Time-optimal synthesis of unitary transformations in a coupled fast and slow qubit system

Abstract

In this paper, we study time-optimal control problems related to 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 much longer time as compared to the time required to produce unitary transformations on the first qubit but 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 SU(2) × SU(2) × 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.

Optimal control in NMR spectroscopy: Numerical implementation in SIMPSON

“…In the recent few years optimal control [1][2][3] has found its way into nuclear magnetic resonance as a powerful tool for efficient design and optimization of experiments for applications in imaging [4][5][6][7], liquid-state NMR [8][9][10][11][12][13], solid-state NMR [14][15][16][17][18][19][20], quantum computation [21][22][23][24][25][26], and dynamic nuclear polarization/electronnuclear interactions [27][28][29][30][31]. This method, originally being introduced for optimizations in engineering and economy, is very attractive for optimization problems dealing with complex internal Hamiltonians and a large number of external manipulation parameters.…”

Time optimal control of coupled spin dynamics: A global analysis

“…We call this SU(2n) SU(n)×SU(n)×U(1) problem. We show for n = 2, this system models the dynamics of coupled electron-nuclear spin system in EPR [13].…”

“…This decomposition of a real semi-simple Lie algebra g = p ⊕ k satisfying (13) is called the Cartan decomposition of the Lie algebra g [27].…”

Preparation of an arbitrary two-qubit quantum gate on two spins with an anisotropic Heisenberg interaction

“…Similar problems for the spin-1 and arbitrary spin were solved in [31] and [32], respectively. Also it was widely studied the implementation of quantum gate on two spins with different types of interaction [33,34,35,36,37,38,39].…”