Control aspects of quantum computing using pure and mixed states

B Y T HOMAS S CHULTE-H ERBRGGEN 2 R AIMUND M ARX 2 A MR F AHMY 1 L OUIS K AUFFMAN 0 S AMUEL L OMONACO 4 N AVIN K HANEJA 3 S TEFFEN J. G LASER 2 0 Department of Mathematics, University of Illinois , 851 S. Morgan Street, Chicago, IL 60607-7045 , USA 1 Biological Chemistry and Molecular Pharmacology, Harvard Medical School , 240 Longwood Avenue, Boston, MA 02115 , USA 2 Department of Chemistry, Technische Universitat Munchen , Lichtenbergstrasse 4, 85747 Garching , Germany 3 Division of Applied Sciences, Harvard University , Cambridge, MA 02138 , USA 4 Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County , 1000 Hilltop Circle, Baltimore, MD 21250 , USA Steering quantum dynamics such that the target states solve classically hard problems is paramount to quantum simulation and computation. And beyond, quantum control is also essential to pave the way to quantum technologies. Here, important control techniques are reviewed and presented in a unified frame covering quantum computational gate synthesis and spectroscopic state transfer alike. We emphasize that it does not matter whether the quantum states of interest are pure or not. While pure states underly the design of quantum circuits, ensemble mixtures of quantum states can be exploited in a more recent class of algorithms: it is illustrated by characterizing the Jones polynomial in order to distinguish between different (classes of) knots. Further applications include Josephson elements, cavity grids, ion traps and nitrogen vacancy centres in scenarios of closed as well as open quantum systems. One contribution of 14 to a Theme Issue 'Quantum information processing in NMR: theory and experiment'. 1. Introduction Controlling quantum dynamics may provide access to efficiently performing computational tasks or to simulating the behaviour of other quantum systems that are beyond experimental handling themselves. In particular, quantum systems can also simulate classical systems efficiently [1,2] sometimes even separating controllable parameters in the quantum analogue that classically cannot be tuned independently. Therefore, both in simulation and in computation, the complexity of a problem may reduce upon going from a classical to a quantum setting [3]. On the computational end, most prominently, there is the exponential speed-up by Shors quantum algorithm of prime factorization [4,5] relating to the ample class of quantum algorithms [6,7] efficiently solving hidden subgroup problems [8,9]. Inspired by topological quantum computation exploring braid groups, recently another type of quantum algorithm has come into focus, to wit the algorithm of Aharonov, Jones and Landau (AJL) [10] for approximating the Jones polynomial, i.e. a central invariant in knot theory. For broader context, see also [11]. While classically it is NP-hard to distinguish two (classes of) knots in terms of their characteristic Jones polynomials, the quantum AJL algorithm, or its predecessor by Kauffman & Lomonaco [12,13], can do so more efficiently with quantum resources. Moreover, as has been experimentally demonstrated by NMR [14,15], these algorithms can be implemented using thermal mixtures of quantum states. Moreover, it suffices to approximate the trace of a controlled unitary encapsulating the information of the Jones polynomial. This class of quantum algorithms is equivalent to deterministic quantum computation with one clean qubit (DQC1) [16], and actually it is even DQC1-complete [17,18], where general belief has it that P DQC1 BQP (e.g. Shor & Jordan [17]). As has nicely been pointed out in Passante et al. [15], note that DQC1 does not require the quantum bit to be in a pure state. While the demands for accuracy (error-correction threshold) in quantum computation may seem daunting at the moment, the quantum simulation end is far less sensitive. Thus, simulating quantum systems [19]in particular at phase transitions [20]has shifted into focus [2124]. Both quantum computation and simulation are challenging quantum engineering tasks requiring high-level manipulations of quantum dynamics. To this end, also among the mathematical tools [25,26] optimal control algorithms have been establishing themselves as indispensable [27,28]. They have matured from principles [29] and early implementations [3032] via spectroscopic applications [3335] to advanced numerical algorithms [36,37] for state-to-state transfer and quantumgate synthesis [3840] alike as will be illustrated in more detail. On the practical end of engineering high-end quantum experiments, progress has been made in many areas, including cold atoms in optical lattice potentials [41,42], trapped ions [4349] and superconducting qubits [5052], to name just a few. At the interface of theory and experiment, optimal control among numerical tools has become increasingly important (see [53] for a recent review). For instance, near time-optimal control may take pioneering realizations of solidstate qubits being promising candidates for a computation platform [54] from their fidelity limit to the decoherence limit [39]. More recently, open systems governed by a Markovian master equation have been addressed [40], and even smaller non-Markovian subsystems can be tackled, if they can be embedded into a larger system that in turn interacts in a Markovian way with its environment [55]. Taking the concept of decoherence-free subspaces [56,57] to more realistic scenarios, avoiding decoherence in encoded subspaces [58] complements recent approaches of dynamic error correction [59,60]. Along these lines, quantum control is anticipated to contribute significantly to bridging the gap between quantum principles demonstrated in pioneering experiments and high-end quantum engineering [27,61]. Many results from controlling spin systems, as can also be found in this Theme Issue in the contributions by the groups of Laflamme at IQC or Jones in Oxford, are paradigmatic for finite-dimensional quantum systems. So their implications reach far beyond spin systems and, in particular, beyond ensembles, which is why we first focus on the general toolbox. To this end, the paper is structured as follows: 2 casts many of the standard quantum optimal control tasks into the framework of bilinear control systems. We show that all of them can conveniently be tackled by a unified program platform DYNAMO comprising concurrent (GRAPE), sequential (K ROTOV-type) as well as hybrid algorithms. In 3, we outline a number of applications to synthesizing quantum gates in closed quantum systems referring to experimental settings such as Josephson charge qubits and cavity grids. Section 4 departs from quantum circuits and shows how control applications help to distinguish classes of knots by way of their Jones polynomials. As demonstrated in 5, also open quantum systems profit from optimal control, e.g. as a means of error avoidance. 2. Algorithmic platform for bilinear quantum control syst (...truncated)