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Optimal Quantum Control by Composite Pulses

Elica S. Kyoseva, Nikolay V. Vitanov

Published Jan 1, 2014• 2 citations
Publisher

Abstract

We present a novel design of high-fidelity composite pulse sequences for dynamical suppression of amplitude noise with applications to NMR and QIP. We derive exact analytic formulas for the composite phases.
DOI
10.1364/cleo_at.2014.jtu4a.40

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We present a systematic SU(2) approach for construction of composite sequences of pulses with smooth temporal shapes that produce high-fidelity two-state excitation profiles. This makes possible the application of composite pulses to quantum control and quantum information processing with short and ultrashort laser pulses. We present an exact analytic formula for the composite phases for arbitrarily accurate broadband pulses and examples of narrowband, passband, and fractional-$\ensuremath{\pi}$ pulses as well as composite pulses with detuning compensation.
Composite pulses---sequences of pulses with well-defined relative phases---are an efficient, robust, and flexible technique for coherent control of quantum systems. Composite sequences can compensate for a variety of experimental errors in the driving field (e.g., in the pulse amplitude, duration, detuning, chirp, etc.) or in the quantum system and its environment (e.g., inhomogeneous broadening, stray electric or magnetic fields, unwanted couplings, etc.). The control parameters are the relative phases between the constituent pulses in the composite sequence, an accurate control over which is required in all composite sequences reported hitherto. In this paper, we introduce two types of composite pulse sequences which, in addition to error compensation in the basic experimental parameters, compensate for systematic errors in the composite phases. In the first type of such composite sequences, which compensate for pulse area errors, relative phase errors of over 10% can be tolerated with reasonably short sequences while maintaining the fidelity above the 99.99% quantum computing benchmark. In the second type of composite sequences, which compensate for simultaneous pulse area and detuning errors, relative phase errors of over 5% can be compensated.
Achieving high-fidelity control of quantum systems is essential for realization of a practical quantum computer. Composite pulse sequences which suppress different types of errors can be nested to suppress a wide variety of errors but the result is often not optimal, especially in the presence of constraints such as bandwidth limitations. Robust smooth pulse shaping provides flexibility but obtaining such analytical pulse shapes is a nontrivial problem and choosing the appropriate parameters typically requires a numerical search in a high-dimensional space. In this work, we extend a previous analytical treatment of robust smooth pulses to allow the determination of pulse parameters without numerical search. We also show that the problem can be reduced to a set of coupled ordinary differential equations which allows for a more streamlined numerical treatment.
Finding control laws (pulse sequences) that can compensate for dispersions in parameters which govern the evolution of a quantum system is an important problem in the fields of coherent spectroscopy, imaging, and quantum information processing. The use of composite pulse techniques for such tasks has a long and widely known history. In this paper, we give several new control law design methods for compensating dispersions in quantum system dynamics. We focus on system models arising in NMR spectroscopy and NMR imaging applications.
We introduce high-fidelity passband (PB) composite pulse sequences constructed by concatenation of recently derived arbitrarily large and arbitrarily accurate broadband B\mathcal{B}B and narrowband N\mathcal{N}N composite sequences. Our PB sequences allow us to produce flexible and tunable nearly rectangular two-state inversion profiles as a function of the individual pulse area because the width and the rectangularity of these profiles can be adjusted at will. Moreover, these PB sequences suppress excitation around pulse area 0 and $2\ensuremath{\pi}$, and suppress deviations from complete population inversion around pulse area $\ensuremath{\pi}$ to arbitrarily high orders. These features make them a valuable tool for high-fidelity qubit operations in the presence of relatively strong amplitude noise. We construct two types of PB pulses: N(B)\mathcal{N}(\mathcal{B})N(B), in which a broadband pulse is nested into a narrowband pulse, and B(N)\mathcal{B}(\mathcal{N})B(N), in which a narrowband pulse is nested into a broadband pulse; the latter sequences deliver narrower profiles. We derive exact analytic formulas for the composite phases of the PB pulses and exact analytic formulas for the inversion profiles. These formulas allow an easy estimation of the experimental resources needed for any desired qubit inversion profile.