Square-Root SWAP Gate

A gate that, applied twice, exchanges two qubits, and applied once produces a half-swap that is a universal entangler.

Definition

The √SWAP gate is the matrix square root of the SWAP gate: √SWAP · √SWAP = SWAP. Physically it arises naturally from the Heisenberg exchange interaction between two spins evolved for half the swap time, which makes it a native two-qubit gate for exchange-coupled quantum dots.

Matrix

The inner block partially exchanges the |01⟩ and |10⟩ amplitudes with a phase. Because it entangles, √SWAP together with single-qubit gates is a universal gate set — unlike the full SWAP, which merely permutes and cannot create entanglement.

Deriving CNOT

A CNOT can be built from two √SWAP gates interleaved with single-qubit Z-rotations and phase gates. This is the standard route to logic on exchange-only spin qubits, where √SWAP is the only two-qubit gate the hardware provides directly.

import numpy as np
def sqrt_swap():
    a, b = (1+1j)/2, (1-1j)/2
    m = np.eye(4, dtype=complex)
    m[1,1]=a; m[2,2]=a; m[1,2]=b; m[2,1]=b
    return m

Uses

√SWAP is the native entangler for Loss-DiVincenzo semiconductor spin qubits and for some superconducting exchange schemes. Its excitation-preserving structure also makes it useful in simulating fermionic hopping, closely related to the XX+YY exchange gate. See XX+YY gate and iSWAP.