Weyl Chamber of Two-Qubit Gates

The tetrahedral map of all inequivalent two-qubit entangling operations, one point per local-equivalence class.

The geometry

The Weyl chamber is a small tetrahedron in three-dimensional coordinate space that contains exactly one point for each local-equivalence class of two-qubit gates. Two gates are locally equivalent — equal up to single-qubit gates on each side — precisely when they map to the same point. Every two-qubit gate lives somewhere in this tetrahedron.

Landmarks

• Origin: the identity and all purely local gates
• One vertex on an edge: CNOT and CZ (maximally entangling, one-CNOT class)
• A corner: iSWAP
• The far vertex: SWAP (permutes but cannot entangle)
• Interior points: generic gates needing three CNOTs
• The B-gate point: two-copies-suffice optimality

Reading off resources

The chamber directly encodes compilation cost. Gates on an edge need one CNOT; gates on a face need two; interior gates need three. Entangling power (the ability to create entanglement from product states) also varies smoothly across the chamber, peaking at the CNOT/iSWAP region and vanishing at the identity and SWAP.

Why it is powerful

The Weyl chamber unifies every two-qubit gate discussion: native-gate selection, minimal CNOT counts, entangling-power comparison, and optimal synthesis all become geometry in one tetrahedron. Hardware entanglers become fixed points, and compiling a target gate is finding a path from a native point to the target using local moves.

# CNOT budget from Weyl coordinates (Vidal-Dawson style)
def cnot_count(cx, cy, cz, tol=1e-9):
    if cx<tol and cy<tol and cz<tol: return 0
    if cz<tol and cy<tol: return 1
    if cz<tol: return 2
    return 3

See canonical gate, magic basis, and Berkeley B gate.