QuLTRA: Hybrid Lumped & Transmission Analyzer

• QuLTRA is a Python package that directly simulates superconducting circuits combining lumped elements and distributed microwave components like CPW lines.
• It employs an admittance matrix formalism with methods such as Brent’s and Newton–Raphson to extract Hamiltonian parameters including frequencies, anharmonicities, and Purcell decay rates.
• By avoiding lumped discretization of CPW lines, QuLTRA bridges the gap between full-wave solvers and reduced lumped-element models, enabling rapid iterative circuit-QED design.

Searching arXiv for QuLTRA and closely related superconducting-circuit analysis tools. First, locating the QuLTRA paper itself. QuLTRA, short for Quantum hybrid Lumped and TRansmission lines circuits Analyzer, is an open-source Python package for simulating superconducting quantum circuits containing both lumped elements and distributed microwave structures such as coplanar waveguide (CPW) transmission lines and multi-line CPW couplers. Its defining feature is the direct treatment of distributed components, without discretizing them into lumped-element equivalents, combined with Hamiltonian extraction via the energy participation ratio (EPR) method. In the formulation presented in "QuLTRA: Quantum hybrid Lumped and TRansmission lines circuits Analyzer" (Zaccaria et al., 3 Sep 2025), this enables the extraction of mode frequencies, anharmonicities, cross-Kerr interactions, and Purcell decay rates while naturally including higher-order modes of distributed structures.

1. Scope and design objective

QuLTRA addresses a practical bottleneck in superconducting circuit design: the need to map a target Hamiltonian onto a realizable physical circuit when that circuit contains both nonlinear lumped elements and distributed microwave components. The design loop is described as iterative: one guesses component values, extracts the Hamiltonian, compares it to the target, and revises the design accordingly (Zaccaria et al., 3 Sep 2025).

Within that workflow, existing tools are divided into two broad categories. Full electromagnetic solvers such as Ansys HFSS + pyEPR are described as very accurate, but computationally expensive and difficult to use iteratively for large or complex layouts. Lumped-element-only Hamiltonian tools such as QuCAT, SQcircuit, and CircuitQ are described as faster, but unable to naturally represent distributed components like CPW resonators, feedlines, and couplers unless these are discretized into many LC sections (Zaccaria et al., 3 Sep 2025).

The central purpose of QuLTRA is to bridge that gap. It retains a circuit-based modeling workflow while handling distributed microwave structures in a physically direct manner. This suggests that QuLTRA is positioned as an intermediate layer between full-wave electromagnetic verification and reduced lumped-network quantization, especially for circuit-QED architectures in which CPW resonators, couplers, and feedlines are structurally central.

2. Quantization framework and Hamiltonian extraction

QuLTRA uses the energy participation ratio method to extract effective Hamiltonian parameters from the linearized circuit (Zaccaria et al., 3 Sep 2025). The Hamiltonian is written as a linear multimode part plus Josephson nonlinearity:

$$\hat{H}=\sum_{m=1}^M{\hbar \omega_m \hat{a}_m^\dagger \hat{a}_m} -\sum_i N_i^2 E_i \left[\cos(\hat{\phi}_i/N_i)+\hat{\phi}_i^2/(2N_i^2)\right],$$

with

$E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$

where $\hat{\phi}_i$ is the reduced flux across the $i$-th Josephson array.

The reduced flux is expanded in normal modes as

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$

and the mode-dependent flux zero-point fluctuation $\phi_{m,i}$ is related to the energy participation ratio $p_{m,i}$ by

$\phi_{m,i}^2 = p_{m,i}\frac{\hbar \omega_m}{2E_i}.$

The participation ratio itself is defined as

$p_{m,i}= \frac{\text{linear inductive energy stored in the $i$-th array in mode$m$}}{\text{total inductive energy stored in mode $m$}}.$

