arxiv: 2604.21463 · v1 · submitted 2026-04-23 · 🪐 quant-ph

Recognition: unknown

Dynamical Regimes of Two Qubits Coupled through a Transmission Line

Fabio Borrelli , Giovanni Miano , Carlo Forestiere

Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:30 UTC · model grok-4.3

classification 🪐 quant-ph

keywords dynamicslinecoupledomegatransmissioncircuitcontinuumdynamical

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The pith

Two qubits coupled by a finite transmission line show non-Markovian dynamics in identifiable parameter regimes when the line is treated as a structured reservoir or few-mode coupler.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Superconducting qubits are tiny circuits that store quantum information. Here two such qubits are linked by a real wire of finite length called a transmission line. The authors start from the basic rules of circuit quantization and split the wire's electromagnetic modes into two groups: even and odd. When the wire is long, these groups act like a continuous bath that the qubits can exchange energy with. They model that bath with a standard Drude-Lorentz shape and solve the resulting equations with a numerical method called hierarchical equations of motion. A standard test for memory effects (the Breuer-Laine-Piilo measure) then shows in which ranges of qubit frequency, wire spacing, and coupling strength the qubits remember their past state instead of relaxing simply. When the wire is short the continuous picture fails and the system reduces to a few discrete modes. The result is a single map that tells an engineer whether the connecting line will behave like a simple coupler or like a complicated memory bath.

Core claim

This establishes a unified cQED picture of the dynamical regimes of finite-length transmission lines in superconducting-circuit architectures.

Load-bearing premise

The continuum Drude-Lorentz spectral density remains accurate for the long-line limit and the Breuer-Laine-Piilo measure correctly flags non-Markovianity without additional corrections from the finite line.

Figures

Figures reproduced from arXiv: 2604.21463 by Carlo Forestiere, Fabio Borrelli, Giovanni Miano.

**Figure 1.** Figure 1: FIG. 1. Two identical superconducting qubits capacitively coupled through a transmission line of length [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Normalized squared coupling strengths [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Schematic classification of the operating regions in the plane spanned by the ratios [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 3.** Figure 3: Two limiting cases are particularly simple. In the comb-edge region, discussed in Sec. III B 1, the qubit is tuned near the low-frequency edge of a dense modal comb, and only a few nearby modes participate significantly in the dynamics. In the short-line region, discussed in Sec. III B 2, the spectrum is sparse on the scale of the coupling and, for typical superconducting-circuit parameters (see Appendix… view at source ↗

**Figure 4.** Figure 4: FIG. 4. Schematic representation of the long-line (contin [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Real and imaginary parts of the correlation function [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Drude–Lorentz spectral density [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: shows the Breuer–Laine–Piilo non-Markovianity measure N (Φ) [32, 33] for the circuit composed of a single qubit capacitively coupled to a finite TL of length d terminated by a short circuit. The BLP measure quantifies information backflow from the environment to the qubit, since it is obtained by integrating the time intervals over which the trace distance between two reduced states increases. In Appendi… view at source ↗

**Figure 8.** Figure 8: shows that, in the two-qubit configuration, the BLP non-Markovianity is still primarily controlled by the same two environmental parameters, namely the bath relaxation rate and the temperature, but its detailed structure is now reshaped by the collective coupling to the TL. In particular, the environment couples to the parity operators Lˆ± so that information backflow is encoded not only in local qubit o… view at source ↗

**Figure 9.** Figure 9: FIG. 9. Comparison between HEOM (solid-line), GKLS (dashed-line) and TCL2 (dash dot line) simulations for the populations [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Comparison between HEOM (solid-line), GKLS (dashed-line) and TCL2 (dash dot line) simulations for the populations [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

**Figure 11.** Figure 11: reports the mode-resolved parameter ηn for these two configurations. In the dispersive case, shown in [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Population [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Population [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14. Population [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15. Schematic of one qubit capacitively coupled to a transmission line closed on a short. [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗

read the original abstract

We investigate the reduced dynamics of two identical superconducting qubits capacitively coupled through a finite-length transmission line. Starting from circuit quantization, we derive a circuit Hamiltonian that naturally separates the line modes into even- and odd-parity sectors coupled to collective qubit operators. Depending on the hierarchy between the qubit frequency $\omega_q$, the mode spacing $\omega_{TL}$, and the coupling scale $\omega_g$, the line acts either as a structured reservoir or as a discrete few-mode coupler. In the long-line continuum limit, each sector is described by a Drude--Lorentz spectral density and the dynamics is solved with the hierarchical equations of motion. Using the Breuer--Laine--Piilo measure, we identify the parameter region in which the reduced dynamics exhibits non-Markovian relaxation. In the short-line limit, the continuum description breaks down and the dynamics becomes respectively multimode or single-mode. This establishes a unified cQED picture of the dynamical regimes of finite-length transmission lines in superconducting-circuit architectures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper maps finite TL qubit regimes via even-odd modes and HEOM with BLP, but the continuum Drude-Lorentz step lacks error bounds.

