Ising Coupling Gate RYY

The YY-generated two-qubit rotation, one of the three Cartan directions that classify every two-qubit entangler.

Definition

RYY(θ) = exp(-i θ (Y⊗Y) / 2) rotates the two-qubit state in the plane picked out by the Y⊗Y interaction. Together with RXX and RZZ it forms the three commuting generators of the two-qubit Cartan subalgebra, so any two-qubit gate reduces to a product of these three plus single-qubit gates.

Matrix

Note the sign pattern differs from RXX: the corner entries carry +i·s rather than -i·s, because Y⊗Y contributes an opposite sign on the |00⟩↔|11⟩ coupling relative to X⊗X.

Combination with RXX

The product RXX(θ)·RYY(θ) acts only within the single-excitation subspace {|01⟩, |10⟩}, giving an excitation-conserving exchange. This is the basis of the XX+YY gate used in hardware-efficient chemistry circuits, since it preserves particle number. See XX+YY gate.

import numpy as np
def ryy(theta):
    c, s = np.cos(theta/2), 1j*np.sin(theta/2)
    return np.array([[c,0,0,s],[0,c,-s,0],[0,-s,c,0],[s,0,0,c]])

Uses

RYY simulates the Y-component of Heisenberg spin couplings in digital quantum simulation. In Trotterized time evolution, one step of a Heisenberg chain is a product RXX(t)·RYY(t)·RZZ(t) on each bond, so RYY is one third of every spin-model timestep.

It also appears in decompositions of iSWAP and its square root, which are exactly excitation-preserving combinations of XX and YY rotations.