Title: Proof of average-case #P- hardness of random circuit sampling with some robustness, and a protocol for blind quantum computation

Date: 09/05/2019

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

A one-parameter unitary-valued interpolation between any two unitary matrices (e.g., quantum gates) is constructed based on the Cayley transformation. We prove that this path induces probability measures that are arbitrarily close to the Haar measure and prove the simplest known average-case # P -hardness of random circuit sampling (RCS). RCS is the task of sampling from the output distribution of a quantum circuit whose local gates are random Haar unitaries, and is the lead candidate for demonstrating quantum supremacy in the "noisy intermediate scale quantum (NISQ)" computing era. Here we also prove exp(-Θ(n^4 )) robustness with respect to additive error. This overcomes issues that arise for extrapolations based on the truncations of the power series representation of the exponential function. (Dis)Proving the quantum supremacy conjecture requires an extension of this analysis to noise that is polynomially small in the system's size. This remains an open problem. Lastly, an efficient and private protocol for blind quantum computation is proposed that uses the Cayley deformations proposed herein for encryption. This is an efficient protocol that only requires classical communication between Alice and Bob.
** The talk is self-contained and does not require any pre-req beyond basic linear algebra (e.g, knowing what a unitary matrix is).

Title: Introduction to Free Products of von Neumann Algebras

Date: 09/10/2019

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

In this learning seminar, I will give an introduction to the free product construction for von Neumann algebras, which is the direct analogue of a free product for groups. Moreover, it defines the non-commutative independence relation most frequently used in free probability. No prior knowledge of von Neumann algebras will be necessary.

Title: Free products of finite-dimensional von Neumann algebras

Date: 09/12/2019

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

I will present joint work with Brent Nelson, where we classify the structure of free products of von Neumann algebras equipped with arbitrary states. Our techniques use our other joint work of assigning a von Neumann algebra associated to a weighted graph. I will discuss this work and how it leads to computing finite-dimensional free products.

19607

Thursday 9/19 11:30 AM

Scott Atkinson, University of California, Riverside

Speaker: Scott Atkinson, University of California, Riverside

Title: Tracial stability and related topics in operator algebras

Date: 09/19/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

We will discuss the notion of tracial stability for operator algebras. Morally, an algebra A is tracially stable if approximate homomorphisms on A are near honest homomorphisms on A. We will discuss several examples and non-examples of tracially stable algebras including certain graph products (simultaneous generalization of free and tensor products) of C*-algebras. We will also discuss properties closely related to tracial stability that provide new characterizations of amenability. Parts of this talk are based on joint work with Srivatsav Kunnawalkam Elayavalli.

Title: The higher dimensional algebra of matrix product operators and quantum spin chains

Date: 09/26/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

In the context of 1D quantum spin chains, matrix product operators provide a way to study non-local operators such as translation in terms of quasi-local information. They have been used to describe a generalized form of symmetry for 1D systems on the boundary of 2D topological phases. In this talk, we will introduce some concepts of higher dimensional algebra, and a broad hypotheses about higher categories and spatially extended quantum systems. We will then explain how the collection of matrix product operators assembles into a higher (symmetric monoidal 2-) category, and discuss some implications of this. Based on joint work with David Penneys.

Speaker: Jeffrey Schenker, Michigan State University

Title: An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states

Date: 10/10/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

Quantum channels represent the most general physical evolution of a quantum system through unitary evolution and a measurement process. Mathematically, a quantum channel is a completely positive and trace preserving linear map on the space of $D\times D$ matrices. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. The repeated composition of these maps along such a sequence could represent the result of repeated application of a given quantum channel subject to arbitrary correlated noise. It is physically natural to assume that such repeated compositions are eventually strictly positive, since this is true whenever any amount of decoherence is present in the quantum evolution. Under such an hypothesis, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one -- “entanglement breaking’’ – channel. We apply this result to describe the thermodynamic limit of ergodic matrix product states and prove that correlations of observables in such states decay exponentially in the bulk. (Joint work with Ramis Movassagh)

We consider the XXZ chain in the Ising phase. The particle number conservation property is used to write the Hamiltonian in a hard-core particles formulation over the $N$-symmetric product of graphs, where $N\in\mathbb{N}_0$ is the number of conserved particle. The droplet regime corresponds to a band at the bottom of the spectrum of the model consisting of a connected set (a droplet) of down-spins, up to an exponential error. It is interesting to know that in the formulation over the $N$-symmetric product graphs, with a fixed $N\geq 1$, the XXZ chain can be seen as a one-dimensional model only when it is restricted to droplet states. This justifies the recent many-body localization indicators proved in the droplet regime by Elgart/Klein/Stolz and Beaud/Warzel for the disordered model, including an area law of arbitrary states in that localized phase. As a first step beyond the droplet regime, we show that the entanglement of arbitrary states above the droplet regime (associated with multiple droplets/clusters) does not follow area laws, and instead, it follows a logarithmically corrected (enhanced) area law. We will comment on the effects of disorder on entanglement, and show how our results hint a phase transition.
(joint work with C. Fischbacher and G. Stolz, arXiv1907.11420)

Title: Localization for the Anderson--Bernoulli model on the integer lattice

Date: 11/07/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

Abstract: I will give a brief mathematical introduction to Anderson localization followed by a discussion of my recent work with Jian Ding. In our work we establish localization near the edge for the Anderson Bernoulli model on the two dimensional lattice. Our proof follows the program of Bourgain--Kenig and uses a new unique continuation result inspired by Buhovsky--Logunov--Malinnikova--Sodin. I will also discuss recent work of by Li and Zhang on the three dimensional case.

In quantum spin systems, the existence of a spectral gap above the ground state has strong implications for the low-energy physics. We survey recent results establishing spectral gaps in various frustration-free spin systems by verifying finite-size criteria. The talk is based on collaborations with Abdul-Rahman, Lucia, Mozgunov, Nachtergaele, Sandvik, Yang, Young, and Wang.

In this learning seminar style talk, I will define the notion of a derivation on a von Neumann algebra. I will also discuss their history and how they factor into modern research in operator algebras.

Title: The graph isomorphism game for quantum graphs

Date: 11/21/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

Non-local games give us a way of observing quantum behaviour through the observation of only classical data. The graph isomorphism game is one such non-local game played by Alice and Bob which involves two finite, undirected graphs. A winning strategy for the game is called quantum if it utilizes some shared resource of quantum entanglement between the players. We say two graphs are quantum isomorphic if there is a winning quantum strategy for the graph isomorphism game. We show that the *-algebraic, C*-algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs X and Y are all equivalent. This is based on joint work with M. Brannan, A. Chirvasitu, S. Harris, V. Paulsen, X. Su, and M. Wasilewski.