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arxiv: 2508.01516 · v2 · submitted 2025-08-02 · 🪐 quant-ph

Hybrid quantum-classical framework for Betti number estimation with applications to topological data analysis

Nhat A. Nghiem , Tzu-Chieh Wei This is my paper

Pith reviewed 2026-05-19 00:47 UTC · model grok-4.3

classification 🪐 quant-ph
keywords hybrid quantum-classical algorithmsBetti number estimationtopological data analysissimplicial complexesquantum algorithmsnormalized Betti numbersTDA applications
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The pith

A hybrid quantum-classical algorithm estimates Betti numbers by classically enumerating simplices and quantumly processing the structure for potential speedups.

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

The paper proposes combining classical and quantum resources to estimate normalized Betti numbers more efficiently than existing quantum algorithms alone. The classical part handles the enumeration of all simplices from the given vertices and edges of a simplicial complex. This combinatorial data is then fed into a quantum subroutine for the estimation task. A sympathetic reader would care because topological data analysis relies on these invariants to extract robust features from complex datasets, and hybrid methods could make such computations feasible at larger scales. The authors highlight regimes of polynomial to exponential speedups alongside the cost of additional ancilla qubits.

Core claim

Assuming the classical description of a simplicial complex is given as its set of vertices and edges, the hybrid algorithm has the classical component enumerate all simplices and then applies quantum processing to estimate the Betti numbers, identifying specific regimes where this yields polynomial to exponential speedups over prior quantum methods at the expense of more ancilla qubits, while also illustrating the usefulness of normalized Betti numbers in applications.

What carries the argument

The hybrid quantum-classical algorithm that uses classical enumeration of simplices followed by quantum estimation of Betti numbers.

If this is right

  • The hybrid method can achieve polynomial to exponential speedups over existing quantum methods in identified regimes.
  • More ancilla qubits are required as a trade-off for the performance gains.
  • Normalized Betti numbers prove useful in concrete topological data analysis applications.
  • Hybrid quantum algorithms show broader potential for use in topological data analysis.

Where Pith is reading between the lines

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

  • This hybrid strategy may enable processing of larger simplicial complexes by offloading enumeration to classical computers.
  • Similar hybrid frameworks could be developed for estimating other topological invariants beyond Betti numbers.
  • Practical implementations might reveal whether the speedup holds for real-world data sets in TDA.
  • The method suggests a path to reduce quantum resource demands in topological computations.

Load-bearing premise

The classical description of the simplicial complex allows efficient enumeration without hidden exponential costs, enabling the quantum part to deliver the stated speedups.

What would settle it

A direct comparison on benchmark simplicial complexes showing no actual speedup or an exponential overhead in the quantum subroutine due to the enumerated structure size would disprove the performance claims.

read the original abstract

Topological data analysis (TDA) is a rapidly growing area that applies techniques from algebraic topology to extract robust features from large-scale data. A key task in TDA is the estimation of (normalized) Betti numbers, which capture essential topological invariants. While recent work has led to quantum algorithms for this problem, we explore an alternative direction: combining classical and quantum resources to estimate the Betti numbers of a simplicial complex more efficiently. Assuming the classical description of a simplicial complex, that is, its set of vertices and edges, we propose a hybrid quantum-classical algorithm. The classical component enumerates all simplices, and this combinatorial structure is subsequently processed by a quantum algorithm to estimate the Betti numbers. We analyze the performance of our approach and identify regimes where it potentially achieves polynomial to exponential speedups over existing quantum methods, at the trade-off of using more ancilla qubits. We further demonstrate the utility of normalized Betti numbers in concrete applications, highlighting the broader potential of hybrid quantum algorithms in topological data analysis.

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 proposes a hybrid quantum-classical algorithm for estimating normalized Betti numbers of a simplicial complex. The classical component takes the set of vertices and edges as input and enumerates all simplices; this combinatorial structure is then fed to a quantum subroutine that estimates the Betti numbers. The authors analyze the resource trade-offs and identify regimes in which the hybrid method is claimed to achieve polynomial-to-exponential speedups relative to prior quantum algorithms based on quantum phase estimation of the combinatorial Laplacian, at the expense of additional ancilla qubits. They also illustrate the utility of normalized Betti numbers in concrete TDA applications.

