Nhat A. Nghiem

Nhat A. Nghiem

We introduce several new quantum algorithms for estimating homological invariants, specifically Betti numbers and persistent Betti numbers, of a simplicial complex given via a structured classical input. At the core of our algorithm lies the ability to efficiently construct the block-encoding of Laplacians (and persistent Laplacians) based on the classical description of the given complex. From such block-encodings, Betti numbers (and persistent Betti numbers) can be estimated. The complexity of our method is polylogarithmic in the number of simplices in both simplex-sparse and simplex-dense regimes, thus offering an advantage over existing works. Moreover, prior quantum algorithms based on spectral methods incur significant overhead due to their reliance on estimating the kernel of combinatorial Laplacians, particularly when the Betti number is small. We introduce a new approach for estimating Betti numbers based on homology tracking and homology property testing, which enables exponential quantum speedups over both classical and prior quantum approaches under sparsity and structure assumptions. We further initiate the study of homology triviality and equivalence testing as natural property testing problems in topological data analysis, and provide efficient quantum algorithms with time complexity nearly linear in the number of simplices when the rank of the boundary operator is large. In addition, we develop a cohomological approach based on block-encoded projections onto cocycle spaces, enabling rank-independent testing of homology equivalence. This yields the first quantum algorithms for constructing and manipulating r-cocycles in time polylogarithmic in the size of the complex. Together, these results establish a new direction in quantum topological data analysis and demonstrate that computing topological invariants can serve as a fertile ground for provable quantum advantage.

Supanut Thanasilp, Samson Wang, Nhat A. Nghiem et al.

A new paradigm for data science has emerged, with quantum data, quantum models, and quantum computational devices. This field, called Quantum Machine Learning (QML), aims to achieve a speedup over traditional machine learning for data analysis. However, its success usually hinges on efficiently training the parameters in quantum neural networks, and the field of QML is still lacking theoretical scaling results for their trainability. Some trainability results have been proven for a closely related field called Variational Quantum Algorithms (VQAs). While both fields involve training a parametrized quantum circuit, there are crucial differences that make the results for one setting not readily applicable to the other. In this work we bridge the two frameworks and show that gradient scaling results for VQAs can also be applied to study the gradient scaling of QML models. Our results indicate that features deemed detrimental for VQA trainability can also lead to issues such as barren plateaus in QML. Consequently, our work has implications for several QML proposals in the literature. In addition, we provide theoretical and numerical evidence that QML models exhibit further trainability issues not present in VQAs, arising from the use of a training dataset. We refer to these as dataset-induced barren plateaus. These results are most relevant when dealing with classical data, as here the choice of embedding scheme (i.e., the map between classical data and quantum states) can greatly affect the gradient scaling.