Limits on Efficient Computation in the Physical World

1
Limits on Efficient Computation in the Physical
World

• Scott Aaronson (MIT)
• www.scottaaronson.com

2
Things we never see
Warp drive
Übercomputer
Perpetuum mobile
The (seeming) impossibility of the first two
machines reflects fundamental principles of
physicsSpecial Relativity and the Second Law
respectively
So what about the third one? What are the
ultimate physical limits on what can be feasibly
computed? And do those limits have any
implications for physics?
3
NP-hardAll NP problems are efficiently reducible
to these
NP-complete
NPEfficiently verifiable
OUR STANDARD MODEL
PEfficiently solvable
4
Does PNP?
The (literally) 1,000,000 question
If there actually were a machine with running
time Kn (or even only with Kn2), this would
have consequences of the greatest
magnitude.Gödel to von Neumann, 1956
5
An important presupposition underlying P vs. NP
is the
The Extended Church-Turing Thesis (ECT) Any
physically-realistic computing device can be
simulated by a deterministic or probabilistic
Turing machine, with at most polynomial overhead
in time and memory
But how sure are we of this thesis?What would a
challenge to it look like?
6
Old proposal Dip two glass plates with pegs
between them into soapy water. Let the soap
bubbles form a minimum Steiner tree connecting
the pegsthereby solving a known NP-hard problem
instantaneously
7
Relativity Computer
DONE
8
Zenos Computer
Time (seconds)
9
Time Travel Computer
S. Aaronson and J. Watrous. Closed Timelike
Curves Make Quantum and Classical Computing
Equivalent, Proceedings of the Royal Society A
465631-647, 2009. arXiv0808.2669.
10
Nonlinear variants of the Schrödinger Equation
Abrams Lloyd 1998 If quantum mechanics were
nonlinear, one could exploit that to solve
NP-complete problems in polynomial time
1 solution to NP-complete problem
No solutions
11
Ah, but what about quantum computing?(you knew
it was coming)
Quantum mechanics Probability theory with minus
signs (Nature seems to prefer it that way)
In the 1980s, Feynman, Deutsch, and others
noticed that quantum systems with n particles
seemed to take 2n time to simulateand had the
amazing idea of building a quantum computer to
overcome that problem
Quantum computing The power of 2n complex
numbers working for YOU
12
Quantum Mechanics in One Slide
13
Journalists BewareA quantum computer is NOT
like a massively-parallel classical computer!
Exponentially-many basis states, but you only get
to observe one of them
Any hope for a speedup rides on the magic of
quantum interference
14
BQP (Bounded-Error Quantum Polynomial-Time) The
class of problems solvable efficiently by a
quantum computer, defined by Bernstein and
Vazirani in 1993
Shor 1994 Factoring integers is in BQP
15
Can QCs Actually Be Built?
Where we are now A quantum computer has factored
21 into 3?7, with high probability (Martín-López
et al. 2012)
Why is scaling up so hard? Because of
decoherence unwanted interaction between a QC
and its external environment, prematurely
measuring the quantum state
A few skeptics, in CS and physics, even argue
that building a QC will be fundamentally
impossible
I dont expect them to be right, but I hope they
are! If so, it would be a revolution in physics
And for me, putting quantum mechanics to the test
is the biggest reason to build QCsthe
applications are icing!
16
Key point factoring is not believed to be
NP-complete! And today, we dont believe quantum
computers can solve NP-complete problems in
polynomial time in general (though not
surprisingly, we cant prove it)
Bennett et al. 1997 Quantum magic wont be
enough
If you throw away the problem structure, and just
consider an abstract landscape of 2n possible
solutions, then even a quantum computer needs
2n/2 steps to find the correct one (That bound
is actually achievable, using Grovers algorithm!)
If theres a fast quantum algorithm for
NP-complete problems, it will have to exploit
their structure somehow
17
Quantum Adiabatic Algorithm(Farhi et al. 2000)
Hi
Hf
Hamiltonian with easily-prepared ground state
Ground state encodes solution to NP-complete
problem
Problem Eigenvalue gap can be exponentially
small
18
Includes P?NP as a special case, but is
stronger No longer a purely mathematical
conjecture, but also a claim about the laws of
physics Could be invoked to explain why
adiabatic systems have small spectral gaps, why
protein folding gets stuck in metastable states,
why the Schrödinger equation is linear, why time
only flows in one direction
19
Conclusion
My suggested research agenda Prove P?NP Prove
that not even quantum computers can solve
NP-complete problems Build a scalable quantum
computer (or even more interesting, show that
its impossible) Clarify whether all of known
physics can be simulated by a quantum
computer Use No-SuperSearch or related
impossibility principles to make progress in
quantum gravity