Methodology of Resonant Equiangular Composite Quantum Gates

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