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A Clifford circuit is composed entirely of Clifford gates. It maps Pauli operators to Pauli operators in the Heisenberg picture and stabilizer states to stabilizer states in the Schrödinger picture.

Daniel Gottesman developed a useful way to describe quantum circuits by following their Pauli operators instead of all their probability amplitudes. An $n$-qubit stabilizer state is specified by $n$ independent compatible commuting Pauli operators whose common $+1$ eigenspace is one-dimensional. A Clifford gate maps Pauli operators to Pauli operators by conjugation, so these generators can be updated efficiently on a computer. This explains why a large family of entangled computations is efficiently simulable, and supplies the language of stabilizer codes.^[ref]

Physicist whose work includes the stabilizer formalism for quantum error correction. His The Heisenberg Representation of Quantum Computers by Daniel Gottesman develops the Pauli group description of Clifford circuits. Verified institutional profile.

In the qubit stabilizer resource theory, a state outside the convex hull of stabilizer states. For one qubit this means outside the stabilizer octahedron. Such a state can supply a non-Clifford resource in a circuit otherwise restricted to Clifford operations and standard measurements.

An $n$-qubit stabilizer state is the unique simultaneous $+1$ eigenstate of an abelian group $G$ of Pauli operators having $2^n$ elements and not containing $-I$. The group $G$ is the state's stabilizer group; equivalently, it is generated by $n$ independent commuting Hermitian Pauli operators.

A stabilizer state on $n$ qubits is the unique simultaneous positive eigenstate of a stabilizer group with $n$ independent generators.

Stabilizer-state preparation starts from a computational-basis state and applies a Clifford circuit, producing a stabilizer state.