Matrix Representation of Arbitrarily Controlled Quantum Gates

Abstract

Controlled operations allow for the entanglement of quantum registers. In particular, a controlled-$U$ gate allows an operation, $U$, to be applied to the target register and entangle the results to certain values in the control register. This can be generalised by making use of the classical notion of conditional statements, where if a value (or state) satisfies some condition then a sequence of operations can be performed. A method is introduced to represent these generalised controlled operations that are based on classical conditional statements. Throughout examples are given to highlight the use of introduced gates.