Quantum Control Machine: The Limits of Quantum Programs as Data

Quantum algorithms for factorization, search, and simulation obtain computational advantage by performing control flow such as branching and iteration based on the value of quantum data in superposition. Complicating realization of these algorithms is the fact that in predominant quantum machine models, all control flow as embodied by the program counter is classical, and cannot exist in superposition. In this work, we identify that an alternative model to enable a program counter in superposition faces an obstacle -- no such machine can correctly support control flow constructs with non-injective semantics, including the conventional conditional jump. In fact, prior attempts to support this instruction cause programs to inappropriately collapse the superposition of data, meaning that quantum advantage is lost. We present a quantum machine model that supports both quantum effects on data and data-dependent control flow, using variants of conditional jump with injective semantics. We identify the necessary condition for programs for such a machine to preserve superposition of data, and show that expressible programs coincide with the unitary quantum circuits.