√SWAP as a Universal Two-Qubit Gate

The square root of SWAP operation, $U_{\text{SWAP}}^{1/2}$, is the gate-level endpoint of isotropic Heisenberg exchange between two physical spins. It is entangling, so repeated $\sqrt{\mathrm{SWAP}}$ pulses plus arbitrary single-qubit rotations form a universal gate set.

Use this note once a calibrated exchange pulse is already available and the question is what gate primitive it produces. Use heisenberg-exchange-in-quantum-dots for the underlying pulse-area Hamiltonian or for exchange projected into singlet-triplet, exchange-only, RX, and AEON logical subspaces. Use exchange-interaction-in-quantum-dots one layer further upstream when the question is how gate voltages, tunneling, and Coulomb repulsion create $J$.

Construction

The SWAP operator exchanges two qubit states: $U_{\text{SWAP}}|ij⟩=|ji⟩$. For

$H_{\mathrm{ex}}(t)=J(t){\vec{S}}_1\cdot {\vec{S}}_2,\Phi =\frac{1}{\hslash }\int J(t)dt,$

the evolution of two spin-$1/2$ particles is, up to a global phase,

$U_{\mathrm{ex}}(\Phi )=\cos \left(\frac{\Phi }{2}\right)I-i\sin \left(\frac{\Phi }{2}\right)U_{\text{SWAP}}.$

A pulse area $\Phi =\pi$ therefore gives SWAP, while $\Phi =\pi /2$ gives $\sqrt{\mathrm{SWAP}}$. The half-SWAP can maximally entangle a suitable product input; for example, it maps $|01⟩$ to $(|01⟩-i|10⟩)/\sqrt{2}$ up to a global phase. Two half-SWAP pulses plus local rotations construct a CNOT-equivalent XOR operation:

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

Why the half-SWAP is the useful primitive

• SWAP merely permutes the two states and cannot entangle a product input.
• $\sqrt{\mathrm{SWAP}}$ coherently superposes “swapped” and “not swapped,” which makes it entangling.
• Both gates still conserve total spin. Universality comes from combining the entangling half-SWAP with noncommuting local rotations, not from breaking spin conservation.

Boundary with iSWAP-family gates

The similar names hide different interactions.

Gate family	Typical interaction	What is conserved	Routing consequence
${\mathrm{SWAP}}^{\alpha }$, including $\sqrt{\mathrm{SWAP}}$	Isotropic $J{\vec{S}}_1\cdot {\vec{S}}_2$	Total spin and excitation number	This note is the direct gate-level explanation.
${\mathrm{iSWAP}}^{\alpha }$, including $\sqrt{\mathrm{iSWAP}}$	XY / flip-flop coupling $g({\sigma }_x{\sigma }_x+{\sigma }_y{\sigma }_y)$	Excitation number, but not the full isotropic-spin symmetry	Use this note only as a contrast; the native gate is not a half-SWAP generated by isotropic exchange.

This distinction matters for flip-flop-qubit and polar-molecule-qubit, whose exchange-like dipolar dynamics are usually described by the iSWAP family. Conversely, loss-divincenzo-qubit is the canonical direct $\sqrt{\mathrm{SWAP}}$ architecture. Encoded qubits such as exchange-only-qubit use the same pairwise Heisenberg resource internally, but their logical gates must be derived after projection into the encoded subspace rather than identified with a single physical half-SWAP pulse.

Routing summary

1. Read exchange-interaction-in-quantum-dots for the device knobs that manufacture $J(t)$.
2. Read heisenberg-exchange-in-quantum-dots for pulse-area evolution and encoded-spin projections.
3. Stay here for the direct two-spin $\sqrt{\mathrm{SWAP}}$ primitive, its entangling action, and its role in a universal gate set.

References

• loss-divincenzo-1998-quantum-dots — direct exchange pulses and the $\sqrt{\mathrm{SWAP}}$-plus-local-rotations construction
• divincenzo-2000-exchange-only — encoded computation using sequences of pairwise exchange operations