 Do physical processes exist which contradict the physical Church-Turing thesis? Apart from analog computation, there exist at least two main kinds of example purporting to show that the notion of recursion, or Turing-computability, is not a natural physical property (Pour-el and Richards 1981; Pitowsky 1990; Hogarth 1994). Although the physical systems involved (a specific initial condition for the wave equation in three dimensions and an exotic solution to Einstein’s field equations, respectively) are somewhat contrived, a thriving school of “hypercomputation” that aspires to extend the limited examples of physical “hypercomputers” and in so doing to physically “compute” the non-Turing-computable has nevertheless emerged (for a review see Copeland (2002); for a criticism: Davis (2003); for a recent proposal and response to criticisms see Andréka et al. (2018)). Quantum hypercomputation is rarely discussed in the literature (see, e.g., Adamyan, Calude, and Pavlov 2004), but the most concrete attempt to harness quantum theory to compute the non-computable is the suggestion to use the quantum adiabatic algorithm (see below) to solve Hilbert’s Tenth Problem (Kieu 2002, 2004)—a Turing-undecidable problem equivalent to the halting problem. Criticism, however, has exposed the unphysical character of the alleged quantum adiabatic hypercomputer (see Hodges 2005; Hagar and Korolev 2007).