Invariant subspaces of two-qubit quantum gates and their application in the verification of quantum computers

Abstract

We present a set of techniques, based on the repeated arbitrary application of cp, cnot, and swap${}^{\ensuremath{\alpha}}$ (power-of-swap) quantum gate operations to an $n$-qubit quantum computer that can be used in its verification. We find isomorphisms between the groups generated by these gate operations and known groups and use techniques from representation theory to determine their invariant subspaces. For the cp operation, we find an isomorphism to the direct product of $n(n\ensuremath{-}1)/2$ cyclic groups of order 2, and determine ${2}^{n}$ one-dimensional invariant subspaces corresponding to the computational-basis vectors. For the cnot operation, we find an isomorphism to $GL(n,2)$, and determine two one-dimensional invariant subspaces and one $({2}^{n}\ensuremath{-}2)$-dimensional invariant subspace. For the swap${}^{\ensuremath{\alpha}}$ operation we find a complex structure of invariant subspaces with varying dimensions and occurrences and present a recursive procedure to construct them. Using knowledge of these invariant subspaces, we propose a hardware verification scheme which tests the correct functioning of a quantum computer.