Quantum permutation puzzles with indistinguishable particles

TL;DR

This paper introduces quantum permutation puzzles using indistinguishable particles, demonstrating how quantum moves create puzzles that cannot be reduced to classical permutations, with a focus on the 2x2 slide puzzle and a brief mention of the Rubik's Cube.

Contribution

It presents the first quantum permutation puzzles with indistinguishable particles and introduces a quantum move, the square root of SWAP, to create genuinely quantum puzzles.

Findings

Quantum permutation puzzles can be mapped to classical puzzles when only particle swaps are used.

Adding the square root of SWAP move creates puzzles that are inherently quantum and not classically reducible.

The study focuses on the quantization of the 2x2 slide puzzle and briefly discusses the 2x2 Rubik's Cube.

Abstract

Permutation puzzles, such as the Rubik's Cube and the 15 puzzle, are enjoyed by the general public and mathematicians alike. Here we introduce quantum versions of permutation puzzles where the pieces of the puzzles are replaced with indistinguishable quantum particles. The moves in the puzzle are achieved by swapping or permuting the particles. We show that simply permuting the particles can be mapped to a classical permutation puzzle, even though the identical particles are entangled. However, we obtain a genuine quantum puzzle by adding a quantum move: the square root of SWAP. The resulting puzzle cannot be mapped to a classical permutation puzzle. We focus predominately on the quantization of the $2\times 2$ slide puzzle and briefly treat the $2\times 2 \times 1$ Rubik's Cube.