Inductive-Energy Participation Ratio (IEPR)

• Inductive-Energy Participation Ratio (IEPR) is a mode-resolved metric that quantifies the fraction of a mode’s inductive energy stored in a specific circuit element, linking simulation data to quantum Hamiltonian construction.
• It underpins parameter extraction, Hamiltonian reconstruction, and device verification in superconducting quantum chips by connecting simulated modal energy distributions to zero-point fluctuations and nonlinear couplings.
• IEPR is computed via electromagnetic simulation and postprocessing, and it is critical for designing both weakly anharmonic systems like transmons and highly anharmonic devices such as fluxonium.

Searching arXiv for papers on IEPR and related EPR methods in superconducting circuits. The inductive-energy participation ratio (IEPR) is a mode-resolved quantity for superconducting circuits that measures what fraction of a mode’s inductive energy is stored in a specified circuit element. In the literature, it functions as a bridge between classical electromagnetic eigenmode simulation and quantum-Hamiltonian construction, and it has been used for parameter extraction, Hamiltonian reconstruction, and device verification in superconducting quantum chips (Yu et al., 2023). Within the broader energy-participation framework, IEPR is the inductive counterpart of participation-based descriptions of field localization; in Josephson-circuit quantization it is the quantity that connects simulated modal energy distributions to zero-point fluctuations, nonlinear couplings, and, in extended formulations, the full nonlinear Hamiltonian of very anharmonic devices such as fluxonium (Minev et al., 2020, Yilmaz et al., 2024).

1. Definition and physical meaning

In the formulation introduced for superconducting quantum chips, the IEPR for normal mode $m$ and element $n$ is defined as

$$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$

where $\mathcal{E}_{mn}^I$ is the average inductive energy associated with element $n$ when normal mode $m$ is excited, and $\mathcal{E}_m^I$ is the total average inductive energy stored in the entire chip in mode $m$ (Yu et al., 2023). In the EPR-based Josephson-junction formalism, the same object is written as

$$p_{mj} = \frac{\langle \psi_m | \frac{1}{2} E_j \varphi_j^2 | \psi_m \rangle}{\langle \psi_m | \frac{1}{2} H_{\text{lin}} | \psi_m \rangle},$$

with $|\psi_m\rangle$ denoting one Fock excitation in mode $n$0, $n$1 the inductive energy of element $n$2, and $n$3 the reduced flux across the element (Yilmaz et al., 2024).

The physical content of the ratio is consistent across these formulations: it answers what fraction of the magnetic or inductive energy in a mode “participates” in a specific element. In the 2023 IEPR formulation, this applies to all inductive elements, not only Josephson junctions, and is used to describe entire elements such as wires, qubits, couplers, and resonators (Yu et al., 2023). In the 2020 EPR quantization framework, the corresponding inductive participation is the key input for nonlinear Hamiltonian construction and is described as the fraction of inductive energy allocated to the Josephson dipole when mode $n$4 is excited (Minev et al., 2020).

A recurrent source of ambiguity is the use of the term “EPR.” In the Hamiltonian-quantization context, the cited literature states that the relevant EPR is the inductive one: capacitive participations are used for dissipation and dielectric-loss budgeting, whereas inductive participations underlie nonlinear Hamiltonian parameters (Minev et al., 2020). This distinction is central when comparing circuit-quantization workflows to loss-analysis workflows.

2. Relation to mode transformations and Hamiltonian structure

A defining feature of the IEPR formalism is that it is not merely a descriptive energy metric. In the 2023 treatment, IEPR is explicitly related to the unitary transformation between the bare-mode description of a circuit and its normal-mode description. The fundamental relation is

$n$5

where $n$6 is an element of the unitary matrix $n$7 connecting bare and normal representations (Yu et al., 2023). With a sign matrix $n$8 extracted from simulated field distributions, the transformation is reconstructed as

$n$9

This representation-theoretic role makes IEPR a bridge between physical layout and measurable modal structure. Bare-mode frequencies may then be extracted through

$$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$0

and coupling strengths through

$$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$1

with $$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$2 denoting normal-mode frequencies (Yu et al., 2023).

