Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians

Abstract: Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. In this paper, we introduce and study the problem of normalized persistence, a practically motivated and easily interpretable version of persistent homology that counts the fraction of holes that persist at different lengthscales. We prove that a variant of normalized persistence is $\mathsf{DQC}_1$-hard and contained in $\mathsf{BQP}$, giving evidence of an exponential quantum speedup for TDA under the standard assumption that $\mathsf{DQC}_1 \not\subseteq \mathsf{BPP}$. These are the first $\mathsf{DQC}_1$-hardness results that are directly applicable to TDA instances. We also find a close connection between normalized persistence and the complexity of estimating spectral quantities in the low-energy subspace of local Hamiltonians. We study a family of such problems, including a low-energy normalized subtrace and spectral density. We show that these are $\mathsf{DQC}_1$-hard for $O(1)$-local Hamiltonians, strengthening previous results that required log-local interactions. We also introduce a variant of $\mathsf{DQC}_1$ with perfect completeness ($\mathsf{SDQC}_1$) to characterize the hardness of problems normalized by an exact kernel. This includes normalized persistence for $O(1)$-local Hamiltonians, which we show is $\mathsf{SDQC}_1$-hard.

• The paper establishes novel DQC1-hardness and SDQC1-hardness results for normalized persistence in both TDA and local Hamiltonian spectral problems.
• It introduces a unified complexity framework linking persistent homology in TDA to quantum Hamiltonian complexity via circuit-to-Hamiltonian reductions and clique complex Laplacians.
• Efficient quantum algorithm containment in BQP is demonstrated, suggesting potential exponential quantum advantage under natural state preparation conditions.

Complexity of Normalized Persistence in TDA and Local Hamiltonians

Introduction and Motivation

Topological Data Analysis (TDA) applies algebraic topology to extract robust structural patterns from data. Persistent homology, a core TDA tool, systematically quantifies the emergence and persistence of high-dimensional topological features (e.g., holes, voids) across varying lengthscales, yielding interpretable invariants with direct significance for machine learning and data analysis. Recent advances have demonstrated connections between TDA and quantum computing, notably revealing potential for quantum advantage in estimating topological invariants.

This paper introduces and rigorously analyzes the computational complexity of normalized persistence: the fraction of topological features (specifically, $d$-dimensional holes) of an initial simplicial complex $X_1$ that persist when embedded into a larger complex $X_2$. The normalized persistence quantifies the robustness of topological features independently of the simplex count, enabling fairer comparison across data sets and scale-invariant interpretation. The work also establishes fundamental connections between TDA problems and low-energy spectral properties of local Hamiltonians, building a unified complexity-theoretic framework that includes classical and quantum settings.

Main Contributions

1. DQC1-Hardness and BQP-Containment for Normalized Persistence

The paper proves that estimating normalized persistence (and several related spectral quantities) is DQC1-hard and contained in BQP under reasonable algorithmic and state-preparation assumptions. This holds for both local Hamiltonians (O(1)-local interactions) and for combinatorial Laplacians arising from clique complexes in TDA. These results establish the first DQC1-hardness results directly for TDA-relevant instances (i.e., clique complexes), filling a key gap in the literature.

2. Extension to SDQC1

For variants involving exact kernels (not just low-energy subspaces), the paper introduces a new complexity class, Subspace DQC1 (SDQC1), generalizing DQC1 to require perfect completeness in a specified subspace of the register. The authors prove SDQC1-hardness for these problems, notably for exact normalized persistence, and conjecture SDQC1-hardness for normalized harmonic persistence in TDA, subject to the existence of kernel-preserving clique-complex simulations.

3. Strengthening Known Hardness Results

By leveraging circuit-to-Hamiltonian reductions and refined spectral simulation tools, the work extends known DQC1-hardness results (previously limited to log-local Hamiltonians or less TDA-relevant chain complexes) to O(1)-local Hamiltonians and to Laplacians of clique complexes, the canonical objects for TDA in practical applications.

