A theoretical preprint proposes a direct way to build quantum-circuit models from impedance and admittance responses, electrical descriptions of how a network reacts. It asks whether those responses can support exact Hamiltonians, the equations used to describe a circuit’s quantum energy and evolution, for superconducting qudits coupled to passive linear environments. The work uses illustrative configurations rather than an empirical cohort, including discrete resonator filters, finite-band metamaterials, nonreciprocal waveguide-QED systems and superconducting giant atoms. The supplied document is arXiv:2608.26027v2, dated 28 September 2026.
The construction starts by breaking the impedance or admittance response into zero-frequency, finite-frequency and infinite-frequency pieces. It then maps those pieces onto a canonical multiport circuit and applies first-order quantization, turning the network description into a Hamiltonian.
From circuit response to quantum equations
The exact construction is gauge-specific. For the impedance class, the paper uses the charge gauge; for the admittance class, it uses the flux gauge. Here, gauge means the choice of circuit variables used in the formulation. The paper also notes that its two endpoint gauges are not interchangeable descriptions for arbitrary infinite-dimensional circuits.
One of the framework’s central claims is that the same causal immittance response can supply the mode structure, frequency shifts, mediated interactions, decay rates and directional cross couplings. It also avoids spurious Lamb-shift divergences, meaning unwanted divergent frequency corrections produced by uncontrolled approximations.
In a strongly anharmonic benchmark, the dispersive Hamiltonian accurately reproduces the full dynamics. The agreement is conditional on the dispersive approximation and the parameters used in that example, so it is evidence for the reported benchmark rather than a guarantee for every circuit.
What the examples reveal
For continuous environments, the study uses weak-coupling and secular or partial-secular reductions to derive open-system dynamics. For passive environments, exact secularization gives positive-semidefinite blocks at degenerate frequencies and a GKSL generator, the standard mathematical form for a reduced open-system equation. The result supplies the required positivity structure for the dissipative coefficients.
A finite-band, left-handed metamaterial example shows how exact capacitive loading changes the picture near a band edge. It regularizes the continuum coupling and transfers spectral weight to a discrete pole below the infrared band. The weak-loading approximation becomes increasingly restrictive as the environment’s impedance grows.
The waveguide examples show that the placement of nonreciprocal circulators matters. Moving them to the injection points changes the local response from an infinite-line form to a semi-infinite-line form and increases local and directional couplings compared with placing them between the injection points.
The braided giant-atom model contains another conditional result. At a propagation phase of one-half pi, local and collective decay rates vanish while coherent exchange remains finite at the single-connection decay-rate scale. The result is obtained within the model’s weak-loading, Markov and reduced two-level description.
The study also highlights how a modeling partition can change a local parameter. In the transmission-line charge-qudit example, the exact charge-gauge Hamiltonian uses the bare junction capacitance rather than the commonly used dressed capacitance.
Useful, but conditional
The reduced master equation is not presented as universally valid. Its Markov reduction requires weak system-bath coupling and an environmental response that changes slowly over the frequency scales generated by the reduced dynamics. The supplied analysis therefore does not establish accuracy when coupling is strong or the environmental response varies rapidly.
Taken together, the preprint offers analytical tools for linking network response to exact and reduced quantum models across discrete, continuous, dissipative and nonreciprocal settings. Its reach remains tied to the paper’s passivity, rank, compactness and gauge conditions, while the best intermediate or mode-dependent gauge remains unresolved.
Paper data and sources
Original title: Immittance formulas for exact blackbox quantization and divergence-free effective models in circuit QED
Authors: Philippe Gigon, Peter Rabl, Adrian Parra-Rodriguez
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text