Scaling the robustness of the solutions for quantum controllable problems (original) (raw)

Robust control in the quantum domain

Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2000

Recent progress in quantum physics has made it possible to perform experiments in which individual quantum systems are monitored and manipulated in real time. The advent of such new technical capabilities provides strong motivation for the development of theoretical and experimental methodologies for quantum feedback control. The availability of such methods would enable radically new approaches to experimental physics in the quantum realm. Likewise, the investigation of quantum feedback control will introduce crucial new considerations to control theory, such as the uniquely quantum phenomena of entanglement and measurement back-action. The extension of established analysis techniques from control theory into the quantum domain may also provide new insight into the dynamics of complex quantum systems. We anticipate that the successful formulation of an input-output approach to the analysis and reduction of large quantum systems could have very general applications in non-equilibrium quantum statistical mechanics and in the nascent field of quantum information theory.

Limitations on Quantum Control

Lattice Statistics and Mathematical Physics - Festschrift Dedicated to Professor Fa-Yueh Wu on the Occasion of His 70th Birthday - Proceedings of APCTP-NANKAI Joint Symposium, 2002

Noise and Controllability: Suppression of Controllability in Large Quantum Systems

Physical Review Letters, 2011

A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise. Can one design control fields such that the effect of noise is negligible on the time-scale of the transformation? Complete controllability in practice requires that the effect of noise can be suppressed for an arbitrary transformation. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases, determined by the dimension of the Hilbert space representation of the algebra. We show that for large quantum systems, generic noise in the controls dominates for a typical class of target transformations i.e., complete controllability is destroyed by the noise. PACS numbers: 32.80.Qk, 03.67.-a, 03.65.Yz, 02.30.Yy

Scalable quantum computation via local control of only two qubits

Physical Review A, 2010

We apply quantum control techniques to control a large spin chain by only acting on two qubits at one of its ends, thereby implementing universal quantum computation by a combination of quantum gates on the latter and swap operations across the chain. It is shown that the control sequences can be computed and implemented efficiently. We discuss the application of these ideas to physical systems such as superconducting qubits in which full control of long chains is challenging. 02.30.Yy Controlling quantum systems at will has been an aspiration for physicists for a long time. Achieving quantum control not only clears the path towards a thorough understanding of quantum mechanics, but it also allows the exploration of novel devices whose functions are based on exotic quantum mechanical effects. Among others, the future success of quantum information processing depends largely on our ability to tame many-body quantum systems that are highly fragile. Although the progress of technology allows us to manipulate a small number of quanta quite well, controlling larger systems still represents a considerable challenge. Unless we overcome difficulties towards the control over large many-body systems, the benefits we can enjoy with the 'quantumness' will be severely limited.

On Some Model Problems in Quantum Control

Communications in Information and Systems, 2009

Control and manipulation of quantum mechanical systems using electromagnetic fields is a widely studied subject in areas of physics and chemistry, including spectroscopy, atomic molecular, and optical physics, and quantum chemistry. This article attempts to provide a glimpse into the rich class of bilinear control systems that are ubiquitous in these problems. In this article, we use control of spin systems in magnetic resonance as a model system to highlight characteristic feature of problems in quantum control. Background information is provided to enable the reader to appreciate new results and developments, where principled use of ideas from control theory have provided new insights into finding optimal ways to control and manipulate quantum mechanical systems. The study of deterministic and stochastic models that arise in problems in measurement and manipulation of quantum mechanical systems may foster new developments in control. * Dedicated to Roger Brockett on the occasion of his 70th birthday.

Quantum control landscapes: a closer look,” ArXiv:1004.3492

2016

Abstract. The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L2(0, T), however, there are critical points for which the fidelity can assume any value in (0, 1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L2(0, T) is larger and includes ...

Quantum Control Landscapes: A Closer Look

2010

The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L2(0,T)L^2(0,T)L2(0,T), however, there are critical points for which the fidelity can assume any value in (0,1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L2(0,T)L^2(0,T)L2(0,T) is larger and includes traps, some of which remain traps even when the target time is allowed to vary.

Quantum control with noisy fields: computational complexity versus sensitivity to noise

Quantum control with noisy fields: computational complexity vs. sensitivity to noise Abstract. A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise, which has to be suppressed to retain controllability. Can one design control fields such that the effect of noise is negligible on the time-scale of the transformation? This question is intimately related to the fundamental problem of a connection between the computational complexity of the control problem and the sensitivity of the controlled system to noise. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases with the dimension of the Hilbert space representation of the algebra. We find two types of control tasks, easy and hard. Easy tasks are characterized by a small variance of the evolving state with respect to the operators of the control operators. They are relatively immune to noise and the control field is easy to find. Hard tasks have a large variance, are sensitive to noise and the control field is hard to find. The influence of noise increases with the size of the system, which is measured by the scaling factor N of the largest weight of the representation. For fixed time and control field as O(N ) for easy tasks and as O(N 2 ) for hard tasks. As a consequence, even in the most favorable estimate, for large quantum systems, generic noise in the controls dominates for a typical class of target transformations, i.e., complete controllability is destroyed by noise.

Complete controllability of quantum systems

Physical Review A, 2001

Sufficient conditions for complete controllability of N-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic oscillator systems, as well as some systems with degenerate energy levels. Controllability of these model systems is of special interest since they have many applications in physics, e.g., Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems.

Quantum Computing by an Optimal Control Algorithm for Unitary Transformations

Physical Review Letters, 2002

Quantum computation is based on implementing selected unitary transformations representing algorithms. A generalized optimal control theory is used to find the driving field that generates a prespecified unitary transformation. The approach is independent of the physical implementation of the quantum computer and it is illustrated for one and two qubit gates in model molecular systems, where only part of the Hilbert space is used for computation.