quantum gate directory

The Pauli group plays the role for quantum information that bit flips play for classical information: $X$ is a bit flip, $Z$ is a phase flip, and $Y$ $= iXZ$ is both at once. Together with the identity they form the elementary alphabet of quantum errors and observables.

The Pauli group on one qubit is generated by the Pauli matrices together with the phases $\pm 1, \pm i$ needed to close it under multiplication:

$$\mathcal{P}_1 := \left\{ i^k \sigma_j \mid k \in \{0, 1, 2, 3\},\ \sigma_j \in \{I, X, Y, Z\} \right\}$$

The $n$-qubit Pauli group consists of all tensor products of single-qubit elements:

$$\mathcal{P}_n = \left\{ i^k \bigotimes\limits_{i=0}^{n-1} P_i \mid P_i \in \{I, X, Y, Z\} \right\}$$

Properties

• Finite, with $4^{n+1}$ elements ($4^n$ Pauli strings times four phases).
• Any two elements either commute or anticommute, and every element squares to $\pm I$. This near-commutativity is what makes the group so tractable.
• The $4^n$ Pauli strings form an orthogonal basis for all $2^n \times 2^n$ matrices under the trace inner product, $\mathrm{Tr}(P^\dag Q) = 2^n \delta_{PQ}$. Hamiltonians, noise channels, and observables are routinely expanded in this basis — this is why "Pauli decomposition" is everywhere in quantum chemistry and error characterization.

Why it matters

• Error correction. Abelian subgroups of $\mathcal{P}_n$ that avoid $-I$ are exactly the stabilizer groups; their joint $+1$ eigenspaces are stabilizer codes. Correcting the discrete errors $X$, $Y$, $Z$ suffices to correct arbitrary errors, by linearity.
• Twirling and benchmarking. Averaging a noise channel over the Pauli group projects it onto a Pauli channel, the basis of randomized benchmarking and error mitigation techniques.
• The Clifford group is defined as the symmetries of this group.

Generalizations

On $d$-level systems the roles of $X$ and $Z$ are played by the shift and clock gates, generating the Weyl–Heisenberg (generalized Pauli) group with $\omega = \mathrm{e}^{2\pi i/d}$ in place of $-1$.