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Physical Review A

Quantum control by decompositions of SU(2)

Viswanath Ramakrishna, Kathryn L. Flores, HERSCHEL A. RABITZ, Raimund J. Ober

Published Oct 13, 2000• 60 citations
Publisher

Abstract

Constructive procedures that make no use of optimization or iterative calculations for the control of many quantum and classical systems, via controls that are required to satisfy constraints on their power or pulse area, are presented. These procedures are based on structured decompositions of SU(2). A general technique for obtaining such structured decompositions is given. Illustrative examples are provided.

DOI
10.1103/physreva.62.053409

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Constructive procedures that make no use of optimization or iterative calculations for the control of many quantum and classical systems, via controls that are required to satisfy constraints on their power or pulse area, are presented. These procedures are based on structured decompositions of SU(2). A general technique for obtaining such structured decompositions is given. Illustrative examples are provided.
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We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique, it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.
The purpose of this paper is four fold: 1) a brief description of how finite dimensional models arise in quantum control is given; 2) many notions in quantum engineering such as universality of logic gates, hard pulse approximation in NMR spectroscopy, etc., are related to controllability of systems on the unitary groups; 3) recent results of the authors, on constructive controllability, in the class of piecewise constant controls with constraints placed on them, for systems with drift on the unitary groups are sketched; and 4) a new decomposition of SU(2) matrices is provided and used to illustrate how problems caused by periodicity arguments, to address the presence of drift, may be circumvented.
Structured decompositions of a desired unitary operator are employed to derive control schemes that achieve certain control objectives for finite-level quantum systems using only sequences of simple control pulses such as square waves with finite rise and decay times or Gaussian wavepackets. The technique is applied to find control schemes that achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of observables.
Development of robust quantum control has been challenging and there are numerous obstacles to applying classical robust control to quantum system including bilinearity, marginal stability, state preparation errors, nonlinear figures of merit. The requirement of marginal stability, while not satisfied for closed quantum systems, can be satisfied for open quantum systems where Lindbladian behavior leads to non-unitary evolution, and allows for nonzero classical stability margins, but it remains difficult to extract physical insight when classical robust control tools are applied to these systems. We consider a straightforward example of the entanglement between two qubits dissipatively coupled to a lossy cavity and analyze it using the classical stability margin and structured perturbations. We attempt, where possible, to extract physical insight from these analyses. Our aim is to highlight where classical robust control can assist in the analysis of quantum systems and identify areas where more work needs to be done to develop specific methods for quantum robust control.
We present an iterative scheme to estimate the minimal duration in which a quantum gate can be realized while satisfying hardware constraints on the control pulse amplitudes. The scheme performs a sequence of unconstrained numerical optimal control cycles that each minimize the gate fidelity for a given gate duration alongside an additional penalty term for the control pulse amplitudes. After each cycle, the gate duration is adjusted based on the inverse of the resulting maximum control pulse amplitudes by re-scaling the dynamics to a new duration where control pulses satisfy the amplitude constraints. Those scaled controls then serve as an initial guess for the next unconstrained optimal control cycle, using the adjusted gate duration. We provide multiple numerical examples that each demonstrate fast convergence of the scheme toward a gate duration that is close to the quantum speed limit, given the control pulse amplitude bound. The proposed technique is agnostic to the underlying system and control Hamiltonian models, as well as the target unitary gate operation, making the time-scaling iteration an easy to implement and practically useful scheme for reducing the durations of quantum gate operations.