Topological eigenvalues braiding and quantum state transfer near a third-order exceptional point

Dongning Zheng; Heng Fan; He Zhang; Kai Xu; Tong Liu; Zhongcheng Xiang

REVIEW 2 major objections 6 minor 56 references

Topological eigenvalues braiding and quantum state transfer near a third-order exceptional point

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a three-dimensional non-Hermitian superconducting system can realize the full braid group $B_3$ of eigenvalues with only three independent parameters, and that fast driving near the third-order exceptional point…

desk verdict A real first: EP3 braid-group control in a superconducting qutrit, but the braid plots are inferred from a fitted model rather than measured spectrally, so the referee should focus on that model-dependence. read the letter →

arxiv 2412.14733 v1 pith:VKDDOBUS submitted 2024-12-19 quant-ph

He Zhang , Tong Liu , Zhongcheng Xiang , Kai Xu , Heng Fan , Dongning Zheng This is my paper · ORCID

classification quant-ph

keywords third-orderexceptionalpointbraidgroupeigenvaluebraidingnon-Hermitianquantumsystemsuperconductingcircuitchiralstatetransferanti-PTsymmetryencirclement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper reports a superconducting-circuit experiment in which a three-level non-Hermitian quantum system is controllably steered around a third-order exceptional point, a degeneracy where three eigenvalues and their eigenstates coalesce. It claims that, thanks to a pseudo-chirality symmetry of the Hamiltonian at zero detuning, three independent control parameters are sufficient to realize the full braid group $B_3$ of the three complex eigenvalues, instead of the generic four parameters. The authors verify the braid generators by quasistatic loops, show that $\sigma_1\sigma_2 \neq \sigma_2\sigma_1$, and that the Yang-Baxter identity $\sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2$ holds, and then observe chiral state transfer under fast driving when no EP is enclosed and time-symmetric transfer when one is. The result matters because it lowers the experimental cost of accessing high-order exceptional-point topology and demonstrates directional quantum-state manipulation in a dissipative quantum processor.

What carries the argument

The load-bearing object is the $3\times 3$ non-Hermitian Hamiltonian in Eq. (1), whose entries are the detuning $\delta_{\rm ef}$, the Rabi amplitude $\Omega$, the cavity-assisted coupling $G$, and the photon-loss rate $\kappa$. At $\delta_{\rm ef}=0$ the Hamiltonian obeys pseudo-chirality and anti-PT symmetry, which reduces the number of independent parameters needed to reach the EP3 from four to three. The paper tracks the discriminant $\Delta$ of the cubic characteristic polynomial: $\Delta=0$ gives the exceptional arcs of EP2s, and the additional condition $p=0$ locates the EP3 at their cusp. The braid group $B_3$ enters through the fundamental group of the eigenvalue-space complement, with each control loop mapping to a permutation of the three eigenvalue strands; this mapping is what lets the authors generate braid words by concatenating loops.

What would settle it

Run the concatenated loops $\sigma_1\sigma_2$ and $\sigma_2\sigma_1$ of Fig. 3e-f with the same initial eigenstate and reconstruct the eigenvalue braids from the measured time traces; if the two resulting braids are topologically equivalent, the loop space does not generate the non-Abelian group $B_3$ and the central claim fails.

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Extended reading notes

Core claim

The central claim is that the Hamiltonian of Eq. (1), built from a transmon qutrit coupled to a lossy resonator, hosts an EP3 at $\delta_{\rm ef}=0$ where all three eigenvalues coalesce, and that the exceptional structure in the three-parameter space $(\delta_{\rm ef}, \Omega, G)$ is rich enough to generate the entire braid group $B_3$. Quasistatic loops around different exceptional arcs produce the generators $\sigma_1$ and $\sigma_2$; concatenating loops yields $\sigma_1^2$, $\sigma_2\sigma_1$, and $\sigma_1\sigma_2$, with the non-commutativity of $\sigma_1$ and $\sigma_2$ directly measured. The same system, driven rapidly along rectangular loops in a plane of fixed $G$, shows directional (chiral) state transfer between the right eigenstates only when the loop encloses no EP, while loops enclosing one or two EPs exhibit time-symmetric transfer. The authors support these claims with eigenvalue coalescence data, retrieved eigenvalues along the control loops, and state-overlap measurements from the superconducting processor.

