quantum gate directory

The Clifford group is the set of unitaries that map Pauli operators to Pauli operators under conjugation — formally, the normalizer of the Pauli group in the unitary group. A Clifford circuit can therefore be tracked entirely by following what happens to a handful of Pauli operators, rather than to exponentially many amplitudes.

$$\mathcal{C}_n := \left\{ U \in \mathsf{U}(2^n) \mid U \mathcal{P}_n U^\dag = \mathcal{P}_n \right\}$$

Properties

• Generated by just three gates: Hadamard, $S$, and CNOT.
• Finite modulo global phase, with order $$|\mathcal{C}_n / \mathsf{U}(1)| = 2^{n^2 + 2n} \prod_{j=1}^{n} (4^j - 1),$$ giving $24$ single-qubit Cliffords and $11520$ two-qubit Cliffords.
• A Clifford is determined (up to Pauli factors and phase) by where it sends the generators $X_i$ and $Z_i$, and this data is just a binary symplectic matrix: $\mathcal{C}_n / \mathcal{P}_n \cong \mathrm{Sp}(2n, \mathbb{F}_2)$. This is why Clifford circuits can be simulated with linear algebra over bits.

Why it matters

• Gottesman–Knill. Circuits of Clifford gates, Pauli measurements, and stabilizer-state inputs are classically simulable in polynomial time (Gottesman). Entanglement alone is therefore not what makes quantum computing hard to simulate — Clifford circuits generate plenty of it.
• Universality boundary. Adding any non-Clifford gate — $T$ being the standard choice — makes the gate set universal. Gates are usefully classified by their level in the Clifford hierarchy: Paulis at level 1, Cliffords at level 2, and gates like $T$, $\mathrm{C}S$, and $\mathrm{CC}Z$ at level 3.
• Error correction. Syndrome extraction, encoding circuits, and logical Paulis of stabilizer codes are all Clifford, which is precisely what lets them be verified and simulated at scale.
• Randomness. The multiqubit Clifford group is a unitary 3-design: random Cliffords reproduce the first three moments of Haar-random unitaries, making them the workhorse of randomized benchmarking and classical shadows.