Schedule & Abstracts

Abstract: This tutorial explains the basics of quantum computing, with a focus on drawing parallels to familiar concepts in topology. It is intended for participants without prior background in quantum computing. 

I will explain the main building blocks of quantum computation, including qubits, quantum gates, quantum circuits, measurements, and their realization in physical hardware. I will then discuss several important quantum algorithms, including an efficient quantum algorithm for approximating the Jones polynomial arising from the relationship between quantum circuits, braid group representations, and braid diagrams.

The tutorial will also introduce basic notions from computational complexity theory, including how one measures the efficiency of a classical or quantum circuit. I will highlight open problems in knot theory whose computational complexity is not known. Finally, I will explain the basics of quantum error correction using the two-dimensional toric code as a central example.

Abstract: As quantum topology and quantum information intersect, our community remains split between experts in quantum mechanics and those fluent in the pure mathematics of topological quantum field theories (TQFTs). This introductory lecture is designed to bridge that gap and build a shared language.

We will explore the connections between state sum TQFTs and topological phases of matter, specifically focusing on commuting projector Hamiltonians with topological ground states. Beginning with an accessible 1+1D warm-up, we will show how the algebraic structure of a TQFT dictates the properties of its associated quantum system. We will then step up to 2+1D to explore how TQFT topology gives rise to richer topological phases, highlighting Levin-Wen models and their connection to Turaev-Viro codes.

Ultimately, this talk will equip the quantum systems audience with the mathematical scaffolding of state sums, while showing the quantum topology audience how their abstract tools map directly onto compelling physical systems. No prior expertise in Hamiltonians or mathematical TQFTs is assumed.

Abstract: An argument can be made that the construction of a useful quantum computer will go through topological fault-tolerance one way or another.  I will discuss a unification of topological quantum computing with non-abelian anyons and traditional fault-tolerance with stabilizer codes via topological quantum field theory and topological phase of matter.

Abstract: We will introduce the link homology theory that categorifies the Jones polynomial and explain its connections to representation theory and 4-dimensional topology. 

Abstract: This talk presents the basic quantum algorithm for calculating homology, and discusses applications of quantum computation to other problems in algebraic topology and geometry.

Abstract: TQFTs (Topological Quantum Field Theories) provide sophisticated tools for the study of low-dimensional topology, with rich connections to representation theory, geometry, and physics. In recent years, non-semisimple constructions have deeply generalized the original semisimple approach to the theory, producing powerful topological invariants and mapping class groups representations with remarkable new properties. In this talk, I will try to provide a gentle introduction to non-semisimple TQFTs, with a focus on those built out of non-semisimple modular categories.

Abstract: We provide a construction of invariants for a pair of a 3-manifold and an embedded framed link based on involutory finite-dimensional Hopf algebras. The invariants unify the Kuperberg invariant and the Hennings invariant, and we give a new proof of the relation between them. We then report some ongoing work in generalizing the construction to the non-semisimple setup and the implications in relating the landscape of different quantum invariants.

Abstract: We introduce a new framework for achieving universal topological quantum computation using Ising anyons via braiding alone. Our approach builds on recent developments in non-semisimple topological quantum field theories in (2+1)-dimensions, which extend the traditional semisimple paradigm. While standard Ising anyons are limited to Clifford operations and thus not computationally universal, we show that their non-semisimple counterparts give rise to new anyon types we call \emph{neglectons}; adding a single neglecton to the traditional Ising theory enables universal quantum computation through braiding alone.

The resulting braid group representations are unitary with respect to an indefinite Hermitian form, with the computational subspace embedded in a positive-definite sector. We identify a range of parameters for which the model supports universal gate sets and, notably, special values of the parameter α where the computational subspace decouples exactly from the negative-norm components of the Hilbert space. This leads to low-leakage, fully unitary evolution on the physical subspace and improved prospects for efficient gate compilation.

Abstract: After almost forty years of effort there are finally several compelling experiments demonstrating anyon braiding in condensed matter systems.   I will review the history of this field as well as the recent results.    I will also discuss ongoing efforts that might be promising but are not yet compelling.

Abstract: The Fermi-Hubbard model is the starting point for the simulation of many interesting materials, including high-temperature superconductors. Their modeling is a key motivation for the development of quantum simulators. However, the detection of superconducting pairing correlations has been challenging, both because of their off-diagonal character and because it is difficult to prepare superconducting states. Quantum computers can in principle measure such correlations but fundamentally operate on qubits rather than fermions.

In this talk I will discuss how borrowing ideas from topological order allowed to for a recent experiment on Quantinuum's 98-qubit "Helios" trapped-ion quantum computer, in which  superconducting pairing correlations were measured in three regimes of the Fermi-Hubbard model, both in and away from equilibrium.

