G. Pauli Matrices

The four letters everything is written in:

I = [[1,0],[0,0+1]]    X = [[0,1],[1,0]]    Y = [[0,−i],[i,0]]    Z = [[1,0],[0,−1]]

Algebra: X²=Y²=Z²=I; XY=iZ (and cyclic: YZ=iX, ZX=iY); anticommute: XY=−YX. Observables: Z measures computational basis (eigen ±1), X measures {|+⟩,|−⟩}, Y the imaginary-axis basis. Hamiltonians: H = Σₖ hₖ Pₖ with Pauli strings Pₖ ∈ {I,X,Y,Z}⊗ⁿ — the universal decomposition (Ch. 47.2). Commutation rule for strings: P, Q commute iff they anticommute on an even number of positions — O(1) check, the engine of Ch. 46.5's optimization. Clifford group: maps Paulis to Paulis under conjugation (UPU† ∈ Paulis) — generated by {H,S,CX}; stabilizer formalism (Ch. 32) is Pauli-string bookkeeping; Gottesman–Knill (Ch. 17.14) makes Clifford circuits classical. QEC: errors are Paulis; syndromes are stabilizer measurements; decoding is Pauli inference (Ch. 36). Stabilizer states: unique +1-eigenstates of maximal commuting Pauli groups — the states QEC can hold. If you internalize one algebra deeply, make it this one.