Heisenberg Exchange in Quantum Dots

Once charge fluctuations are integrated out, neighboring quantum-dot spin qubits reduce to the exchange Hamiltonian

$H_{\mathrm{ex}}(t)=J(t){\vec{S}}_1\cdot {\vec{S}}_2.$

This note is intentionally the Hamiltonian-first companion to exchange-interaction-in-quantum-dots. Use it when the question is what rotation an exchange pulse implements, how encoded spin manifolds inherit logical axes, or why $\sqrt{\text{SWAP}}$ keeps reappearing. For barrier-vs-detuning control, Hubbard-model intuition, and charge-noise tradeoffs, go back to the companion note.

Routing heuristic

• Start here only after accepting that a calibrated exchange coupling $J(t)$ already exists as a control resource.
• Stay here when the question is about unitaries and encoded manifolds: pulse area, singlet-triplet phase accumulation, logical axes, or how pairwise exchange projects into a reduced qubit basis.
• Go back to exchange-interaction-in-quantum-dots when the question turns back into device physics: where $J$ came from, why symmetric barrier control helps, or how materials and charge admixture set the noise floor.

Effective-model boundary

This note deliberately starts after the charge sector has been integrated out. The upstream question, “what gate-voltage move produced this particular $J(t)$ and how noisy is that control surface?”, belongs in exchange-interaction-in-quantum-dots. The downstream question, “what unitary or encoded logical axis follows from this pulse area?”, belongs here.

Quick split from the device note

If the question is…	Start here?	Why
What does an exchange pulse do inside a two-spin or encoded-spin manifold?	Yes	This note is about the unitary action after projection.
Why is $\sqrt{\mathrm{SWAP}}$ the native entangler of the Loss-DiVincenzo picture?	Yes	The pulse-area algebra lives here.
Why does singlet-triplet treat exchange as a logical splitting while exchange-only turns it into non-collinear logical axes?	Yes	Those are projection questions, not fabrication questions.
How do barrier gates, detuning, and charge admixture determine the usable $J(t)$ in the first place?	No, go to exchange-interaction-in-quantum-dots	That microscopic control story belongs in the companion note.

Two-spin algebra

Exchange conserves total spin and splits the singlet from the triplet manifold by an amount set by $J$. That symmetry makes the interaction simultaneously useful and constrained: a pulse changes the relative phase between singlet and triplet sectors, so exchange alone does not give arbitrary two-qubit control, but it naturally generates the SWAP / sqrt-swap-as-universal-gate family once combined with local phases.

For a pulse of duration $t$ or, more generally, pulse area $\Phi =\int J(t)dt/\hslash$,

$U_{\mathrm{ex}}(\Phi )=\exp \left[-i\Phi {\vec{S}}_1\cdot {\vec{S}}_2\right].$

The important router is not the exact matrix form, but the fact that pulse area is the abstraction boundary. Once $J(t)$ has been reduced to a calibrated area, short pulses give partial swaps, the canonical half-swap pulse gives the native entangler of the Loss-DiVincenzo picture, and the same area logic survives after projection into larger encoded-spin manifolds.

Encoded-spin projections

Branch	What exchange becomes after projection	Best entry points
Direct two-spin branch	The native nearest-neighbor entangling resource, usually discussed through $\sqrt{\mathrm{SWAP}}$ pulses	loss-divincenzo-qubit, spin-qubit, sqrt-swap-as-universal-gate
Two-spin encoded branch	A logical splitting in the ${	S\rangle,
Three-spin encoded branch	Non-collinear logical axes built from pairwise terms like $J_{12}{\mathbf{S}}_1\cdot {\mathbf{S}}_2+J_{23}{\mathbf{S}}_2\cdot {\mathbf{S}}_3$	exchange-only-qubit
Always-on resonant branch	The projected exchange-defined splitting becomes the object a resonant drive addresses	rx-qubit
Always-on protected branch	The same projected structure is parked at a sweet spot so control means steering always-on couplings without reintroducing detuning sensitivity	aeon-qubit, charge-noise-sweet-spot

Scope boundary

• Keep microscopic origin, barrier-vs-detuning tuning, and materials-specific charge sensitivity in exchange-interaction-in-quantum-dots.
• Keep operating-point protection and detuning-symmetry arguments in charge-noise-sweet-spot.
• Use this note as the compact algebraic bridge between those device stories and gate-level control.

Key relationships

• sqrt-swap-as-universal-gate — gate-level consequence of exchange pulses
• singlet-triplet-qubit — logical $Z$ splitting from exchange in the $m_S=0$ subspace
• exchange-only-qubit — universal control from non-collinear exchange axes in a three-spin encoding
• rx-qubit — always-on exchange operated in a resonantly driven regime
• aeon-qubit — always-on exchange at a double sweet spot
• exchange-interaction-in-quantum-dots — microscopic origin and tuning knobs for $J$ before projection to an effective spin Hamiltonian

References

• loss-divincenzo-1998-quantum-dots — direct two-spin exchange as a native entangler
• petta-2005-singlet-triplet — exchange control in the two-spin encoded manifold
• divincenzo-2000-exchange-only — three-spin encoded control built from pairwise exchange
• shim-2016-aeon — always-on exchange at a double sweet spot