Abstract: I will describe a map that turns problems in symmetric function theory into problems in quantum many-body physics. This correspondence recasts questions from algebraic geometry, representation theory, and combinatorics in the language of interacting quantum systems. This leads to new classical and quantum algorithms for computing these quantities. I will close with open problems and directions.

Abstract: The Jones polynomial is a link invariant, important in the field of knot theory. Remarkably, it constitutes a quantum-native computational problem; it can be evaluated efficiently on quantum computers, while classically the resource cost scales exponentially in the worst case. We optimise the compilation of the quantum algorithm to Quantinuum's trapped-ion high-fidelity quantum processors and introduce problem-tailored error-mitigation protocols. In parallel, we improve on the state-of-the-art classical algorithms using tensor networks. Further, we introduce a benchmark which leverages the structure inherent in the problem which allows us to extrapolate locate the instance sizes for which quantum computational advantage is expected. Finally, we unify known results about the cases where the problem is classically tractable in terms of the ZX-calculus.

Abstract: It is broadly accepted that unitary modular tensor categories (UMTCs) classify 2D topological order, up to stacking with invertible states. In joint work with A. Kiteav, D. Ranard, and I. Kim, we have developed a new argument for this result, with a focus on completeness - why two states with the same UMTC are circuit-equivalent up to stacking with an invertible state. Our proof is formulated within a modified version of the entanglement bootstrap program. In this talk I will present the argument at the physical level of rigor, giving only the new physical ideas involved.

Abstract: I discuss how to implement logical gates in topological codes using code deformation and protected gates. In D=2 dimensions, examples of the first approach arise from the mapping class group representation of the associated TQFT. For local stabilizer codes in D dimensions, gates associated with the second approach are constrained to lie in the D-th level of the Clifford hierarchy. Similar restrictions apply to locality-preserving logical unitaries in TQFTs. I illustrate these ideas using Kitaev’s toric code and the Turaev-Viro code.

This talk is based on the following papers:

R. Koenig, G. Kuperberg, B. W. Reichardt, Quantum computation with Turaev-Viro codes, Annals of Physics 325, 2707-2749 (2010)

S. Bravyi and R. Koenig, Classification of topologically protected gates for local stabilizer codes, Phys. Rev. Lett. 110, 170503 (2013)

M. Beverland, O. Buerschaper, R. Koenig, F. Pastawski, J. Preskill, S. Sijher, Protected gates for topological quantum field theories, J. Math. Phys. 57, 022201 (2016)

Abstract: TBD

Abstract: Perhaps every algebra meeting should have one analysis talk (and vice versa), lest we forget that the other exists. In my role as the outsider, I will tell you today about the other - perturbative - evaluation of path integrals, where instead of hoping that nature will help us compute faster, we approximate nature by things we already can compute quickly.

Specifically I will tell you how in the Chern-Simons-Witten theory you can perturb the base Lie algebra from where it's easy towards where it's strong, leading to the strongest genuinely computable knot invariant we presently have.

I wish I could give my talk in the language of the Kabbalah, but I ain't smart enough for that. So I'll highlight the Kabbalistic points that we're still missing, and then stick to the Talmud.

https://www.math.toronto.edu/~drorbn/Talks/MonteVerita-2604/

Abstract: Gaussian Boson Sampling (GBS) is a special purpose quantum computing platform, which can only sample in a certain probability distribution, similarly to Boson Sampling, which was first proposed by Aronson and Arkhipov and proven to be impossible on a classical computer (unless the polynomial hierarchy collapses).

Hamilton and coauthors subsequently showed the same for GBS. We will offer a new practical application of GBS, namely, to estimate Gaussian Expectation Value problems, for which we have established provable exponential speedup over Monte Carlo Simulations under certain assumptions. Computing Gaussian Expectation Values has a number of industrial applications including financial, and various optimization problems, including energy production, drug design and logistics. We will address its advantage, even if our assumptions for exponential speedup do not apply.

Abstract: The Andrews-Curtis (AC) conjecture is a classical open problem in combinatorial group theory with rich ties to low-dimensional topology. Despite its simple definition given in terms of moves that can be applied to a presentation of a trivial group, it is super-exponentially complex. These two properties - game-like definition and complexity limiting classical search, make it a perfect ground for Reinforcement Learning (RL). After stating the conjecture and introducing the basics of RL, I will explain how incorporating the underlying mathematical structure and symmetries of the problem into the model allows to find shorter solution paths for AC-trivial presentations than state-of-art classical methods, and generate vast training curriculum. Joint with Lucas Fagan, Michele Tarquini, Ali Shehper, Angus Gruen, Coco Huang, Giorgi Butbaia, Davide Passaro and Sergei Gukov.