Quantum circuits for permutation matrices

TL;DR

This work develops quantum circuit constructions that realize permutation matrices on the 2^n computational basis using multi-controlled Toffoli gates. It provides two complementary strategies: a one-ancilla method that can realize any permutation by decomposing it into transpositions, and a no-ancilla method that uses bit-wise adjacent transpositions (with Hamming distance 1) to achieve the same goal. The authors prove that any permutation admits such a bit-wise adjacent decomposition, and they offer reduction strategies to minimize the number of transpositions, along with gate-count analyses showing O(n 2^n) scaling. Together, these results broaden the toolkit for implementing permutation-based unitaries in quantum circuits, with explicit constructions and trade-offs between ancilla usage and circuit depth/gate count.

Abstract

Two different algorithms are presented for generating a quantum circuit realization of a matrix representing a permutation on $2^n$ letters. All circuits involve $n$ qubits and only use multi--controlled Toffoli gates. The first algorithm constructs a circuit from any decomposition of the permutation into a product of transpositions, but uses one ancilla line. The second, which uses no ancillae, constructs a circuit from a decomposition into a product of transpositions that have a Hamming distance of one. We show that any permutation admits such a decomposition, and we give a strategy for reducing the number of transpositions involved.