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Multilinear Formulas and Skepticism of Quantum Scott Aaronson

Summary: Multilinear Formulas and Skepticism of Quantum
Scott Aaronson
Several researchers, including Leonid Levin, Gerard 't Hooft,
and Stephen Wolfram, have argued that quantum mechan-
ics will break down before the factoring of large numbers be-
comes possible. If this is true, then there should be a natural
set of quantum states that can account for all quantum com-
puting experiments performed to date, but not for Shor's
factoring algorithm. We investigate as a candidate the set of
states expressible by a polynomial number of additions and
tensor products. Using a recent lower bound on multilinear
formula size due to Raz, we then show that states arising in
quantum error-correction require n(log n)
additions and ten-
sor products even to approximate, which incidentally yields
the first superpolynomial gap between general and multilin-
ear formula size of functions. More broadly, we introduce a
complexity classification of pure quantum states, and prove


Source: Aaronson, Scott - Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT)


Collections: Physics; Computer Technologies and Information Sciences