Load-bearing premise

The entire construction assumes the system is exactly described by Eq. (1) with $\delta_{\rm ef}=0$ pseudo-chirality, meaning no residual coupling between $|e,0\rangle$ and $|g,1\rangle$ and a single loss channel on $|g,1\rangle$; if these idealizations fail, the exceptional arcs shift and the measured braids would not be the predicted $B_3$ words.

Editorial extensions

If this is right

Full $B_3$ braiding of a three-level non-Hermitian system becomes accessible with only three tunable parameters, eliminating the generic four-parameter requirement whenever the pseudo-chirality symmetry is present.
Concatenating control loops supplies a physical implementation of the braid generators and of the Yang-Baxter relation $\sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2$, tying circuit control to non-Abelian topology.
Chiral state transfer without an enclosed exceptional point shows that fast loops near, but not around, an EP can select the final eigenstate by direction and starting point.
Loops that enclose one or two EPs suppress the chirality, confirming that the number of enclosed EPs, not just their proximity, determines the time asymmetry.
The same platform can realize higher-order EP physics with engineered dissipation, since the resonator loss provides a tunable non-Hermitian channel.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One natural extension beyond the paper would be a four-level symmetry-protected system, where pseudo-chirality might realize a subset of $B_4$ with fewer than the generic six parameters; the paper does not address this case.
Because the braid group is non-Abelian, concatenated loops could in principle implement braid-based operations on states encoded in the eigenstate manifold, but the authors stop at demonstrating the braids.
The direction-dependent chiral transfer without EP encirclement may serve as a fast dissipative state-preparation tool, a use the paper leaves implicit.
A time-domain test comparing final states after $\sigma_1\sigma_2\sigma_1$ and $\sigma_2\sigma_1\sigma_2$ would dynamically confirm the Yang-Baxter identity, going beyond the paper's topological demonstration.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a legitimate experimental advance—EP3 in a driven superconducting qutrit, braid-group B3 operations via three parameters, and chiral state transfer—and the data support the central narrative. The main caveat is that the eigenvalue braids are not directly measured; they come from fitting the population dynamics to the same 3x3 Hamiltonian and then diagonalizing it. That makes the braid plots model-dependent, and the paper gives no error bars that propagate calibration uncertainties into the homotopy class of the loops.

What's actually new: prior EP3 experiments were classical (Patil et al., Tang et al.) and prior quantum EP work was mostly EP2 (Liu et al., Abbasi et al.). This paper does EP3 braiding and the symmetry-driven reduction from four to three parameters in a quantum circuit, plus a clear demonstration that loops outside EP2 arcs give chiral transfer and loops enclosing them give time-symmetric transfer. The theoretical scaffold—pseudo-chirality at delta_ef=0, discriminant analysis, braid concatenation, Yang-Baxter identity—is standard but correctly handled. The population dynamics in Fig. 4 match simulation well, and the methods for state preparation and tomography are plausible.

Soft spots, in order. First, model-dependent eigenvalue extraction. Fig. 3 "experimental data" are eigenvalues of Eq. (1) computed with parameters fitted to the populations. This is an indirect measurement, not an independent spectral probe. It is not fatal—fitting all three populations is a reasonable way to infer the effective Hamiltonian—but it means the braid plots inherit every assumption of Eq. (1). The paper should say this explicitly and quantify fit uncertainties. Second, the symmetry-breaking coupling. The entire "three parameters suffice" argument sits on the absence of a direct |e,0>-|g,1> coupling at delta_ef=0. The manuscript mentions this and gives an argument about phase invariance, but no bound on residual off-resonant coupling is reported. If eta were non-negligible relative to kappa, the EP2 arcs would separate and the loop homotopy could change. This may be negligible in the actual device given the ~1 GHz detuning, but the authors should put a number on it. Third, no data or code release. For a claim that depends on fitted parameters and simulation comparisons, releasing raw populations and fit procedures would let a referee check the braid inference.

