Scott Aaronson (MIT)

The Limits of Computation:Quantum Computers and Beyond

Things we never see…

Warp drive Perpetuum mobile

GOLDBACH CONJECTURE: TRUE

NEXT QUESTION

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…

=

Is there any feasible way to solve NP-complete problems, consistent with the laws of physics?

And it’s conjectured that thousands of interesting problems are inherently

intractable for Turing machines…

(Why is it so hard to prove PNP? We know a lot about that today, most recently from algebrization [A.-Wigderson 2007])

Relativity Computer

DONE

Zeno’s Computer

STEP 1

STEP 2

STEP 3STEP 4

STEP 5

Tim

e (s

econ

ds)

Time Travel Computer

R CTC R CR

C

0 0 0

Answer

“Causality-Respecting Register”

“Closed Timelike

Curve Register”

Polynomial Size Circuit

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.

A quantum state of n “qubits” takes 2n complex numbers to describe:

0,1n

x

x

x

Chemists and physicists knew that for decades, as a major practical problem!

In the 1980s, Feynman, Deutsch, 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 QMShor 1994: Such a machine could also factor integers

Interesting

The practical problem: decoherence.

What we’ve learned from quantum computers so far:

21 = 3 × 7(with high probability)

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

Using quantum techniques to understand classical computing better [A. 2004] [A. 2005] [A. 2011]

Quantum Money that anyone can verify, but that’s physically impossible to counterfeit [A.-Christiano 2012]

Quantum Generosity … Giving back because we careTM

The Information Content of Quantum StatesFor 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]

Thank you for your support!

NP

NP-complete

P

Factoring

BQPBoson

Sampling