Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation

The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds. We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting. The algorithm is motivated by the relationship between topological quantum computers and (2+1)-D topological quantum field theories. Its accuracy is shown to be nontrivial, as the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer. Thus approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation. This establishes a novel relation between the task of distinguishing non-homeomorphic 3-manifolds and the power of a general quantum computer.Comment: 4 pages, 3 figure

Spartan Daily, October 8, 1948

Volume 37, Issue 8https://scholarworks.sjsu.edu/spartandaily/12996/thumbnail.jp

The Crescent Student Newspaper, April 20, 1998

Student Newspaper of George Fox University.https://digitalcommons.georgefox.edu/the_crescent/2196/thumbnail.jp

The Crescent Student Newspaper, April 20, 1998

Student Newspaper of George Fox University.https://digitalcommons.georgefox.edu/the_crescent/2196/thumbnail.jp

Spartan Daily, April 7, 1959

Volume 46, Issue 99https://scholarworks.sjsu.edu/spartandaily/3878/thumbnail.jp

Invariants and TQFT's for cut cellular surfaces from finite groups

We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elements of a finite group, subject to a "flatness" condition for each 2-cell. These invariants are also described in a TQFT setting, which is not the same as the usual 2-dimensional TQFT framework. We study the properties of functions which arise in this context, associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology, including properties which come from invariance under the Hatcher-Thurston moves on pants decompositions.Comment: 28 pages, 27 figures. Revised version, including a topological proof of the property: the number of conjugacy classes of a finite group G equals the commuting fraction of G times the order of G. To appear in Boletim da Sociedade Portuguesa de Matem\'atic

Cleaning up the Mess and Messing up the Clean: a response to ‘Happiness Lessons in Schools'

First paragraph: The fact that certain educators and psychologists (assuming that there is any difference between the two ) believe that happiness lessons are a good idea is simultaneously bizarre whilst being utterly in keeping with current orthodoxies. I make this claim, because during my experience as an English teacher, we were pretty much told that every lesson ought to be a happiness lesson. By this, I mean that every lesson ought to be fast paced (preventing boredom), that it should accommodate different kinds of learner kinaesthetic, auditory and visual (those students with ants in their pants will get to shake off those ants). All areas being taught should be scaffolded so that students would feel comfortable with what they were learning in order that emotional scarring would not result from their confusion and they could learn more effectively. If possible learning should be like a game, in fact turning certain areas into games is widely held to be good practice-it is important that learning should never be a slow, difficult or onerous activity. Students should also be rewarded whenever possible (in the case of disaffected students you might reward them for not doing certain things-swearing, beating each other up etc.). Ultimately if an inspector came to watch your lesson she would hope to see most if not all of these things. Indeed, not only would the inspector not want to see mayhem within the classroom, rows of quiet attentive children would be almost as unacceptable-the students should be champing at the bit paralytic with excitement at the thought of being able to learn kinaesthetically or answer questions in brief speedy question and answer sessions