 
Summary: Multilinear Formulas and Skepticism of Quantum
Computing
Scott Aaronson
ABSTRACT
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 errorcorrection 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
