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REVIEW 3 major objections 6 minor 25 references

Signature kernels boost some quantum CNN architectures on binary MNIST

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2026-07-09 04:17 UTC pith:T6454EBN

load-bearing objection Signature-kernel QCNN is a reasonable proof-of-concept; the empirical claim needs error bars and a qubit-count control before it's taken seriously. the 3 major comments →

arxiv 2607.07634 v1 pith:T6454EBN submitted 2026-07-08 quant-ph cs.AI

QCNN with Rough Path Signature Kernels

Leonardo Nogueira Falabella , Vasily Sazonov This is my paper
classification quant-ph cs.AI
keywords rough path signaturessignature kernelquantum convolutional neural networksvariational quantum linear solvertime series classificationhybrid quantum-classical computingNISQ algorithmsMNIST
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proposes a hybrid quantum-classical pipeline for time series classification that feeds signature kernel features — derived from rough path theory, which captures geometric structure of sequential data in a way that is invariant to time reparametrization — into a Quantum Convolutional Neural Network (QCNN). The signature kernel between two paths is defined as the inner product of their (infinite) path signatures and can be computed by solving a second-order hyperbolic PDE (a Goursat problem) rather than by explicitly evaluating the exponentially large signature objects. The authors explore two ways to integrate these kernels into the QCNN: a pure substitution architecture where the full kernel matrix replaces the image input, and an auxiliary-qubit fusion architecture where a single scalar summary of the kernel is appended as a ninth qubit alongside the standard eight-qubit image encoding. They also attempt to compute the kernel PDE solution on quantum hardware using the Variational Quantum Linear Solver (VQLS), which variationally prepares a quantum state proportional to the solution vector of the discretized linear system. The paper evaluates the full suite of architectures on a binary 0-vs-1 MNIST stroke-sequence classification task using noiseless statevector simulators, comparing multiple convolutional ansatz configurations (U_TTN, U5–U15, U_SO(4), U_SU(4)). The central empirical claim is that introducing signature kernel features yields a distinct, architecture-dependent performance enhancement, with the auxiliary-qubit fusion approach consistently outperforming pure kernel substitution. On the quantum-computing side, the paper documents that VQLS can solve the kernel linear system for paths of 3–4 time steps but fails to scale beyond 5–6 steps due to condition-number inflation and non-convex optimization landscapes, making it impractical for realistic training pipelines that require thousands of kernel evaluations.

Core claim

The paper's central finding is twofold. First, signature kernel features — which encode the reparametrization-invariant geometry of sequential data — can serve as a useful structural prior for quantum neural networks, producing architecture-dependent accuracy improvements on a binary classification task when integrated via an auxiliary-qubit fusion strategy. Second, the attempt to compute these kernels end-to-end on quantum hardware via VQLS hits a hard scalability wall at roughly 5–6 time steps, because the condition number of the discretized PDE's linear system inflates rapidly with path length, trapping the variational optimizer in local minima and degrading solution fidelity. The paper's

What carries the argument

Signature kernel (inner product of infinite path signatures, computed via a Goursat PDE solution); Variational Quantum Linear Solver (VQLS, a NISQ-compatible variational algorithm for linear systems); Quantum Convolutional Neural Network (QCNN, a hierarchical quantum circuit with O(log n) depth and parameter count); amplitude encoding and angle encoding for quantum data loading; hardware-efficient variational ansatz with 4 layers of RY rotations and CNOT entanglement for VQLS; multiple two-qubit convolutional ansatzes (U_TTN, U5–U15, U_SO(4), U_SU(4)) for QCNN

Load-bearing premise

The assumption that binary 0-vs-1 MNIST stroke classification — a task so easy that simple baselines achieve high accuracy — is a meaningful testbed for evaluating whether signature kernel features provide genuine advantages for quantum classifiers, compounded by the absence of error bars or multiple random seeds that would let a reader assess whether the observed architecture-dependent improvements are statistically real.

