Quantum Circuits for Isometries

Raban Iten, Roger Colbeck, Ivan Kukuljan, Jonathan Home, Matthias Christandl

Published 2015-01-27Version 1

Every quantum operation can be decomposed into a sequence of single-qubit and Controlled-NOT (C-NOT) gates. In many implementations, single-qubit gates are simpler to perform than C-NOTs, and it is hence desirable to minimize the number of C-NOT gates required to implement a circuit. Previous work has looked at C-NOT-efficient synthesis of arbitrary unitaries and state preparation. Here we consider the generalization to arbitrary isometries from m qubits to n qubits. We derive a theoretical lower bound on the number of C-NOT gates required to decompose an isometry for arbitrary m and n, and give an explicit gate decomposition that achieves this bound up to a factor of about two in the leading order. We also perform some bespoke optimizations in the case of small m and n. In addition, we show how to apply our result for isometries to give a decomposition scheme for an arbitrary quantum operation via Stinespring's theorem, and derive a lower bound on the number of C-NOTs in this case too.