### 2023–2024 Academic Year

**Organized by: **George Androulakis (giorgis@math.sc.edu)

Unless otherwise noted, the seminar will be held on Fridays from 3:30pm to 4:30pm in LeConte 348.

This page will be updated as new seminars are scheduled. Make sure to check back each week for information on upcoming seminars.

We encourage all participants to attend in person as that fosters a greater academic community. In the case that the speaker cannot be with us in person, then the seminar will be delivered and attended online via Zoom. In this case, the Zoom meeting information is the following:

Join Zoom Meeting**Meeting ID:** 818 2090 1820**Passcode:** 123456

**When:** POSTPONED - check back later for an updated time.

**Where:** LeConte 348

**Speaker:** Ralph Howard

**Abstract: **We show that for a "generic" norm on a 3 dimensional vector space, that no pair of two
dimensional subspaces of the space are linearly isometric. This is joint work with
Maria Girardi.

**When: **January 26^{th} at 3:30 p.m.

**Where:** LeConte 348

**Speaker:** Stephen Dilworth

**Abstract:** I will define the transportation cost space associated to a finite metric space M.
It is a finite-dimensional normed space whose dual is the space of Lipschitz functions
on M. I will present some examples which motivate an open question on the structure
of the transportation cost space. Analysis of the `invariant' projections (from
the edge space onto the space of Lipschitz functions) which commute with a group
of isometries of the edge space has yielded useful information for the families of
diamond and Laakso graphs. Recent results (with Kutzarova and Ostrovskii) for discrete
tori and Hamming graphs which use this method will appear in a volume in honor of
Professor Per Enflo. I will also discuss the limitations of the method, in particular
the disappointing fact that it cannot answer the open question.

**When:** December 8^{th} at 2:15 p.m.

**Where:** LeConte 348

**Speaker:** Ralph Howard

**Abstract: **We review and give proofs of theorems of Blaschke, Lagunov, and Pestov and Ionin which
give lower bounds on the inradius of a domain in terms of the boundary curvature.
An example result, due to Pestov and Ionin, is that a simple closed curve in the
plane which has curvature with respect to the inner normal at most 1 surrounds a disk
of radius 1.

**When:** December 1^{st} at 2:15 p.m.

**Where:** LeConte 348

**Speaker:** Haonan Zhang

**Abstract:** A fundamental problem from computational learning theory is to well-reconstruct an
unknown function on the discrete hypercubes. One classical result of this problem
for the random query model is the low-degree algorithm of Linial, Mansour, and Nisan
in 1993. This was improved exponentially by Eskenazis and Ivanisvili in 2022 using
a family of polynomial inequalities going back to Littlewood in 1930. Recently, quantum
analogs of such polynomial inequalities were conjectured by Rouzé, Wirth, and Zhang
(2022). This conjecture was resolved by Huang, Chen, and Preskill (2022) without knowing
it when studying learning problems of quantum dynamics. In this talk, I will discuss
another proof of this conjecture that is simpler and gives better estimates. As an
application, it allows us to recover the low-degree algorithm of Eskenazis and Ivanisvili
in the quantum setting. This is based on arXiv:2210.14468, joint work with Alexander Volberg (MSU).

**When: **November 17^{th} 2023 from 2:15pm - 3:15pm

**Where:** LeConte 348

**Speaker:** Frank (Peng) Fu (USC, Dept. of Computer Science and Engineering)

**Abstract:** In this talk, I will first talk about how to model the teleportation protocol with
concepts from compact closed categories. If time permit, I will give another brief
introduction to dagger categories and sketch how they can be used to model common
concepts in quantum computing.

References:

* "A categorical semantics of quantum protocols" by Samson Abramsky and Bob Coecke

* "Dagger compact closed categories and completely positive maps" by Peter Selinger.

**When:** November 10^{th} 2023, from 3:30pm to 4:30pm

**Where:** LeConte 348

**Speaker:** Frank (Peng) Fu, (USC, Dept. of Computer Science and Engineering)

**Abstract:** Compact closed categories and their extensions can be used

as a framework to model many concepts in quantum computing

(e.g. scalars, inner products, even completely positive maps).

In this talk, I will give a brief introduction to compact closed categories and

their graphical language. As an example, I will sketch how they can be used to

understand the teleportation protocol.