The corresponding linear Hamiltonian in the bare representation is written as

$$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$3

with similar treatment for inductive coupling (Yu et al., 2023). This allows the same simulation-derived IEPR data to support both bare-mode and normal-mode Hamiltonians, which the source contrasts with approaches that only reconstruct normal-mode properties.

A plausible implication is that IEPR offers a compact way to encode both energy localization and basis transformation. The cited papers state this connection directly in terms of $$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$4, and emphasize that orthogonality and the block-diagonal unitary transformation naturally conserve quantum commutation relations (Yu et al., 2023).

3. Extraction from electromagnetic simulation

The practical IEPR workflow is simulation-driven. In the superconducting-chip procedure, the first stage is a classical electromagnetic simulation of the three-dimensional chip layout, for example in HFSS, from which one extracts each eigenmode frequency $$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$5 and electric-field distribution $$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$6 (Yu et al., 2023). The second stage is postprocessing.

The total inductive energy is obtained from a global field integration: $$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$7 For each element, the voltage across the component is computed along its path,

$$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$8

from which the peak flux is

$$r_{mn} = \frac{\mathcal{E}_{mn}^I}{\mathcal{E}_m^I},$$9

and the element inductive energy is

$\mathcal{E}_{mn}^I$0

The IEPR is then evaluated in practice through

$\mathcal{E}_{mn}^I$1

The sign matrix $\mathcal{E}_{mn}^I$2 is determined from the voltage sign and reference orientation in the field simulation (Yu et al., 2023).

In the EPR quantization framework for Josephson circuits, the workflow is expressed somewhat differently but serves the same end. The full circuit is linearized by replacing nonlinear elements with linear inductances, electromagnetic eigenmodes are computed, and for each mode one evaluates the inductive energy in each nonlinear or dissipative element and divides by the total inductive energy of the mode (Minev et al., 2020). For a lumped Josephson inductance,

$\mathcal{E}_{mn}^I$3

where $\mathcal{E}_{mn}^I$4 is the current through junction $\mathcal{E}_{mn}^I$5 in mode $\mathcal{E}_{mn}^I$6 (Minev et al., 2020).

These procedures make IEPR computable from classical simulation data. The 2023 paper emphasizes that this enables extraction of crucial characteristic parameters in both bare and normal representations, including cases involving distributed structures and modes that challenge conventional lumped approaches (Yu et al., 2023).

4. Zero-point fluctuations and nonlinear quantization

IEPR enters quantization through the phase operator of each nonlinear element. In the EPR formalism, the reduced phase is expanded in modal ladder operators as

$\mathcal{E}_{mn}^I$7

and the zero-point fluctuation amplitudes are determined by

$\mathcal{E}_{mn}^I$8

The literature explicitly identifies this as the core bridge from classically calculated inductive participation to quantum zero-point fluctuations (Minev et al., 2020, Yilmaz et al., 2024).

Once $\mathcal{E}_{mn}^I$9 is known, the full Hamiltonian for a multi-mode, multi-junction circuit is written as

$n$0

with $n$1 (Minev et al., 2020). For weakly anharmonic systems, nonlinear parameters such as Kerr terms can then be derived. The 2020 EPR paper gives

$n$2

while the 2023 IEPR paper presents a related quartic-expansion treatment in terms of the transformation coefficients $n$3: $n$4

$n$5

$n$6

These expressions make explicit that the nonlinear inheritance of a mode is controlled by its inductive participation in the underlying nonlinear elements (Yu et al., 2023).

The literature also states universal constraints for the EPRs in the Josephson-circuit setting: $n$7 and $n$8 (Minev et al., 2020). This suggests that participation data are not arbitrary fitting parameters but constrained quantities tied to the circuit’s modal structure.

5. Extension to very anharmonic circuits

A major limitation of the classic EPR method is that it relies on a low-order expansion of nonlinear terms and is therefore most effective in weakly anharmonic systems such as transmons (Minev et al., 2020). The 2024 work on very anharmonic superconducting circuits extends the framework to fluxonium, a circuit for which high-order nonlinear terms are crucial (Yilmaz et al., 2024).

