This work introduces quantum topological data encoding (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution, and indicates several areas of application where the framework can be used to provide a more efficient and reliable data representation.
Abstract
Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.
Characterizing quantum topological phases requires measuring non-local string order parameters, demanding access to the full system, which is often experimentally unfeasible. In this work, we introduce a data-efficient supervised learning framework that circumvents this limitation by recognizing quantum phases from small subsystems. Our protocol utilizes a quantum kernel constructed from the reduced density matrices of these subsystems, which can be efficiently estimated experimentally. We benchmark our framework with the classification of the phase diagrams of two spin models on one-dimensional lattices, namely the generalized cluster-Ising spin-1/2 chain and the anisotropic Haldane spin-1 chain. Remarkably, our approach achieves high accuracy in phase classification when operations are limited to as few as one to four sites, and it also generalizes to longer chains even when trained on moderate system sizes. These findings demonstrate that local reduced density matrices preserve vital signatures of global topological phases, offering a practical route to characterize rich phase diagrams of quantum many-body systems.
Mehran Khosrojerdi, Sougato Bose, A. Cuccoli et al.· 0 citations
This work establishes a scalable and physics guided candidate prioritization framework for inverse discovery of TMs beyond brute force electronic structure calculations and topological studies.
Tasneem U. Rehman, N. Yaqoob, A. A. Ganaie et al.· Physica Scripta· 0 citations
Exact fermion to qubit transformations are conventionally regarded as algorithmic tools that translate many-body Hamiltonians into qubit representations for quantum simulation. Here we show that they also define intrinsic geometric representations whose structure encodes physically meaningful information beyond spectral equivalence. We develop a geometric framework based on weighted hypergraphs and coupling space representations constructed from the Bravyi--Kitaev (BK) and Xia--Bian--Kais (XBK) encodings. Within the BK representation, we introduce a geometric observable that compares the algebraic connectivities of the kinetic and interaction hypergraphs, derive its exact analytical dependence on interaction strength, and uncover two geometric universality classes together with an exact spectral organization originating from the binary tree architecture of the encoding. The complementary XBK representation describes the evolution of encoded Hamiltonians through probability measures in coupling space, where optimal transport quantifies interaction-driven reorganization independently of the spectral analysis. Applications to the Hubbard, spinless tV , single impurity Anderson, and Kitaev models demonstrate that these connectivity and transport based geometric descriptions consistently capture the structural evolution of encoded quantum Hamiltonians across distinct classes of many-body systems. Our results establish hypergraph geometry as a new framework for understanding fermion-to-qubit encodings,revealing that they serve not only as computational mappings but also as geometric representations of quantum many-body Hamiltonians.
Lakshya Nagpal, Nishith Reen, S. R. Hassan· 0 citations
Results indicate that the topology-aligned inductive bias is the active ingredient driving parameter efficiency at QM9 scale, with implications for matched-baseline benchmarking in quantum machine learning.
This paper investigates foundational quantum computing protocols from the intuitive perspective of categorical tensor-graph semantics within the category \textbf{FHilb}. While conventional Hilbert-space formalisms often conceal the structural nature of quantum algorithms behind high-dimensional matrix operations, the topological framework directly encodes algorithmic functionalities into their graphical skeletons. We provide a comprehensive topological reinterpretation of the Bernstein--Vazirani and Simon algorithms, demonstrating how topological transformations distill their core mathematical essence and clarify the operational mechanisms of oracles. Going beyond the standard qubit model, we construct explicit representations for the qutrit-adapted topological Deutsch--Jozsa and single-shot Grover algorithms. In particular, we establish a necessary and sufficient condition for the single-shot Grover search. We further implement CNOT gates via complementary Frobenius structures and investigate a diagrammatic decomposition scheme for the W-state preparation protocol. By bridging tensor category theory with practical quantum algorithmic design, this work furnishes a composable, scalable diagrammatic toolkit essential for automated circuit optimization across the evolving quantum hardware ecosystem.
This work examines when structured long-range connectivity provides a useful resource, focusing on sparse power-of-two (PWR2) coupling graphs, and identifies circuit geometry and qubit reconfigurability as task-dependent resources for variational algorithms.
Helene M. Losl, Aydin Deger, Andrew J. Daley· 0 citations