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Quantum programming languages aim to reduce the burden of manipulatinghardware-level logic gates when implementing a quantum algorithm. A hurdle tothis goal is the difficulty of expressing control flow, such as branching anditeration, that depends on the value of data in quantum superposition. Toimplement algorithms for factorization, search, and simulation that containcontrol flow, quantum languages often require the use of bit-level logic gatesas opposed to the high-level constructs provided by classical languages.  The reason for this gap is that whereas a classical computer supportsimperative abstractions for control flow via a program counter that can dependon data and functional abstractions via terms in the $\lambda$-calculus, thetypical architecture of a quantum computer does not provide a program counterthat can depend on data in superposition, nor a physical representation of$\lambda$-terms in superposition. In principle, a quantum architecturesupporting such abstractions would simplify the implementation of control flowin quantum programs.  However, in this work, we identify a fundamental obstacle to control flow inquantum programming, which is that a quantum computer cannot correctly supportthe conventional conditional jump instruction in superposition, nor the$\beta$-reduction of $\lambda$-terms in superposition. We formally prove thatprogramming abstractions with non-injective state transition semantics, such asthe above, produce incorrect results in superposition.  As a way forward, we present the necessary and sufficient conditions forcontrol flow in superposition to be correctly realizable in a program. Weintroduce the quantum control machine, an instruction set architecture thatsatisfies these conditions, and show how it enables the use of control flow toimplement algorithms such as phase estimation, quantum walk, and physicalsimulation.

Quantum Control Machine: The Limits of Control Flow in Quantum Programming