Quantum circuits for permutation matrices

TL;DR

This paper introduces two algorithms for constructing quantum circuits that realize permutation matrices on 2^n elements, utilizing multi-controlled Toffoli gates, with one algorithm requiring an ancilla qubit and the other not.

Contribution

It presents novel algorithms for quantum circuit synthesis of permutation matrices, including methods that minimize ancilla usage and optimize transposition decompositions.

Findings

Both algorithms use n qubits and multi-controlled Toffoli gates.

Any permutation can be decomposed into transpositions with Hamming distance one.

Strategies are provided to reduce the number of transpositions needed.

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.