In this paper, we study the design of pulse sequences for NMR spectroscopy as a problem of time optimal control of the unitary propagator. Radio frequency pulses are used in coherent spectroscopy to implement a unitary transfer of state. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. Moreover, the general proofs are no more difficult than the specific problems of current interest. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group, what is the minimum time required to steer the system from some initial point to a specified final point? In NMR spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin systems.
A parameterization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parameterization we find the region of permissible vectors which represent a density operator. The inequalities which specify the region are shown to involve the Casimir invariants of the group. In particular cases, this allows the determination of degeneracies in the spectrum of the operator. The identification of the Casimir invariants also provides a method of constructing quantities which are invariant under local unitary operations. Several examples are given which illustrate the constraints provided by the positivity requirements and the utility of the coherence vector parameterization.
Finding control fields ͑pulse sequences͒ that can compensate for the dispersion in the parameters governing the evolution of a quantum system is an important problem in coherent spectroscopy and quantum information processing. The use of composite pulses for compensating dispersion in system dynamics is widely known and applied. In this paper, we make explicit the key aspects of the dynamics that makes such a compensation possible. We highlight the role of Lie algebras and noncommutativity in the design of a compensating pulse sequence. Finally, we investigate three common dispersions in NMR spectroscopy, namely the Larmor dispersion, rf inhomogeneity, and strength of couplings between the spins.Many applications in control of quantum systems involve controlling a large ensemble by using the same control field. In practice, the elements of the ensemble could show variation in the parameters that govern the dynamics of the system. For example, in magnetic resonance experiments, the spins of an ensemble may have large dispersion in their natural frequencies ͑Larmor dispersion͒, strength of applied rf field ͑rf inhomogeneity͒, and the relaxation rates of the spins ͓1͔. In solid-state NMR spectroscopy of powders, the random distribution of orientations of internuclear vectors of coupled spins within an ensemble leads to a distribution of coupling strengths ͓12͔. A canonical problem in control of quantum ensembles is to develop external excitations that can simultaneously steer the ensemble of systems with variation in their internal parameters from an initial state to a desired final state. These are called compensating pulse sequences as they can compensate for the dispersion in the system dynamics. From the standpoint of mathematical control theory, the challenge is to simultaneously steer a continuum of systems between points of interest with the same control signal. Typical applications are the design of excitation and inversion pulses in NMR spectroscopy in the presence of Larmor dispersion and rf inhomogeneity ͓1-10͔ or the transfer of coherence or polarization in a coupled spin ensemble with variations in the coupling strengths ͓13͔. In many cases of practical interest, one wants to find a control field that prepares the final state as some desired function of the parameter, for example slice selective excitation and inversion pulses in magnetic resonance imaging ͓14-17͔. The problem of designing excitations that can compensate for dispersion in the dynamics is a well studied subject in NMR spectroscopy, and extensive literature exists on the subject of composite pulses that correct for dispersion in system dynamics ͓1-7͔. Composite pulses have recently been used in quantum information processing to correct for systematic errors in single-and two-qubit operations ͓18-23͔. The focus of this paper is not to construct a new compensating pulse sequence but rather to highlight the aspects of system dynamics that make such a compensation possible and give proofs of the existence of a compensating pulse sequence. Our...
In this paper, we demonstrate that optimal control algorithms can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristic ratio of the time scales for local controls vs non-local (i.e. coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity.In the limit of fast local controls (as e.g. in NMR), time-optimised realisations are shown for the quantum Fourier transform (QFT) and the multiply controlled not-gate (c n−1 not) in various coupling topologies of n qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and swap gates, while the c n−1 not-gate for completely coupled network of six qubits is nearly seven times faster.
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