimal for this task, while weak measurement with biased basis could realize the best protection of two nonorthogonal states. Furthermore, the inefficiencies in the feedback process are also discussed.
We exploit how to overcome phase damping decoherence when the phase damping parameters are uncertain. It is shown that the “relaxed” control goal may be achieved by combing simple open-loop coherent control with periodic projective measurements. Both the sufficient conditions and constructive design procedures are presented. The control design is reduced to selecting the period of projective measurements and fidelity-truncated time based on two lower-bound functions for both probability and fidelity performance. When both probability and fidelity performances are specified, the stabilization time is further maximized by searching the “optimal” period of projective measurements. It is also exemplified that larger uncertainties may lead to shorter stabilization time for the given performance level.
For any specified pure state of quantum open system, we can construct a kind of quantum generalized measurement (QGM) that the state of the system after measurement will be deterministically collapsed into the specified pure state from any initial state. In other words, any pure state of quantum open system is reachable by QGM. Subsequently, whether the qubit is density matrix controllable is discussed in the case of pure dephasing. Our results reveal that combining QGM with coherent control will enhance the ability of controlling the quantum open system. Furthermore, it is found that the ability to perform QGM on the quantum open system, combined with the ability of coherence control and conditions of decoherence-free subspace, allows us to suppress quantum decoherence.
We address the experimentally relevant problem of robust mitigation of dephasing noise acting on a qubit. We first present an extension of the method of Kuopanportti et al. [Phys. Rev. A 77, 032334 (2008)] for representing $1/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$ noise to the efficient representation of arbitrary Markovian noise. We then add qubit control pulses to enable the design of numerically optimized, two-dimensional, bounded amplitude control functions capable of decoupling the qubit from the dephasing effects of a broad variety of Markovian noise spectral densities during one- and two-qubit quantum operations. We illustrate the method with development of numerically optimized control pulse sequences that minimize decoherence due to a combination of $1/\ensuremath{\omega}$ and constant-offset noise sources. Comparison with the performance of standard dynamical decoupling protocols shows that the numerically optimized pulse sequences are considerably more robust with respect to zero-frequency noise. Application to the mitigation of dephasing noise on spin qubits in silicon indicates that high-fidelity quantum gates may, in principle, be implemented for such qubits with the assistance of current pulse-generation technology.