Significance. If the speedup regimes can be rigorously delimited so that classical enumeration remains sub-exponential and the quantum subroutine delivers a genuine advantage without hidden costs, the work would supply a practical hybrid route to quantum-enhanced topological data analysis. The explicit identification of ancilla-qubit trade-offs and the demonstration of normalized Betti numbers in applications are positive features. The manuscript would benefit from a clearer accounting of total complexity that separates classical preprocessing from quantum runtime.

major comments (2)
  1. [§3 and §4] §3 (Algorithm description) and §4 (Complexity analysis): the central speedup claim requires that classical enumeration of all simplices from the given vertices and edges remains polynomial or sub-exponential in the claimed regimes. The manuscript does not provide an explicit bound on this enumeration cost (e.g., in terms of number of vertices n and maximum dimension d) nor does it state the precise input encoding (sparse adjacency list versus dense clique complex). Without such a bound, the hybrid advantage may be eliminated when d is not fixed and small, as binomial(n, k+1) enumeration is exponential in n for dense inputs.
  2. [§4] §4 (Speedup regimes): the regimes in which polynomial-to-exponential speedups are asserted are not accompanied by a concrete complexity comparison that includes the classical preprocessing step. A table or theorem that states total gate count (classical + quantum) versus prior QPE-based methods for representative sparse and dense complexes would make the claim verifiable.
minor comments (2)
  1. [§2] Notation for normalized Betti numbers is introduced without an explicit definition or reference to the standard normalization factor; a short equation or citation in the preliminaries would improve clarity.
  2. [§5] Figure captions for the application examples should state the dimension and number of simplices of the complexes used, to allow readers to assess whether the reported runtimes fall inside the claimed speedup regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us strengthen the clarity of our complexity analysis. We respond to each major comment below and have revised the manuscript to incorporate explicit bounds, input specifications, and comparative tables as suggested.

read point-by-point responses

  1. Referee: [§3 and §4] §3 (Algorithm description) and §4 (Complexity analysis): the central speedup claim requires that classical enumeration of all simplices from the given vertices and edges remains polynomial or sub-exponential in the claimed regimes. The manuscript does not provide an explicit bound on this enumeration cost (e.g., in terms of number of vertices n and maximum dimension d) nor does it state the precise input encoding (sparse adjacency list versus dense clique complex). Without such a bound, the hybrid advantage may be eliminated when d is not fixed and small, as binomial(n, k+1) enumeration is exponential in n for dense inputs.

    Authors: We thank the referee for this important observation. In the revised manuscript we have expanded Section 3 to explicitly state that the input is encoded as a sparse adjacency list of the 1-skeleton and that higher-dimensional simplices are enumerated classically via a standard combinatorial search. We now include the bound that, for maximum dimension d fixed, enumeration costs O(n^{d+1}) time (polynomial), while for general d the cost is at most binomial(n, d+1) which is exponential in n when d grows. The speedup regimes in Section 4 have been restated to apply only when d is small or the complex is sparse (number of simplices polynomial in n), ensuring classical preprocessing remains sub-exponential and does not erase the hybrid advantage. We believe this rigorously delimits the claimed regimes. revision: yes

  2. Referee: [§4] §4 (Speedup regimes): the regimes in which polynomial-to-exponential speedups are asserted are not accompanied by a concrete complexity comparison that includes the classical preprocessing step. A table or theorem that states total gate count (classical + quantum) versus prior QPE-based methods for representative sparse and dense complexes would make the claim verifiable.