read the letter

The punchline is that the paper gives a unified regime map for two superconducting qubits capacitively coupled through a finite transmission line. They use even-odd mode separation from the circuit Hamiltonian, model the long-line limit with Drude-Lorentz spectral densities solved by hierarchical equations of motion, and apply the Breuer-Laine-Piilo measure to identify non-Markovian relaxation regions. What it does well is connect the circuit quantization directly to the open-system treatment. The even and odd sectors couple to symmetric and antisymmetric qubit operators, which simplifies the problem nicely. Depending on whether the mode spacing is large or small compared to the other scales, the line switches from acting like a few-mode coupler to a continuous bath. This kind of classification is useful for people designing superconducting circuits where transmission lines are common. The main soft spot is the continuum limit step. The real mode frequencies are discrete multiples of pi v over L with couplings that scale as one over square root of n. Approximating the sum by an integral to get the Lorentzian form is standard, but the paper does not quantify the error this introduces into the BLP non-Markovianity calculation. If finite-length corrections move the boundaries, the map could be off in the reported windows. Without numerical comparisons to the exact discrete case or bounds on the approximation, that part stays a bit loose. Overall, this is aimed at researchers in circuit quantum electrodynamics who work with transmission-line mediated interactions. Someone already comfortable with HEOM and non-Markovian measures will get the most from the parameter diagrams. It deserves a serious referee because the question is well-defined and the methods are appropriate, though the authors should add some validation of the long-line approximation before publication. I would recommend sending it out for peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript derives the circuit Hamiltonian for two identical superconducting qubits capacitively coupled through a finite-length transmission line, separating the line modes into even- and odd-parity sectors coupled to collective qubit operators. It analyzes dynamical regimes based on the hierarchy of qubit frequency ω_q, mode spacing ω_TL, and coupling ω_g. In the long-line continuum limit, each sector is modeled by a Drude-Lorentz spectral density, with dynamics solved via hierarchical equations of motion (HEOM); the Breuer-Laine-Piilo (BLP) measure is then used to identify regions of non-Markovian relaxation. Short-line limits are treated as multimode or single-mode discrete couplers, yielding a unified cQED picture of finite-length transmission lines.

Significance. If the central results hold, the work supplies a practical framework for distinguishing reservoir-like versus coupler-like behavior of transmission lines in superconducting circuits, directly relevant to cQED device design where finite-length effects control coherence and entanglement dynamics.

major comments (2)

Long-line continuum limit: The replacement of the exact discrete sum over modes (ω_n = nπv/L, g_n ∝ 1/√n) by a Drude-Lorentz spectral density is invoked to demarcate non-Markovian regimes via HEOM+BLP, yet no explicit error bound or finite-L validation is provided for the BLP integral. This approximation is load-bearing for the reported parameter regions; without a quantified L→∞ error estimate or direct comparison to the discrete-mode BLP value inside the claimed window, the demarcation of non-Markovianity cannot be confirmed as robust.
Validation of HEOM+BLP pipeline: The abstract and derivation outline the sequence from circuit quantization to HEOM to BLP but supply no explicit checks against known limits (e.g., Markovian decay for weak coupling or exact single-mode Rabi dynamics in the short-line limit). Such benchmarks are required to establish that the numerical method correctly reproduces the expected crossover between regimes.

minor comments (1)

Notation: The symbols ω_TL and ω_g are introduced without immediate explicit definitions or relations to circuit parameters (capacitances, inductances); a dedicated paragraph or table early in the text would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting these important points regarding the robustness of our approximations and numerical pipeline. We address each major comment below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

Referee: Long-line continuum limit: The replacement of the exact discrete sum over modes (ω_n = nπv/L, g_n ∝ 1/√n) by a Drude-Lorentz spectral density is invoked to demarcate non-Markovian regimes via HEOM+BLP, yet no explicit error bound or finite-L validation is provided for the BLP integral. This approximation is load-bearing for the reported parameter regions; without a quantified L→∞ error estimate or direct comparison to the discrete-mode BLP value inside the claimed window, the demarcation of non-Markovianity cannot be confirmed as robust.

