Three-step implementation of any nxn unitary with a complete graph of n qubits

Abstract: Quantum computation with a complete graph of superconducting qubits has been recently proposed, and applications to amplitude amplification, phase estimation, and the simulation of realistic atomic collisions given [Phys. Rev. A 91, 062309 (2015)]. This single-excitation subspace (SES) approach does not require error correction and is practical now. Previously it was shown how to implement symmetric nxn unitaries in a single step, but not general unitaries. Here we show that any element in the unitary group U(n) can be executed in no more than three steps, for any n. This enables the implementation of highly complex operations in constant time, and in some cases even allows for the compilation of an entire algorithm down to only three operations. Using this protocol we show how to prepare any pure state of an SES chip in three steps, and also how to compute, for a given SES state rho, the expectation value of any nxn Hermitian observable O in a constant number of steps.