arXiv subjects

Frederic Cadet

Publications and source records attributed to Frederic Cadet.

13 recordsLinked to original sources

arXiv2026

From Block-encoding to Generalized Quantum Signal Processing: Principles, Algorithms and Applications

Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.

quant-ph↗

arXiv2026

Numerical Evaluation of ZX Calculus Optimization for Solovay Kitaev Quantum Circuit Synthesis

Fault-tolerant architectures implement non-Clifford T gates through magic-state distillation, so the T-count of a synthesized circuit dominates its physical cost. The Solovay-Kitaev algorithm approximates any single-qubit unitary from a finite gate set with a sequence length that grows only polylogarithmically in the inverse target error, but it optimizes for numerical convergence rather than circuit economy, and its output carries structural redundancy that a gate-level compiler cannot see. We report a measurement of what diagrammatic post-processing recovers from that redundancy. Twelve hundred random single-qubit targets, spanning the three Pauli rotation families and the general gate U(theta, phi, lambda), are synthesized over Clifford+T at three recursion depths, translated into graph-like ZX-diagrams, simplified by automated rewriting, and extracted back to circuits. Post-processing removes 26.6-30.1% of the total gate count and 18.5-22.2% of the T-count. The absolute saving grows with recursion depth, from about 60 to about 1600 gates, while the fractional saving does not: it rises slightly from the shallowest setting and is then flat across a twenty-five-fold change in circuit length, and by the deepest setting the four target families are no longer distinguishable from one another. Because the rewrite rules preserve the implemented linear map, the approximation error is unchanged. The compile-time cost of the rewriting layer, by contrast, grows sharply with depth and comes to dominate the synthesis itself.

quant-ph↗

arXiv2026

Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations

Protein function may depend not only on endpoint conformations but also on the ordered deformation histories through which they are reached. This distinction is relevant to allostery, conformational switching, mutation-induced rearrangements, and epistatic effects, where different perturbation sequences may produce similar visible structures while retaining distinct internal transport histories. Current state-centered or endpoint-centered representations do not always preserve this order-sensitive information. The practical motivation is therefore to provide a foundation for future descriptors of protein deformation trajectories that can distinguish ordered histories even when endpoint conformations are similar. We propose a deformation-first geometric framework based on quaternionic frame transport along the protein backbone. Local backbone frames are lifted to quaternionic variables, with infinitesimal rotation encoded by $\Omega(\ell)=2\,q(\ell)^{-1}\partial_\ell q(\ell).$ Ordered concatenation of admissible deformation paths generates a noncommutative transport algebra, recording that deformation A followed by B need not be equivalent to B followed by A. From this ordered transport layer, we construct a spectral-response layer comprising a global Dirac-type operator, local spectral germs, a renormalized spectral density, and a mixed response form. A minimal realization on an idealized $\alpha$-helix shows how localized pitch and bending perturbations can yield similar endpoint descriptors while producing a nonzero endpoint-derived ordered-transport discrepancy. At the formal level, the framework separates an order-sensitive transport-memory sector, lost under a commutative shadow, from a spectral-response sector that remains visible.

q-bio.BM↗

arXiv2026

Algebraic Representability as the Limiting Regime of Grokking: An Exactly Solvable Model with Holomorphic Activations

Neural networks trained on modular arithmetic exhibit grokking, a delayed transition from memorisation to generalisation known to depend on model capacity: too little and the network memorises slowly or not at all, too much and it generalises almost immediately. What happens at the extreme of this spectrum, when the architecture's expressible function class collapses to a finite-dimensional algebraic variety? We study two-layer networks with a holomorphic monomial activation sigma(z)=z^k, trained on modular tasks encoded via roots of unity. Here the network output, regardless of hidden width, is confined to a (k+1)-dimensional subspace of characters of (Z_p)^2, an O(k/p^2) slice of the full function space. We give a complete algebraic characterisation of this subspace: a task is representable if and only if its discrete Fourier support lies on the diagonal u+v = k (mod p), which for linear-phase targets reduces to the arithmetic criterion m+n=k. This is not merely a constraint on eventual generalisation but on memorisation itself: because the outputs are algebraically confined, a non-representable target cannot be fit even on the training set, and we prove a positive lower bound on the training loss, independent of width. Across 585 runs the algebraic prediction matches the observed outcome with 99.8% accuracy, with no memorisation regime and no grokking; outcomes split cleanly into instant success and outright failure. This binary behaviour is the limiting case of the capacity-grokking relationship: when the expressible class shrinks to a fixed algebraic object, the question of when a network will grok dissolves into whether it can represent the target at all. A bottleneck ablation connects this extreme to standard networks, tracing a continuous path from representational failure, through memorisation without generalisation, to grokking with a shrinking gap as capacity grows.

