101 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups
It is, by now, classical that lattices in higher rank semisimple groups have
various rigidity properties. In this work, we add another such rigidity
property to the list: uniform stability with respect to the family of unitary
operators on finite-dimensional Hilbert spaces equipped with submultiplicative
norms. Namely, we show that for (most) high-rank lattices, every
finite-dimensional unitary "almost-representation" of is a small
deformation of a (true) unitary representation. This extends a result of
Kazhdan (1983) for amenable groups and of Burger-Ozawa-Thom (2013) for SL(n,Z)
(for n>2). Towards this goal, we first build an elaborate cohomological theory
capturing the obstruction to such stability, and show that the vanishing of
second cohomology implies uniform stability in this setting. This cohomology
can be roughly thought of as an asymptotic version of bounded cohomology, and
sheds light on a question raised in Monod (2006) about a possible connection
between vanishing of second bounded cohomology and Ulam stability.Comment: 71 pages, 3 figures, Added clarifications in Section 4.2, Corrected
typo
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Algebraic Topology for Data Scientists
This book gives a thorough introduction to topological data analysis (TDA),
the application of algebraic topology to data science. Algebraic topology is
traditionally a very specialized field of math, and most mathematicians have
never been exposed to it, let alone data scientists, computer scientists, and
analysts. I have three goals in writing this book. The first is to bring people
up to speed who are missing a lot of the necessary background. I will describe
the topics in point-set topology, abstract algebra, and homology theory needed
for a good understanding of TDA. The second is to explain TDA and some current
applications and techniques. Finally, I would like to answer some questions
about more advanced topics such as cohomology, homotopy, obstruction theory,
and Steenrod squares, and what they can tell us about data. It is hoped that
readers will acquire the tools to start to think about these topics and where
they might fit in.Comment: 322 pages, 69 figures, 5 table
Universal Hamiltonians for quantum simulation and their applications to holography
Recent work has demonstrated the existence of universal Hamiltonians – simple spin lattice models that can simulate any other quantum many body system. These universal Hamiltonians have applications for developing quantum simulators, as well as for Hamiltonian complexity, quantum computation, and fundamental physics. In this thesis we extend the theory of universal Hamiltonians. We begin by developing a new method for proving that a given family of Hamiltonians is indeed universal. We then use this method to construct two new universal models – both of which consist of translationally invariant interactions acting on a 1D spin chain.
But the benefit of our method doesn’t just lie in the simple universal models it allows us to construct. It also gives deeper insight into the origins of universality – and demonstrates a link between the universality and complexity. We make this insight rigorous, and derive a complexity theoretic classification of universal Hamiltonians which encompasses all known universal models. This classification provides a new, simplified route to checking whether a particular family of Hamiltonians meets the conditions to be a universal simulator.
We also consider the practical use of analogue Hamiltonian simulation. Under- standing the effect of noise on Hamiltonian simulation is a key issue in practical implementations. The first step to tackling this issue is characterising the noise processes affecting near term quantum devices. Motivated by this, we develop and numerically benchmark an algorithm which fits noise models to tomographic data from quantum devices to enable this process. This algorithm has applicability beyond analogue simulators, and could be used to investigate the physical noise processes in any quantum computing device.
Finally, we apply the theory of universal Hamiltonians to high energy physics by using them to construct toy models of holographic duality which capture more of the expected features of the AdS/CFT correspondence
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022
The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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