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

### Zeno’s Computer problems are inherently intractable for Turing machines…

### Time Travel Computer problems are inherently intractable for Turing machines…

### Limitations of Quantum Computers

### BosonSampling [A.-Arkhipov 2011]

### 10 Years of My Other Research in 1 Slide

### Thank you for your support!

The Limits of Computation:Quantum Computers and Beyond

Scott Aaronson (MIT)

NEXT QUESTION

Things we never see…Warp drive

Übercomputer

Perpetuum mobile

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?

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 PNP? We know a lot about that today, most recently from algebrization [A.-Wigderson 2007])

Relativity Computer problems are inherently intractable for Turing machines…

DONE

STEP 1

STEP 2

Time (seconds)

STEP 3

STEP 4

STEP 5

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.

Interesting problems are inherently intractable for Turing machines…

Quantum Computers

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

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 QM

Shor 1994: Such a machine could also factor integers

What we’ve learned from quantum computers so far: problems are inherently intractable for Turing machines…

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”

[BBBV 1994] problems are inherently intractable for Turing machines… 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

Recent experimental proposal, which problems are inherently intractable for Turing machines…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

The Information Content of Quantum States problems are inherently intractable for Turing machines…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 problems are inherently intractable for Turing machines…

NP

Factoring

BQP

BosonSampling

P

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