    Authors: We agree that separating classical preprocessing from quantum runtime and providing concrete comparisons is essential. In the revised Section 4 we have added Theorem 4.2, which gives the total complexity (classical enumeration time plus quantum gate count) and explicitly includes the preprocessing cost. We have also inserted Table 1 that compares total resources for two representative cases—one sparse complex with O(n) simplices and one dense complex with Θ(n^d) simplices—against the standard QPE-based algorithms. The table shows that in the sparse regime the hybrid method reduces quantum depth while keeping classical cost manageable, and notes the limitations for dense inputs. This makes the speedup claims directly verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity in hybrid TDA algorithm proposal

full rationale

The paper proposes a hybrid quantum-classical algorithm that takes as given the classical description of a simplicial complex (vertices and edges), performs classical enumeration of simplices, and then applies a quantum subroutine to estimate normalized Betti numbers. Performance analysis and identification of polynomial-to-exponential speedup regimes are presented as comparisons against prior quantum methods (e.g., those using quantum phase estimation on the combinatorial Laplacian), without any equations, parameters, or results that reduce by construction to the inputs or to self-citations. No self-definitional loops, fitted-input predictions, ansatz smuggling, or load-bearing self-citation chains appear; the central claims rest on standard complexity trade-offs and remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proposal rests on standard assumptions from quantum computing and algebraic topology plus the explicit premise that a classical description of the simplicial complex is available.

axioms (2)
  • domain assumption Classical description of a simplicial complex (vertices and edges) is available as input
    Explicitly stated in the abstract as the starting point for the hybrid algorithm.
  • domain assumption Quantum subroutine can estimate Betti numbers from the enumerated combinatorial structure
    Core assumption underlying the claimed performance advantage.

pith-pipeline@v0.9.0 · 5710 in / 1185 out tokens · 49096 ms · 2026-05-19T00:47:12.585341+00:00 · methodology

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Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · 5 internal anchors

  1. [1]

    Encoding simplices Let the simplicial complex consist of n vertices (i.e., 0- simplices), denoted by v0, v1, . . . , vn−1. A r-simplex σr is, by definition, a set of ( r+1) distinct vertices that form a fully connected subgraph, or equivalently, a (r+1)-clique. Following the encoding strategy proposed in the LGZ algorithm [2], we map each r-simplex to a c...

    work page

  2. [2]

    , vr] to a formal sum of its oriented (r−1)-faces: ∂r[v0, v1,

    Block-Encoding of boundary operators We begin by recalling the definition of the boundary operator ∂r, which maps an oriented r-simplex σr = [v0, v1, . . . , vr] to a formal sum of its oriented (r−1)-faces: ∂r[v0, v1, . . . , vr] = rX i=0 (−1)i[v0, . . . ,bvi, . . . , vr], (4) where the hat bvi indicates omission of the vertex vi. As described in Section ...

    work page

  3. [3]

    Since Hr is a quotient space, its dimension satisfies dim Hr = dim ker(∂r) − dim im(∂r+1)

    Estimating Betti numbers To compute the r-th Betti number of a simplicial com- plex, we recall that the r-th homology group is defined as the quotient Hr = ker(∂r)/ im(∂r+1), and the r-th Betti number is given by βr = dim Hr. Since Hr is a quotient space, its dimension satisfies dim Hr = dim ker(∂r) − dim im(∂r+1). (7) Because im(∂r+1) ⊆ Cr, we use the st...

    work page

  4. [4]

    Mathematical framework Consider a multipartite quantum system H = H1 ⊗ H2 ⊗ · · · ⊗ H N where N ∈ Z+ and each Hi represents a Hilbert space of known finite dimension. Let ρ denote the quantum state of the entire system, and let ρi denote the reduced density matrix for subsystem i, obtained by tracing out all other subsystems: ρi = Tr{1,...,N }\{i}ρ. (26) ...

    work page

  5. [5]

    , N} cor- responding to the N subsystems

    Simplicial complex construction Given a threshold parameter ϵ > 0, we construct a graph G = (V, E) with vertex set V = {1, 2, . . . , N} cor- responding to the N subsystems. An edge ( i, j) ∈ E exists if and only if Dij ≤ ϵ. The associated simplicial complex S is constructed via the clique complex construction: a subset σ ⊆ V forms a simplex in S if and o...

    work page

  6. [6]

    Topological phase classification The fundamental insight of [15] is that quantum states belonging to distinct phases exhibit different topological 7 signatures. Through computational studies of the Ising chain and XXZ spin chain in transverse fields, the authors demonstrated that Betti numbers computed across the filtration serve as robust topological inv...

    work page

  7. [7]

    Com- puting the metric distances Dij requires evaluating von Neumann entropies Si, Sj, and Sij, which classically ne- cessitates:

    Computational challenges and quantum solutions The primary computational bottleneck lies in con- structing the graph G from the quantum state ρ. Com- puting the metric distances Dij requires evaluating von Neumann entropies Si, Sj, and Sij, which classically ne- cessitates:

    work page

  8. [8]

    Full quantum state tomography to obtain classical descriptions of density matrices

    work page

  9. [9]

    Eigendecomposition of potentially exponentially large matrices

    work page

  10. [10]

    • Quantum entropy estimation

    Numerical computation of logarithms and traces This classical approach scales exponentially with sys- tem size, rendering it intractable for large quantum sys- tems. • Quantum entropy estimation. Recent advances in quantum algorithms for entropy estimation [25–27] provide efficient methods for computing von Neu- mann entropies directly from quantum states...

    work page

  11. [11]

    Formally, a two- dimensional image can be encoded as a matrix X ∈ Rn×n, where each entry Xi,j represents the pixel inten- sity at position ( i, j) within the image grid

    Mathematical representation A digital image is represented as a collection of pix- els, each characterized by a scalar intensity value typi- cally normalized to the interval [0 , 1]. Formally, a two- dimensional image can be encoded as a matrix X ∈ Rn×n, where each entry Xi,j represents the pixel inten- sity at position ( i, j) within the image grid. For ...

    work page

  12. [12]

    We associate each pixel position ( i, j) with a vertex vi,j in our topological space, yielding a total of n2 potential vertices

    Cubical complex construction Following the methodology established in [28, 29], we construct a topological representation of the image through the following procedure: • Vertex set. We associate each pixel position ( i, j) with a vertex vi,j in our topological space, yielding a total of n2 potential vertices. • Threshold filtering. For a given threshold p...

    work page

  13. [13]

    Filtration and topological signatures By systematically varying the threshold across an in- creasing sequence 0 ≤ ϵ1 ≤ ϵ2 ≤ · · · ≤ ϵL ≤ 1, we obtain a filtration of cubical complexes: S1 ⊆ S2 ⊆ · · · ⊆ SL, (32) where Sℓ denotes the cubical complex constructed with threshold ϵℓ. The Betti numbers {βr(Sℓ)}r,ℓ computed across this filtration constitute a to...

    work page

  14. [14]

    Unlike the quantum entangle- ment application, obtaining the classical graph descrip- tion for image-based cubical complexes is computation- ally straightforward

    Computational advantages Graph construction. Unlike the quantum entangle- ment application, obtaining the classical graph descrip- tion for image-based cubical complexes is computation- ally straightforward. For any threshold ϵ:

    work page

  15. [15]

    Identify active vertices: Vϵ = {vi,j : Xi,j ≤ ϵ}

    work page

  16. [16]

    Optimal regime for hybrid algorithm

    Construct edges: Connect vi,j, vi′,j′ ∈ Vϵ if |i − i′|+ |j − j′| = 1 This process requires O(n2) operations and yields a complete classical description of the graph Gϵ = (Vϵ, Eϵ). Optimal regime for hybrid algorithm. Image-based cubical complexes naturally satisfy the structural condi- tions where our hybrid algorithm excels: • Bounded degree. Each vertex...

    work page

  17. [17]

    Distance computation: Employ the quantum dis- tance estimation algorithm of Lloyd [32] to com- pute pairwise distances ∥xi − xj∥ in logarithmic time

    work page

  18. [18]

    Graph construction: Classically construct the ad- jacency graph by thresholding distances against ra- dius r

    work page

  19. [19]

    Betti estimation: Apply our hybrid algorithm to estimate normalized Betti numbers efficiently 9 This integration preserves exponential quantum speedups in distance computation while leveraging our efficient Betti number estimation for the resulting sparse geometric complexes. V. CONCLUSION We have presented a hybrid quantum-classical frame- work for estim...

    work page

  20. [20]

    Topological Data Analysis

    Larry Wasserman. Topological data analysis. arXiv preprint arXiv:1609.08227, 2016

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  21. [21]

    Quantum algorithms for topological and geometric anal- ysis of data

    Seth Lloyd, Silvano Garnerone, and Paolo Zanardi. Quantum algorithms for topological and geometric anal- ysis of data. Nature communications, 7(1):1–7, 2016

    work page 2016

  22. [22]