After expanding the cosine to fourth order and applying the rotating-wave approximation, the effective Hamiltonian becomes

$$\hat{H} \simeq \sum_{m=1}^M \left[ \hbar(\omega_{m}- \Delta_{m}) \hat{a}_{m}^\dagger \hat{a}_{m} - \frac{1}{2} \hbar \alpha_{m} \hat{a}_{m}^{\dagger 2} \hat{a}_{m}^2 \right] - \sum_{n \ne m} \frac{1}{2} \hbar \chi_{mn} \hat{a}_{m}^\dagger \hat{a}_{m} \hat{a}_{n}^\dagger \hat{a}_{n},$$

where $E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$0 is the bare mode frequency, $E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$1 the Lamb shift, $E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$2 the self-Kerr or anharmonicity, and $E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$3 the cross-Kerr coupling (Zaccaria et al., 3 Sep 2025). The relations

$E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$4

and

$E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$5

show that once the linear mode frequencies $E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$6 and the EPRs $E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$7 are known, the nonlinear Hamiltonian parameters follow directly (Zaccaria et al., 3 Sep 2025).

This organization is consequential. It separates the problem into a linear eigenmode computation and a nonlinear parameter extraction stage, which is the same broad philosophy associated with EPR-based quantization. A plausible implication is that the method is especially attractive when the linear microwave environment is structurally complicated but the nonlinearity remains localized in Josephson elements.

3. Linear-mode computation through admittance matrices

QuLTRA represents the circuit as a network of multi-port subcircuits described by admittance matrices (Zaccaria et al., 3 Sep 2025). After selecting a reference node, Kirchhoff’s current law yields

$E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$8

and the mode frequencies are obtained from the roots of

$E_i=\left(\frac{\hbar}{2e}\right)^2\frac{1}{L_i},$9

For a lossless circuit, $\hat{\phi}_i$0. For a lossy circuit, the paper uses

$\hat{\phi}_i$1

where $\hat{\phi}_i$2 is the mode frequency and $\hat{\phi}_i$3 is the mode linewidth or dissipation rate (Zaccaria et al., 3 Sep 2025).

The root-finding procedure is split by circuit class. For lossless circuits, QuLTRA scans the target frequency interval in overlapping sub-intervals and uses Brent’s method to find roots of the real or imaginary part of the determinant, depending on matrix-size parity. Candidate roots are filtered by checking that $\hat{\phi}_i$4 (Zaccaria et al., 3 Sep 2025). For lossy circuits, it first finds the lossless roots by shorting resistors and then refines each root with Newton–Raphson using the lossy circuit as the target. This yields both resonant frequencies and linewidths.

After obtaining a root $\hat{\phi}_i$5, QuLTRA computes the corresponding node-voltage eigenvector $\hat{\phi}_i$6 from the null space of $\hat{\phi}_i$7. Those node voltages are then used to compute inductive energies and therefore participation ratios:

$\hat{\phi}_i$8

For lumped inductors and Josephson junctions, the inductive energy uses

$\hat{\phi}_i$9

For distributed CPW lines and couplers, the current distribution is computed analytically and the inductive energy is integrated over the structure (Zaccaria et al., 3 Sep 2025).

This admittance-matrix formalism is one of the package’s general design principles. The paper states that the method is not fundamentally limited to CPW lines; it can be generalized to any component described by an admittance matrix (Zaccaria et al., 3 Sep 2025).

4. Direct modeling of CPW transmission lines and multi-line couplers

A central contribution of QuLTRA is the direct treatment of straight, lossless CPW lines as genuine transmission-line elements rather than as LC ladders (Zaccaria et al., 3 Sep 2025). For a line, the voltage and current profiles are written as

$i$0

with

$i$1

and

$i$2

A line of length $i$3 is represented by the two-port admittance matrix

$i$4

Given the endpoint voltages for a mode, QuLTRA reconstructs $i$5, then $i$6, then the full current profile $i$7, and computes the line inductive energy analytically:

$i$8

Because the CPW is treated as a continuum, QuLTRA automatically captures the fundamental mode, higher-order harmonics, and the correct mode structure of distributed resonators (Zaccaria et al., 3 Sep 2025).

