2,329 research outputs found
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
Reduction and reconstruction of multisymplectic Lie systems
A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra. In this work, multisymplectic forms are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples from physics, mathematics, and control theory
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Remarks on mod-2 elliptic genus
For physicists: For supersymmetric quantum mechanics, there are cases when a
mod-2 Witten index can be defined, even when a more ordinary
-valued Witten index vanishes. Similarly, for 2d supersymmetric
quantum field theories, there are cases when a mod-2 elliptic genus can be
defined, even when a more ordinary elliptic genus vanishes. We study such mod-2
elliptic genera in the context of supersymmetry, and show
that they are characterized by mod-2 reductions of integral modular forms,
under some assumptions.
For mathematicians: We study the image of the standard homomorphism for or
, by relating them to the mod-2 reductions of integral modular forms.Comment: 31 page
Interfaces and Quantum Algebras, I: Stable Envelopes
The stable envelopes of Okounkov et al. realize some representations of
quantum algebras associated to quivers, using geometry. We relate these
geometric considerations to quantum field theory. The main ingredients are the
supersymmetric interfaces in gauge theories with four supercharges, relation of
supersymmetric vacua to generalized cohomology theories, and Berry connections.
We mainly consider softly broken compactified three dimensional theories. The companion papers will discuss applications of this
construction to symplectic duality, Bethe/gauge correspondence, generalizations
to higher dimensional theories, and other topics.Comment: 152 pages; v2: references added, various explanations improve
Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems
We study non-interacting electrons in disordered one-dimensional materials
which exhibit a spectral gap, in each of the ten Altland-Zirnbauer symmetry
classes. We define an appropriate topology on the space of Hamiltonians so that
the so-called strong topological invariants become complete invariants yielding
the one-dimensional column of the Kitaev periodic table, but now derived
without recourse to K-theory. We thus confirm the conjecture regarding a
one-to-one correspondence between topological phases of gapped non-interacting
1D systems and the respective Abelian groups
in the spectral gap regime. The
main tool we develop is an equivariant theory of homotopies of local unitaries
and orthogonal projections. Moreover, we extend the unitary theory to partial
isometries, thus providing a perspective towards the understanding of
strongly-disordered, mobility-gapped materials.Comment: 45 page
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