4. Unified Complexity Landscape

The paper delineates dependencies between classical homology decision problems (e.g., #P-hardness/PSPACE-hardness for computing Betti numbers), quantum witnesses (QMA, BQP), and DQC1/SDQC1, charting how various TDA and local Hamiltonian spectral estimation problems reduce to each other. Specializations to pure-state or unique-kernel cases recover established overlap and energy-estimation problems, connecting TDA with quantum Hamiltonian complexity in a principled formalism. The paper emphasizes that for all these DQC1-hard or SDQC1-hard tasks, efficient quantum algorithms exist under natural state preparation and overlap conditions, confirming containment in BQP.

Technical Approach

The analysis revolves around reductions from DQC1/SDQC1 decision problems to subspace-normalized spectral density or persistence-like estimators:

• Low-Energy Normalized Subtrace (LENS): Given an O(1)-local Hamiltonian, estimating the average energy in a low-energy subspace (normalized by its dimension) is shown to be DQC1-hard. The reduction adapts circuit-to-Hamiltonian constructions and leverages a modified unary clock encoding for precise control of locality and normalization.
• Reduction to Laplacians of Clique Complexes: By employing spectral simulation results [Ray24, CMP18], the authors show any O(1)-local Hamiltonian can be spectrally simulated in the low-energy sector of an explicitly constructed clique complex Laplacian, enabling direct reductions for TDA instances.
• Normalized Quasi-Persistence: Introduced as a robust relaxation of normalized persistence (allowing overlap with low-energy subspaces rather than exact kernels), this problem is also proved DQC1-hard for local Hamiltonians and clique complexes.
• SDQC1 and Exact Kernel Problems: For exact normalized persistence, where the persistence is measured with respect to kernels of Hamiltonians (zero-energy states), perfect subspace completeness is required. SDQC1 is introduced, and the corresponding normalized persistence problem is shown to be SDQC1-hard for O(1)-local Hamiltonians.
• Containment in BQP: Efficient estimation is demonstrated theoretically for all problems whenever one can efficiently prepare uniform mixtures over the relevant subspaces and ensure suitable overlap; in particular, containment is shown for low-energy, quasi-kernel, and kernel normalized estimators.

Key Theoretical and Practical Implications

• Exponential Quantum Advantage: Under the standard assumption that DQC1 $\not\subset$ BPP, these results substantiate regimes admitting robust exponential quantum speedup for TDA and related Hamiltonian problems, conditional on efficient state preparation and overlap.
• Bridging Quantum Complexity with TDA: The results establish a direct mapping of important TDA primitives (such as persistent Betti number ratios) onto canonical spectral problems in quantum Hamiltonian complexity, providing a pathway for rigorous quantum complexity analysis of machine learning pipelines rooted in TDA.
• SDQC1 as a Natural Class for Kernel-Normalized Problems: The introduction and analysis of SDQC1 highlight the necessity of new quantum complexity classes to accommodate nuances in quantum spectral estimation tasks arising from the TDA context, in particular exact-kernel normalization.

Open Problems and Future Work

• SDQC1-Completeness and Classical Simulatability: Whether SDQC1 is classically simulable remains open, as does the completeness of the proposed normalized persistence problems for SDQC1.
• Containment in DQC1 and Intermediate Classes: The authors suggest possible refinements via the intermediate class $½$-BQP (half-BQP) and advocate further exploration of perfect-completeness analogues.
• Fine-Grained Circuit-to-Hamiltonian Tradeoffs: More granular analyses of locality versus Hilbert space dimension in the circuit-to-Hamiltonian construction for TDA-specific tasks are identified as an important avenue.
• General Kernel-Preserving Simulation: The ability to construct clique-complex Laplacians that simulate target Hamiltonian kernels with exactness and spectral gap preservation remains conjectural; resolving this would cement the SDQC1-hardness conjecture for normalized harmonic persistence in TDA.

Conclusion

This work provides a rigorous foundation for the quantum computational complexity of normalized persistence problems in both TDA and local Hamiltonian frameworks. By establishing DQC1- and SDQC1-hardness for a suite of normalized persistence and spectral density problems—precisely those which arise in practical TDA and quantum physics—the paper offers strong theoretical evidence for exponential quantum advantage in topological data analysis tasks. Furthermore, by framing TDA quantities as special cases of general low-energy and kernel subspace spectral problems, the results bridge the gap between quantum algorithms for data analysis and the complexity theory of quantum Hamiltonians, enriching both domains with versatile new technical tools and avenues for future inquiry.