What would settle it

If, on a harder sequential classification task (e.g., multi-class or temporally complex data) with proper error bars and multiple seeds, the auxiliary-qubit signature kernel fusion architecture showed no statistically significant improvement over the baseline image-only QCNN across all ansatz types, the central claim that signature kernels serve as a useful structural prior for quantum neural networks would be undermined.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If signature kernels genuinely provide a structural prior for quantum classifiers, the benefit should become more pronounced on harder sequential classification tasks where temporal structure matters more and raw spatial features are less discriminative.
  • The VQLS scalability bottleneck (failure beyond 5–6 time steps due to condition-number inflation) suggests that any quantum-linear-system approach to PDE-based kernel computation will need either preconditioning strategies or fundamentally different ansatz architectures to handle realistic path lengths.
  • The auxiliary-qubit fusion architecture — appending a single scalar feature as an extra qubit — offers a parameter-efficient template for integrating domain-specific invariants into quantum circuits more generally, beyond signature kernels specifically.
  • The gap between theoretical exponential speedup (HHL) and practical NISQ feasibility (VQLS) for this class of problems quantifies how far current variational solvers are from realizing the theoretical advantages of quantum linear algebra for structured PDE problems.

Where Pith is reading between the lines

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

  • The binary 0-vs-1 MNIST task is near-trivially separable, so the reported architecture-dependent improvements may reflect ansatz optimization variance rather than kernel-induced structural gains; the absence of error bars or multiple seeds makes the statistical significance of the improvements unassessable from the paper alone.
  • If the VQLS bottleneck is fundamentally about condition-number scaling rather than circuit depth, then structured preconditioners exploiting the lower-triangular form of the kernel PDE system could potentially extend the feasible range — but this is not explored in the paper.
  • The choice to downsample stroke sequences to 16 points (15 intervals) for QCNN input may interact with the signature kernel's ability to capture fine geometric distinctions, confounding the comparison between kernel-based and image-based architectures.
  • The architecture-dependent (rather than parameter-count-dependent) nature of the improvement suggests the benefit may stem from how specific ansatz entanglement structures align with the kernel feature geometry, which could be tested by systematically varying entanglement topology while holding parameter count fixed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This manuscript proposes a hybrid quantum-classical architecture for time series classification that combines rough path signature kernels with Quantum Convolutional Neural Networks (QCNNs). The signature kernel between a reference path and a target path is computed either classically (via the PDE-based method of Salvi et al.) or variationally (via VQLS), and the resulting features are fed into a QCNN for downstream classification. The authors evaluate several QCNN ansatz configurations on a binary MNIST 0-vs-1 stroke classification task, comparing a standard image baseline, a pure kernel-substitution architecture, and an auxiliary-qubit fusion architecture. They also characterize the scalability limits of the VQLS component, finding it impractical beyond 5-6 time steps. The mathematical framework (§2) correctly restates known results on signature kernels and their PDE formulation. The VQLS formulation (§3) is standard. The central empirical claim is that signature kernel features yield a 'distinct performance enhancement for specific QCNN architectures' (§5.2).

Significance. The paper presents an interesting and novel combination of rough path theory with quantum machine learning. The mathematical framework is correctly stated and properly attributed to prior work (Salvi et al. [21], Bravo-Prieto et al. [2], Hur et al. [10]). The VQLS scalability analysis (§5.1, Fig. 4) provides a useful and honest characterization of the practical limitations of near-term quantum linear solvers for this application. The exploration of multiple QCNN ansatz configurations is thorough. However, the central empirical claim regarding performance enhancement rests on experiments that lack statistical rigor, which limits the significance of the contribution in its current form.

