arXiv subjects

Yoshi-aki Shimada

Publications and source records attributed to Yoshi-aki Shimada.

5 recordsLinked to original sources

arXiv2026

Hodge Spectral Surrogates for Topology-Constrained Optimization

Topology-constrained optimization arises in applications such as medical image segmentation, geometric design, and network generation, where global structural correctness matters. Yet Betti numbers and persistent homology are discrete and combinatorial, making them difficult to control directly by gradient-based optimization. We propose a differentiable framework that replaces discrete topological counting with a soft spectral relaxation of the Hodge Laplacian. Candidate simplices are embedded in a fixed ambient complex, while simplex inclusion is represented by continuous activations, producing a smooth spectral path between different hard complexes. The soft states are used only as optimization surrogates; in the hard limit, the ordinary Hodge Laplacian and its homological zero modes are recovered. We establish operator-norm and spectral perturbation bounds connecting hard Betti numbers to soft near-zero modes, and derive differentiable objectives using heat, resolvent, and polynomial low-pass filters. Controlled experiments on Vietoris--Rips point clouds show more spatially distributed gradients, reduced scale-normalized derivative variation near persistence-pairing changes, and geometry-aware update directions in the tested settings. Experiments on graph clique complexes further show controllable shifts in sampled hard Betti regimes and compatibility with standard graph objectives. The framework provides a complementary Hodge-spectral approach to persistent-homology-based topology optimization.

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arXiv2026

Gauge Geometry of Hodge Zero-Mode Transport in Parameter-Dependent Topological Data Analysis

We propose a practical computational framework for detecting structural changes in parameter-dependent topological data. In many applications, such as time-series data analysis, anomaly detection, and monitoring of systems under changing control parameters, persistence diagrams describe the birth and death of topological features at each parameter value, but they do not fully capture how these features are reorganized over time. To address this limitation, we represent homological features by zero modes of the ordinary combinatorial Hodge Laplacian and track the corresponding feature spaces in a common ambient chain space. This allows us to compute curvature and holonomy as descriptors of local reorganization and accumulated memory in evolving topological structures. Curvature highlights parameter regions where homological features mix or change rapidly, while holonomy summarizes the net effect of such changes after a closed cycle. We also establish stability estimates showing that these descriptors are robust under perturbations of the Hodge Laplacian on regular regions. Numerical experiments on controlled time-dependent point-cloud data show that the proposed method detects tracking instability, distinguishes systems with nearly identical persistence diagrams, and captures cycle-level memory invisible to pointwise feature matching. These results suggest that zero-mode transport geometry can serve as a useful computational tool for analyzing dynamic topological data.

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arXiv2026

Quantum-accelerated conjugate gradient method via spectral initialization

Solving large-scale linear systems problems is a cornerstone in scientific and industrial computing. Classical iterative solvers face increasing difficulty as the number of unknowns becomes large, while fully quantum linear solvers require fault-tolerant resources that remain far beyond near-term feasibility. Here we propose a quantum-accelerated conjugate gradient (QACG) method in which a fault-tolerant quantum algorithm is used exclusively to construct a spectrally informed initial guess for a classical conjugate gradient (CG) solver. We estimate the total runtime and resource requirements of an integrated quantum-HPC platform for the 3D Poisson equation. A central feature of QACG is the controllable decomposition of the condition number between the quantum and the classical solver, enabling flexible allocation of computational effort. Under explicit architectural assumptions, we identify regimes in which QACG yields a runtime advantage over purely classical approaches while requiring substantially fewer quantum resources than end-to-end quantum linear solvers. These results illustrate a concrete pathway toward the scientific and industrial use of early-stage fault-tolerant quantum computing and point to a integrated paradigm in which quantum devices act as accelerators within high-performance computing workflows.

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arXiv2026

Spectral Codes: A Geometric Formalism for Quantum Error Correction

We present a new geometric perspective on quantum error correction based on spectral triples in noncommutative geometry. In this approach, quantum error correcting codes are reformulated as low energy spectral projections of Dirac type operators that separate global logical degrees of freedom from local, correctable errors. Locality, code distance, and the Knill Laflamme condition acquire a unified spectral and geometric interpretation in terms of the induced metric and spectrum of the Dirac operator. Within this framework, a wide range of known error correcting codes including classical linear codes, stabilizer codes, GKP type codes, and topological codes are recovered from a single construction. This demonstrates that classical and quantum codes can be organized within a common geometric language. A central advantage of the spectral triple perspective is that the performance of error correction can be directly related to spectral properties. We show that leakage out of the code space is controlled by the spectral gap of the Dirac operator, and that code preserving internal perturbations can systematically increase this gap without altering the encoded logical subspace. This yields a geometric mechanism for enhancing error correction thresholds, which we illustrate explicitly for a stabilizer code. We further interpret Berezin Toeplitz quantization as a mixed spectral code and briefly discuss implications for holographic quantum error correction. Overall, our results suggest that quantum error correction can be viewed as a universal low energy phenomenon governed by spectral geometry.

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arXiv2025

Quantum spectroscopy of topological dynamics via a supersymmetric Hamiltonian

Topological data analysis (TDA) characterizes complex dynamics through global invariants, but classical computation becomes prohibitive for high-dimensional data. We reinterpret time-domain dynamics as the eigenvalue spectrum of a supersymmetric (SUSY) Hamiltonian and thereby estimate topological descriptors through quantum spectroscopy. While zero modes correspond to Betti numbers, we show that low-lying excited states quantify the stability of topological features. Using a Takens embedding of the Lorenz system together with a resource-efficient quantum phase estimation implemented on IBM quantum hardware, we observe that the spectral gap of the SUSY Laplacian tracks the persistence of homological structures. Notably, the minimum of this spectral gap coincides with the onset of chaos, whereas its reopening reflects the geometric maturation of the attractor. Validated on small complexes yet offering an exponential advantage over classical diagonalization (from $O(N^3)$ to $\mathrm{poly}(\log N)$), this framework suggests that quantum hardware can function as a spectrometer for data topologies beyond classical reach.

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