Purdue Quantum Theory Seminar

I will present an overview of a research program dubbed "entanglement bootstrap," which aims to be a mathematical theory for defining topological phases on a lattice and classifying them. If time permits, I will describe a recent progress in making the framework invariant under constant-depth quantum circuits, as well as several classification results for topological phases in two spatial dimensions.

Jacob Lurie's classification of fully extended topological field theories applies in a broad and abstract setting across all dimensions. However, concrete examples remain scarce. In three dimensions, the Turaev-Viro theory has long been conjectured to fit within this framework. In this talk, we will establish that it necessarily does, outline a sketch of the proof, and explore the essential role of skein theory in this context.

The surface code is one of the leading approaches to building a fault-tolerant quantum computer, actively pursued by groups at Google, IBM, Microsoft, and elsewhere. A central challenge for the surface code is implementing non-Clifford -- unitary operations that map Pauli operators to non-Pauli operators. In this talk, I will introduce group surface codes, which are a natural generalization of the usual \(\mathbb{Z}_2\) surface code and can be understood as quantum double models with specific boundary conditions. I will argue that group surface codes, for suitably chosen groups, can be leveraged to perform non-Clifford gates in the \(\mathbb{Z}_2\) surface code. Time permitting I will describe three strategies for completing a universal gate set using group surface codes: (1) through magic state preparation, (2) using transversal non-Clifford gates, and (3) by sliding group surface codes. These strategies extend recent efforts in performing universal quantum computation in topological orders without the braiding of anyons.

This talk is based on upcoming work with Vieri Mattei, Naren Manjunath, and Apoorv Tiwari.

In this talk, I will discuss the gauging of Abelian modulated symmetries -- finite counterparts of subsystem symmetries -- that act non-uniformly across space. In two dimensions, this provides a simple route to constructing symmetry-enriched topological phases (SETs), where crystalline symmetries enrich the charges carried by topological anyons. Using local Hamiltonians of spin chains as a concrete setting, I will show how the dual theories feature exact modulated 1-form symmetries accompanied by mixed anomalies.

As Moore's Law slows, quantum computers promise to accelerate computationally intensive machine learning tasks like training and data generation. However, a fundamental disconnect exists between the continuous, analog nature of modern ML models (defined by real-valued parameters and data) and the discrete, digital logic of the dominant quantum circuit model. This mismatch creates an unnatural and often unintuitive foundation for designing new quantum algorithms and frameworks. In this talk, I argue that the Hamiltonian formalism, a computational model equivalent to quantum circuits, provides a natural and powerful alternative. I will demonstrate how Hamiltonians serve as a unifying language for designing new quantum algorithms for key ML tasks, such as optimization and sampling, leading to provable speedups. Crucially, because Hamiltonians directly describe the low-level physics of quantum hardware, this approach also enables vastly more efficient algorithm compilation. This hardware-aware framework bypasses many overheads of the circuit model, paving the way for deploying large-scale quantum ML applications on near-term devices.

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