major comments (3)
  1. §5.2, Fig. 6: The central claim that signature kernel features yield 'a distinct performance enhancement for specific QCNN architectures' is supported solely by single-run accuracy values in a bar chart with no error bars, confidence intervals, or multiple random seeds. The binary 0-vs-1 MNIST task is near-trivially separable, and accuracy differences of a few percentage points across ansatzes are well within the range of optimizer stochasticity and initialization variance. Without statistical significance testing, the observed architecture-dependent improvements cannot be distinguished from noise. This is the load-bearing evidence for the paper's main claim and must be addressed.
  2. §4.4, §5.2: The SigKernel aux qubit architecture uses 9 qubits (8 for image + 1 auxiliary), while the Standard baseline uses 8 qubits. The improvement attributed to the signature kernel scalar could instead arise from the additional qubit/circuit capacity, or from the generic benefit of fusing any auxiliary feature with the image, rather than from signature kernel information specifically. The paper does not include a control where a random or constant scalar replaces the kernel value on the 9th qubit. This confound undermines the attribution of the improvement to signature kernel features and thus undermines the central claim.
  3. §5.2: The choice of binary 0-vs-1 classification—a task where even simple baselines achieve >99% accuracy—limits the ability to assess whether signature kernel features provide genuine advantages for quantum classifiers. The authors themselves acknowledge (§6) that 'applying this signature-based learning pipeline to complex datasets where conventional classical methods perform poorly may better isolate and reveal the true advantages.' The current benchmark does not adequately test the central claim and a more challenging task would strengthen the evidence.
minor comments (6)
  1. §2, Eq. (8): The approximation formula for the double integral appears to have a notational issue with the factor (u−s)(s−v), which should likely involve grid spacing. The authors should clarify the notation.
  2. §3, Eq. (17): The decomposition of A into unitaries is described, but the specific structure of the matrix A arising from the signature kernel discretization (its sparsity pattern, condition number scaling) could be discussed more explicitly to support the scalability argument.
  3. §5.1: The condition number inflation is mentioned qualitatively but no quantitative data on κ vs. path length is provided, though Fig. 4(b) shows fidelity degradation. A table or additional plot would strengthen the scalability argument.
  4. Fig. 6: The bar chart would benefit from a table of exact numerical values and from clearer labeling of the ansatz types.
  5. §5: The reference path X_ref is stated to be 'a fixed reference sample representing a digit 0' but the sensitivity of the results to this choice is not discussed. A brief comment on this would be useful.
  6. The abstract states 'potential advantages' which is more cautious than the claim in §5.2 of 'distinct performance enhancement.' The language should be consistent with the strength of the evidence.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies that the mathematical framework is properly stated and attributed, that the VQLS scalability analysis is honest and useful, and that the combination of rough path theory with quantum machine learning is novel and interesting. The referee's three major comments all concern the empirical evidence supporting our central claim of architecture-dependent performance enhancement from signature kernel features. We agree with all three comments and will revise the manuscript accordingly. Specifically: (1) we will rerun all experiments across multiple random seeds and report mean accuracy with error bars and significance tests; (2) we will add a control experiment using a random/constant scalar on the auxiliary qubit to isolate the contribution of signature kernel information from generic additional circuit capacity; and (3) we will add a more challenging classification task (multi-class MNIST or a harder binary task) to better test whether signature kernel features provide genuine advantages. We discuss each point below.

read point-by-point responses

  1. Referee: §5.2, Fig. 6: The central claim that signature kernel features yield 'a distinct performance enhancement for specific QCNN architectures' is supported solely by single-run accuracy values in a bar chart with no error bars, confidence intervals, or multiple random seeds. The binary 0-vs-1 MNIST task is near-trivially separable, and accuracy differences of a few percentage points across ansatzes are well within the range of optimizer stochasticity and initialization variance. Without statistical significance testing, the observed architecture-dependent improvements cannot be distinguished from noise.

    Authors: The referee is entirely correct. The single-run results presented in Figure 6 are insufficient to support the claim of a 'distinct performance enhancement.' We will rerun all experiments across at least 5 random seeds, report mean accuracy with standard deviation error bars, and apply pairwise significance tests (e.g., paired t-tests or Wilcoxon signed-rank tests) between the Standard, SigKernel, and SigKernel aux qubit configurations for each ansatz. We will also soften the language of the central claim to reflect what the statistical evidence supports, removing the word 'distinct' if the corrected experiments do not warrant it. revision: yes

  2. Referee: §4.4, §5.2: The SigKernel aux qubit architecture uses 9 qubits (8 for image + 1 auxiliary), while the Standard baseline uses 8 qubits. The improvement attributed to the signature kernel scalar could instead arise from the additional qubit/circuit capacity, or from the generic benefit of fusing any auxiliary feature with the image, rather than from signature kernel information specifically. The paper does not include a control where a random or constant scalar replaces the kernel value on the 9th qubit. This confound undermines the attribution of the improvement to signature kernel features and thus undermines the central claim.

