Time optimal control in coupled spin systems: a second order analysis

TL;DR

This paper analyzes time optimal control in coupled spin systems, using a second order approach to characterize the reachable set and optimize control strategies in NMR and quantum computing.

Contribution

It introduces a second order analysis method for time optimal control in coupled spin systems, leveraging a Cartan decomposition and convexity ideas.

Findings

Complete characterization of the reachable set

Second order optimality conditions derived

Enhanced control synthesis strategies proposed

Abstract

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.