**When:** November 3^{rd} 2023 from 3:30pm to 4:30pm

**Where:** LeConte 348

**Speaker:** Rabins Wosti (USC, Dept. of Computer Science and Engineering)

**Abstract: **Consider a general quantum source that emits at discrete time steps quantum pure states
which are chosen from a finite alphabet according to some probability distribution
which may depend on the whole history. Also, fix two positive integers \(m\) and
\(l\). We encode any tensor product of \(ml\) many states emitted by the quantum source
by breaking it into \(m\) many blocks where each block has length \(l\), and considering
sequences of \(m\) many isometries so that each isometry encodes one of these blocks
into the Fock space, followed by the concatenation of their images. We only consider
certain sequences of such isometries that we call ``special block codes" in order
to ensure that the the string of encoded states is uniquely decodable. We compute
the minimum average codeword length of these encodings which depends on the quantum
source and the integers \(m\), \(l\), among all possible special block codes. Our
result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports
7.1 (2017): 14765] where the minimum was computed for one block, i.e. for \(m=1\).
This is a joint work with G. Androulakis.

**When:** October 27^{th} 2023 from 3:30pm to 4:30 pm

**Where: **LeConte 348

**Speaker: **George Androulakis

**Abstract: **We will review the use of Nussbaum-Szkoła distributions in quantum information and
in particular in computing quantum divergences. The talk will be based on joint works
with T.C. John.

**When:** October 13^{th} 2023 from 3:30pm to 4:30pm

**Where:** LeConte 348

**Speaker:** Stephen A. Fenner (USC, Dept. of Computer Science and Engineering)

**When:** October 6^{th} 2023 from 3:30pm to 4:30pm

**Where:** LeConte 348

**Speaker:** Stephen A. Fenner (USC, Dept. of Computer Science and Engineering)

**When:** September 29^{th} 2023 from 3:30pm to 4:30pm

**Where:** LeConte 348

**Speaker:** Stephen A. Fenner (USC, Dept. of Computer Science and Engineering)

**When: **September 8^{th} 2023 from 3:30pm to 4:30pm

**Where: **LeConte 348

**Speaker:** George Androulakis (USC)

**Abstract:** The hyperfinite II_{1} factor is an infinite von Neumann algebra which is very similar to the nxn matrix
algebra since it has a finite faithful tracial state. We will describe its construction
and indicate some of its uses in physics.

### Previous Seminars

### 2021 – 2022 Academic Year

**Organized by:** Daniel Dix ( dix@math.sc.edu )

This is a traditional in-person seminar. No recordings are planned. Come and participate!

Organizational Meeting

- Friday, Feb 4
- 3pm
- COL 1015

**Daniel Dix**

- Friday, Feb 11
- 2:15pm
- COL 1015

**Abstract**: This will be an overview of how an interesting groupoid can be derived from a molecular
system of three identical nuclei plus some number of electrons. The structure of the
groupoid will be fully determined, and that will significantly constrain the electronic
energy eigenvalue intersection patterns for the molecule.

**Daniel Dix**

- Friday, Feb 18
- 2:15pm
- COL 1015

**Abstract**: We will show how a groupoid arises from the tangent mapping of a section of an associated
bundle to the \(C^2\) invariant subspace bundle (that we derived from a triatomic
molecular system in Part 1) at a triple eigenvalue intersection point that has maximal
\(S_3\) symmetry. By linearization and passage to the range of the tangent mapping
we arrive at a computable groupoid that gives information about the eigenvalue intersections
of the molecular system.