    Authors: This is a valid and important confound that we had not adequately addressed. We will add a control experiment in which the 9th auxiliary qubit receives either a random scalar or a constant value in place of the signature kernel value, using the same 9-qubit architecture. This will allow us to disentangle the contribution of the signature kernel information from the generic effect of additional circuit capacity or auxiliary feature fusion. If the improvement persists only with the signature kernel scalar and not with the control, the attribution is supported; if not, we will revise our claims accordingly. revision: yes

  3. Referee: §5.2: The choice of binary 0-vs-1 classification—a task where even simple baselines achieve >99% accuracy—limits the ability to assess whether signature kernel features provide genuine advantages for quantum classifiers. The authors themselves acknowledge (§6) that 'applying this signature-based learning pipeline to complex datasets where conventional classical methods perform poorly may better isolate and reveal the true advantages.' The current benchmark does not adequately test the central claim and a more challenging task would strengthen the evidence.

    Authors: We agree that the 0-vs-1 task is too easy to serve as a strong test of the central claim. We will add experiments on a more challenging classification task. Candidates include a multi-class MNIST setting (e.g., 3-way or 5-way classification) or a harder binary task (e.g., distinguishing digits with similar stroke trajectories such as 3-vs-8 or 4-vs-9). The harder task will be selected to ensure that simple baselines do not achieve near-perfect accuracy, so that any improvement from signature kernel features is more informative. We will retain the 0-vs-1 results as a proof-of-concept but will frame the new task as the primary evidence for the central claim. revision: yes

Circularity Check

0 steps flagged

No circularity found. The derivation chain is built entirely on independent prior work, and the central empirical claim is experimental, not derived from a fitted input.

full rationale

The paper's derivation chain is self-contained against external sources at every load-bearing step. The signature kernel PDE (Eq. 6) and its finite-difference discretization (Eqs. 8–9) are cited from Salvi et al. [21], an independent prior work. The VQLS algorithm (Eqs. 16–21) is from Bravo-Prieto et al. [2], also independent. The QCNN framework follows Hur et al. [10, 11], again independent. The linear system formulation (Eqs. 10–15) is a straightforward algebraic rewriting of the discretization in Eq. 9 — no hidden assumption is smuggled in. None of the 25 references are authored by the present paper's authors (Falabella, Sazonov), so there is no self-citation chain whatsoever. The central empirical claim — that signature kernel features yield performance improvements for specific QCNN architectures (§5.2) — is supported by experimental results on the external MNIST dataset, not by a derivation that reduces to a fitted parameter. The absence of error bars and the qubit-count confound are legitimate correctness and statistical-validity concerns, but they are not circularity: no prediction or first-principles result is equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities, particles, forces, or mathematical objects. All mathematical machinery (signatures, kernels, PDE, VQLS, QCNN) is drawn from cited prior work. The free parameters are architectural and optimization choices typical of variational quantum algorithms, not fundamental constants. The main concern is the ad hoc choice of benchmark task and reference path, which are not independently motivated.

free parameters (5)
  • VQLS ansatz depth L_VQLS = 4
    Number of variational layers in the VQLS hardware-efficient ansatz, chosen by hand (§3, Eq. 18).
  • QCNN convolutional ansatz type = various (U_TTN, U5-U15, U_SO(4), U_SU(4))
    Choice of two-qubit unitary for convolutional layers, selected empirically (§4.2).
  • Path downsampling length = 16 points (15 intervals)
    Chosen to fit 8-qubit QCNN register constraint, not derived from data analysis (§5.1).
  • Reference path X_ref = a digit 0 sample
    Fixed reference sample for kernel evaluation, chosen ad hoc (§5.1).
  • Number of QCNN qubits = 8 (or 9 with auxiliary)
    Hardware constraint, not derived from information-theoretic considerations (§4.4).
axioms (4)
  • standard math The signature kernel is the unique solution to the Goursat PDE (Eq. 6)
    Invoked in §2, cited from Salvi et al. [21]. This is a proven mathematical result used as the foundation for the kernel computation.
  • domain assumption VQLS can variationally approximate solutions to linear systems on NISQ devices
    Invoked in §3, cited from Bravo-Prieto et al. [2]. The paper relies on VQLS being a viable near-term solver, then shows it fails for their use case.
  • domain assumption QCNN hierarchical architecture is suitable for classification of amplitude-encoded features
    Invoked in §4, following Cong et al. [5] and Hur et al. [10]. The paper assumes this architecture transfers to signature kernel features without modification.
  • ad hoc to paper Binary 0-vs-1 MNIST classification is a meaningful benchmark for evaluating quantum ML methods
    The choice of this specific toy task as the sole evaluation is not justified against harder benchmarks; it is the only experimental validation in the paper (§5).

