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Quantum algorithms for invariants of triangulated manifolds

Alagic, Gorjan and Bering IV, Edgar A. (2012) Quantum algorithms for invariants of triangulated manifolds. Quantum Information and Computation, 12 (9-10). pp. 843-863. ISSN 1533-7146. https://resolver.caltech.edu/CaltechAUTHORS:20121109-123028617

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Abstract

One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a manifold, and the structure of the tensors ensures that the overall tensor is independent of the choice of internal triangulation. This leads to quantum algorithms for additively approximating certain invariants of triangulated manifolds. We discuss the details of this construction in two specific cases. In the first case, we consider triangulated surfaces, where the triangle tensor is defined by the multiplication operator of a finite group; the resulting invariant has a simple closed-form expression involving the dimensions of the irreducible representations of the group and the Euler characteristic of the surface. In the second case, we consider triangulated 3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci anyon model; the resulting invariant is the well-known Turaev-Viro invariant of 3-manifolds.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://www.rintonpress.com/journals/qiconline.html#v12n910PublisherUNSPECIFIED
http://arxiv.org/abs/1108.5424v1arXivUNSPECIFIED
Additional Information:© 2012 Rinton Press. Received May 12, 2011. Revised May 31, 2012. Communicated by: I Cirac & R de Wolf. G.A. is indebted to Alex Russell and Cris Moore for first telling us about the idea of approximating topological invariants by contracting tensor networks. We thank Stephen Jordan, Robert König and Dylan Thurston for useful conversations. G.A. acknowledges the support of NSERC, MITACS, and the U.S. ARO. E.B. thanks Ashwin Nayak and the University of Waterloo Combinatorics & Optimization department for accepting him into the summer URA program during which his contributions were conducted. E.B. acknowledges the support of NSERC.
Funders:
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
MITACSUNSPECIFIED
Army Research Office (ARO)UNSPECIFIED
Issue or Number:9-10
Record Number:CaltechAUTHORS:20121109-123028617
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20121109-123028617
Official Citation:Quantum algorithms for invariants of triangulated manifolds (pp0843-0863) Gorjan Alagic and Edgar A. Bering IV
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:35390
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Nov 2012 21:28
Last Modified:03 Oct 2019 04:27

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