 
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 mechanics will break down before the factoring of large numbers becomes
possible. If this is true, then there should be a natural set of quantum states that can account for
all quantum computing 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 tensor
products even to approximate, which incidentally yields the first superpolynomial gap between
general and multilinear formula size of functions. More broadly, we introduce a complexity
classification of pure quantum states, and prove many basic facts about this classification. Our
goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses
that might be experimentally testable in the near future.
1 Introduction
QC of the sort that factors long numbers seems firmly rooted in science fiction . . . The
present attitude would be analogous to, say, Maxwell selling the Daemon of his famous
thought experiment as a path to cheaper electricity from heat. Leonid Levin [34]