pith-pipeline@v1.1.0-glm · 14344 in / 2859 out tokens · 488770 ms · 2026-07-09T04:17:08.045789+00:00 · methodology

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read the original abstract

Time series analysis plays a vital role across a wide range of scientific and engineering domains but poses substantial computational challenges. A major difficulty arises from the time reparameterization invariance of time series data, which complicates the extraction of meaningful temporal features. In this work, we address the problem of time series classification by exploring the application of quantum computation techniques. We propose a hybrid quantum-classical architecture that integrates recent advances in quantum neural networks with the mathematical framework of path signatures, mitigating the impact of time reparametrization invariance. The architecture employs feature layers that compute a signature kernel between pairs of input paths, consisting of a reference path and a target path for classification, using either classical or quantum variational linear solvers (VQLS). These feature layers are followed by a Quantum Convolutional Neural Network (QCNN) to perform downstream learning tasks. We evaluate several realizations of the proposed architecture, differing in QCNN configurations, on a binary classification task involving time series representations of handwritten digits. Our experiments demonstrate the potential advantages of implementing path signature kernel layers within quantum circuits and provide an analysis of the computational limitations associated with the VQLS component.

Figures

Figures reproduced from arXiv: 2607.07634 by Leonardo Nogueira Falabella, Vasily Sazonov.

Figure 1
Figure 1. Figure 1: Schematic diagram for the hybrid quantum-classical classification pipeline. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the QCNN architecture applied to an 8-qubit system. The [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Circuit scheme of the extra qubit kernel architecture, exemplified with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scalability benchmarks for the VQLS framework across varying path compression levels: [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visual analysis of the MNIST sequence dataset: (a) Characteristic downsampled pen-tip [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Binary classification accuracy for the 0-vs-1 task benchmarks comparing the baseline [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 6 internal anchors

  1. [1]

    PennyLane: Automatic differentiation of hybrid quantum-classical computations

    Ville Bergholm, Josh Izaac, Maria Schuld, et al. Pennylane: Automatic differentiation of hybrid quantum-classical computations.arXiv preprint arXiv:1811.04968, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  2. [2]

    Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J

    Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles. Variational quantum linear solver.Quantum, 7:1188, 2023

    work page 2023

  3. [3]

    A primer on the signature method in machine learning

    Ilya Chevyrev and Andrey Kormilitzin. A primer on the signature method in machine learning. arXiv preprint arXiv:1603.03788, 2016

    work page arXiv 2016

  4. [4]

    Childs, Robin Kothari, and Rolando D

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

    work page 1920

  5. [5]

    Iris Cong, Soonwon Choi, and Mikhail D. Lukin. Quantum convolutional neural networks. Nature Physics, 15(12):1273–1278, 2019

    work page 2019

  6. [6]

    Ginn & Company, 1916

    Edouard Goursat and Earle Raymond Hedrick.A Course in Mathematical Analysis. Ginn & Company, 1916

    work page 1916

  7. [7]

    A Quantum Approximate Optimization Algorithm

    Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm.arXiv preprint arXiv:1411.4028, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  8. [8]

    Deep learning for time series classification: a review.Data Mining and Knowledge Discovery, 33(4):917–963, 2019

    Hassan Ismail Fawaz, Germain Forestier, Jonathan Weber, Lhassane Idoumghar, and Pierre- Alain Muller. Deep learning for time series classification: a review.Data Mining and Knowledge Discovery, 33(4):917–963, 2019

    work page 2019

  9. [9]

    Harrow, Avinatan Hassidim, and Seth Lloyd

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations.Physical Review Letters, 103(15):150502, 2009

    work page 2009

  10. [10]

    Tak Hur, Leeseok Kim, and Daniel K. Park. Quantum convolutional neural network for classical data classification.Quantum Machine Intelligence, 4(1):3, 2022

    work page 2022

  11. [11]

    Tak Hur, Leeseok Kim, and Daniel K. Park. Quantum convolutional neural network (qcnn), 2021.https://github.com/takh04/QCNN. Implementation of quantum convolutional neural networks for classical data classification

    work page 2021

  12. [12]

