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Department of Mathematics

Quantum Information & Analysis Seminars

Informal seminars that encourage learning, collaboration, and social activities between the participants, are held weakly. Regular participants include graduate students and faculty from the Departments of Computer Science and Engineering, Mathematics, and Physics.

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 26th 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 8th 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 1st 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 17th 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 10th 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 3rd 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 27th 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 13th 2023 from 3:30pm to 4:30pm

Where: LeConte 348

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

When: October 6th 2023 from 3:30pm to 4:30pm

Where: LeConte 348

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

When: September 29th 2023 from 3:30pm to 4:30pm

Where: LeConte 348

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

Seminar Notes [PDF]

When: September 8th 2023 from 3:30pm to 4:30pm

Where: LeConte 348

Speaker: George Androulakis (USC)

Abstract: The hyperfinite II1 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:

Symmetry laws showcase the elegant relationship between mathematics and physical systems. Noether’s theorem, which relates symmetries in a Hamiltonian with conserved physical quantities, is one of the most impactful theorems throughout physics. As such, describing this property in a Hamiltonian is of the utmost importance in many applications–from determining state transition laws to expressing resource theories. In this talk, I give algorithms to determine if a Hamiltonian is symmetric with respect to a discrete, finite group \(G\) and its associated unitary representation \(\{U(g)\}_{g\in G}\). Furthermore, I directly relate the acceptance probability of these algorithms with the typical commutation relationship for symmetry in quantum mechanics. I show that one of the algorithms can efficiently compute the normalized commutator of the group representation and Hamiltonian. 
 
Joint work with Mark M. Wilde and available as arXiv:2203.10017

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


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