**Daniel Dix**

- Friday, Feb 25
- 2:15pm
- COL 1015

**Abstract**: If \(f\colon \mathbb R^n\to M\) is a \(C^2\) mapping, where \(M\) is an \(m\)-dimensional
manifold, equipped with an atlas of homeomorphisms \(\phi_\mu\colon U_\mu\to V_\mu\),
where \(U_\mu\subset\mathbb R^m\) and \(V_\mu\subset M\) are open sets, with \(C^2\)
overlap mappings, and \(\mathbf l_0\in\mathbb R^n\), then there is a natural groupoid
defined as follows. The objects are pairs \((\mu,A)\), where \(f(\mathbf l_0)\in V_\mu\)
and \(A=D(\phi_\mu^{-1}\circ f)(\mathbf l_0)\). An arrow between objects \((\mu,A)\)
and \((\nu,B)\) is determined by a triple \((\mu,G_{\nu,\mu},\nu)\), where \(G_{\nu,\mu}\)
is a linear isomorphism so that \(B=G_{\nu,\mu}A\), i.e. \(G_{\nu,\mu}=D(\phi_\nu^{-1}\circ\phi_\mu)(\phi_\mu^{-1}(f(\mathbf
l_0)))\). This groupoid is another way of presenting the tangent mapping (differential)
of \(f\) at \(\mathbf l_0\). We apply this construction where \(n=3\) and \(M=\mathfrak
B\) and \(f(\mathbf l) =(\mathbf l,\Pi\breve{\mathcal H}(\mathbf l))\), where \(\Pi\breve{\mathcal
H}\) is the trace-free projection of the molecular electronic Hamiltonian restricted
to a 3-dimensional invariant subspace \(\mathcal F(\mathbf l)\), and where \(\mathbf
l_0\) is an equilateral triangle configuration at which the three lowest eigenvalues
of \(\breve{\mathcal H}(\mathbf l_0)\) coincide. This construction, combined with
certain functorial (groupoid homomorphism) images, leads to a groupoid we can completely
compute.

** **

**Ralph Howard**

- Friday, Mar 18
- 2:15pm
- COL 1015

**Abstract**: For curves in the plane which have linearly independent velocity and acceleration
vectors there a notion of affine arclength and affine curvature which is invariant
under area preserving affine maps of the plane. In terms of the Euclidean arclength
\(s\) and curvature \(\kappa\) the affine arclength is

\(\int_a^b \kappa^{1/3} ds\)

We will outline the basic theory of the differential geometryof affine curves and give some new results which estimate the area bounded by the curve and the segment between the endpoints of the curve in terms of the affine arclength of the curve and its affine curvature.

Most of the proofs do not involve any mathematics not in in Math 241 and 242 (or Math 550 and 520).

**Ralph Howard**

- Friday, Mar 18
- 2:15pm
- COL 1015

**Abstract**: For curves in the plane which have linearly independent velocity and acceleration
vectors there a notion of affine arclength and affine curvature which is invariant
under area preserving affine maps of the plane. In terms of the Euclidean arclength
\(s\) and curvature \(\kappa\) the affine arclength is

\(\int_a^b \kappa^{1/3} ds\)

We will outline the basic theory of the differential geometryof affine curves and give some new results which estimate the area bounded by the curve and the segment between the endpoints of the curve in terms of the affine arclength of the curve and its affine curvature.

Most of the proofs do not involve any mathematics not in in Math 241 and 242 (or Math 550 and 520).

** **

**Stephen Fenner**

- Friday, Apr 8
- 2:15pm
- COL 1015

**Abstract**: The quantum fanout gate has been used to speed up quantum algorithms such as the
quantum Fourier transform used in Shor's quantum algorithm for factoring. Fanout
can be implemented by evolving a system of qubits via a simple Hamiltonian involving
pairwise interqubit couplings of various strengths. We characterize exactly which
coupling strengths are sufficient for fanout: they are sufficient if and only if they
are odd multiples of some constant energy value J. We also investigate when these
couplings can arise assuming that strengths vary inversely proportional to the squares
of the distances between qubits.

This is joint work with Rabins Wosti.

**Rabins Wosti**, Computer Science and Engineering Department

- Friday, Apr 15
- 2:15pm
- COL 1015

**Abstract**: The quantum fanout gate has been used to speed up quantum algorithms such as the quantum
Fourier transform used in Shor's quantum algorithm for factoring. Fanout can be implemented
by evolving a system of qubits via a simple Hamiltonian involving pairwise interqubit
couplings of various strengths. We characterize exactly which coupling strengths
are sufficient for fanout: they are sufficient if and only if they are odd multiples
of some constant energy value J. We also investigate when these couplings can arise
assuming that strengths vary inversely proportional to the squares of the distances
between qubits.

This is joint work with Stephen Fenner.

**Margarite Laborde**

- Friday, Apr 22
- 2:15pm
- COL 1015

**Abstract**:

Previous Quantam Seminar information can be found on Dr. George Androulakis's website.