QuLTRA also directly models multi-line CPW couplers, including two-line and three-line structures. For an $i$9-conductor geometry,

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$0

with

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$1

where

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$2

and the matrices satisfy

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$3

The capacitance matrix is obtained via conformal mapping / Schwarz–Christoffel transformations. The paper gives the mapping

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$4

with the $$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$5 chosen so that grounded conductors map to a continuous boundary, and then obtains the capacitance matrix elements from the mapped geometry as

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$6

For a two-line or three-line coupler of length $$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$7, QuLTRA constructs a 4-port admittance matrix $$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$8 by exciting one port at a time with 1 V while grounding the others, then calculating the resulting port currents from the propagated-wave solution. The stored inductive energy is

$$\phi_i = \sum_{m=1}^M \phi_{m,i}(a_m^\dagger + a_m),$$9

The significance of this construction is methodological rather than merely implementational. It removes the need to approximate distributed structures with a large number of lumped sections and therefore avoids the associated growth in node count and the convergence issues of LC discretization. This suggests that the package is especially advantageous when higher distributed modes, coupler geometry, or line-mediated environmental effects are central to the device physics.

5. Higher-order modes, accuracy, and validation

QuLTRA treats distributed components as objects with infinitely many modes, and includes as many as needed by finding additional roots of $\phi_{m,i}$0 (Zaccaria et al., 3 Sep 2025). The paper emphasizes that this is especially important for $\phi_{m,i}$1 and $\phi_{m,i}$2 resonators, ultra-strong coupling systems, Purcell filters using higher resonator poles, and multiplexed readout circuits where harmonics matter.

The validation program compares QuLTRA against Ansys HFSS + pyEPR, QuCAT, and literature benchmarks (Zaccaria et al., 3 Sep 2025). The reported examples are summarized below.

Validation case	Reported comparison	Reported outcome
Transmon + $\phi_{m,i}$3 resonator	HFSS vs QuLTRA	Largest discrepancies below about 8%
$\phi_{m,i}$4 resonator inductively coupled to a feedline	HFSS vs QuLTRA	Very good linewidth agreement; QuLTRA a few seconds on a laptop, HFSS almost one hour on a server
Lumped-element test cases	QuCAT vs QuLTRA	Almost perfect agreement
Notch Purcell filter	Literature/HFSS-inspired benchmark	$\phi_{m,i}$5, notch behavior, resonance peaks, and Purcell suppression reproduced

For a transmon capacitively coupled to a $\phi_{m,i}$6 readout resonator, the reported values are: qubit mode frequency HFSS 5.76 GHz vs QuLTRA 5.85 GHz, resonator mode frequency 9.36 GHz vs 9.34 GHz, qubit anharmonicity 348 MHz vs 321 MHz, and cross-Kerr 1.8 MHz vs 1.93 MHz (Zaccaria et al., 3 Sep 2025). The largest discrepancies are stated to be below about 8%, mainly in anharmonicity and cross-Kerr, with the likely explanation that QuLTRA uses a capacitance-matrix model for the qubit rather than a full electromagnetic simulation.

For a $\phi_{m,i}$7 resonator inductively coupled to a feedline, QuLTRA is reported to match HFSS very well for resonator linewidth versus coupler length, while requiring only a few seconds on a laptop, compared with almost one hour on a server for HFSS (Zaccaria et al., 3 Sep 2025). For lumped-element test cases, the comparison with QuCAT is described as almost perfect agreement.

A literature-inspired notch Purcell filter benchmark is used to compare impedance and dissipation-related quantities. At the optimal resonator length, the reported values are a qubit dissipation rate of $\phi_{m,i}$8 Hz, a Purcell decay time of 9.59 s, a readout mode linewidth of $\phi_{m,i}$9 MHz, and a dispersive shift of $p_{m,i}$0 MHz (Zaccaria et al., 3 Sep 2025).

