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--- quantum-gate-two-qubit [August 26, 2026 at 18:03] – Ivan Janevski
+++ quantum-gate-two-qubit [August 26, 2026 at 18:03] (current) – Ivan Janevski
@@ Line 19: / Line 19: @@
   - [[quantum-gate-zz|ZZ (Ising coupling)]]
-## Bipartite entanglement
-Two-qubit entanglement comes in only one kind: entangled or not, and the only question is how much. A gate $U$ is **local** if $U = A \otimes B$ for $A, B \in \mathrm{SU}(2)$, forming the subgroup $\mathrm{SU}(2) \otimes \mathrm{SU}(2) \subset \mathrm{SU}(4)$; it can always be absorbed into single-qubit compilation, so it carries no entangling cost on hardware.
-$$\mathrm{CX} \left( \frac{|0\rangle + |1\rangle}{\sqrt{2}} \otimes |0\rangle \right) = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$$
-CX turns this product state into a Bell pair: measuring either qubit fixes the other's outcome. Every entangled two-qubit state is local-equivalent to some Bell pair, so one number classifies it: **concurrence**, from 0 (local) to 1 (maximally entangled). CX, CZ, SWAP, and $\text{XX}(\pi/2)$ reach 1; $R_Z(\theta) \otimes R_Z(\theta)$ stays at 0 for any $\theta$. This single-number classification is why two-qubit gates admit the clean KAK form below.
 ## Matrix representations
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 Note: $\mathrm{SWAP} = \mathrm{CX}_{12} \, \mathrm{CX}_{21} \, \mathrm{CX}_{12}$, and $\mathrm{iSWAP} = \text{XX}(\pi/2)\,\text{YY}(\pi/2)$ up to single-qubit phases.
-## Clifford vs non-Clifford
-**Clifford gates**: CX, CY, CZ, SWAP, iSWAP. For any Pauli string $P \otimes Q$, $U (P \otimes Q) U^\dagger$ is again a Pauli string up to sign, e.g.:
-- $\mathrm{CX} (X \otimes I) \mathrm{CX} = X \otimes X$
-- $\mathrm{CY} (X \otimes I) \mathrm{CY} = X \otimes Y$
-- $\mathrm{CZ} (X \otimes I) \mathrm{CZ} = X \otimes Z$
-Together with single-qubit Cliffords, these generate the [[quantum-gate-clifford|two-qubit Clifford group]], still efficiently simulatable via the stabilizer formalism. CY is rarely native because most hardware's native entangler is CZ or an Ising-type coupling, and CY doesn't fall directly out of either; it's synthesized by conjugating CX with single-qubit S gates ($\mathrm{CY} = (I \otimes S) \mathrm{CX} (I \otimes S^\dagger)$) rather than built from scratch.
-**Non-Clifford**: $\text{XX}$/$\text{YY}$/$\text{ZZ}(\theta)$ are Clifford only at $\theta = \pi/2$, mirroring how [[quantum-gate-t|T]] is a special-angle $R_Z$. At generic angle, conjugating a Pauli string no longer returns a Pauli string, it returns a genuine superposition of Paulis, which is exactly why classical stabilizer simulation stops working. At generic angle they're what most hardware natively implements.
 ## Group structure