√SWAP as a Universal Two-Qubit Gate

The square root of SWAP operation, $U_{\text{sw}}^{1/2}$, combined with single-qubit rotations, forms a universal gate set for quantum computation. This is the native entangling gate for exchange-coupled spin qubits.

Construction

The SWAP operator exchanges two qubit states: $U_{\text{sw}}|ij⟩=|ji⟩$. It arises naturally from the Heisenberg exchange $H=J{\vec{S}}_1\cdot {\vec{S}}_2$ when pulsed for $\int Jdt=\pi$.

The $\sqrt{\text{SWAP}}$ is obtained by pulsing for half that duration. It creates maximal entanglement from a product state and, together with single-qubit $Z$-rotations, constructs CNOT:

$U_{\text{XOR}}=e^{i\frac{\pi }{2}{S}_1^z}{e}^{-i\frac{\pi }{2}{S}_2^z}{U}_{\text{sw}}^{1/2}{e}^{i\pi S_1^z}{U}_{\text{sw}}^{1/2}$

Significance

• SWAP alone conserves total spin angular momentum → not universal by itself
• $\sqrt{\text{SWAP}}$ breaks this conservation → universal when combined with local operations
• Native to any architecture with heisenberg-exchange-in-quantum-dots

References

• loss-divincenzo-1998-quantum-dots
• exchange-only-qubit — uses only exchange interactions (no single-qubit rotations needed)