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Time optimal control of a right invariant system on a compact Lie group

Jody Lynn Storm

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Abstract. In this paper we will study the pulse sequences in NMR spec-troscopy and quantum computing as a time control problem. Radio frequency pulses are used to execute a unitary transfer of state. Sequences of pulses should be as short as possible to minimize decoherence. We model the problem as a controllable right invariant system on a compact Lie group. We investigate the minimum time required to steer the system from an initial point to a specified final point. 1.

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Abstract. In this paper we will study the pulse sequences in NMR spec-troscopy and quantum computing as a time control problem. Radio frequency pulses are used to execute a unitary transfer of state. Sequences of pulses should be as short as possible to minimize decoherence. We model the problem as a controllable right invariant system on a compact Lie group. We investigate the minimum time required to steer the system from an initial point to a specified final point. 1.
In this paper, we study the design of pulse sequences for nuclear magnetic resonance 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 between states. 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. 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 nuclear magnetic resonance (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 system.
Radio-frequency pulses are used in nuclear-magnetic-resonance spectroscopy to produce unitary transfer of states. 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. Many coherence-transfer experiments in NMR, involving a network of coupled spins, use temporary spin decoupling to produce desired effective Hamiltonians. In this paper, we demonstrate that significant time can be saved in producing an effective Hamiltonian if spin decoupling is avoided. We provide time-optimal pulse sequences for producing an important class of effective Hamiltonians in three-spin networks. These effective Hamiltonians are useful for coherence-transfer experiments in three-spin systems and implementation of indirect swap and ${\ensuremath{\Lambda}}_{2}(U)$ gates in the context of NMR quantum computing. It is shown that computing these time-optimal pulses can be reduced to geometric problems that involve computing sub-Riemannian geodesics. Using these geometric ideas, explicit expressions for the minimum time required for producing these effective Hamiltonians, transfer of coherence, and implementation of indirect swap gates, in a three-spin network are derived (Theorems 1 and 2). It is demonstrated that geometric control techniques provide a systematic way of finding time-optimal pulse sequences for transferring coherence and synthesizing unitary transformations in quantum networks, with considerable time savings (e.g., 42.3% for constructing indirect swap gates).
Abstract In this paper we study the relationship between factorization problems on SU(2n) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We study a quantum spin system acting on a single quantum bit. The evolution of this system is governed by the Schrodinger equation which takes the form of a right-invariant system on the special unitary group SU(2) with two control inputs. Using a suitable version of Pontryagin's Principle which is tailor-made for control problems on Lie groups, the optimal controls are derived in two cases: the energy-optimal case (in which the control effort is minimized for a specified end time) and the time-optimal case (in which the control duration is minimized for given constraints on the size of the controls).
Given a compact, connected Lie group G with Lie algebra � . We discuss time-optimal control of bilinear systems of the form (I) where H d , H j ∈ � , U ∈ G , and the v j act as control variables. The case G = SU (2 n ) has found interesting appli- cations to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n -particle system with fixed drift Hamiltonian H d , which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v j . The question of interest here is to transfer the sys- tem from a given initial state U 0 to a prescribed final state U 1 in least possible time. Denote bythe Lie algebra spanned by H 1 , …, H m , and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G / K we discuss in detail time-optimal control strategies for system (I) in the case where G / K carries the structure of a Riemannian sym- metric space.
For a right-invariant system on a compact Lie group G , I present two methods to design a control to drive the state from the identity to any element of the group. The first method, under appropriate assumptions, achieves exact control to the target but requires estimation of the ‘size’ of a neighborhood of the identity in G and solution of a nonlinear algebraic equation. The second method does not involve any mathematical difficulty and obtains control to a desired target with arbitrary accuracy . A third method is then given combining the main ideas of the previous methods. This is also very simple in its formulation and turns out to be generically more efficient as illustrated by one of the examples I consider. The methods described in the paper provide arbitrary constructive control for any right-invariant system on a compact Lie group. In particular, the results can be applied to the coherent control of general multilevel quantum systems to an arbitrary target.