Qubit ZooTwo-qubit interaction layer of circuit QED
A shared resonator becomes a quantum bus when two qubits couple to the same cavity mode but both stay detuned from it. Each qubit has its own jaynes-cummings-in-circuits coupling to the mode, and after adiabatically eliminating the cavity the two interactions combine into an effective qubit-qubit exchange:
J \approx \frac{g_1 g_2}{2}\left(\frac{1}{\Delta_1} + \frac{1}{\Delta_2}\right).$$ In the symmetric case $\Delta_1 \approx \Delta_2 = \Delta$, this reduces to the familiar estimate $J \sim g_1 g_2/\Delta$. The cavity stays only **virtually** occupied, so the resonator mediates the interaction without storing a real photon during the gate. ## What this note is for Use this note when the live question is: - how two qubits talk to each other through one cavity mode, - why early circuit-QED architectures could entangle qubits separated by centimeter distances, or - why shared-mode coupling naturally creates an always-on interaction budget. Do **not** use this note as the primary entry point for: - one-qubit cavity hybridization, which belongs in [[jaynes-cummings-in-circuits]], or - qubit-state-dependent cavity shifts for measurement, which belong in [[dispersive-readout-mechanism]]. ## Why the bus mattered historically 1. **It broke the nearest-neighbor wiring constraint** - The resonator let superconducting qubits interact over distances far beyond direct capacitive-coupling range. - That made circuit QED feel like a real architecture, not just an isolated qubit plus a readout cavity. 2. **It turned one cavity resource into two jobs** - The same resonator family that supports dispersive readout can also support entangling interactions. - This is why [[circuit-qed]] became the shared infrastructure layer for measurement, coupling, and later bosonic encoding. 3. **It exposed the cost of fixed interactions** - In the simplest bus picture, the effective exchange and associated residual $ZZ$ terms are always there once the qubits share a mode. - That pushed the field toward architectures that preserve the long-range-coupling intuition but localize the aggressive tunability in a separate interaction element. ## Boundary with newer coupling architectures - The bus picture here is the clean conceptual ancestor of modern superconducting multi-qubit wiring. - [[tunable-coupler]] is the later answer to its biggest systems problem: keep the data qubits coherence-friendly while switching the interaction nearly off at idle. - [[gmon]] is the historically specific Google branch where Xmon-like qubits and a tunable coupler became one scalable processor architecture. ## Circuit-QED abstraction ladder Read this cluster in order: 1. [[vacuum-rms-field-scaling]] for why the cavity fields are large enough to make $g$ useful. 2. [[jaynes-cummings-in-circuits]] for the one-qubit one-mode interaction. 3. the bus layer in this note for the two-qubit effective interaction that follows from sharing that mode. Source: [[blais-2004-circuit-qed]] Related: [[circuit-qed]], [[transmon]], [[jaynes-cummings-in-circuits]], [[dispersive-readout-mechanism]], [[tunable-coupler]], [[gmon]]