cs.LG↗

arXiv2026

Assessing Cost Hamiltonian Reliability in Quantum Protein Structure Prediction

In variational quantum algorithms, QAOA, and quantum annealing, the cost Hamiltonian defines the optimization landscape explored by the quantum hardware; however, in many application-driven formulations, this Hamiltonian is a simplified proxy for the true task-level objective. Using lattice-based quantum protein structure prediction as a case study, we investigate whether the contact-energy cost Hamiltonian commonly used in this setting is sufficiently aligned with structural accuracy, as measured by RMSD against experimentally determined structures. Through this specific problem, we show the importance of studying the reliability of the cost Hamiltonian independently from the quantum approach used. This work shows that, for small peptides and on average, the energy landscape of the considered cost Hamiltonian is not correlated well enough to the actual error to provide meaningful predictions. Moreover, this correlation was estimated through Monte-Carlo sampling for larger instances. It shows an increase of said correlation for larger problem instances and when more interaction shells are considered. This investigation illustrates the meaningfulness of investigating cost Hamiltonian relevance independently from the quantum algorithm used.

quant-ph↗

arXiv2026

Wavelet Variance Equipartition as a Threshold for World-Model Quality and Quantum Kernel TN-Simulability

While world models learn compact representations of complex environments, they lack a physics-grounded metric to assess the structural fidelity of their latent spaces. We identify the wavelet scaling exponent $\alpha$ as a critical diagnostic, proposing optimal representations satisfy variance equipartition ($\alpha \approx 1/2$) -- mirroring Kolmogorov's inertial range. We establish $\alpha = 1/2$ as a sharp transition boundary for the classical simulability of amplitude-encoded quantum kernels. Using tensor-network theory, we prove latents with $\alpha > 1/2$ reside in an area-law phase admitting efficient classical emulation, while $\alpha < 1/2$ triggers a volume-law phase where the Matrix Product State bond dimension $\chi$ grows exponentially with qubit count $n$. Analyzing pre-trained VideoMAE latents reveals a dichotomy: spatial tokens approach the equipartition limit ($\alpha \approx 0.423$), but permutation-invariant feature channels exhibit unstructured disorder ($\alpha \approx -0.123$). This forces real-world latents deep into the volume-law phase, providing a data-driven necessary condition for simulation hardness. Finally, we apply Weingarten calculus to derive the exact variance of the scrambled transition probability under a 2-design ensemble. We prove this variance scales strictly as $\Var[X] = \Theta(d^{-2})$. We confirm this numerically with a log-log slope of $-1.881$ ($R^2 = 0.999$), identifying a formidable shot-noise wall demanding a measurement budget of $M = \Omega(d^2)$ that constrains quantum machine learning scalability.

quant-ph↗

arXiv2025

Hermitian Matrix Function Synthesis without Block-Encoding

Implementing polynomial functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advanced quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding and avoids the compounding post-selection overheads characteristic of LCU-based constructions, achieving a stable, degree-independent success probability. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead and opens new pathways for quantum algorithm design for functions of Hermitian matrices, particularly in settings where the Hermitian operator arises naturally from symmetric combinations of unitaries.

quant-ph↗

arXiv2025

WAN3DNS: Weak Adversarial Networks for Solving 3D Incompressible Navier-Stokes Equations

The 3D incompressible Navier-Stokes equations model essential fluid phenomena, including turbulence and aerodynamics, but are challenging to solve due to nonlinearity and limited solution regularity. Despite extensive research, the full mathematical understanding of the 3D incompressible Navier-Stokes equations continues to elude scientists, highlighting the depth and difficulty of the problem. Classical solvers are costly, and neural network-based methods typically assume strong solutions, limiting their use in underresolved regimes. We introduce WAN3DNS, a weak-form neural solver that recasts the equations as a minimax optimization problem, allowing learning directly from weak solutions. Using the weak formulation, WAN3DNS circumvents the stringent differentiability requirements of classical physics-informed neural networks (PINNs) and accommodates scenarios where weak solutions exist, but strong solutions may not. We evaluated WAN3DNS's accuracy and effectiveness in three benchmark cases: the 2D Kovasznay, 3D Beltrami, and 3D lid-driven cavity flows. Furthermore, using Galerkin's theory, we conduct a rigorous error analysis and show that the $L^{2}$ training error is controllably bounded by the architectural parameters of the network and the norm of residues. This implies that for neural networks with small loss, the corresponding $L^{2}$ error will also be small. This work bridges the gap between weak solution theory and deep learning, offering a robust alternative for complex fluid flow simulations with reduced regularity constraints. Code: https://github.com/Wenran-Li/WAN3DNS

physics.flu-dyn↗

arXiv2025

Quantum Autoencoder: An efficient approach to quantum feature map generation

Quantum machine learning methods often rely on fixed, hand-crafted quantum encodings that may not capture optimal features for downstream tasks. In this work, we study the power of quantum autoencoders in learning data-driven quantum representations. We first theoretically demonstrate that the quantum autoencoder method is efficient in terms of sample complexity throughout the entire training process. Then we numerically train the quantum autoencoder on 3 million peptide sequences, and evaluate their effectiveness across multiple peptide classification problems including antihypertensive peptide prediction, blood-brain barrier-penetration, and cytotoxic activity detection. The learned representations were compared against Hamiltonian-evolved baselines using a quantum kernel with support vector machines. Results show that quantum autoencoder learned representations achieve accuracy improvements ranging from 0.4\% to 8.1\% over Hamiltonian baselines across seven datasets, demonstrating effective generalization to diverse downstream datasets with pre-training enabling effective transfer learning without task-specific fine-tuning. This work establishes that quantum autoencoder architectures can effectively learn from large-scale datasets (3 million samples) with compact parameterizations ($\sim$900 parameters), demonstrating their viability for practical quantum applications.

