Permutation-invariant quantum circuits

TL;DR

This paper introduces a method to incorporate permutation symmetry into quantum circuits, reducing parameters and complexity, and enabling the creation of permutation-invariant quantum algorithms.

Contribution

It extends permutation symmetry into quantum circuit design via Lie algebra exponentiation, providing a systematic way to build permutation-invariant circuits.

Findings

Parameter scaling is reduced to O(n^3) with symmetry integration.

Permutation symmetry can be incorporated into existing circuits through modifications.

Permutation-invariant circuits can be systematically constructed using the proposed method.

Abstract

The implementation of physical symmetries into problem descriptions allows for the reduction of parameters and computational complexity. We show the integration of the permutation symmetry as the most restrictive discrete symmetry into quantum circuits. The permutation symmetry is the supergroup of all other discrete groups. We identify the permutation with a $\operatorname{SWAP}$ operation on the qubits. Based on the extension of the symmetry into the corresponding Lie algebra, quantum circuit element construction is shown via exponentiation. This allows for ready integration of the permutation group symmetry into quantum circuit ansatzes. The scaling of the number of parameters is found to be $\mathcal{O}(n^3)$, significantly lower than the general case and an indication that symmetry restricts the applicability of quantum computing. We also show how to adapt existing circuits to be invariant under a permutation symmetry by modification.