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Unformatted text preview: A New Hierarchical Genetic Algorithm Approach to Determine Pulse Sequences in NMR Ashok Ajoy 1, 2, * and Anil Kumar 2 1 Birla Institute of Technology and Science - Pilani, Zuarinagar, Goa - 403726, India. 2 NMR Research Centre, Indian Institute of Science, Bangalore - 560012, India. Nuclear Magnetic Resonance (NMR) spectroscopy provides a valuable tool by which one can control a spin ensemble. Control is achieved by using radio-frequency (RF) pulses. Pulse sequence design has been an active research area for many years. Recently, optimal control theory has been successfully applied to the design of pulse sequences, so as to minimize their total duration and improve their efficiency. In this paper, we develop a new class of genetic algorithm that computationally determines efficient pulse sequences to implement a quantum gate U in a three-qubit system. The method is shown to be quite general, and the same algorithm can be used to derive efficient sequences for a variety of target matrices. We demonstrate this by implementing the inversion-on-equality gate efficiently when the spin-spin coupling constants J 12 = J 23 = J and J 13 = 0. We also propose new pulse sequences to implement the parity gate and fanout gate, which are about 50% more efficient than the previous best efforts. Moreover, these sequences are shown to require significantly less RF power for their implementation. The proposed algorithm introduces several new features in the conventional genetic algorithm framework. We use matrices instead of linear chains, and the columns of these matrices have a well defined hierarchy. The algorithm is a genetic algorithm coupled to a fast local optimizer, and is hence a hybrid GA. It shows fast convergence, and running on a MATLAB platform takes about 20 minutes on a standard personal computer to derive efficient pulse sequences for any target 8X8 matrix U . PACS numbers: * Electronic address: [email protected] arXiv:0911.5465v2 [quant-ph] 4 Dec 2009 2 I. INTRODUCTION In recent years, there has been considerable interest in formulating time optimal pulse sequences in NMR. Various efforts have focused on replacing traditionally well known se- quences (for example, sequences to transfer coherence between coupled spins in multidimen- sional NMR experiments ) by their time optimal counterparts. The advantages of time optimal sequences are many. By reducing the time required to perform a desired unitary operation, they reduce the impact of undesirable effects due to decoherence or relaxation. The efficiency in achieving the desired operation can be improved drastically (in some cases it can be doubled ). It is becoming clear that any serious at- tempts at quantum computing  using NMR would require such time optimal sequences at their foundation....
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- Unitary matrix