The citation pattern looks fine; the relevant prior EP3 and EP2 experiments are cited. This deserves a serious referee. I'd send it out, but the referee should be asked to focus on the model-dependence of the braid extraction and on the residual-coupling question. If those are answered, the paper is a solid PRL/PRX-type result. Readership: non-Hermitian topology and circuit-QED quantum control.

Referee Report

2 major / 6 minor

Summary. The paper reports an experimental study of a three-level non-Hermitian system realized in a superconducting transmon-resonator device. The central claims are (i) the observation of a third-order exceptional point (EP3) whose surrounding parameter-space topology yields eigenvalue braiding described by the full braid group B3 using only three independent control parameters, and (ii) the observation of chiral quantum state transfer when dynamical loops enclose no exceptional points, with time-symmetric behavior when EPs are enclosed. The theoretical model is the three-level Hamiltonian in Eq. (1), whose pseudo-chirality symmetry at δef=0 is invoked to reduce the parameter-space dimension. The experimental evidence for the EP structure is based on fitting measured population dynamics to this model and then computing eigenvalues from the fitted Hamiltonian, while the state-transfer results are direct overlap measurements compared with numerical simulations.

Significance. If fully supported, the braid-group result would be a notable advance in quantum simulation of non-Hermitian topology, being the first demonstration of non-Abelian B3 braiding near an EP3 in a solid-state quantum system and showing that symmetry can reduce the number of tuning parameters from four to three. The theoretical analysis, including the Cardano-form discriminant and the pseudo-chirality/anti-PT symmetry arguments, is clean and explicit. The state-overlap measurements in Fig. 4 are a genuine experimental result and match the numerics well. However, the eigenvalue-braid data are not directly measured, and the central claim therefore rests on a model-dependent extraction that needs to be made falsifiable. The paper lacks a quantitative sensitivity analysis of the zero-coupling assumption on which the symmetry and the braid labels rely.

major comments (2)

[Results, 'Complex eigenvalues braiding' (Fig. 3); also Results, EP characterization (Fig. 2)] The eigenvalue braids shown in Fig. 3 are not measured directly. The paper's procedure (described for the EP characterization and presumably used for the braid data) is to fit the measured population dynamics to the model Hamiltonian in Eq. (1), extract parameters (δ_ef, Ω, G), and then compute eigenvalues from that same Hamiltonian. The 'experimental data' in Fig. 3 therefore reduce to a re-evaluation of the theoretical model with fitted parameters; they cannot independently confirm the existence of the EP2 arcs or the braid group B3. To make the central claim falsifiable, the authors should either extract the complex eigenvalues directly from the time-resolved complex amplitudes (including phases) without assuming Eq. (1), or, at minimum, perform a parameter-uncertainty analysis showing that the homotopy class of each loop is stable under the calibration and fitting errors. Without such an analysis, the statement 'we experimentally investigate the eigenvalues braiding' is not supported by the data as presented.
[Methods, 'Symmetries of non-Hermitian Hamiltonian'; Eq. (1)] The pseudo-chirality symmetry that enables the EP3 and the reduction to three parameters holds only for the exact Hamiltonian of Eq. (1), in which the coupling between |e,0> and |g,1> is exactly zero. In the physical device, the transmon-resonator coupling J/2π=48 MHz is always present; although the direct coupling is far off-resonant (resonator at 6.697 GHz versus transmon at 5.684 GHz), it will generate a small but finite effective coupling η between |e,0> and |g,1> through higher-order processes. The paper neither estimates η nor provides an upper bound. A nonzero η generically deforms the discriminant locus: the two EP2 arcs in Fig. 2a may no longer meet at the EP3 cusp, and the loops labeled σ1 and σ2 in Fig. 3 may encircle different arcs or none at all. The authors should quantify η for their device parameters and demonstrate that the braid-group claim and the specific braid labels are robust to this coupling and to the calibration uncertainties in δ_ef, Ω, and G. This is a load-bearing assumption that currently lacks any error analysis.

