The XOR swap trick also features in the compilation/synthesis of quantum algorithms, where the XOR instruction (in the form of a CNOT gate) is fundamental in many architectures, and where native swapping need not be available.

One extension that I ran into, and which I think forms a nice problem is the following:

Just like the XOR swap trick can be used to swap to variables (and let's just say that they're bools), it can be extended to implement any permutation of the variables: suppose that the permutation is written as a composition of n transpositions (i.e., swaps of pairs), and that is the minimal number of transpositions that let's you do that. Each transposition can be implemented by 3 XORs, by the XOR swap trick for pairs, and so the full permutation can be implemented by 3n XORs. Now here's the question: Is it possible to come up with a way of doing it with less than 3n, or can we find a permutation that has a shortcut through XOR-land (not allowing any other kinds of instructions)? In other words, is XOR-swapping XOR-optimal?

I'm not going to spoil it, but only last year a paper was published in the quantum information literature that contains an answer [0]. I ended up making a little game where you get to play around with XOR-optimizing not only permutations, but general linear reversible circuits. [1]

[0] https://link.springer.com/article/10.1007/s11128-025-04831-5

[1] https://swapple.fuglede.dk