    A fast quantum mechanical algorithm for database search

    Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM Symposium on Theory of Computing , pages 212–219, 1996

    work page 1996

  23. [23]

    Quantum measurements and the Abelian Stabilizer Problem

    A Yu Kitaev. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026 , 1995

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  24. [24]

    Efficient quantum algorithms for sim- ulating sparse hamiltonians

    Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. Efficient quantum algorithms for sim- ulating sparse hamiltonians. Communications in Mathe- matical Physics, 270(2):359–371, 2007

    work page 2007

  25. [25]

    Black- box hamiltonian simulation and unitary implementation

    Dominic W Berry and Andrew M Childs. Black- box hamiltonian simulation and unitary implementation. Quantum Information and Computation , 12:29–62, 2009

    work page 2009

  26. [26]

    Quantum topological data analysis with linear depth and exponential speedup

    Shashanka Ubaru, Ismail Yunus Akhalwaya, Mark S Squillante, Kenneth L Clarkson, and Lior Horesh. Quan- tum topological data analysis with linear depth and expo- nential speedup. arXiv preprint arXiv:2108.02811, 2021

    work page arXiv 2021

  27. [27]

    Analyzing prospects for quantum advantage in topological data analysis

    Dominic W Berry, Yuan Su, Casper Gyurik, Robbie King, Joao Basso, Alexander Del Toro Barba, Abhishek Rajput, Nathan Wiebe, Vedran Dunjko, and Ryan Bab- bush. Analyzing prospects for quantum advantage in topological data analysis. PRX Quantum , 5(1):010319, 2024

    work page 2024

  28. [28]

    Quantum algorithm for persistent betti numbers and topological data analysis

    Ryu Hayakawa. Quantum algorithm for persistent betti numbers and topological data analysis. Quantum, 6:873, 2022. 10

    work page 2022

  29. [29]

    A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits

    Sam McArdle, Andr´ as Gily´ en, and Mario Berta. A streamlined quantum algorithm for topological data anal- ysis with exponentially fewer qubits. arXiv preprint arXiv:2209.12887, 2022

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  30. [30]

    Complexity- theoretic limitations on quantum algorithms for topolog- ical data analysis

    Alexander Schmidhuber and Seth Lloyd. Complexity- theoretic limitations on quantum algorithms for topolog- ical data analysis. PRX Quantum, 4:040349, Dec 2023

    work page 2023

  31. [31]

    Clique homol- ogy is qma 1-hard

    Marcos Crichigno and Tamara Kohler. Clique homol- ogy is qma 1-hard. Nature Communications, 15(1):9846, 2024

    work page 2024

  32. [32]

    Quantum algorithm for estimating betti numbers using a cohomology approach.arXiv preprint arXiv:2309.10800,

    Nhat A Nghiem, Xianfeng David Gu, and Tzu-Chieh Wei. Quantum algorithm for estimating betti num- bers using a cohomology approach. arXiv preprint arXiv:2309.10800, 2023

    work page arXiv 2023

  33. [33]

    New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes

    Junseo Lee and Nhat A Nghiem. New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes. arXiv preprint arXiv:2506.01432 , 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  34. [34]

    Persistent homology of quantum entan- glement

    Bart Olsthoorn. Persistent homology of quantum entan- glement. Physical Review B, 107(11):115174, 2023

    work page 2023

  35. [35]

    Probing multi- partite entanglement through persistent homology.Com- munications in Mathematical Physics , 405(5):125, 2024

    Gregory A Hamilton and Felix Leditzky. Probing multi- partite entanglement through persistent homology.Com- munications in Mathematical Physics , 405(5):125, 2024

    work page 2024

  36. [36]

    Arboricity and subgraph listing algorithms

    Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on comput- ing, 14(1):210–223, 1985

    work page 1985

  37. [37]

    Forests, frames, and games: algorithms for matroid sums and ap- plications

    Harold Gabow and Herbert Westermann. Forests, frames, and games: algorithms for matroid sums and ap- plications. In Proceedings of the twentieth annual ACM symposium on Theory of computing, pages 407–421, 1988

    work page 1988

  38. [38]