    Hyndman and George Athanasopoulos.Forecasting: principles and practice

    Rob J. Hyndman and George Athanasopoulos.Forecasting: principles and practice. OTexts, 2nd edition, 2018

    work page 2018

  13. [13]

    Quantum computing with Qiskit

    Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J. Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D. Nation, Lev S. Bishop, Andrew W. Cross, Blake R. John- son, and Jay M. Gambetta. Quantum computing with qiskit.arXiv preprint arXiv:2405.08810, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  14. [14]

    Kiraly and Harald Oberhauser

    Franz J. Kiraly and Harald Oberhauser. Kernels for Sequentially Ordered Data.Journal of Machine Learning Research, 20(31):1–45, 2019

    work page 2019

  15. [15]

    MNIST handwritten digit database

    Yann LeCun and Corinna Cortes. MNIST handwritten digit database. http://yann.lecun.com/exdb/mnist/, 2010

    work page 2010

  16. [16]

    Time-series forecasting with deep learning: a survey.Philo- sophical Transactions of the Royal Society A, 379(2194):20200209, 2021

    Bryan Lim and Stefan Zohren. Time-series forecasting with deep learning: a survey.Philo- sophical Transactions of the Royal Society A, 379(2194):20200209, 2021. 14

    work page 2021

  17. [17]

    Identifying Bottlenecks of NISQ-friendly HHL algorithms

    Giulia Marfany, Ievgeniia Sakhnenko, and Rainer Lorenz. Identifying bottlenecks of NISQ- friendly HHL algorithms.arXiv preprint arXiv:2406.06288, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  18. [18]

    Otto, Nicholas Zolman, J

    Samuel E. Otto, Nicholas Zolman, J. Nathan Kutz, and Steven L. Brunton. A unified frame- work to enforce, discover, and promote symmetry in machine learning.Journal of Machine Learning Research, 26(248):1–83, 2025

    work page 2025

  19. [19]

    Vasilakos

    Deepak Ranga, Aryan Rana, Sunil Prajapat, Pankaj Kumar, Kranti Kumar, and Athanasios V. Vasilakos. Quantum machine learning: Exploring the role of data encoding techniques, challenges, and future directions.Mathematics, 12:3318, 2024

    work page 2024

  20. [20]

    Dynamic programming algorithm optimization for spoken word recognition.IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(1):43– 49, 1978

    Hiroaki Sakoe and Seibi Chiba. Dynamic programming algorithm optimization for spoken word recognition.IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(1):43– 49, 1978

    work page 1978

  21. [21]

    The Signature Kernel is the solution of a Goursat PDE.SIAM Journal on Mathematics of Data Science, 3(3):873–899, 2021

    Cristopher Salvi, Thomas Cass, James Foster, Terry Lyons, and Weixin Yang. The Signature Kernel is the solution of a Goursat PDE.SIAM Journal on Mathematics of Data Science, 3(3):873–899, 2021

    work page 2021

  22. [22]

    Johnson, and Al´ an Aspuru-Guzik

    Sukin Sim, Peter D. Johnson, and Al´ an Aspuru-Guzik. Expressibility and capability of pa- rameterized quantum circuits for hybrid quantum-classical algorithms.Advanced Quantum Technologies, 2(12):1900070, 2019

    work page 2019

  23. [23]

    Introduction to rough paths theory

    Mazyar Ghani Varzaneh and Sebastian Riedel. Introduction to rough paths theory.arXiv preprint arXiv:2302.04653, 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  24. [24]

    Expanding data encod- ing patterns for quantum algorithms.2021 IEEE 18th International Conference on Software Architecture Companion (ICSA-C), 2021

    Manuela Weigold, Johanna Barzen, Frank Leymann, and Marie Salm. Expanding data encod- ing patterns for quantum algorithms.2021 IEEE 18th International Conference on Software Architecture Companion (ICSA-C), 2021

    work page 2021

  25. [25]

    Solving Linear Systems on Quantum Hardware with Hybrid HHL++

    Rodrigo Yalovetzky, Patrick Minssen, Daniel Herman, and Marco Pistoia. Solving linear systems on quantum hardware with hybrid HHL++.arXiv preprint arXiv:2110.15958, 2021. 15

    work page internal anchor Pith review Pith/arXiv arXiv 2021