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arxiv: 2604.26914 · v1 · submitted 2026-04-29 · 🪐 quant-ph

Digital Simulation of Non-Hermitian Knotted Bands on Quantum Hardware

Truman Yu Ng , Yuzhu Wang , Wei Jie Chan , Ruizhe Shen , Tianqi Chen , Ching Hua Lee This is my paper

Pith reviewed 2026-05-07 11:13 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-Hermitian topologyknotted bandsquantum simulationbraid invariantsknot polynomialssuperconducting quantum processorspectral topologytwister models
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The pith

Non-Hermitian knotted bands are simulated on quantum hardware by mapping eigenstate windings to braid topology.

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

This paper establishes a method to simulate and characterize multi-band non-Hermitian systems with knotted structures using programmable quantum processors. It introduces twister models and a non-variational protocol that uses eigenstate winding measurements to extract braid words and knot invariants. This approach avoids full tomography and optimization steps that are difficult on current hardware. If successful, it provides a practical route for studying exotic non-Hermitian topologies on near-term devices. The experimental reconstruction of knots like the Hopf chain demonstrates the method's capability.

Core claim

We introduce a family of non-Hermitian multi-band twister models and implement a non-variational protocol on a superconducting quantum processor to characterize their complex braided band structures. By mapping the winding of eigenstates to the spectral topology, the protocol extracts braid information including braid words and knot invariants such as the Alexander and Jones polynomials without full spectral tomography or repeated optimization. We experimentally reconstruct complicated knots and links such as the Hopf chain and Solomon's knot.

What carries the argument

The mapping from eigenstate winding to spectral topology in non-Hermitian multi-band twister models, which enables extraction of braid words and knot invariants from quantum measurements.

Load-bearing premise

The winding of eigenstates in these models accurately reflects the spectral topology and braid information even in the presence of noise on the quantum hardware.

What would settle it

Running the protocol on the quantum processor for the Solomon's knot and finding that the extracted Jones polynomial does not match the expected mathematical value for that knot.

Figures

Figures reproduced from arXiv: 2604.26914 by Ching Hua Lee, Ruizhe Shen, Tianqi Chen, Truman Yu Ng, Wei Jie Chan, Yuzhu Wang.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 view at source ↗
Figure 4
Figure 4. Figure 4: ), all eight inequivalent knots/links shown can be distinctly identified view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 view at source ↗
Figure 9
Figure 9. Figure 9: illustrates how the k-independent winding number Wij can be determined without performing an explicit integration by leveraging the geometric interpretation introduced above. Consider the leftmost column as an example: the phase traced by the vector Ei(k)−Ej (k) initially increases, subsequently decreases, and ultimately returns to zero. Here, we treat complex numbers in C and their corresponding two-dimen… view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 view at source ↗
read the original abstract

Knots and links represent a fundamental motif of non-local connectivity that permeates the physical sciences from string theory to protein folds. While spectral braiding has been explored in two-band non-Hermitian models across various platforms, its direct simulation and characterization on programmable quantum hardware, particularly beyond two strands, remains a formidable challenge due to the limitations of variational optimization in these systems. Here, we introduce a family of non-Hermitian multi-band twister models and implement a non-variational protocol to characterize their complex braided band structures on a programmable superconducting quantum processor. By mapping the winding of eigenstates to the spectral topology, we devise an efficient measurement strategy that extracts braid information, including braid words and knot invariants like the Alexander and Jones polynomials, without requiring full spectral tomography or repeated optimization. We experimentally demonstrate the reconstruction of complicated knots and links such as the Hopf chain and Solomon's knot. Our approach provides a general framework for investigating exotic non-Hermitian topology on near-term quantum devices, opening a route to simulate more sophisticated topological structures in knot theory.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces a family of non-Hermitian multi-band twister models and a non-variational protocol to simulate their complex braided band structures on a programmable superconducting quantum processor. By mapping the winding of eigenstates to spectral topology, the protocol extracts braid words and knot invariants (Alexander and Jones polynomials) without full tomography or optimization, and the authors report an experimental demonstration of reconstructing structures such as the Hopf chain and Solomon's knot.

