05 · Silicon spin

Silicon spin qubits

A spin inside a quantum dot.

SCHEMATIC GEOMETRY

CALCULATED · IDEAL MODEL

Hᵣ/ℏ = [ΔZ + Ω(cos φ X + sin φ Y)]/2; Hₑ/ℏ = δ(ZI − IZ)/4 + j(XX + YY + ZZ)/4; j = J/ℏ; U(t) = exp(−iHt/ℏ)

Assumptions and limits

One gate-defined electron spin starts in |↓⟩, or a selected pair starts in |↑↓⟩. Basis order is ↑, ↓; Z|↑⟩ = +|↑⟩. Unselected dots are not simulated.
Resonance uses a rotating frame and the rotating-wave approximation. Exchange uses the mean-Zeeman rotating frame and a constant rectangular exchange pulse; no simultaneous microwave drive.
Controls use MHz and μs; the solver converts frequency to radians per μs. Values are illustrative effective parameters, not calibrated gate voltages.
Ideal unitary dynamics exclude valley/orbital leakage, charge and nuclear noise, heating, relaxation, crosstalk and readout errors. Displayed probabilities are Born probabilities, not measured hardware data.
Geometry is schematic and not to scale. Quantum-dot connectivity is separate from silicon’s diamond-cubic atomic structure. Rings mark the selected subsystem; spin is not a classical rotation of an electron.

Sources · 7 references

Loss & DiVincenzo · Quantum computation with quantum dots (1998)Phys. Rev. A 57, 120 (1998) · The spin-qubit-in-a-dot proposal, cited for the Frontier chapter.
Xue et al. · Quantum logic with spin qubits crossing the surface code threshold (2022)Nature 601, 343 (2022), DOI resolved through Crossref 13 September 2026 · Cited in the Frontier chapter for silicon two-qubit gates reported above the surface-code threshold.
Noiri et al. · Fast universal quantum gate above the fault-tolerance threshold in silicon (2022)Nature 601, 338 (2022), DOI resolved through Crossref 13 September 2026 · Cited in the Frontier chapter alongside Xue et al.
Unseld et al. · 2D silicon spin control (2025)1 July 2025 · Experimental 2×2 Si/SiGe array, EDSR, nearest-neighbor exchange and the effective exchange Hamiltonian.
Philips et al. · Six-qubit silicon processor (2022)28 September 2022 · Linear-dot control, barrier-pulsed exchange and suppression of flip-flops by Zeeman differences.
Veldhorst et al. · Addressable quantum-dot spin (2014)12 October 2014 · Silicon electron-spin resonance, Rabi control and gate tunability.
Intel · Tunnel Falls research chip (2023)15 June 2023 · Dated manufacturer context: 12 quantum dots and electron-spin encoding. The scene does not replicate this chip.

Starting 3D…