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The Kolmogorov option
Andrey Nikolaevich Kolmogorov was one of the giants of 20th-century mathematics. I’ve always found it amazing that the same man was responsible both for establishing the foundations of …
Scott Aaronson
I'm Schlumberger Centennial Chair of Computer Science at The University of Texas at Austin , and director of its Quantum Information Center . My research interests center around the capabilities and limits of quantum computers, and computational complexity theory more generally.
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Knuth on Huang's Sensitivity Proof: “I've got the proof down to one page” [pdf]
The Sensitivity Conjecture, which I blogged about here, says that, for every Boolean function f:{0,1}n→{0,1}, the sensitivity of f—that is, the maximum, over all 2n input strings x∈{0,1}n, of…
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Eigenmorality
This post is about an idea I had around 1997, when I was 16 years old and a freshman computer-science major at Cornell. Back then, I was extremely impressed by a research project called CLEVER, wh…
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P vs. NP
For those who just want the survey itself, not the backstory, it’s here. (Note: Partly because of the feedback I’ve gotten on this blog, it’s now expanded to 121 pages!) Upda…
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Google, D-Wave, and the case of the factor-10^8 speedup
Update (Dec. 16): If you’re still following this, please check out an important comment by Alex Selby, the discoverer of Selby’s algorithm, which I discussed in the post. Selby querie…
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How Shor's Algorithm works (2007)
I’ve been talking a lot recently about how quantum algorithms don’t work. But last week JR Minkel, an editor at Scientific American, asked me to write a brief essay about how quantum al…
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NSA in P/poly: The Power of Precomputation
Even after the Snowden revelations, there remained at least one big mystery about what the NSA was doing and how. The NSA’s classified 2013 budget request mentioned, as a priority item, R…
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D-Wave: Truth finally starts to emerge
Wrap-Up (June 5): This will be my final update on this post (really!!), since the discussion seems to have reached a point where not much progress is being made, and since I’d like to oblige …
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Your yearly dose of is-the-universe-a-simulation
Yesterday Ryan Mandelbaum, at Gizmodo, posted a decidedly tongue-in-cheek piece about whether or not the universe is a computer simulation. (The piece was filed under the category “LOL.…
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Who Can Name the Bigger Number?
[This essay in Spanish] [This essay in French] [This essay in Chinese] In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces "Eighty-three!" The second, mightily impressed, replies "You win." A biggest number contest is clearly pointless when the…
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Quantum mechanics as a generalization of probability (2007)
There are two ways to teach quantum mechanics. The first way -- which for most physicists today is still the only way -- follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then you…
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The 8000th Busy Beaver number eludes ZF set theory (2016)
I’ve supervised a lot of great student projects in my nine years at MIT, but my inner nerdy teenager has never been as personally delighted by a project as it is right now. To…
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Matrix multiplication in O(n^2.373)
For twenty years, the fastest known algorithm to multiply two n-by-n matrices, due to Coppersmith and Winograd, took a leisurely O(n2.376) steps. Last year, though, in his PhD thesis, Andrew Stot…
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The Complete Idiot’s Guide to the Independence of the Continuum Hypothesis
A global pandemic, apocalyptic fires, and the possible descent of the US into violent anarchy three days from now can do strange things to the soul. Bertrand Russell—and if he’d done no…
15
Why Philosophers Should Care About Computational Complexity
Update (August 11, 2011): Thanks to everyone who offered useful feedback! I uploaded a slightly-revised version, adding a “note of humility” to the introduction, correcting the footnot…
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Quasi-Polynomial Algorithm for Graph Isomorphism
Update (Dec. 14): The long wait is over! Here’s Laci’s paper on the arXiv. So far, I’ve read it only deeply enough to note that it contains the following sentence: A group G ≤ S…
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Progress on longstanding open problems
For those who haven’t seen it: Aubrey de Grey, better known to the world as a radical life extension researcher, on Sunday posted a preprint on the arXiv claiming to prove that the chromatic …
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The Scientific Case for P≠NP
Out there in the wider world—OK, OK, among Luboš Motl, and a few others who comment on this blog—there appears to be a widespread opinion that P≠NP is just “a fashionable dogma of…
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Ten Signs a Claimed Mathematical Breakthrough is Wrong
Yesterday several people asked my opinion of a preprint claiming to solve the Graph Isomorphism problem in deterministic polynomial time. I responded: If I read all such papers, then I wouldn’t hav…
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Time Travel (2006)
Last time we talked about free will, superintelligent predictors, and Dr. Evil planning to destroy the earth from his moon base. Today I'd like to talk about a more down-to-earth topic: time travel. The first point I have to make is one that Carl Sagan made: we're all time travelers—at the rate of one second per…
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Insert D-Wave Post Here
In the two months since I last blogged, the US has continued its descent into madness. Yet even while so many certainties have proven ephemeral as the morning dew—the US’…
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Bell inequality violation finally done right
A few weeks ago, Hensen et al., of the Delft University of Technology and Barcelona, Spain, put out a paper reporting the first experiment that violates the Bell inequality in a way that closes off…
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Scott Aaronson explains his $200k bet (8 Signs A Claimed P≠NP Proof Is Wrong)
As of this writing, Vinay Deolalikar still hasn’t retracted his P≠NP claim, but a clear consensus has emerged that the proof, as it stands, is fatally flawed. The first reason is that weR…
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Why are amplitudes complex?
[By prior agreement, this post will be cross-posted on Microsoft’s Q# blog, even though it has nothing to do with the Q# programming language. It does, however, contain many examples th…
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P vs. NP for Dummies
A reader named Darren commented on my last post: I have this feeling that this whole P and NP thing is not only a profound problem that needs solving, but something that can be infinitely curious t…
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PHYS771 Quantum Computing Since Democritus
University of Waterloo, Fall 2006 Tuesdays and Thursdays, 1:00-2:30pm BFG Building, 2nd floor seminar room (BFG2125)