Shawn Skelton

Evolution in adiabatic quantum computing is famously polynomial-equivalent to evolution in the circuit model of quantum computation. Herein, we are not interested in translating an arbitrary unitary from one model to another, but instead in describing a particular class of evolutions in both models. Specifically, we are interested in when an evolution with quantum signal processing (QSP) is arbitrarily close to a unitary adiabatic evolution, without assuming a small qubit overhead, and when the reverse statement is true. The QSP circuit will be determined by a set of real phase factors, for some given signal operator. The quantum adiabatic evolution will be determined by the scalar functions $\Vec{x}$ used to parameterize $H(\Vec{x}(s))$. We show that in the high-degree and small-signal limit, a class of QSP circuits applied to an eigenstate of $H(0)$ and the result of the adiabatic evolution can be made close, with a direct functional relationship between the adiabatic schedule and the set of phase factors that determine the QSP circuit.