Silicon Spins

Silicon spin qubits use electron or hole spins in semiconductor quantum dots for quantum information processing. Exchange-only (EO) qubits encode a logical qubit in three electron spins, using only exchange interactions (controlled by gate voltages) for universal single-qubit control.

Coming Soon

The SiliconSpinSystem template is under development. The Hamiltonian and API below describe the planned implementation.

Exchange-Only Qubit

Hamiltonian

Three electron spins in a triple quantum dot with pairwise Heisenberg exchange:

\[H = J_{12}(\epsilon)\, \boldsymbol{S}_1 \cdot \boldsymbol{S}_2 \;+\; J_{23}(\epsilon)\, \boldsymbol{S}_2 \cdot \boldsymbol{S}_3\]

where $J_{ij}(\epsilon)$ are the exchange couplings controlled by detuning voltages $\epsilon$. In the $S = 1/2$ subspace (total spin $S_{\text{tot}} = 1/2$), this gives a logical qubit with two states:

\[|0_L\rangle = |S_{12}\rangle|\!\uparrow\rangle, \qquad |1_L\rangle = \sqrt{\tfrac{1}{3}}|T_{12}\rangle|\!\downarrow\rangle - \sqrt{\tfrac{2}{3}}|T_{12}^0\rangle|\!\uparrow\rangle\]

where $|S_{12}\rangle$ and $|T_{12}\rangle$ are singlet and triplet states of the first two spins.

Projected Hamiltonian

Projecting into the logical subspace, the effective Hamiltonian is:

\[H_{\text{eff}} = \frac{J_{12} + J_{23}}{4}\, I + \frac{J_{12} - J_{23}}{4}\, \sigma_z + \frac{\sqrt{3}}{4}\, J_{23}\, \sigma_x\]

The exchange couplings $J_{12}$ and $J_{23}$ serve as control parameters, providing full single-qubit control via $\sigma_x$ and $\sigma_z$ rotations.

Two Exchange-Only Qubits

For two EO qubits (six electrons in six dots), the coupling between logical qubits arises from exchange between the adjacent dots of each triple:

\[H = H_{\text{EO},1} + H_{\text{EO},2} + J_{36}\, \boldsymbol{S}_3 \cdot \boldsymbol{S}_4\]

where $J_{36}$ is the inter-qubit exchange coupling (between dot 3 of qubit 1 and dot 4 of qubit 2).

Planned API

# Single EO qubit
sys = ExchangeOnlyQubit(
    J12_bounds = (0.0, 10.0),   # Exchange coupling range (GHz)
    J23_bounds = (0.0, 10.0),
)

# Two coupled EO qubits
sys = TwoExchangeOnlyQubits(
    J12_bounds = (0.0, 10.0),
    J23_bounds = (0.0, 10.0),
    J45_bounds = (0.0, 10.0),
    J56_bounds = (0.0, 10.0),
    J34_bounds = (0.0, 5.0),    # Inter-qubit coupling
)

Typical Parameters

Parameter	Typical Value	Unit
Exchange $J$	0.1–10	GHz
Gate time	1–100	ns
$T_2^*$ (dephasing)	1–10	μs
$T_1$ (relaxation)	0.1–10	ms

References

DiVincenzo et al., "Universal quantum computation with the exchange interaction," Nature408, 339 (2000)
Weinstein et al., "Universal logic with encoded spin qubits in silicon," Nature615, 817 (2023)