Information about summer 2022 seminars can be found
here.
Fall 2022 schedule
September 13rd, 2022
Deni Salja, Dalhousie University
Title: Pseudocolimits of filtered diagrams of internal categories
Abstract:
Pseudocolimits are formal gluing constructions used to combine objects
in a category that are indexed by a pseudofunctor. When the objects
are categories and the domain of the pseudofunctor is a small filtered
category it is known from Exercise 6.6 of Exposé 6 of SGA4 that the
pseudocolimit can be computed by taking the Grothendieck construction
of the pseudofunctor and then inverting the class of cartesian arrows
with respect to the canonical fibration. In my masters thesis I described a
set of conditions on an ambient category E for constructing an internal
Grothendieck construction and another set of conditions on E along
with conditions on an internal category, ℂ, in Cat(E) and a map
w : W → ℂ₁ that allow us to translate the axioms for a category of
(right) fractions, and construct an internal category of (right)
fractions. These can be combined in a suitable context to compute the
pseudocolimit of a (filtered) diagram of internal categories.
September 20th, 2022
Robert Raphael, Concordia University
Title: Applications to two reflectors to the ring C1
Abstract:
This is a progress report on joint work with Walter Burgess of the University
of Ottawa.
We procede from work with Barr and Kennison which appeared in TAC 30(2015)
229-304. I will resume this work which concerns embedding a subcategory of
rings into a complete one.
The ring C1 of continuously differentiable functions from the reals to the
reals is not complete for the two instances that interest us. Thus we seek the
completions of C1. This leads us into questions in analysis for which we have
progress but lack definitive answers to date.
October 4th, 2022
Andre Kornell, Dalhousie University
Title: Structured objects in categories of relations
Abstract:
We will look at several examples of biproduct dagger
compact closed categories, which are also known as
strongly compact closed categories with biproducts. Such
a category is a dagger category that is equipped with two
symmetric monoidal structures $\oplus$ and $\otimes$,
where $\oplus$ is a biproduct and $\otimes$ has dual
objects. A biproduct dagger compact closed category is
automatically enriched over commutative monoids. In some
natural examples, these monoids are cancellative, but in
other natural examples they are idempotent. This talk
will focus on the latter class.
Specifically, we will look at several examples of
biproduct dagger compact closed categories that are
enriched over the category of bounded lattices and
join-homomorphisms. We will define poset objects and
group objects in this setting, recovering a number of
familiar examples including discrete quantum
groups. Thus, the last part of this talk will focus on
the category of quantum sets and their binary relations.
October 11th, 2022 Robert Pare, Dalhousie University
Title: The Double Category of Abelian Groups
Abstract:
What, indeed, is THE double category of Abelian groups?
There is ample empirical evidence that THE double category
of sets has functions and spans as arrows, THE double category
of categories functors and profunctors, and perhaps to a lesser
extent, THE double category of rings homomorphisms and
bimodules, but there doesn’t seem to be an obvious choice
for THE double category of Abelian groups. Yet, Ab was the
base category for almost all of category theory for the first 25 years
of the subject and has seen a resurgence in recent years.
In this talk I will interview some of the main candidates, evaluate
their qualifications and see if they work well with others, with the
view of nominating one for the position of “Double Category of
Abelian Groups".
October 18th, 2022 Michael Lambert, University of Massachusetts-Boston
Title: Cartesian Equipments for Data Manipulation
Abstract:
The aim of this talk is to illustrate how double
categories and in particular cartesian equipments provide
a framework for databases, knowledge representation, and
data manipulation. This work combines the functional
approach developed by Robert Kent and David Spivak with
the relational and bicategorical approach of Evan
Patterson.
Title: A biset-enriched categorical model for Proto-Quipper with dynamic lifting
Abstract:
Quipper and Proto-Quipper are a family of quantum
programming languages that, by their nature as circuit
description languages, involve two runtimes: one at which
the program generates a circuit and one at which the
circuit is executed, normally with probabilistic results
due to measurements. Accordingly, the language
distinguishes two kinds of data: parameters, which are
known at circuit generation time, and states, which are
known at circuit execution time. Sometimes, it is
desirable for the results of measurements to control the
generation of the next part of the circuit. Therefore,
the language needs to turn states, such as measurement
outcomes, into parameters, an operation we
call dynamic lifting.