Authors: We agree that an explicit quantification of the continuum approximation error would improve the manuscript. The Drude-Lorentz form is obtained by taking the L→∞ limit of the exact mode sum (with the standard linear dispersion and 1/√n coupling), which is a standard procedure in the literature for transmission-line reservoirs. Nevertheless, in the revised manuscript we will add a direct numerical comparison: for representative large but finite L values inside the claimed long-line regime, we compute the BLP measure both from the exact discrete-mode sum (truncated at high frequencies) and from the Drude-Lorentz spectral density, thereby providing a quantified error bound on the non-Markovianity demarcation. revision: yes
Referee: Validation of HEOM+BLP pipeline: The abstract and derivation outline the sequence from circuit quantization to HEOM to BLP but supply no explicit checks against known limits (e.g., Markovian decay for weak coupling or exact single-mode Rabi dynamics in the short-line limit). Such benchmarks are required to establish that the numerical method correctly reproduces the expected crossover between regimes.

Authors: We acknowledge that explicit validation benchmarks were not included. In the revised version we will add two sets of checks: (i) in the weak-coupling, large-L Markovian limit we will demonstrate that the HEOM+BLP pipeline recovers the expected exponential decay with rate given by the Fermi golden rule expression; (ii) for the short-line single-mode regime we will compare the discrete-mode dynamics (solved exactly via the multimode or single-mode Hamiltonian) against the known Rabi oscillation solution, confirming that the overall framework correctly captures the crossover from reservoir-like to coupler-like behavior. These additions will be placed in a new subsection on numerical validation. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from circuit quantization through standard continuum limit to established open-system methods

full rationale

The paper begins with circuit quantization to obtain the Hamiltonian, partitions the transmission-line modes into even/odd sectors coupled to collective qubit operators, and applies the long-line continuum limit to arrive at a Drude-Lorentz spectral density. The subsequent use of hierarchical equations of motion and the Breuer-Laine-Piilo measure follows directly from that derived spectral density. None of these steps reduces the final demarcation of non-Markovian regimes to a parameter fitted from the same data, a self-definition, or a load-bearing self-citation whose content is itself unverified. The chain remains independent of the target result and is self-contained against external benchmarks such as standard circuit quantization and established HEOM/BLP techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard circuit quantization and open-quantum-system techniques rather than new postulates; no free parameters are introduced beyond the physical frequencies already present in the model.

axioms (2)

domain assumption Circuit quantization yields a Hamiltonian separable into even- and odd-parity sectors
Invoked at the start of the derivation from the lumped-element circuit.
domain assumption Drude-Lorentz spectral density accurately represents the continuum limit of the transmission line
Used to close the long-line description before applying HEOM.

pith-pipeline@v0.9.0 · 5472 in / 1386 out tokens · 27398 ms · 2026-05-09T22:30:48.570192+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Dynamical Regimes of Two Qubits Coupled through a Transmission Line

and by Devoret [14], who established practical quan- tization rules for microwave circuits. These foundations were subsequently refined and systematized to encom- pass widely used classes of superconducting quantum circuits [15–18], as well as “black-box” approaches that quantize complex electromagnetic environments through their normal modes or effective...

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Comb-edge discrete region It is important to emphasize that condition (24a) alone does not suffice to justify a continuum description. This is already apparent in Fig. 2(c): even whenω TL ≪ω g, ifω q ≲ω TL the qubit probes the spectral region near the lower comb edge, where the density of states is not locally uniform and the full discrete-mode Hamiltonia...
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Short-line single-mode region We now consider the limit in which the separation be- tween adjacent modes is much greater than the coupling frequencyω g, i.e. ωg ≪ω TL.(32) This region corresponds to the left-hand side of Fig. 3. As we have seen from Fig. 5, in this case a continuum frequency description of the spectral density fails to de- scribe the corr...
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Two identical qubits coupled by a TL Having established this behavior in the single-qubit case, we now turn to the two-qubit setup. This compari- son is useful because it allows us to unravel which features of the BLP landscape are inherited from the structured TL environment and which instead arise from the collec- tive nature of the qubits–bath coupling...
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