The methodology of composite quantum gates

Guang Hao Low, Theodore J. Yoder, Isaac L. Chuang

Published 2016-03-13Version 1

The creation of composite quantum gates that implement quantum response functions $\hat{U}(\theta)$ dependent on some parameter of interest $\theta$ has historically been more of an art than a science. Through inspired design, a sequence of $L$ primitive gates also depending on $\theta$ can engineer a highly nontrivial $\hat{U}(\theta)$ that enables myriad precision metrology, spectroscopy, and control techniques. However, discovering new useful examples of $\hat{U}(\theta)$ requires great intuition to perceive the possibilities, and often brute-force to find optimal implementations. These demands hobble our imagination of new applications. We present a systematic and efficient methodology for composite gate design of arbitrary length, where phase-controlled primitive gates all rotating by $\theta$ act on a single spin. We fully characterize the realizable family of $\hat{U}(\theta)$, provide an efficient algorithm that decomposes a choice of $\hat{U}(\theta)$ into its shortest sequence of gates, and show how to efficiently choose achievable $\hat{U}(\theta)$ that for fixed $L$, are optimal approximations to objective functions on its quadratures. A strong connection is forged with \emph{classical} discrete-time signal processing, allowing us to swiftly construct, as examples, compensated gates with optimal bandwidth that implement arbitrary single spin rotations with optimal sub-wavelength spatial selectivity.