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Combinatorics & Optimization
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext 33038
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Zoom (for information email emma.watson@uwaterloo.ca)
Title: Permanent Hardness from Linear Optics
Speaker: | Daniel Grier |
Affiliation: | University of Waterloo |
Location: | Online (Zoom) |
Abstract:
One of the great accomplishments in complexity theory was Valiant's 1979 proof that the permanent of a matrix is #P-hard to compute. Subsequent work simplified Valiant's ideas and even began to recast them as problems in quantum computing. In 2011, this culminated in a striking proof by Aaronson, based solely on quantum linear optics, of the #P-hardness of the permanent. Although this simplified (at least for physicists) aspects of Valiant's proof by off-loading its difficulty onto central and well-known theorems in linear optics, the question remained: what else was gained by converting Valiant's combinatorial proof into a linear optical one?
In this talk I'll give one possible answer to this question--namely, that these quantum techniques are useful for proving hardness of permanents for matrices that obey classical Lie group structure. In particular, I will prove that computing the permanent of real orthogonal matrices is #P-hard. The hardness result translates to permanents of orthogonal matrices over finite fields of characteristic not equal to 2 or 3.
No prior knowledge of linear optics is necessary.
Joint work with Luke Schaeffer.
Combinatorics & Optimization
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext 33038
PDF files require Adobe Acrobat Reader.
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.