Two-Qubit KAK Decomposition

The Cartan factorization that splits any two-qubit gate into local rotations around a canonical entangling core.

Statement

The KAK decomposition — from the Cartan KAK theorem of Lie theory — states that any two-qubit unitary U can be written U = (A1 ⊗ A2) · N(c1, c2, c3) · (B1 ⊗ B2), where the A and B factors are single-qubit unitaries (local gates) and N is a canonical entangling core built from XX, YY, and ZZ interactions.

The canonical core

The core is N = exp(i (c1 X⊗X + c2 Y⊗Y + c3 Z⊗Z)). The three coefficients (c1, c2, c3) — the Cartan coordinates — capture everything about the gate that local operations cannot change. Two gates are locally equivalent exactly when they share the same coordinates up to the Weyl-chamber symmetry.

CNOT count from coordinates

The number of CNOTs a two-qubit gate needs is read directly off its Cartan coordinates: zero if it is purely local, one if it lies on a single edge of the Weyl chamber, two for a face, and three for a generic interior point. This is how a compiler decides the optimal entangler budget before synthesizing.

# scipy-based sketch of extracting Cartan coordinates
# 1. move to the magic basis M
# 2. U' = M^dag U M ; form U'^T U'
# 3. eigenphases of U'^T U' give the (c1,c2,c3)
def kak_cnot_budget(edges_touched):
    return {'local':0,'edge':1,'face':2,'interior':3}[edges_touched]

Uses

KAK underlies every optimal two-qubit compiler, the classification of native gates by their canonical class, and the mapping of hardware entanglers (iSWAP, √iSWAP, CZ, MS) onto a common coordinate system. See canonical gate, Weyl chamber, and magic basis.