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arxiv: 2605.28927 · v1 · pith:BEZ2VSCLnew · submitted 2026-05-27 · 🪐 quant-ph · cs.CG· math.AT

Quantum encodings that preserve persistent homology

Arthur J. Parzygnat , Andrew Vlasic This is my paper

Pith reviewed 2026-06-29 11:43 UTC · model grok-4.3

classification 🪐 quant-ph cs.CGmath.AT
keywords quantum encodingspersistent homologytopological data analysisquantum TDApoint cloudsfiltered complexessimplicial complexes
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The pith

Certain quantum encodings of classical point clouds leave their persistent homology unchanged.

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

The paper asks which quantum encodings applied directly to classical datasets preserve the persistent homology that would be computed from the original data. This matters because standard quantum topological data analysis begins with pre-built combinatorial objects that consume substantial classical resources to construct. A direct encoding route would let quantum algorithms operate on the raw input while still recovering the same topological invariants. The authors identify conditions under which an encoding induces an action on filtered complexes that leaves the homology unchanged or at least recoverable from the quantum output.

Core claim

Quantum encodings acting directly on classical datasets are admissible for applying quantum algorithms to extract topological features from those datasets without first constructing combinatorial objects, provided the encoding preserves the persistent homology of the associated filtered complexes.

What carries the argument

Quantum encodings of the raw data whose induced action on the associated filtered complexes leaves the persistent homology unchanged.

If this is right

  • Quantum TDA algorithms can begin from the classical dataset directly rather than from pre-constructed simplicial complexes.
  • The resource cost of building combinatorial objects can be bypassed for encodings that preserve the filtration.
  • Topological invariants remain recoverable from quantum measurements when the encoding condition holds.
  • The direct-encoding route extends existing quantum TDA methods that start from combinatorial objects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If such encodings are found, quantum speed-ups in TDA could apply to larger point clouds than current methods allow.
  • The same preservation condition might be tested on other topological invariants beyond persistent homology.
  • Amplitude or angle encodings could be checked first on low-dimensional point clouds to see whether they satisfy the condition.

Load-bearing premise

There exist quantum encodings of the raw data whose induced action on the associated filtered complexes leaves the persistent homology unchanged or at least computable from the quantum output.

What would settle it

An explicit quantum encoding applied to a simple point cloud whose output filtration yields a different persistent homology barcode than the classical filtration would disprove that preserving encodings exist.

Figures

Figures reproduced from arXiv: 2605.28927 by Andrew Vlasic, Arthur J. Parzygnat.

Figure 1
Figure 1. Figure 1: FIG. 1. A dataset obtained by sampling from probability [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. This is the binary tree decomposition of the data [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Heat maps of the Euclidean distance on the interval [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The top image shows a finite metric space [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The top image is one of a centered data set with disks [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. A sequence of three affine transformations, each of [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. A centered data set in [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The set [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. A heatmap of the Bures fidelity distance matrix [PITH_FULL_IMAGE:figures/full_fig_p036_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. This is the same as Figure [PITH_FULL_IMAGE:figures/full_fig_p037_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The top row depicts the heatmaps of the Bures fidelity distance matrix after applying angle encoding to the dataset [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The dataset [PITH_FULL_IMAGE:figures/full_fig_p038_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The top row depicts heatmaps of the Bures fidelity distance matrix after applying dense angle encoding to the dataset [PITH_FULL_IMAGE:figures/full_fig_p039_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Heatmaps of the Bures fidelity distance matrix af [PITH_FULL_IMAGE:figures/full_fig_p039_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Randomly perturbed circle with 200 synthetic data [PITH_FULL_IMAGE:figures/full_fig_p040_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Barcodes, persistence diagrams, and MDS reconstructions for the four maps applied to the circle data: uniform [PITH_FULL_IMAGE:figures/full_fig_p041_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. The function [PITH_FULL_IMAGE:figures/full_fig_p042_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The Bloch sphere together with an MDS encoding [PITH_FULL_IMAGE:figures/full_fig_p043_22.png] view at source ↗
read the original abstract

