Talks

For any vertex operator algebra V, Y. Zhu constructed an associative algebra Zhu(V) that captures its representation theory (more generally, given a finite order automorphism g of V, there exists an algebra Zhu_g(V) that captures g-twisted representation theory of V). 

To a 4d N=2 superconformal theory T, one assigns a vertex algebra V[T] by the construction of Beem et al. We explain one role of Zhu algebra in this context. Namely, we show that a certain quotient of the Zhu algebra describes what happens to the Schur sector of the theory T under the dimensional reduction on S^1. This connects the VOA construction in 4d N=2 SCFT to the topological quantum mechanics construction in 3d N=4 SCFT, with the latter being given by the aforementioned quotient of the Zhu algebra. In the process, we will discuss how to reformulate the VOA construction on an S^3 x S^1 geometry.

We discuss a classical analogue to the singular value transformation framework for quantum linear algebra algorithms developed by Gilyén et al. The proofs underlying this classical framework have natural connections to well-known ideas in randomized numerical linear algebra. Namely, our proofs are based on one key observation: approximating matrix products is easy in the quantum-inspired input model. This primitive's simplicity makes finding classical analogues to quantum machine learning algorithms straightforward and intuitive. We present sketches of the key proofs of our sketches as well as of its applications to dequantizing low-rank quantum machine learning. Joint work with Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang.

A small random sample of rows/columns of any matrix is a decent proxy for the matrix, provided sampling probabilities are proportional to squared lengths. Since the early theorems on this from the 90’s, there has been a substantial body of work using sampling (random projections and probabil-ties based on leverage scores are two examples) to reduce matrix sizes for Linear Algebra computations. The talk will describe theorems, applications and challenges in the area.

Variational Quantum Circuits, which include examples of quantum approximate optimization algorithms (QAOA),  variational quantum eigensolver (VQE), and quantum neural-networks (QNN), are predicted to be one of the most important near-term quantum applications, not only because of their potential promises as classical neural-networks but also because of their feasibility on near-term noisy intermediate size quantum (NISQ) machines. This talk reports some progress toward a principled understanding of the training of variational quantum circuits. First, I will demonstrate the difficulty of training by constructing an example with exponentially many local optima,  however,  due to a differential nature from classical neural-networks. Second, I will explain how to facilitate the training by incorporating the optimal-transport norm in the context of quantum generative adversarial networks (GANs), as well as its application in compressing quantum circuits in practical uses.

We will review recent work on quantum Machine Learning with a focus on longer-term quantum algorithms. We will discuss challenges and prospects for such algorithms and ways of bringing them closer to practical solutions.