quant-ph↗

arXiv2025

EHCube4P: Learning Epistatic Patterns Through Hypercube Graph Convolution Neural Network for Protein Fitness Function Estimation

Understanding the relationship between protein sequences and their functions is fundamental to protein engineering, but this task is hindered by the combinatorially vast sequence space and the experimental noise inherent in fitness measurements. In this study, we present a novel framework that models the sequence landscape as a hypercube $H(k,2)$ and integrates wavelet-based signal denoising with a graph convolutional neural network (GCN) to predict protein fitness across rugged fitness landscapes. Using a dataset of 419 experimentally measured mutant sequences of the Tobacco 5-Epi-Aristolochene Synthase (TEAS) enzyme, we preprocess the fitness signals using a 1-D discrete wavelet transform with a Daubechies-3 basis to suppress experimental noise while preserving local epistatic patterns. Our model comprises two GCN layers, allowing for beyond pairwise aggregation, followed by a multi-layer perceptron (MLP). We show that our approach, EHCube4P, generalizes well across different enzyme activity datasets and effectively captures higher-order mutational interactions. Performance varies with the ruggedness of the fitness landscape, with smoother signals yielding higher test set $r^2$ scores. These results demonstrate that combining wavelet preprocessing with graph-based deep learning enhances the robustness and generalization of fitness prediction, particularly for sparse and noisy biological datasets. The approach provides a scalable and interpretable framework for protein fitness estimation applicable to a broad range of combinatorial biological systems.

q-bio.QM↗

arXiv2025

From thermodynamics to protein design: Diffusion models for biomolecule generation towards autonomous protein engineering

Protein design with desirable properties has been a significant challenge for many decades. Generative artificial intelligence is a promising approach and has achieved great success in various protein generation tasks. Notably, diffusion models stand out for their robust mathematical foundations and impressive generative capabilities, offering unique advantages in certain applications such as protein design. In this review, we first give the definition and characteristics of diffusion models and then focus on two strategies: Denoising Diffusion Probabilistic Models and Score-based Generative Models, where DDPM is the discrete form of SGM. Furthermore, we discuss their applications in protein design, peptide generation, drug discovery, and protein-ligand interaction. Finally, we outline the future perspectives of diffusion models to advance autonomous protein design and engineering. The E(3) group consists of all rotations, reflections, and translations in three-dimensions. The equivariance on the E(3) group can keep the physical stability of the frame of each amino acid as much as possible, and we reflect on how to keep the diffusion model E(3) equivariant for protein generation.

q-bio.QM↗

arXiv2024

Non-Hemolytic Peptide Classification Using A Quantum Support Vector Machine

Quantum machine learning (QML) is one of the most promising applications of quantum computation. However, it is still unclear whether quantum advantages exist when the data is of a classical nature and the search for practical, real-world applications of QML remains active. In this work, we apply the well-studied quantum support vector machine (QSVM), a powerful QML model, to a binary classification task which classifies peptides as either hemolytic or non-hemolytic. Using three peptide datasets, we apply and contrast the performance of the QSVM, numerous classical SVMs, and the best published results on the same peptide classification task, out of which the QSVM performs best. The contributions of this work include (i) the first application of the QSVM to this specific peptide classification task, (ii) an explicit demonstration of QSVMs outperforming the best published results attained with classical machine learning models on this classification task and (iii) empirical results showing that the QSVM is capable of outperforming many (and possibly all) classical SVMs on this classification task. This foundational work paves the way to verifiable quantum advantages in the field of computational biology and facilitates safer therapeutic development.

quant-ph↗

arXiv2003

Deformation quantization using groupoids. Case of toric manifolds

In the framework of C*-algebraic deformation quantization we propose a notion of deformation groupoid which could apply to known examples e.g. Connes' tangent groupoid of a manifold, its generalisation by Landsman and Ramazan, Rieffel's noncommutative torus, and even Landi's noncommutative 4-sphere. We construct such groupoid for a wide class of T^n-spaces, that generalizes the one given for C^n by Bellissard and Vittot. In particular, using the geometric properties of the moment map discovered in the '80s by Atiyah, Delzant, Guillemin and Sternberg, it provides a \cstar-algebraic deformation quantization for all toric manifolds, including the 2-sphere and all complex projective spaces.

math.OA↗