Lightweight Detection of a Small Number of Large Errors in a Quantum Circuit

Noah Linden, Ronald de Wolf
2021 Quantum  
Suppose we want to implement a unitary U, for instance a circuit for some quantum algorithm. Suppose our actual implementation is a unitary U~, which we can only apply as a black-box. In general it is an exponentially-hard task to decide whether U~ equals the intended U, or is significantly different in a worst-case norm. In this paper we consider two special cases where relatively efficient and lightweight procedures exist for this task.First, we give an efficient procedure under the
more » ... under the assumption that U and U~ (both of which we can now apply as a black-box) are either equal, or differ significantly in only one k-qubit gate, where k=O(1) (the k qubits need not be contiguous). Second, we give an even more lightweight procedure under the assumption that U and U~ are Clifford circuits which are either equal, or different in arbitrary ways (the specification of U is now classically given while U~ can still only be applied as a black-box). Both procedures only need to run U~ a constant number of times to detect a constant error in a worst-case norm. We note that the Clifford result also follows from earlier work of Flammia and Liu, and da Silva, Landon-Cardinal, and Poulin.In the Clifford case, our error-detection procedure also allows us efficiently to learn (and hence correct) U~ if we have a small list of possible errors that could have happened to U; for example if we know that only O(1) of the gates of U~ are wrong, this list will be polynomially small and we can test each possible erroneous version of U for equality with U~.
doi:10.22331/q-2021-04-20-436 fatcat:fwwr57lhhrhg5mhn7453bk256a