minor comments (6)

[Results, EP characterization paragraph after Eq. (1)] The text contains a typo: 'ac Start shift' should read 'ac Stark shift'.
[Eq. (1) and the preceding paragraph] The sign convention for δef and the precise definition of the rotating frame are not fully specified; please define the frame and list all terms (including the ac Stark compensation) so that Eq. (1) can be reproduced unambiguously.
[Fig. 3a caption and discussion] Two exceptional lines with Ω=0, G=±κ/4 are mentioned as being omitted because 'they do not impact our results'; a one-sentence justification of why they do not affect the homotopy classes of the shown loops would improve clarity.
[Abstract and Conclusion] The phrase 'entire braid group B3' is stronger than what the experiment verifies; the paper demonstrates generators σ1 and σ2 and two concatenations. Please qualify the claim as 'generated by the demonstrated loops in the model' until an independent measurement of the braids is provided.
[Conclusion, last sentence before acknowledgements] The statement that the state transfer vanishes when the starting point is set at (Ω_m,0) is presented without an experimental data point; please indicate whether this is a numerical prediction or an experimental observation.
[Throughout] The claim that the standard error of the mean is smaller than the plotted points is not supported by any quantitative number; please provide the actual error values for the overlap measurements in Fig. 4 and for the fitted eigenvalues in Figs. 2–3.

Circularity Check

1 steps flagged · score 6.0 of 10

Eigenvalue braiding 'data' are eigenvalues of the fitted Hamiltonian, so the experimental braid observation is model-internal; the theoretical B3 generation remains an independent Hamiltonian property.

fitted input called prediction [Results, 'Determine EP2s and EP3' (Fig. 2 caption and parameter-extraction paragraph); also Figs. 3b-f]
"The retrieved eigenvalues are obtained from the non-Hermitian Hamiltonian using the extracted experiment parameters ( δexp ef , Ωexp, Gexp). These parameters are determined by fitting the population dynamics of three states simultaneously."

The 'experimental data' in the eigenvalue plots are not directly measured eigenvalues; they are eigenvalues of Eq. (1) evaluated at parameters fitted to the same population dynamics. The 'theoretic results' shown as solid lines are also eigenvalues of Eq. (1). Therefore the agreement between dots and lines is assured by construction, and the observed EP2/EP3 crossings and braids in Figs. 2b-d and 3b-f are generated by the fitted Hamiltonian rather than independently confirmed. This makes the experimental demonstration of eigenvalue braiding model-internal, though the mathematical claim that Eq. (1) with three parameters realizes B3 remains an independent derivation.

full rationale

The paper's central theoretical claim—that the three-parameter Hamiltonian of Eq. (1) supports an EP3 at δef=0 and that loops in this parameter space generate the braid group B3—is derived from the characteristic polynomial, Cardano's formula, and the pseudo-chirality symmetry at δef=0. That derivation is self-contained and does not reduce to a fit or to a self-citation; the symmetry argument is not imported from the authors' own prior work, and the cited uniqueness/symmetry results are from independent groups. However, the experimental validation of the eigenvalue braiding is partially circular: the 'retrieved eigenvalues' plotted as experimental dots are computed from Eq. (1) using parameters obtained by fitting the population dynamics to the same Hamiltonian. Hence the agreement with the theoretical eigenvalue curves is tautological, and the reported braid patterns are not independently observed quantities. The chiral state-transfer measurements, by contrast, use tomographic state overlaps and are not circular in the same way. On balance, the braid-group generation claim retains independent mathematical content, but one of the paper's main experimental demonstrations reduces by construction to the fitted model, giving a score of 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The main fitted quantities are the three Hamiltonian parameters extracted from the population dynamics; the eigenvalues and braids are computed from these fitted parameters, which creates a circularity burden. The pseudo-chirality symmetry is an assumption necessary to reduce the parametric dimension. No new physical entities are introduced.