In that formulation, the full Hamiltonian is decomposed as

$n$9

with

$m$0

and after diagonalization,

$m$1

For a Josephson junction in a fluxonium loop, the nonlinear term is retained as a full cosine rather than truncated: $m$2 where $m$3 accounts for external flux (Yilmaz et al., 2024).

The phase operator is expanded mode by mode, for example in a qubit–resonator system as

$m$4

with the coefficients again extracted from IEPR through

$m$5

For transmon-like regimes a quartic approximation yields

$m$6

but for fluxonium the source states that no analytic expression is possible; instead the full Hamiltonian is diagonalized numerically, and the dispersive shift is extracted from eigenenergies according to

$m$7

This extension preserves the IEPR-to-ZPF bridge while removing the low-order truncation that limits the classic method in very anharmonic devices (Yilmaz et al., 2024).

6. Applications, validation, and relation to other participation frameworks

IEPR has been applied as a practical characterization and verification tool for superconducting quantum chips. The 2023 paper reports its use in coupler architectures, nonlinear parameter extraction, high-order modes, and transmon–3D-cavity systems, including extraction of bare frequencies, couplings, renormalized frequencies, self-Kerr and cross-Kerr parameters, and the impact of parasitic or box modes (Yu et al., 2023). In the coupler example, the method was used to verify tunability of qubit–qubit coupling and to identify sign change in the coupling as a function of coupler Josephson inductance; the source notes that IEPR allowed direct extraction of sign and of the full Hamiltonian, whereas NMS cannot determine sign (Yu et al., 2023).

The 2024 fluxonium study provides experimental validation for the extended EPR treatment. A fluxonium qubit capacitively coupled to a coplanar waveguide resonator was simulated using Qiskit Metal with ANSYS HFSS, and the extended EPR framework processed these results to extract participation ratios and build the full quantum Hamiltonian (Yilmaz et al., 2024). The reported outcomes are that resonator and qubit frequencies as functions of external flux show excellent agreement between EPR Hamiltonian and measured data, and that experimentally measured dispersive shifts closely match the EPR-derived Hamiltonian over a wide flux range, surpassing the accuracy of a simple lumped element model, especially near $m$8 and $m$9–$\mathcal{E}_m^I$0 flux quanta (Yilmaz et al., 2024). The paper attributes this to better accounting for the distributed nature of the device and renormalization of effective parameters due to mode structure.

IEPR also sits alongside another participation-based literature centered on dielectric loss. In the 2026 SesQ work, the main quantity of interest is the electric-field participation ratio

$\mathcal{E}_m^I$1

used to assess dielectric losses at interfaces such as substrate-metal, metal-air, and substrate-air (Wang et al., 30 Mar 2026). That paper then gives the corresponding inductive generalization,

$\mathcal{E}_m^I$2

and states that the same surface-integral-equation principles extend naturally to magnetostatic cases, with excitation current density and dyadic Green’s functions enabling computation of inductance and magnetic field distribution (Wang et al., 30 Mar 2026). The paper focuses on electrostatic EPR rather than IEPR, but its discussion indicates that multiscale numerical bottlenecks in magnetic-field participation-ratio computation may be mitigated by the same SIE strategy.

Taken together, these works delineate three related but distinct uses of participation ratios. First, IEPR in circuit quantization characterizes inductive energy localization and feeds directly into Hamiltonian construction (Minev et al., 2020, Yu et al., 2023). Second, extended EPR/IEPR methods support accurate modeling of highly nonlinear circuits by retaining full nonlinear terms rather than truncating them (Yilmaz et al., 2024). Third, participation-ratio methods in dielectric-loss simulation emphasize capacitive energy localization, with magnetostatic IEPR appearing as a formal generalization rather than the primary computational target (Wang et al., 30 Mar 2026).

A common misconception is that participation-ratio methods are confined either to Josephson junctions or to dielectric-loss estimates. The cited literature shows a more differentiated picture: standard Josephson-circuit EPR focuses on inductive participation in nonlinear elements (Minev et al., 2020), the IEPR method extends participation analysis to all inductive elements and to representation transformation on full chips (Yu et al., 2023), and surface-integral formulations developed for capacitive EPR are presented as naturally extensible to magnetic-field participation ratios in magnetostatic settings (Wang et al., 30 Mar 2026).