quantum gate directory

Permutation gates do the one thing a classical reversible computer can do: shuffle computational basis states. No superpositions are created from basis states, no phases appear — every entry of the matrix is a $0$ or a $1$. This group is the precise sense in which classical reversible logic sits inside quantum computing.

$$\mathrm{S}_N \cong \left\{ P \in \mathsf{U}(N) \mid P|x\rangle = |\pi(x)\rangle \right\}$$

where $\pi$ ranges over the permutations of $\{0, \ldots, N-1\}$.

For $n$ qubits, $N = 2^n$, and the group is the symmetric group on $2^n$ letters — finite, with $N!$ elements, and a subgroup of the orthogonal group.

Properties

• The classical logic dictionary: $X$ is NOT, CNOT is XOR, and the Toffoli computes AND. The Toffoli alone (with ancilla bits) generates every permutation — it is universal for classical reversible computation.
• Basis states map to basis states, so a permutation circuit acting on a basis-state input is simulable by literally running the classical program. All the quantumness comes from superposed inputs: a CNOT applied after a Hadamard entangles precisely because the permutation acts on both branches at once.
• This is how classical functions enter quantum algorithms: an oracle $|x, y\rangle \mapsto |x, y \oplus f(x)\rangle$ is a permutation gate, queried in superposition.
• Closed under composition and inverse (a permutation matrix's inverse is its transpose), and every element has finite order.

Relations to other groups

Permutation gates are exactly the gates that are simultaneously real and "phase-free": the intersection of the orthogonal group with matrices of non-negative entries. Signed permutations like $Z$, $\mathrm{C}Z$, or the Margolus gate sit just outside, in the orthogonal group proper.