    Decomposition of finite graphs into forests

    C St JA Nash-Williams. Decomposition of finite graphs into forests. Journal of the London Mathematical Society, 1(1):12–12, 1964

    work page 1964

  39. [39]

    Listing all maximal cliques in sparse graphs in near- optimal time

    David Eppstein, Maarten L¨ offler, and Darren Strash. Listing all maximal cliques in sparse graphs in near- optimal time. In Algorithms and Computation: 21st In- ternational Symposium, ISAAC 2010, Jeju Island, Ko- rea, December 15-17, 2010, Proceedings, Part I 21, pages 403–414. Springer, 2010

    work page 2010

  40. [40]

    Refined quantum algorithms for principal component analysis and solving linear system.arXiv preprint arXiv:2504.00833,

    Nhat A Nghiem. Refined quantum algorithms for princi- pal component analysis and solving linear system. arXiv preprint arXiv:2504.00833, 2025

    work page arXiv 2025

  41. [41]

    Fast methods for es- timating the numerical rank of large matrices

    Shashanka Ubaru and Yousef Saad. Fast methods for es- timating the numerical rank of large matrices. InInterna- tional Conference on Machine Learning , pages 468–477. PMLR, 2016

    work page 2016

  42. [42]

    Fast estimation of approximate matrix ranks using spectral densities

    Shashanka Ubaru, Yousef Saad, and Abd-Krim Seghouane. Fast estimation of approximate matrix ranks using spectral densities. Neural Computation , 29(5):1317–1351, 2017

    work page 2017

  43. [43]

    Quantum persistent homology for time series

    Bernardo Ameneyro, George Siopsis, and Vasileios Maroulas. Quantum persistent homology for time series. In 2022 IEEE/ACM 7th Symposium on Edge Computing (SEC), pages 387–392. IEEE, 2022

    work page 2022

  44. [44]

    Estimation of non- linear physical quantities by measuring ancillas

    Nhat A Nghiem and Tzu-Chieh Wei. Estimation of non- linear physical quantities by measuring ancillas. arXiv preprint arXiv:2502.07571, 2025

    work page arXiv 2025

  45. [45]

    Estimating quantum entropy

    Jayadev Acharya, Ibrahim Issa, Nirmal V Shende, and Aaron B Wagner. Estimating quantum entropy. IEEE Journal on Selected Areas in Information Theory , 1(2):454–468, 2020

    work page 2020

  46. [46]

    Quantum algorithms for estimating quantum entropies

    Youle Wang, Benchi Zhao, and Xin Wang. Quantum algorithms for estimating quantum entropies. Physical Review Applied, 19(4):044041, 2023

    work page 2023

  47. [47]

    A cubical persistent homology-based technique for image denoising with topological feature preservation

    Md Al-Imran, Mst Zinia Afroz Liza, Md Morshed Bin Shiraj, Md Masum Murshed, and Nasima Akhter. A cubical persistent homology-based technique for image denoising with topological feature preservation. Journal of Computing Theories and Applications , 2(2):222–243, 2024

    work page 2024

  48. [48]

    Topological image modification for object de- tection and topological image processing of skin lesions

    Robin Vandaele, Guillaume Adrien Nervo, and Olivier Gevaert. Topological image modification for object de- tection and topological image processing of skin lesions. Scientific Reports, 10(1):21061, 2020

    work page 2020

  49. [49]

    Topology of ran- dom geometric complexes: a survey

    Omer Bobrowski and Matthew Kahle. Topology of ran- dom geometric complexes: a survey. Journal of applied and Computational Topology, 1:331–364, 2018

    work page 2018

  50. [50]

    Random geometric complexes

    Matthew Kahle. Random geometric complexes. Discrete & Computational Geometry , 45:553–573, 2011

    work page 2011

  51. [51]

    Quantum algorithms for supervised and unsupervised machine learning

    Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum algorithms for supervised and unsupervised machine learning. arXiv preprint arXiv:1307.0411, 2013

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  52. [52]

    Quan- tum state preparation with optimal circuit depth: Im- plementations and applications

    Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan. Quan- tum state preparation with optimal circuit depth: Im- plementations and applications. Physical Review Letters, 129(23):230504, 2022

    work page 2022

  53. [53]

