fSim Gate

A two-parameter fermionic-simulation gate that captures a tunable exchange plus a controlled phase in one native operation.

Definition

The fSim(θ, φ) gate combines an iSWAP-type exchange of strength θ with a controlled-phase of strength φ in a single tunable two-qubit gate. It was introduced to match the natural interaction of tunable-coupler superconducting hardware, where both effects appear together and can be dialed independently.

Matrix

The θ term mixes the single-excitation states like a partial iSWAP; the φ term phases |11⟩ like a controlled-phase. Special cases: fSim(π/2, 0) = iSWAP, fSim(π/4, 0) = √iSWAP, fSim(0, φ) = controlled-phase.

Why fermionic

Under the Jordan-Wigner mapping, a fermionic hopping term plus an on-site interaction on two adjacent modes is exactly an fSim gate. Simulating the Hubbard model or molecular Hamiltonians therefore maps naturally onto fSim, avoiding the overhead of expressing the same physics in CNOT-based gates.

import numpy as np
def fsim(theta, phi):
    c, s = np.cos(theta), np.sin(theta)
    m = np.eye(4, dtype=complex)
    m[1,1]=c; m[2,2]=c; m[1,2]=-1j*s; m[2,1]=-1j*s
    m[3,3]=np.exp(-1j*phi)
    return m

Calibration and the Sycamore gate

Real tunable couplers realize an fSim with device-specific θ and φ that must be characterized rather than assumed. Google's Sycamore gate is a particular calibrated fSim used in its quantum-advantage experiments. Because the whole family lives in one native operation, compilers can target the exact point the hardware achieves. See √iSWAP and canonical gate.