In this talk, I will first sketch a $\textit{biset}$-enriched category
$\overline{\mathbf{C}}$ that can model
many aspects of Proto-Quipper. If time permits, I will
also talk about how to refine $\overline{\mathbf{C}}$ so
that we have an interpretation for dynamic lifting.
This is a joint work with Kohei Kishida, Neil J. Ross
and Peter Selinger.
November 1st, 2022 Xiaoning Bian, Dalhousie University
Title: 3-qubit Clifford+CS Operators is an amalgamated product of two finite groups
Abstract:
This is a work in progress. We found a finite
presentation of the group G of 3-qubit Clifford+CS operators in terms
of generators and relations. A corollary is that G is an amalgamated
product. The proof is easy — applying a known method to a known
result. The calculation is non-trivial, which involves simplifying
hundreds of long equations. Our main contribution is the
simplification method. Its idea is factoring a group into a product of
cosets, in other words, finding a "nice" tower of group extensions
starting from the trivial group to the group in focus. The ongoing
part is to check our result in proof assistant Agda.
November 15th, 2022 Dongho Lee, Dalhousie University
Title: Formal methods for quantum programming languages
Abstract:
In this talk, I would like to discuss the semantics of a quantum
programming language called Proto-Quipper-L. The language formalizes a
quantum circuit description language with dynamic lifting, or
measurement, which allows us to use the quantum measurement inside the
computation. This work has some limitations while providing new
questions. For the second part of the talk, I wish to briefly summarize
some thoughts on an on-going project on semantics of dependent type
systems for quantum computation.
November 29th, 2022 Julien Ross, Dalhousie University
Title: Catalytic embeddings: theory and applications
Abstract:
Let $C$ be a quantum circuit and let $G$ be a set of quantum gates. A
catalytic embedding of $C$ over $G$ is a pair $(D,v)$ consisting of a state
$v$ and a circuit $D$ over $G$ such that for every state $u$ we have
$D(u \otimes v) = (Cu) \otimes v$.
Because the state $v$ is left unchanged by the application of $D$, it is
known as a catalyst. Catalytic embeddings are useful when the circuit
$C$ cannot be exactly represented over the gate set $G$. In such cases,
one can leverage the catalyst to implement (any number of occurrences
of) $C$ using circuits over $G$.
In this talk, I will present the theory of catalytic embeddings and
discuss applications to the exact and approximate synthesis of quantum
circuits.
December 6th, 2022 Sarah Li, Institute for Quantum Computing,
University of Waterloo
Title: Visualize fault-tolerant CSS codes using ZX calculus
Abstract:
How to correctly exchange messages in a noisy room? In a
quantum system, the information storage unit (aka qubit)
is incredibly sensitive to interference. As a result,
quantum information gets degraded. To this end, quantum
error correcting codes (QECC) and stabilizer theory were
developed to make large-scale quantum computation
practical. CSS codes are a special type of stabilizer
codes constructed from classical codes, yielding
substantial benefits for QECC developments.
As an intuitive graphical language for quantum computation,
ZX-calculus can be formulated as a category of natural numbers with
addition as a monoidal structure. It consists of ZX diagrams (string
diagrams to represent any linear map, i.e., $(2^n)*(2^m)$ complex
matrix) and a set of rewrite rules. In recent years, researchers
started to study how stabilizer theory and ZX-calculus are related. By
examining QECC through the lens of ZX, we wish to answer a more
generic question about QECC construction and its fault-tolerant
implementations. In 2022, Kissinger provided a recipe to visualize the
encoder for any CSS codes using ZX.
In this talk, I will present our ongoing research efforts to visualize
fault-tolerant CSS codes. We generalized Kissinger’s recipe to find a
physical implementation for any logical operations in CSS codes, and
provided a ZX solution to circumvent the no-go on transversal gates.