Composing Quantum Algorithms

Composition is something we take for granted in classical algorithms design, and in particular, we take it as a basic axiom that composing ``efficient'' algorithms should result in an ``efficient'' algorithm -- even using this intuition to justify our definition of ``efficient.'' Composing quantum algorithms is a much more subtle affair than composing classical algorithms. It has long been known that zero-error quantum algorithms \emph{do not} compose, but it turns out that, using the right algorithmic lens, bounded-error quantum algorithms do. In fact, in the bounded-error setting, quantum algorithms can even avoid the log factor needed in composing bounded-error randomized algorithms that comes from amplifying the success probability via majority voting. In this article, aimed at a general computer science audience, we try to give some intuition for these results: why composing quantum algorithms is tricky, particularly in the zero-error setting, but why it nonetheless works \emph{better} than classical composition in the bounded-error setting.