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The Limits of Computation:
Quantum Computers and Beyond
Scott Aaronson (MIT)
Things we never see…
GOLDBACH
CONJECTURE:
TRUE
NEXT QUESTION
Warp drive
Perpetuum mobile
Übercomputer
The (seeming) impossibility of the first two machines
reflects fundamental principles of physics—Special
Relativity and the Second Law respectively
Does physics also put limits on computation?
Moore’s Law
Extrapolating: Robot uprising?
But even a killer robot would still be
“merely” a Turing machine, operating on
principles laid down in the 1930s…
=
And it’s conjectured that thousands of
interesting problems are inherently
intractable for Turing machines…
(Why is it so hard to prove PNP? We know a
lot about that today, most recently from
algebrization [A.-Wigderson 2007])
Is there any feasible way to solve NP-complete
problems, consistent with the laws of physics?
Relativity Computer
DONE
Zeno’s Computer
Time (seconds)
STEP 1
STEP 2
STEP 3
STEP 4
STEP 5
Time Travel Computer
S. Aaronson and J. Watrous. Closed Timelike
Curves Make Quantum and Classical
Computing Equivalent, Proceedings of the Royal
Society A 465:631-647, 2009. arXiv:0808.2669.
Answer
Polynomial
Size Circuit
C
“Closed
Timelike
Curve
Register”
R CTC
R CR
0 0 0
“CausalityRespecting
Register”
A quantum state of n “qubits” takes 2n complex
numbers to describe:

x x
x0,1
Chemists and physicists knew that for decades, as a
major practical problem!
n
In the 1980s, Feynman, Deutsch,Interesting
and others had the
amazing idea of building a new type of computer that
could overcome the problem, by itself exploiting the
exponentiality inherent in QM
Shor 1994: Such a machine could also factor integers
What we’ve learned from
quantum computers so far:
21 = 3 × 7
(with high probability)
The practical problem: decoherence.
A few people think scalable QC is fundamentally
impossible ... but that would be even more
interesting than if it’s possible!
[A. 2004]: Theory of “Sure/Shor separators”
Limitations of Quantum Computers
[BBBV 1994] explained why quantum computers probably
don’t offer exponential speedups for the NP-complete
problems
[A. 2002] proved the first lower bound (~N1/5) on the time
needed for a quantum computer to find collisions in a long
list of numbers from 1 to N—thereby giving evidence that
secure cryptography should still be possible even in a
world with QCs
4 2 1 3 2 5 4 5 1 3
BosonSampling [A.-Arkhipov 2011]
Recent experimental proposal, which
involves generating n identical photons,
passing them through a network of
beamsplitters, then measuring where they
end up
Almost certainly wouldn’t yield a universal
quantum computer—and indeed, it seems
easier to implement
Nevertheless, our experiment would sample a certain
probability distribution, which we give strong evidence is hard to
sample with a classical computer
Jeremy O’Brien’s group at the University of Bristol has built our
experiment with 4 photons and 16 optical modes on-chip
10 Years of My Other Research in 1 Slide
The Information Content of Quantum States
For many practical purposes, the “exponentiality” of
quantum states doesn’t actually matter—there’s a
shorter classical description that works fine
Describing quantum states on efficient measurements
only [A. 2004], “pretty-good tomography” [A. 2006]
Using quantum techniques to understand classical
computing better [A. 2004] [A. 2005] [A. 2011]
Quantum Generosity … Giving back because we careTM
Quantum Money that anyone can verify, but that’s
physically impossible to counterfeit [A.-Christiano 2012]
NP-complete
NP
BQP
Factoring
P
Boson
Sampling
Thank you for your support!