Significance. If the winding-to-topology mapping remains faithful under realistic device noise for multi-band systems, the work supplies a general, scalable framework for hardware simulation of non-Hermitian knotted topology that extends beyond two-band models and avoids variational methods, thereby opening routes to study more intricate topological structures in knot theory on near-term quantum devices.

major comments (2)
  1. [Abstract] Abstract: the central experimental claim that the protocol reconstructs the Hopf chain and Solomon's knot is presented without quantitative fidelity metrics, error bars on extracted invariants, or explicit checks that finite-shot winding measurements preserve the braid word under typical superconducting-device noise (decoherence, readout errors, imperfect non-Hermitian dilation).
  2. [Protocol description] The non-variational mapping from measured eigenstate windings to braid words and invariants is asserted to be accurate for multi-band twister models, yet the manuscript supplies no noise-robustness analysis or comparison against full tomography that would confirm the mapping does not shift topology on noisy hardware.
minor comments (2)
  1. [Model definition] Notation for the multi-band twister models and the precise definition of the winding extraction procedure could be clarified with an explicit algorithmic pseudocode or flowchart to aid reproducibility.
  2. [Abstract and Introduction] The abstract and introduction would benefit from a brief statement of the number of qubits, circuit depth, and shot counts used in the hardware runs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have revised the manuscript to strengthen the presentation of the experimental results and protocol validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central experimental claim that the protocol reconstructs the Hopf chain and Solomon's knot is presented without quantitative fidelity metrics, error bars on extracted invariants, or explicit checks that finite-shot winding measurements preserve the braid word under typical superconducting-device noise (decoherence, readout errors, imperfect non-Hermitian dilation).

    Authors: We agree that the abstract would benefit from these quantitative details. The original version was kept concise, but we have revised the abstract to report fidelity metrics for the reconstructed invariants along with error bars obtained from finite-shot statistics. We have also added explicit discussion and supporting numerical checks in the main text and supplementary material confirming that the braid word is preserved under realistic levels of decoherence, readout errors, and imperfect dilation for the device parameters employed. revision: yes

  2. Referee: [Protocol description] The non-variational mapping from measured eigenstate windings to braid words and invariants is asserted to be accurate for multi-band twister models, yet the manuscript supplies no noise-robustness analysis or comparison against full tomography that would confirm the mapping does not shift topology on noisy hardware.

    Authors: The referee correctly notes the absence of a dedicated robustness study in the initial submission. While the mapping is derived exactly in the ideal case, we have now added a direct comparison of the protocol outputs against full state tomography on both simulated and experimental data, showing agreement within statistical uncertainties. We have further included noise-robustness analysis in the revised methods and supplementary sections using realistic superconducting-device noise models, which demonstrates that the extracted braid words and knot invariants remain topologically stable. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental extraction of braid invariants from measured eigenstate windings is self-contained

full rationale

The paper introduces non-Hermitian multi-band twister models and a non-variational protocol that maps measured eigenstate windings directly to spectral topology on quantum hardware. Braid words and knot invariants (Alexander/Jones polynomials) are extracted from finite-shot measurements without optimization or full tomography, and the reported reconstructions of structures such as the Hopf chain and Solomon's knot are presented as direct experimental outcomes. No derivation step reduces by construction to a fitted parameter, self-citation chain, or definitional equivalence; the central mapping is a newly devised measurement strategy whose validity is tested against hardware data rather than assumed tautologically. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the newly postulated multi-band twister models and the validity of the eigenstate-winding-to-spectral-topology mapping; these are introduced without independent external benchmarks in the provided abstract.

axioms (1)
  • standard math Standard non-Hermitian Hamiltonian formalism and eigenstate topology
    The models rely on complex eigenvalues and eigenstate winding numbers as in prior non-Hermitian topology literature.
invented entities (1)
  • non-Hermitian multi-band twister models no independent evidence
    purpose: To generate controllable knotted and linked band structures beyond two bands
    Introduced in the paper as the core modeling framework for the simulation.

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