IVADO's Quantum Seminars

About Aram

Many-Body Entanglement in Quantum Computing

Abstract: The idea of quantum computers is that "More is different" when it comes to qubits. One qubit is not so interesting but many of them together can create exotic and computationally powerful forms of many-body entanglement.

Given this, we might expect that many-body physics and quantum information would often be related. I will describe two recent examples.

1. The Ising model is a simple model of a magnet. But it turns out to also describe the competition between quantum interactions creating entanglement and measurements destroying entanglement. This can tell us how about the power of near-term quantum computers.

2. How can a closed system reach thermal equilibrium? There have been many answers to this question dating back to the 19th century. I will explain how entanglement is a plausible source of thermalization.

About Alexander

Quantum Algorithms from Algebraic Topology

Abstract: One of the central challenges in quantum computing is to identify new computational problems for which quantum algorithms offer exponential speedups over any classical method. In this talk, I’ll argue that algebraic topology provides a surprisingly rich source of such problems. I will focus on two recent developments in this directions.

The first concerns Khovanov homology [1], a categorification of the Jones polynomial and a powerful knot invariant that also appears as a physical observable in 4D supersymmetric Yang-Mills theory. I’ll present a quantum algorithm that combines techniques from quantum algorithms for the Jones polynomial and recent advances in quantum homology computation.

The second is the estimation of persistent Betti numbers [2,3] — a core subroutine of Topological Data Analysis (TDA) that captures the shape of data across scales. I’ll discuss quantum algorithms for estimating these invariants, as well as recent complexity-theoretic results that clarify the limitations and potential of quantum approaches in TDA.

No prior background in algebraic topology is required.

Based on:

• [1] A quantum algorithm for Khovanov homology, Schmidhuber et al., arXiv:2501.12378

• [2] Complexity-theoretic limitations on quantum algorithms for topological data analysis, Schmidhuber & Lloyd, arXiv:2209.14286

• [3] Quantum computing and persistence in topological data analysis, Gyurik, Schmidhuber, et al., arXiv:2410.21258

About Risi

Computational Aspects of Equivariant Neural Networks

Abstract: Equivariant networks have become the de-facto standard architecture for modeling physical systems with neural networks from learning force fields for molecular dynamics to learning the wave function of quantum systems. In addition to operations encountered in neural nets, equivariant networks involve a couple of additional operators, specifically, the Clebsch-Gordan product of the underlying symmetry group. In practical implementations these operators often become the computational bottleneck and ultimately constrain the expressive power of the entire network. In this talk, I will describe a combination of mathematical and computational approaches for accelerating equivariant architectures.

About Martin

Quantum Algorithms for Representation-Theoretic Multiplicities

Abstract: Kostka, Littlewood-Richardson, Plethysm and Kronecker coefficients are the multiplicities of irreducible representations in the decomposition of representations of the symmetric group that play an important role in representation theory, geometric complexity and algebraic combinatorics. We give quantum algorithms for computing these coefficients whenever the ratio of dimensions of the representations is polynomial and study the computational complexity of this problem. We show that there is an efficient classical algorithm for computing the Kostka numbers under this restriction and conjecture the existence of an analogous algorithm for the Littlewood-Richardson coefficients. We argue why such a classical algorithm does not straightforwardly work for the Plethysm and Kronecker coefficients and conjecture that our quantum algorithms lead to superpolynomial speedups for these problems. The conjecture about Kronecker coefficients was disproved by Greta Panova in [arXiv:2502.20253] with a classical solution which, if optimal, points to a O(n^{4+2k}) vs Ω(n^{4k^2+1}) polynomial gap in quantum vs classical computational complexity for an integer parameter k.

Reference: https://arxiv.org/abs/2407.17649

About Mark

Quantum Thermodynamics and Semi-Definite Optimization

Abstract: In quantum thermodynamics, a system is described by a Hamiltonian and a list of non-commuting charges representing conserved quantities like particle number or electric charge, and an important goal is to determine the system's minimum energy in the presence of these conserved charges. In optimization theory, a semi-definite program involves a linear objective function optimized over the cone of positive semi-definite operators intersected with an affine space. These problems arise from differing motivations in the physics and optimization communities and are phrased using very different terminology, yet they are essentially identical mathematically. By adopting Jaynes' mindset motivated by quantum thermodynamics, we review how minimizing free energy in the aforementioned thermodynamics problem, instead of energy, leads to an elegant solution in terms of a dual chemical potential maximization problem that is concave in the chemical potential parameters. As such, one can employ standard (stochastic) gradient ascent methods to find the optimal values of these parameters, and these methods are guaranteed to converge quickly. At low temperature, the minimum free energy provides an excellent approximation for the minimum energy. After reviewing this background, we discuss how Jaynes' approach can be used in both classical and quantum algorithms for minimizing energy, and equivalently, how it can be used to solve semi-definite programs, with guarantees on the runtimes of the algorithms. The approach discussed here is well grounded in quantum thermodynamics and, as such, provides physical motivation underpinning why algorithms published fifty years after Jaynes' seminal work, including the matrix multiplicative weights update method, the matrix exponentiated gradient update method, and their quantum algorithmic generalizations, perform well at semi-definite optimization tasks. Joint work with Nana Liu, Dhrumil Patel, and Michele Minervini.