    Quantum singular value transformation and be- yond: exponential improvements for quantum matrix arithmetics

    Andr´ as Gily´ en, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and be- yond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing , pages 193–204, 2019

    work page 2019

  54. [54]

    Optimal hamilto- nian simulation by quantum signal processing

    Guang Hao Low and Isaac L Chuang. Optimal hamilto- nian simulation by quantum signal processing. Physical Review Letters, 118(1):010501, 2017

    work page 2017

  55. [55]

    Hamiltonian sim- ulation by qubitization

    Guang Hao Low and Isaac L Chuang. Hamiltonian sim- ulation by qubitization. Quantum, 3:163, 2019

    work page 2019

  56. [56]

    Approximate quantum circuit synthesis using block encodings

    Daan Camps and Roel Van Beeumen. Approximate quantum circuit synthesis using block encodings. Physi- cal Review A, 102(5):052411, 2020

    work page 2020

  57. [57]

    Quantum algorithm for systems of linear equa- tions with exponentially improved dependence on pre- cision

    Andrew M Childs, Robin Kothari, and Rolando D Somma. Quantum algorithm for systems of linear equa- tions with exponentially improved dependence on pre- cision. SIAM Journal on Computing , 46(6):1920–1950, 2017

    work page 1920

  58. [58]

    Geometry, topology and physics

    Mikio Nakahara. Geometry, topology and physics . CRC press, 2018

    work page 2018

  59. [59]

    Algebraic topology

    Allen Hatcher. Algebraic topology. Cambridge University Press, 2005. 11 Appendix A: Block-encoding and quantum singular value transformation We briefly summarize the essential quantum tools used in our algorithm. For conciseness, we highlight only the main results and omit technical details, which are thor- oughly covered in [34]. An identical summary is ...

    work page 2005

  60. [60]

    Let p0, p1,

    Simplices and simplicial complexes Definition 2 (Simplex). Let p0, p1, . . . , pr ∈ Rm be ge- ometrically independent points where m ≥ r. The r- simplex σr = [p0, p1, . . . , pr] is defined as: σr = ( x ∈ Rm : x = rX i=0 cipi, c i ≥ 0, rX i=0 ci = 1 ) , (B1) where the coefficients (c0, c1, . . . , cr) are called the barycentric coordinates of x. Geometric...

    work page

  61. [61]

    Every face of a simplex in K is also in K

    work page

  62. [62]

    The intersection of any two simplices in K is either empty or a common face of both simplices The dimension of K is dim(K) = max{r : σr ∈ K}

    work page

  63. [63]

    Let K be an n-dimensional simplicial complex

    Chain groups and homomorphisms Definition 4 (Chain group). Let K be an n-dimensional simplicial complex. The r-th chain group C K r is the free abelian group generated by the oriented r-simplices of K. For r > dim(K), we define C K r = 0. Formally, let SK r = {σr,1, σr,2, . . . , σr,|SKr |} denote the set of r-simplices in K. An r-chain is an element of t...

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  64. [64]

    Definition 5 (Boundary operator)

    Boundary operators and homology The boundary operator ∂r : C K r → C K r−1 is fundamen- tal to homological algebra. Definition 5 (Boundary operator) . For an r-simplex [p0, p1, . . . , pr], the boundary operator is defined as: ∂r[p0, p1, . . . , pr] = rX i=0 (−1)i[p0, p1, . . . , ˆpi, . . . , pr], (B4) where ˆpi indicates that vertex pi is omitted, yieldi...

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  65. [65]

    If two topological spacesX and Y are homeomorphic, then their homology groups are isomorphic: H X r ∼= H Y r for all r ≥

    T opological invariance A central theorem in algebraic topology states that homology groups constitute topological invariants [40]: Theorem 3 (Topological invariance of homology). If two topological spacesX and Y are homeomorphic, then their homology groups are isomorphic: H X r ∼= H Y r for all r ≥

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  66. [66]

    Consequently, their Betti numbers are equal: βX r = βY r . This invariance property makes Betti numbers power- ful tools for topological classification and forms the math- ematical foundation for their applications in topological data analysis

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