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Bulletin of the American Physical Society

Optimal control of logical operations in the presence of decoherence: A two-spin model

Matthew Grace, Constantin Brif, HERSCHEL A. RABITZ, Ian A. Walmsley, Robert L. Kosut, Daniel A. Lidar

Published Mar 16, 2006• 2 citations
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Abstract A protocol is proposed to generate singlet states of three logical qubits constructed by pairs of spins. Single and multiple operations of logical qubits are studied for the construction of an effective Hamiltonian, with which robust control fields are derived with invariant‐based reverse engineering and optimal control. Moreover, systematic errors are further compensated by periodic modulation for better robustness. Furthermore, resistance to random noise and decoherence of the protocol is also shown with numerical simulations. Therefore, the protocol may provide useful perspectives for the generation of logical qubit entanglement in spin systems.
We present a formalism for encoding the logical basis of a qubit into subspaces of multiple physical levels. The need for this multilevel encoding (MLE) arises naturally in situations where the speed of quantum operations exceeds the limits imposed by the addressability of individual energy levels of the qubit physical system. A basic feature of the MLE formalism is the logical equivalence of different physical states and correspondingly, of different physical transformations. This logical equivalence is a source of a significant flexibility in designing logical operations, while the multilevel structure inherently accommodates fast and intense broadband controls thereby facilitating faster quantum operations. Another important practical advantage of MLE is the ability to maintain full quantum-computational fidelity in the presence of mixing and decoherence within encoding subspaces. The formalism is developed in detail for single-qubit operations and generalized for multiple qubits. As an illustrative example, we perform a simulation of closed-loop optimal control of single-qubit operations for a model multilevel system, and subsequently apply these operations at finite temperatures to investigate the effect of decoherence on operational fidelity.
In this paper, we study some control problems related to the control of coupled spin dynamics in the presence of relaxation and decoherence in nuclear magnetic resonance spectroscopy. The decoherence is modelled through a master equation. We study some model problems, whereby, through an appropriate choice of state variables, the system is reduced to a control system, where the state enters linearly and controls quadratically. We study this quadratic control system. Study of this system gives us explicit bounds on how close a coupled spin system can be driven to its target state and how much coherence and polarization can be transferred between coupled spins. Optimal control for the quadratic control system can be understood as the separation of closed cones, and we show how the derived results on optimal efficiency can be interpreted in this formulation. Finally, we study some finite-time optimal control problems for the quadratic control system.This article is part of the themed issue 'Horizons of cybernetical physics'.