Exchange Interaction in Quantum Dots

The Heisenberg exchange interaction $J$ between electron spins in adjacent quantum dots is the workhorse two-qubit coupling for semiconductor spin qubits. It arises microscopically from the interplay of quantum tunneling and Coulomb repulsion: when two electrons occupy neighboring dots separated by a tunable tunnel barrier, virtual hopping events — where one electron briefly visits the other’s dot — lower the energy of the spin-singlet state relative to the spin-triplet. In the simplest Hubbard model, the exchange splitting is $J=4t^2/U$, where $t$ is the interdot tunnel coupling and $U$ is the on-site Coulomb energy. This perturbative result holds when $t\ll U$, the regime where the dots are singly occupied and the Heisenberg Hamiltonian $H=J{\vec{S}}_1\cdot {\vec{S}}_2$ faithfully describes the low-energy physics.

The power of the exchange interaction lies in its purely electrical tunability. Two complementary knobs exist: barrier control adjusts the height of the potential barrier between dots, modulating $t$ directly while keeping the charge distribution symmetric; detuning control tilts the double-well potential, shifting one dot’s energy relative to the other, which changes $J$ through the effective $U$ in the denominator. Barrier control is generally preferred in modern devices because it preserves the charge symmetry point, reducing sensitivity to charge noise. In practice, $J$ can be tuned over many orders of magnitude — from effectively zero (dots fully isolated, $J/h<1\text{kHz}$) to tens of GHz (strong tunnel coupling) — by adjusting a single gate voltage on a timescale of nanoseconds.

The exchange interaction generates the $\sqrt{\text{SWAP}}$ and SWAP gates that underpin two-qubit operations across the spin qubit family. A pulse of duration $\tau$ with ${\int }_0^{\tau }J(t^{\prime })d{t}^{\prime }=\pi /2$ produces $\sqrt{\text{SWAP}}$, which combined with single-qubit rotations gives a universal gate set. For singlet-triplet qubits, exchange directly rotates between the logical $|0⟩=|S⟩$ and $|1⟩=|T_0⟩$ states. For exchange-only qubits, $J$ couplings between three dots provide both single-qubit and two-qubit control without any magnetic field gradients — the entire gate set is exchange-based. The AEON qubit uses an always-on exchange variant to operate at a sweet spot. In every case, the fidelity of exchange-based gates is limited by charge noise coupling through $\partial J/\partial ϵ$ (detuning sensitivity) or $\partial J/\partial V_B$ (barrier sensitivity), electrical noise on the control lines, and residual exchange when the interaction should be off.

The exchange interaction also appears in extended forms: superexchange couples next-nearest-neighbor dots through a virtual intermediate occupation (relevant for linear arrays), and the Hubbard parameters can be engineered in Si/SiGe, GaAs/AlGaAs, and Si-MOS platforms with quantitatively different $t$ and $U$ scales. Understanding and controlling $J$ at the few-percent level is the central materials and device challenge for scaling spin qubit processors.

Key relationships

• loss-divincenzo-qubit — original proposal using $J$-based $\sqrt{\text{SWAP}}$ as the native entangling gate
• singlet-triplet-qubit — uses $J$ to rotate between $|S⟩$ and $|T_0⟩$ within the two-electron subspace
• exchange-only-qubit — all control via exchange; no magnetic field gradients needed
• aeon-qubit — always-on exchange variant operating at a sweet spot for noise protection
• rx-qubit — resonant exchange qubit driven at the $S$–$T_{+}$ anticrossing, exchange-mediated
• hybrid-qubit — combines charge-like and spin-like states; exchange sets the energy scales
• silicon-spin-qubit — Si/SiGe and Si-MOS platforms where exchange is the primary 2-qubit mechanism
• sqrt-swap-as-universal-gate — the gate operation generated by exchange pulses
• heisenberg-exchange-in-quantum-dots — complementary note on the Hamiltonian formalism

References

• Loss & DiVincenzo (1998) — original proposal for exchange-based quantum computation in quantum dots
• Petta et al. (2005) — first coherent manipulation of singlet-triplet states via exchange in GaAs double dots
• Reed et al. (2016) — reduced sensitivity to charge noise using symmetric barrier control
• Martins et al. (2016) — noise-insensitive exchange operation via symmetric operating point