Improved classical simulation of quantum circuits dominated by Clifford gates: Supplemental Material

In this Section we state some facts concerning stabilizer groups and stabilizer states. Let Pn be the n-qubit Pauli group. Any element of Pn has the form iP1⊗ · · · ⊗Pn, where each factor Pa is either the identity or a singlequbit Pauli operator X,Y, Z and m ∈ Z4. An abelian subgroup G ⊆ Pn is called a stabilizer group if −I / ∈ G. Each stabilizer group has the form G = 〈G1, . . . , Gr〉 for some generating set of pairwise commuting self-adjoint Pauli operators G1, . . . , Gr ∈ G such that |G| = 2. The integer r is called the dimension of G and is denoted r = dim (G). A state |ψ〉 is said to be stabilized by G if P |ψ〉 = |ψ〉 for all P ∈ G. States stabilized by G span a “codespace” of dimension 2n−r. A projector onto a codespace has the form