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IntechOpen eBooks

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

Navin Khaneja

Published Feb 7, 2019• 0 citations
Publisher

Abstract

In this chapter, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation.Time optimal control helps to minimize relaxation losses.In a two qubit system, the ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems.The generators of unitary evolution, g, are decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g ¼ p ⊕ k.Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems.The main contribution of the chapter is, we carry out a global analysis of time optimality.

DOI
10.5772/intechopen.80567

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In this chapter, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation.Time optimal control helps to minimize relaxation losses.In a two qubit system, the ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems.The generators of unitary evolution, g, are decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g ¼ p ⊕ k.Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems.The main contribution of the chapter is, we carry out a global analysis of time optimality.
In this paper, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. The ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of evolution, $\g$, are decomposed into fast generators $\k$ (local Hamiltonians) and slow generators $\p$ (couplings) as a Cartan decomposition $\g = \p \oplus \k$. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. In this paper, we carry out a second order analysis of time optimality.
In this paper, we study time-optimal control problems related to a system of two coupled qubits where the time scales involved in performing unitary transformations on each qubit are significantly different. In particular, we address the case where unitary transformations produced by evolutions of the coupling take a much longer time as compared to the time required to produce unitary transformations on the first qubit, but a much shorter time as compared to the time to produce unitary transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the subgroup $\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$, which is natural in understanding the time-optimal control problem of such a coupled qubit system with significantly different time scales. A typical setting involves dynamics of a coupled electron-nuclear spin system in pulsed electron paramagnetic resonance experiments at high fields. Using the proposed canonical decomposition, we give time-optimal control algorithms to synthesize various unitary transformations of interest in coherent spectroscopy and quantum-information processing.
In this paper, we give a characterization of all the unitary transformations that can be synthesized in a given time for a two-qubit system in the presence of general time varying coupling tensor. This characterization helps to compute the minimum time and the shortest pulse sequence for generating a general two-qubit transformation under nonstationary interactions. The methods presented here can be applied in design of time optimal pulse sequences for transferring coherence and polarization between coupled spins with time varying couplings as in solid-state NMR under magic angle spinning.
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In this thesis, we study methods for optimal manipulation of coupled spin dynamics in the presence of relaxation. We use these methods to compute analytical upper bounds for the efficiency of coherence and polarization transfer between coupled nuclear spins in multidimensional nuclear magnetic resonance (NMR) experiments, under the presence of relaxation. We derive relaxation optimized pulse sequences which achieve or approach these bounds and maximize the sensitivity of the experiments. The two systems examined in detail in this thesis, arising in the context of protein NMR spectroscopy, are a coupled heteronuclear spin pair and an Ising spin chain. New schemes for coherence and polarization transfer are found, which yield substantial gains in transfer efficiency, compared to the traditionally used pulse sequences. From the standpoint of mathematical control theory, the problems arising in connection with optimal manipulation of dissipative quantum dynamics, motivate the study of a class of dissipative bilinear control systems and the computation of their reachable sets. For these systems we show that the optimal solution, corresponding to the maximization of transfer efficiency, and the reachable set can be found by solving a semidefinite program.
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