 
Summary: NPcomplete Problems and Physical Reality
Scott Aaronson #
Abstract
Can NPcomplete problems be solved e#ciently in the physical universe? I survey proposals
including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia
batic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation,
analog computing, MalamentHogarth spacetimes, quantum gravity, closed timelike curves, and
``anthropic computing.'' The section on soap bubbles even includes some ``experimental'' re
sults. While I do not believe that any of the proposals will let us solve NPcomplete problems
e#ciently, I argue that by studying them, we can learn something not only about computation
but also about physics.
1 Introduction
``Let a computer smearwith the right kind of quantum randomnessand you
create, in e#ect, a `parallel' machine with an astronomical number of processors . . . All
you have to do is be sure that when you collapse the system, you choose the version
that happened to find the needle in the mathematical haystack.''
From Quarantine [31], a 1992 sciencefiction novel by Greg Egan
If I had to debate the science writer John Horgan's claim that basic science is coming to an
end [48], my argument would lean heavily on one fact: it has been only a decade since we learned
that quantum computers could factor integers in polynomial time. In my (unbiased) opinion, the