Given a data set with a notion of distance, such as a point cloud in Euclidean space, topological data analysis (TDA) uses techniques from algebraic topology and metric geometry to infer the topology of a hypothetical manifold from which the data are sampled. This inference is achieved by calculating topological invariants, some of which are difficult to compute classically. Meanwhile, quantum TDA utilizes quantum processes to extract the invariants used in making such inferences in an attempt to speed up the computations. Because applying transformations to the original classical dataset could alter the associated topological invariants, we investigate which quantum encodings would best preserve the invariants of the original dataset. This line of inquiry is distinct from standard approaches in quantum TDA, whose typical starting point is not from the classical dataset directly, but rather from the associated combinatorial objects, such as simplicial complexes, which typically demand a lot of resources to construct. We take the first step at a more direct approach by focusing on which quantum encodings acting directly on the data are admissible for applying quantum algorithms to extract topological features from classical datasets.

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

1 major / 0 minor

Summary. The manuscript proposes to investigate quantum encodings of classical datasets such as point clouds that preserve persistent homology, thereby permitting quantum algorithms to extract topological features directly from the raw data without first constructing combinatorial objects like simplicial complexes. It positions this direct approach as distinct from standard quantum TDA pipelines that begin with pre-built filtered complexes.

Significance. If concrete encodings were supplied together with proofs that they leave the filtration (e.g., Vietoris-Rips) and its persistent homology unchanged, the work would open a route to quantum TDA that avoids classical complex construction overhead. The present text, however, contains only the statement of the investigative goal and supplies neither explicit maps nor any verification, so the potential significance remains unrealized.

major comments (1)
  1. [Abstract] Abstract: the central claim that admissible quantum encodings exist whose induced action on filtered complexes leaves persistent homology unchanged is asserted as an existence statement but is not supported by any explicit encoding (amplitude, angle, or variational), any derivation showing commutation with a filtration function, or any demonstration that Betti numbers or persistence diagrams can be recovered from the quantum output. This unproven assertion is load-bearing for the claimed distinction from prior quantum TDA work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The manuscript frames an investigative goal rather than asserting solved constructions, and we address the comment on the abstract below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that admissible quantum encodings exist whose induced action on filtered complexes leaves persistent homology unchanged is asserted as an existence statement but is not supported by any explicit encoding (amplitude, angle, or variational), any derivation showing commutation with a filtration function, or any demonstration that Betti numbers or persistence diagrams can be recovered from the quantum output. This unproven assertion is load-bearing for the claimed distinction from prior quantum TDA work.

    Authors: The abstract does not assert existence of admissible encodings or supply proofs/derivations; it states the goal as investigating 'which quantum encodings would best preserve the invariants' and taking 'the first step at a more direct approach by focusing on which quantum encodings acting directly on the data are admissible'. The distinction from prior work is in the starting point (raw classical dataset vs. pre-built complexes), which the text makes explicit without claiming completed verification. We agree the work is preliminary and would benefit from future explicit examples, but the current scope is accurately reflected and no change to the abstract is required. revision: no

Circularity Check

0 steps flagged

No circularity; paper is prospective investigation without derivations or fitted results

full rationale

The manuscript states an intent to investigate admissible quantum encodings that preserve persistent homology invariants but supplies no explicit maps, equations, proofs, or parameter fits. The abstract and provided text frame the work as taking 'the first step' toward a direct approach, with no load-bearing claims that reduce by construction to inputs, self-citations, or ansatzes. This matches the reader's assessment of a purely prospective text with no derivation chain to analyze.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract; the ledger is therefore empty.

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Reference graph

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    Square-root encoding for ideal circle For square-root encoding as defined in Example II.17, we first must apply the uniform transformation (IV.30) to map then th roots of unity onto the standard 2-simplex Y Y 1 0 distance FIG. 13. A heatmap of the Bures fidelity distance matrix associated with the quantum encoding given by first applying the uniform trans...

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    IQP encoding for ideal circle For IQP encoding, we refer back to Example II.22. Un- der this encoding, the elementx k = (rck, rsk) of the set Xgets sent to the 2-qubit pure quantum state |xk⟩=e i rckσ1⊗12+rsk12⊗σ1+(π−rck)(π−rsk)σ1⊗σ1 |00⟩. (V.11) In this case, the explicit calculation of the inner prod- uct⟨x j|xk⟩is not particularly illuminating and is q...

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