free parameters (3)

δ_ef (detuning) = fitted per experimental point
Determined by fitting the measured state-population dynamics to the Hamiltonian in Eq. (1); used to compute the retrieved eigenvalues.
Ω (Rabi drive amplitude) = fitted per experimental point
Same fitting procedure; central to the loop parameters.
G (cavity-assisted coupling) = fitted per experimental point
Same fitting procedure; controls the coupling between |f,0> and |g,1>.

assumptions (4)

domain assumption The transmon-resonator system is accurately truncated to the three states |e,0>, |f,0>, and |g,1>, with photon loss as the only dissipation channel.
Underlies the derivation of Eq. (1); higher transmon levels and other decoherence are neglected.
domain assumption At δef=0 the Hamiltonian satisfies pseudo-chirality and anti-PT symmetry, reducing the number of constraints for an EP3 from four to two.
This is the key symmetry enabling three-parameter braid generation; appears in Methods and Results.
domain assumption The parameter loops are traversed slowly enough for adiabatic eigenvalue tracking in the braiding experiments, and with the specified fast ramps in the state-transfer experiments.
The dynamical regime determines whether eigenvalues braid or states transfer chirally.
standard math The fundamental group of the parameter-space complement of the exceptional set is the braid group B3 (from Refs. [14-16]).
Used to justify that concatenating σ1 and σ2 loops generates all of B3.

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Cite this review

Pith. "Pith review of Topological eigenvalues braiding and quantum state transfer near a third-order exceptional point." pith.science (2026). https://pith.science/paper/VKDDOBUS

@misc{pith2026241214733,
  author       = {Pith},
  title        = {Pith review of: Topological eigenvalues braiding and quantum state transfer near a third-order exceptional point},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VKDDOBUS}},
  note         = {Machine review of arXiv:2412.14733}
}

abstract

Non-Hermitian systems exhibit a variety of unique features rooted in the presence of exceptional points (EP). The distinct topological structure in the proximity of an EP gives rise to counterintuitive behaviors absent in Hermitian systems, which emerge after encircling the EP either quasistatically or dynamically. However, experimental exploration of EP encirclement in quantum systems, particularly those involving high-order EPs, remains challenging due to the difficulty of coherently controlling more degrees of freedom. In this work, we experimentally investigate the eigenvalues braiding and state transfer arising from the encirclement of EP in a three-dimensional non-Hermitian quantum system using superconducting circuits. We characterize the second- and third-order EPs through the coalescence of eigenvalues. Then we reveal the topological structure near the EP3 by quasistatically encircling it along various paths with three independent parameters, which yields the eigenvalues braiding described by the braid group $B_3$. Additionally, we observe chiral state transfer between three eigenstates under a fast driving scheme when no EPs are enclosed, while time-symmetric behavior occurs when at least one EP is encircled. Our findings offer insights into understanding non-Hermitian topological structures and the manipulation of quantum states through dynamic operations.

Figures

Figures reproduced from arXiv: 2412.14733 by the authors.

**Figure 2.** FIG. 2. Determine EP2s and EP3. (a) The phase diagram in the first quadrant of the plane [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Braid of eigenvalues. (a) EPs and control loops. Blue solid lines represent EPs. Red, blue, green, brown, and purple [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Dynamically encircle EPs. (a) The sketch of encirclement loops on the plane [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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