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Scott Aaronson (MIT) The Limits of Computation: Quantum Computers and Beyond

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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 physicsSpecial Relativity and the Second Law respectively Does physics also put limits on computation?

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Moores Law

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Extrapolating: Robot uprising?

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But even a killer robot would still be merely a Turing machine, operating on principles laid down in the 1930s… =

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

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Relativity Computer DONE

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Zenos Computer STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 Time (seconds)

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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.

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A quantum state of n qubits takes 2 n 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 Interesting

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The practical problem: decoherence. What weve 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 its possible! [A. 2004]: Theory of Sure/Shor separators

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Limitations of Quantum Computers [BBBV 1994] explained why quantum computers probably dont offer exponential speedups for the NP-complete problems [A. 2002] proved the first lower bound (~N 1/5 ) on the time needed for a quantum computer to find collisions in a long list of numbers from 1 to Nthereby 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

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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 wouldnt yield a universal quantum computerand 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 OBriens group at the University of Bristol has built our experiment with 4 photons and 16 optical modes on-chip

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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 thats physically impossible to counterfeit [A.-Christiano 2012] Quantum Generosity … Giving back because we care TM The Information Content of Quantum States For many practical purposes, the exponentiality of quantum states doesnt actually mattertheres a shorter classical description that works fine Describing quantum states on efficient measurements only [A. 2004], pretty-good tomography [A. 2006]

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Thank you for your support! NP NP-complete P Factoring BQP Boson Sampling

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The Learnability of Quantum States Scott Aaronson University of Waterloo.

The Learnability of Quantum States Scott Aaronson University of Waterloo.

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