Heisenberg Exchange in Quantum Dots

The fundamental two-qubit interaction in semiconductor spin qubits. When the tunnel barrier between two single-electron quantum dots is lowered, virtual tunneling produces a Heisenberg exchange coupling:

This note is intentionally the Hamiltonian-first abstraction. Use it when the question is what unitary an exchange pulse implements, how encoded spin states rotate, or why $\sqrt{\text{SWAP}}$ appears so often. For the microscopic origin of $J$, the barrier-vs-detuning tuning knobs, and device-level noise tradeoffs, see exchange-interaction-in-quantum-dots.

$H_s(t)=J(t){\vec{S}}_1\cdot {\vec{S}}_2$

where the exchange constant $J(t)=4t_0^2(t)/u$ depends on the tunneling matrix element $t_0$ and the on-site charging energy $u$.

Key Properties

• Purely electrical control: Modulated by gate voltages on the tunnel barrier, not by magnetic fields or microwave drives.
• Encoded-spin backbone: The same $J_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ form underlies loss-divincenzo-qubit, singlet-triplet-qubit, exchange-only-qubit, rx-qubit, and aeon-qubit; what changes across platforms is the encoding and operating point, not the core exchange algebra.
• Always-on problem: In practice, residual exchange $J$ is never perfectly zero — motivates aeon-qubit and dynamical decoupling schemes.
• Validity conditions: Requires single-band approximation ($u>t_0$), adiabatic pulsing (${\tau }_s>\hslash /\Delta E$), and $\Delta E\gg kT$.
• Swap and $\sqrt{\text{SWAP}}$: Pulsing $J$ for $\int Jdt=\pi$ gives SWAP; half that gives sqrt-swap-as-universal-gate.

Superexchange Variant

Three aligned dots with a higher-energy middle dot provide superexchange: $J=4t_0^4(1/{ϵ}^{2}u+1/2{ϵ}^3)$, enabling longer-range coupling.

References

• loss-divincenzo-1998-quantum-dots — original proposal
• loss-divincenzo-qubit
• exchange-only-qubit