Quantum Topological Data Analysis Beyond Betti Numbers: Complexity Hardness & An Algorithm for Torsion Witness

arXiv:2609.28112 · quant-ph · Submitted 2026-09-23 · Read on arXiv

quant-ph

Submitted: 2026-09-23

Updated: 2026-09-23

Terminology

• Block Encoding — 15× in this paper · explained in 4 episodes
• Ancilla Qubits — 6× in this paper · explained in 1 episode
• Trace — 6× in this paper · explained in 2 episodes
• Accuracy — 4× in this paper · explained in 1 episode
• Generalization — 3× in this paper · explained in 13 episodes
• Adam — 2× in this paper · explained in 2 episodes
• Axioms — 2× in this paper · explained in 2 episodes
• Topological Data Analysis (TDA) — 2× in this paper · explained in 2 episodes
• University of Economics Ho Chi Minh City — 2× in this paper
• Additive error estimation — 1× in this paper · explained in 2 episodes
• Alignment — 1× in this paper · explained in 1 episode
• BQP — 1× in this paper · explained in 2 episodes
• Embeddings — 1× in this paper · explained in 2 episodes
• Quantum Singular Value Transformation (QSVT) — 1× in this paper · explained in 4 episodes
• Verification — 1× in this paper · explained in 2 episodes

Sources

• Quantum algorithms for supervised and unsupervised machine learning
• Quantum embeddings for machine learning
• Quantum Topological Data Analysis with Linear Depth and Exponential Speedup
• A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits
• Computing Homotopy Types of Directed Flag Complexes
• Towards quantum topological data analysis: torsion detection
• A note on independence complexes of chordal graphs and dismantling
• Verifiable Quantum Advantage via Optimized DQI Circuits
• Review of a Quantum Algorithm for Betti Numbers
• A new quantum ripple-carry addition circuit
• High Performance Quantum Modular Multipliers
• Quantum Amplitude Amplification and Estimation
• Torsion detection in clique complexes is conditionally QMA 1-hard
• New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes
• On the rank of random matrices over finite fields
• Quantum Arithmetic on Galois Fields

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