Feynman Meets Turing: The Uncomputability of Quantum Gate-Circuit Emulation and Concatenation

We investigate the feasibility of computing quantum gate-circuit emulation (QGCE) and quantum gate-circuit concatenation (QGCC) on digital hardware. QGCE serves the purpose of rewriting gate circuits comprised of gates from a varying input gate set to gate circuits formed of gates from a fixed target gate set. Analogously, QGCC serves the purpose of finding an approximation to the concatenation of two arbitrary elements of a varying list of input gate circuits in terms of another element from the same list. Problems of this kind occur regularly in quantum computing and are often assumed an easy task for the digital computers controlling the quantum hardware. Arguably, this belief is due to analogical reasoning: The classical Boolean equivalents of QGCE and QGCC are natively computable on digital hardware. In the present paper, we present two insights in this regard: Upon applying a rigorous theory of computability, QGCE and QGCC turn out to be uncomputable on digital hardware. The results remain valid when we restrict the set of feasible inputs for the relevant functions to one parameter families of fixed gate sets. Our results underline the possibility that several ideas from quantum-computing theory may require a rethinking to become feasible for practical implementation.