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arXiv:quant-ph/0103022 · Physical Review A

Quantum control without access to the controlling interaction

Dominik Janzing, Frederik Armknecht, Robert Zeier, T. Beth

Published Jan 10, 2002• 13 citations
arXivHTMLPDF

Abstract

In our model a fixed Hamiltonian acts on the joint Hilbert space of a quantum system and its controller. We show conditions under which measurements, state preparations, and unitary implementations on the system can be performed by quantum operations on the controller only. It turns out that a measurement of the observable $A$ and an implementation of the one-parameter group $\mathrm{exp}(\mathrm{iAr})$ can be performed by almost the same sequence of control operations. Furthermore, measurement procedures for $A+B,$ for $(AB+BA),$ and for $i\ensuremath{\lfloor}A,B\ensuremath{\rfloor}$ can be constructed from measurements of $A$ and $B.$ This shows that the algebraic structure of the set of observables can be explained by the Lie group structure of the unitary evolutions on the joint Hilbert space of the measuring device and the measured system. A spin chain model with nearest-neighborhood coupling shows that the border line between controller and system can be shifted consistently.

DOI
10.1103/physreva.65.022104

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In our model a fixed Hamiltonian acts on the joint Hilbert space of a quantum system and its controller. We show conditions under which measurements, state preparations, and unitary implementations on the system can be performed by quantum operations on the controller only. It turns out that a measurement of the observable AAA and an implementation of the one-parameter group exp(iAr)\mathrm{exp}(\mathrm{iAr})exp(iAr) can be performed by almost the same sequence of control operations. Furthermore, measurement procedures for A+B,A+B,A+B, for (AB+BA),(AB+BA),(AB+BA), and for i\ensuremath⌊A,B\ensuremath⌋i\ensuremath{\lfloor}A,B\ensuremath{\rfloor}i\ensuremath⌊A,B\ensuremath⌋ can be constructed from measurements of AAA and B.B.B. This shows that the algebraic structure of the set of observables can be explained by the Lie group structure of the unitary evolutions on the joint Hilbert space of the measuring device and the measured system. A spin chain model with nearest-neighborhood coupling shows that the border line between controller and system can be shifted consistently.
An n-qubit quantum register can in principle be completely controlled by operating on a single qubit that interacts with the register via an appropriate fixed interaction. In this paper, we consider a hypothetical system consisting of n spin-1/2 nuclei that interact magnetically with an electron spin (control and probe particle). We describe control algorithms that implement premeasurements on nontrivial joint observables. We then devote a section to the implementation of unitary gates for the register by acting only on the control spin. Although this model of universal quantum computation is not efficient for large n, it can become efficient if we allow n possible positions for the control particle. This ``toy'' model of premeasurements shows how specific interactions between the register and the probe particle would support specific types of joint measurements. This means that some joint observables can be measured by performing a simple sequence of operations on the probe particle. In our model, some measurements of joint observables are simpler to perform than for one-qubit observables.
The ability to navigate and control the state space of a quantum system is fundamental to quantum computing, quantum information processing, and advanced spectroscopy. This paper explores a rigorous mathematical framework for quantifying state space accessibility by leveraging the theory of operator commutation relations, particularly within the context of Lie algebras. We demonstrate that the span of iterated commutators of system Hamiltonians and control operators forms a dynamical Lie algebra, whose dimension and structure directly dictate the reachable states and unitary transformations. For closed quantum systems, the Lie algebra rank condition provides a powerful criterion for complete controllability, asserting that if the dynamical Lie algebra spans the entire space of anti-Hermitian operators, then any state can be reached from any other, and any unitary operation can be implemented. For open quantum systems, while environmental interactions introduce decoherence and dissipation, the Lie algebraic approach can still offer insights into the attainable subspaces and the limits of control, often involving Lie semigroups and Lie wedges. We present methods to analyze the accessibility of both pure and mixed states, illustrating how the algebraic properties encoded in commutation relations provide a precise measure of the manipulability of quantum systems. The implications extend to the design of robust quantum control protocols, optimized state preparation, and the development of error-resilient quantum technologies.
Abstract — The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finitedimensional submanifold of the unit sphere in Hilbert space – and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples. I.
It is unrealistic to control all of the degrees of freedom of a high-dimensional quantum system. Here, we consider a scenario where our direct access is restricted to a small subsystem S that is constantly interacting with the rest of the system E. What we investigate is the fundamental structures of the Hilbert space and the algebra of hamiltonians that are caused solely by the restrictedness of the direct control. One key finding is that hamiltonians form a Jordan algebra, and this leads to a significant observation that there is a sharp distinction between the cases of dimHS ≥ 3 and dimHS = 2 in terms of the nature of possible operations in E. Since our analysis is totally free from specific properties of any physical systems, it would form a solid basis for obtaining deeper insights into quantum control related issues, such as controllability and observability.
The control of spin chains represents a very interesting problem from the point of view of quantum computation. The problem consists in defining a procedure to obtain any possible evolution operator of the spin chain by means of in external magnetic field. The set of possible evolution operators of the system corresponds to the unitary group SU(2/sup N/) (where N is the number of atoms in the chain) and the interactions involved can he set to correspond to elements in the corresponding Lie algebra. As a consequence, the whole problem can be formulated in Lie algebraic terms and the design issues be reduced to a suitable decomposition of the group elements. The goal of this paper is to introduce a Cartan decomposition for the unitary group based on a minimal switching decomposition of the special orthogonal group. We analyze its implications from the point of view of time optimality in the construction of a program as a sequence of quantum gates.