Fig. 4: Two-qubit gate family from nearest-neighbour exchange.
From: Pipeline quantum processor architecture for silicon spin qubits
a Bloch sphere two-qubit dynamics on the mz = 0 subspace under nearest-neighbour exchange. The mz = 0 states rotate around an axis defined by the relative magnitudes of exchange strength and Zeeman energy difference. b–e Circuit identities for the unitary time evolution UNNE(ϵ, φ, χ) Eq. (3), describing nearest-neighbour exchange in the presence of Zeeman energy differences. Multiples of 2πn are left out of the rotation angles for simplicity. b, c Choice of rotation angle φ = π + 2πn realises the phase gates (b) CPhase and (c) Ising ZZ-rotation gate. d Choice of φ = π/2 + 2πn realises a gate close to the Givens rotation, where the rotation angle χ depends on the ratio ΔEZ/Jij. e The SWAP-rotation gate can be constructed from the native unitary gate with φ = π/2 + 2πn and χ = π/4, as two such native operations separated by single-qubit Z-rotation gates. The phases of the mz = ± 1 components are fixed by subsequent application of another phase gate and single-qubit Z-rotations. f Bloch sphere representation of the SWAP-rotation identity of (e). g–i Fidelities for the native gates with (g) ϕ = π + 2πn, realising the Ising ZZ-rotation gate, and (h) ϕ = π/2 + 2πn, realising the Givens(χ) SWAP operation, which, for χ = π/4 is used in the composition of SWAP(θ) (see e), and (i) the composite SWAP(θ) rotation gate as a function of rotation angles and tunnel coupling variance \({\sigma }_{{t}_{ij}}/{t}_{ij}\).
