Square-Root iSWAP Gate

A partial exchange gate, the entangler that many superconducting processors calibrate as their fundamental two-qubit operation.

Definition

The √iSWAP gate is the matrix square root of iSWAP: applied twice it produces a full iSWAP. It performs a half-exchange of the single-excitation amplitudes, and it is a favored native gate because it is fast, high-fidelity, and, with single-qubit dressing, sufficient to build any two-qubit gate.

Matrix

The inner block is a balanced beam-splitter-like mixing of |01⟩ and |10⟩ with an i phase. It sits at a face of the Weyl chamber, so a generic two-qubit gate needs two √iSWAP applications, while many useful gates need only one.

Why hardware prefers it

A √iSWAP calibrated interaction is often shorter and less error-prone than a full CZ or iSWAP on the same coupler. Because two √iSWAP gates plus single-qubit rotations synthesize an arbitrary two-qubit unitary, compilers can express everything in this one primitive, simplifying calibration to a single two-qubit gate type.

import numpy as np
def sqrt_iswap():
    r = 1/np.sqrt(2)
    m = np.eye(4, dtype=complex)
    m[1,1]=r; m[2,2]=r; m[1,2]=1j*r; m[2,1]=1j*r
    return m

Uses

√iSWAP is used as the entangling layer in hardware-efficient variational circuits and in fermionic simulation, where a partial hop is exactly the physical operation of interest. It also generalizes into the tunable fSim gate family that spans a whole continuum of two-qubit interactions. See fSim gate and iSWAP.