Taken together, these comparisons support two narrower conclusions stated in the source material: first, QuLTRA preserves agreement with lumped-circuit tools on problems those tools already cover; second, it extends direct simulation to distributed structures while avoiding the computational cost of full electromagnetic solvers.

6. Application domains in circuit QED

The paper discusses several application classes that illustrate the package’s intended operational regime (Zaccaria et al., 3 Sep 2025).

Purcell filters

QuLTRA is used to model a notch Purcell filter formed by two $p_{m,i}$1 CPW resonators coupled through a multi-line CPW structure. In that setting, it computes impedance spectra, predicts the placement of impedance notches, and evaluates qubit Purcell decay rates directly from the full circuit. The example also shows extreme sensitivity of Purcell suppression to qubit-frequency detuning from the notch.

A second Purcell-filter example exploits higher-frequency poles of a $p_{m,i}$2 resonator as an intrinsic notch filter. By tuning the resonator length to align a pole with the qubit frequency, QuLTRA yields a qubit dissipation rate of $p_{m,i}$3 Hz and a Purcell time of 9.59 s, while maintaining the reported readout linewidth and dispersive shift (Zaccaria et al., 3 Sep 2025).

Multimode ultra-strong coupling

In a transmon coupled to a $p_{m,i}$4 resonator in the multimode ultra-strong coupling regime, QuCAT is described as requiring a Foster ladder with at least 10 LC sections for accurate qubit frequency and anharmonicity (Zaccaria et al., 3 Sep 2025). QuLTRA instead treats the resonator as a true CPW line and directly obtains a qubit frequency of 8.02 GHz and an anharmonicity of 352 MHz. It also computes the Lamb shift as higher resonator modes are included, showing the expected gradual saturation.

Multiplexed qubit readout

The package is also applied to a shared CPW readout line connected to multiple qubit-resonator subsystems. The analysis demonstrates coupling-position-dependent linewidths, standing-wave effects on a non-terminated line, and maxima in linewidth at positions corresponding to integer multiples of $p_{m,i}$5 for capacitive coupling and integer multiples of $p_{m,i}$6 for inductive coupling (Zaccaria et al., 3 Sep 2025). The paper presents this for two structures: QSTR1, comprising qubit + readout resonator + Purcell filter with capacitive coupling, and QSTR2, comprising qubit + readout resonator with inductive coupling through a CPW coupler.

These examples indicate the intended niche of QuLTRA: architectures in which distributed microwave engineering and Hamiltonian-level quantum modeling cannot be cleanly separated. A plausible implication is that the package is most useful when one needs both layout-aware microwave structure and rapid iteration at the circuit-model level.

7. Advantages, assumptions, and present limitations

The paper enumerates several advantages of QuLTRA (Zaccaria et al., 3 Sep 2025). It models CPW transmission lines and couplers directly instead of approximating them as many lumped elements; it supports fast iterative design relative to full electromagnetic simulation; it extracts mode frequencies, anharmonicities, cross-Kerr couplings, Lamb shifts, and Purcell decay rates through EPR; it includes higher-order resonator modes naturally; and it is built on a general multi-port admittance-matrix formalism.

Those advantages, however, are paired with explicit constraints. The current implementation supports only ideal, lossless, straight CPW transmission lines and straight multi-line couplers (Zaccaria et al., 3 Sep 2025). The paper also notes that bends or curvature may introduce small deviations, that the substrate permittivity is currently hard-coded for silicon in the interface, and that future work may extend the method to more general distributed components and stronger nonlinearity.

These limitations are important for interpretation. QuLTRA is not presented as a replacement for final full-wave validation in arbitrary geometries. Rather, it occupies a specific methodological role in the superconducting-circuit design stack: a circuit-QED analysis tool for hybrid lumped-distributed networks that is substantially more direct than lumped discretization and substantially lighter than full electromagnetic simulation (Zaccaria et al., 3 Sep 2025). In that sense, its principal contribution is to make distributed microwave structures first-class objects in Hamiltonian extraction workflows for superconducting quantum hardware.