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

We investigate the feasibility of computing quantum gate-circuit emulation (QGCE)functions and quantum gate-circuit concatenation (QGCC) functions on digital hardware. QGCE functions serve the purpose of rewriting gate-circuits comprised of gates from a varying (possibly universal) input gate-set to gate-circuits comprised of gates from a fixed target gate set. Analogously, QGCC functions serve 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 considered an easy task for the digital computers controlling the quantum hardware. However, recent results employing a rigorous mathematical theory of computability indicate that this may not be the case. This paper extends the aforementioned theory, providing two relevant insights: Upon applying a rigorous theory of computability, QGCE functions and QGCC functions turn out to be uncomputable on digital hardware. The results remain valid when we restrict the set of feasible inputs for these functions to one-parameter families of fixed gate sets, which is applicable even to standard one-qubit systems. Our insights underline the possibility that several ideas from the theory of quantum computing may require a rethinking in order to become feasible for practical implementation.