Tunable-Coupler Zero Exchange and Residual ZZ
Node role: Concept.
Reader task: report a tunable-coupler operating point without confusing exchange cancellation, a response splitting, and residual conditional shift.
Why cancellation is not decoupling
An exchange term transfers an excitation within a chosen model. A conditional shift compares transition energies after the nonlinear Hamiltonian is solved. Cancelling the first coefficient need not cancel the second quantity. Begin with the cosine-to-spectrum distinction if classical coupling and quantum energy shifts are still unfamiliar.
Two different quantities
Internal definitions / engineering choice. The equations below define the selected exchange zero and signed conditional shift. Their meaning requires the declared model and state continuation; they do not quote a universal coupler law.
For a declared topology, basis, model, and control input \(\lambda\), a zero-exchange condition has the form
\[ J_{\mathrm{eff}}(\lambda_0)=0. \]
It is a statement about the particular effective exchange coefficient named by that model. It does not by itself remove every interaction from the physical circuit, prove a zero observed splitting, or define a residual nonlinear interaction.
For computational-state energies \(E_{ij}\), define the eigenangular frequencies \(\omega_{ij}=E_{ij}/\hbar\). One conditional-shift convention is then
\[ \zeta_{ZZ} \equiv (\omega_{11}-\omega_{01})-(\omega_{10}-\omega_{00}). \]
This is a conditional eigenangular-transition-frequency difference. Its sign, units, Hamiltonian/model layer, state labeling, control point, and extraction method must be stated with every value. It is not inferred from a zero of \(J_{\mathrm{eff}}\).
What must be declared
| Quantity | Required qualifiers |
|---|---|
| \(J_{\mathrm{eff}}\) | Topology, coordinate/basis, representation or extraction, control domain, and units. |
| \(\lambda_0\) | Named control quantity, operating branch, and the exact zero-exchange coefficient it refers to. |
| \(\zeta_{ZZ}\) | Computational-state convention, Hamiltonian/model layer, control point, sign convention, extraction, and units. |
The general symbol convention supplies the qualifier pattern and concise coupling definitions tied to a stated canonical basis, normalization and approximation. A full quadratic exchange/pairing block derivation is not developed in this selected material. The Josephson cosine page explains why a conditional shift requires nonlinear quantum information.
Evidence boundary
Floating tunable coupler for scalable quantum computing architectures is the current source-backed entrypoint for direct, mediated, net, and zero-coupling vocabulary (Sete et al. 2021, secs. IV–V). Its residual-ZZ numerical value remains unpromoted pending its own source verification. This page sets neither an acceptable residual nor a Gate.
Connections
- source-owner parameter tables records the artifact-facing zero-exchange control and residual-ZZ rows.
- System Objectives and Trade-offs routes a target-specific trade-off to its proper owner.
| Field | Value |
|---|---|
| Status | Source-backed |
| Used by | source-owner parameter tables and future tunable-coupler Design Targets. |
| Semantic state | CONVERGING; no threshold, acceptance condition, or observed-splitting equivalence is asserted. |
| Open review question | Which model layer and extraction will a future target use for \(J_{\mathrm{eff}}\) and \(\zeta_{ZZ}\)? |