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IEEE Conference on Decision and Control (CDC)

Constructive controllability for systems with drift motivated by and applied to quantum control

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

Published Nov 11, 2002• 1 citation
Publisher

Abstract

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.

DOI
10.1109/cdc.2000.912896

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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.
One of the long-standing problems in the field of computational quantum control is to devise algorithms that converge to high yield controls with a desired structure; particularly, controls that can be reproduced by laboratory laser pulse shapers. In this paper we propose an approach that is a considerable step forward toward the solution of this problem. Our treatment employs piecewise constant controls. For these, the concept of “desired structure” is addressed by including appropriate quadratic forms in the performance functionals. Thereafter, correct extremal control equations are derived, whereas the equations appearing in the literature are correct only to first or second order. We illustrate numerically that our approach can give extremely high yield controls with a desired structure. Our results are valid not only in the context of quantum control, but also for similiar bilinear systems arising in different areas. In particular, they are suitable for controlling linear flows with drift on unitary matrix groups.
Abstract One of the main theoretical challenges in quantum computing is the design of explicit schemes that enable one to effectively factorize a given final unitary operator into a product of basic unitary operators. As this is equivalent to a constructive controllability task on a Lie group of special unitary operators, one faces interesting classes of bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In this paper we give a review on recent Lie‐theoretical developments in finite‐dimensional quantum control that play a key role for solving such factorization problems on a compact Lie group. After a brief introduction to basic terms and concepts from quantum mechanics, we address the fundamental control theoretic issues for bilinear control systems and survey standard techniques fromLie theory relevant for quantum control. Questions of controllability, accessibility and time optimal control of spin systems are in the center of our interest. Some remarks on computational aspects are included as well. The idea is to enable the potential reader to understand the problems in clear mathematical terms, to assess the current state of the art and get an overview on recent developments in quantum control‐an emerging interdisciplinary field between physics, control and computation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
This paper considers right-invariant and controllable driftless quantum systems with state X(t) evolving on the unitary group U(n) and m inputs u = (u1, …, um). The T-sampling stabilisation problem is introduced and solved: given any initial condition X0 and any goal state , find a control law u = u(X, t) such that for the closed-loop system. The purpose is to generate arbitrary quantum gates corresponding to . This is achieved by the tracking of T-periodic reference trajectories of the quantum system that pass by using the framework of Coron’s return method. The T-periodic reference trajectories are generated by applying controls that are a sum of a finite number M of harmonics of sin (2πt/T), whose amplitudes are parameterised by a vector a. The main result establishes that, for M big enough, X(jT) exponentially converges towards for almost all fixed a, with explicit and completely constructive control laws. This paper also establishes a stochastic version of this deterministic control law. The key idea is to randomly choose a different parameter vector of control amplitudes a = aj at each t = jT, and keeping it fixed for t ∈ [jT, (j + 1)T). It is shown in the paper that X(jT) exponentially converges towards almost surely. Simulation results have indicated that the convergence speed of X(jT) may be significantly improved with such stochastic technique. This is illustrated in the generation of the C–NOT quantum logic gate on U(4).
Constructive algorithms are presented for controlling quantum systems evolving on the SU (1, 1) Lie group. These procedures are performed via structured decomposition of SU (1, 1), which achieve precise controls without any approximations or iterative computations, under the sufficient condition that examines the existence of such decomposition. The technique is applied to controlling transitions between SU (1, 1) coherent states. These results open up new perspectives on the control design of infinite-dimensional quantum systems involving discrete or continuous spectra.
This paper considers right-invariant and controllable quantum systems with m inputs u = (u1,… , um) and state X(t) evolving on the unitary Lie group U(n). The ε-steering problem is introduced and solved for systems with drift: given any initial condition X0 at the initial time instant t0 ≥ 0, any goal state Xgoal∈ U(n) and ε > 0, find a control law such that dist(X(t0+t‾),Xgoal)≤ϵ, where t‾>0 is big enough and dist(X1, X2) is a convenient right-invariant notion of distance between two elements X1, X2 ∈ U(n). The purpose is to approximately generate arbitrary quantum gates corresponding to Xgoal. This is achieved by solving a tracking problem for a special kind of reference trajectories W‾: [t0, ∞) → U(n), which are here called c-universal reference trajectories. It is shown that, for this special kind of trajectories, the tracking problem can be solved up to an error ε for any reference trajectory X‾(t)=W‾(t)R which is a right-translation of W‾(t), at least when dist(X(t0),X‾(t0)) is finite. Furthermore, it is shown that dist(X(t),X‾(t)) converges uniformly exponentially to zero in the sense that the rate of convergence is independent of t0, R and X0. The approach considered here for showing such convergence is a generalisation of the results of a previous paper of the authors, which is mainly based on the central ideas of Jurdjevic and Quinn and Coron's return method. Taking a right-translation R such that X‾(t0+t‾)=Xgoal, one may solve the ε-steering problem by solving the tracking problem for the reference trajectory X‾(t), at least when dist(X(t0),X‾(t0))≤c. When dist(X(t0),X‾(t0))≥c, it is shown that the ε-steering problem can be globally solved in a two-iteration procedure. The underlying algorithmic complexity to get the steering control is essentially equivalent to the numerical integration of the Cauchy problem governing X(t). A numerical example considering a Toffoli quantum gate on U(8) for a chain of three coupled qubits that are controlled only locally is presented.