Clifford gate

The Clifford group on n qubits is the set of unitary operators that normalize the Pauli group: for any Pauli operator P, the conjugate U P U† is again a Pauli operator (up to a global phase).¹ This property makes Clifford gates efficiently simulable classically, as they map stabilizer states to stabilizer states.

Common examples include the Hadamard gate H, the phase gate S (also called P), and the controlled-NOT (CNOT) gate. Together, these generate the entire Clifford group on any number of qubits.² The group is finite, with the order formula given in the metrics.

Clifford gates are the backbone of stabilizer codes, the most widely studied class of quantum error-correcting codes. In these codes, logical operations are often implemented by transversal Clifford gates, which are naturally fault-tolerant.³ For example, the Steane code and the surface code support transversal H, S, and CNOT gates.

However, no stabilizer code can implement a universal set of logical gates transversally; this is known as the Eastin–Knill theorem.⁴ Consequently, fault-tolerant quantum computing requires additional techniques such as magic state distillation to implement non-Clifford gates like the T gate.

The Gottesman–Knill theorem states that any quantum circuit consisting solely of Clifford gates, with initial states in the computational basis and measurements in the computational basis, can be efficiently simulated on a classical computer.⁵ This result, published in 1998, highlights a sharp boundary between classically simulable and potentially powerful quantum computation.

Clifford circuits are also used in randomized benchmarking, a technique to estimate gate fidelity by averaging over random Clifford sequences.⁶ The efficient classical simulation of Clifford circuits is exploited in this protocol to extract error rates.

Beyond the standard generators, the Clifford group has a rich mathematical structure. It is isomorphic to the symplectic group Sp(2n, F₂) modulo the Pauli group, a fact used in the classification of stabilizer states.¹ The single-qubit Clifford group has 24 elements, which correspond to the symmetries of the cube.

Clifford gates are also central to the study of quantum contextuality and non-stabilizerness (also called magic), which are resources for quantum advantage. The T gate, a non-Clifford gate, is often used as a measure of this resource.⁷ Additionally, Clifford circuits appear in the study of quantum machine learning, where they serve as a benchmark for classical simulability.

Pauli group

The group generated by the Pauli matrices X, Y, Z and the identity, with phases.

Stabilizer state

A quantum state that is the unique eigenstate of a set of commuting Pauli operators.

Transversal gate

A logical gate implemented by applying physical gates independently to each qubit of a code block.

Magic state distillation

A process to produce high-fidelity non-Clifford states from noisy ones, enabling universal fault-tolerant quantum computing.