[HN Gopher] Shor, I'll do it (2007) ___________________________________________________________________ Shor, I'll do it (2007) Author : monort Score : 285 points Date : 2022-07-31 09:38 UTC (13 hours ago) (HTM) web link (scottaaronson.blog) (TXT) w3m dump (scottaaronson.blog) | Eddy_Viscosity2 wrote: | This was pretty good, though I would still like to know 'how' the | quantum computer gets all those amplitude (and how many are | needed) to interfere with each other to get the answer. | FartyMcFarter wrote: | > and how many are needed | | According to this paper, one can use 2n+3 qubits to factor an | integer with n bits: | | https://arxiv.org/abs/quant-ph/0205095 | | Those would be perfect qubits, if you use noisy qubits you'd | need many more (maybe a factor of 1000x more), since quantum | error correction imposes a lot of overhead. | Strilanc wrote: | Counting the number of amplitudes isn't really the right way to | approach it. Individual amplitudes are basically irrelevant to | a large quantum computation. It's all about large scale | patterns in the amplitudes. Little exceptions to these patterns | don't matter much. | | Suppose I gave you the power to negate a billion amplitudes of | your choosing in the middle of a 2000 bit quantum factoring | computation. You might think this could destroy the | computation, but a billion is way way _way_ less than 2^2000 so | the computation would for all intents and purposes be | completely unaffected. | | The things the quantum computers operate on are the qubits, not | the amplitudes. Noise processes also operate on qubits, not | amplitudes. It's the quantity and quality of the qubits that | matter. | | You can factor 2048 bit RSA integers if you have 20 million | qubits each having a 0.1% gate error rate [1]. | | 1: https://quantum-journal.org/papers/q-2021-04-15-433/ | semi-extrinsic wrote: | If I understand your question on "how" correctly: all objects | are interfering with each other at the quantum level all the | time. But this effect is vanishingly tiny compared to e.g. | thermal noise, we don't notice it at all. So the quantum | computer isn't making the qubits interfere, that always | happens, but it is working very hard to remove all other | sources of noise such that only the quantum interactions | remain. | [deleted] | FartyMcFarter wrote: | I don't fully understand all the details, but even I can tell | that this algorithm is beautiful. | | Mixing together the concepts of modular sequences, periods, and | Fourier transforms, plus doing this fast with computers that | barely exist in order to find factors or numbers is just an | amazing construction. | | There's a video featuring Peter Shor about the invention of this | algorithm: https://www.youtube.com/watch?v=6qD9XElTpCE | [deleted] | H8crilA wrote: | How does one implement the QFT? The same way as one would do FFT, | but with quantum gates? I.e. taking O(n*log(n)) gates with two | inputs and two outputs? I'm imagining that there's some sort of a | vector of quantum (complex) amplitudes left after the | exponentiation that needs to be transformed. | Strilanc wrote: | The QFT is the Cooley-Tukey FFT algorithm [1] expressed as a | tensor network [2]. | | Cooley-Tukey has two main steps that are repeated recursively: | bulk replacing a,b with a+b,a-b along bit boundaries, and | applying twiddle factors. The bit-boundary a,b->a+b,a-b part | becomes a Hadamard gate and the bit twiddling becomes a set of | phase gates. Also there's some re-ordering but that's not the | meat. | | The actual quantum circuit: [4]. | | The quantum circuit is simple enough that it's a really solid | mnemonic for remembering Cooley-Tukey, if you know how to | translate it. There are also various ways to optimize the gate | count or gate depth of this circuit, and these optimizations | translate into changes to the classical FFT (though they are | not always optimizations after translation) [5]. | | 1: | https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algor... | | 2: https://en.wikipedia.org/wiki/Tensor_network | | 3: https://en.wikipedia.org/wiki/Hadamard_transform | | 4: | https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B%2... | | 5: https://algassert.com/2016/06/14/qft-by-multiply.html | dandanua wrote: | No, QFT requires only O(log(n)^2) gates. It can be described by | using a recursion. That is, QFT on 2^n values (n qubits) can be | computed using QFT on 2^(n-1) values (n-1 qubits). | Sniffnoy wrote: | (2007) | benreesman wrote: | Sidebar: I absolutely adore Scott "I don't give a fuck anymore" | Aaronson. | | My personal aspirations to this kind of biting wit are futile and | vain, but my admiration for it is even greater and some people | can bring it off with style to spare, and ever since that uh, | thing, Scott is just playing the ten minute guitar solo. | | Into to Number Theory with periodicity and modular math might be | stretching "nothing more than arithmetic" a bit, but fuck it, | this is the best accessible discussion of Shor I've ever read and | if there's any justice in the multiverse it will become the go-to | link on the topic, which will mean that 10^500 laypeople will | roll their eyes at the next 10^500 "quantum computers are the | next step after digital computers" Aeon fluff pieces floating by. | [deleted] | [deleted] | carver wrote: | Is this the "uh, thing"? | | https://www.theatlantic.com/politics/archive/2015/01/the-blo... | aorth wrote: | I think it was the doxxing against his will by NY Times. | | https://www.newyorker.com/culture/annals-of-inquiry/slate- | st... | | Edit: users pointed out this is a different Scott. My | mistake! | Chinjut wrote: | This is a different Scott. | fossuser wrote: | Yeah, though I wonder if Aaronson's writing tone is | somewhat influenced by Alexander's. At least he reminded | me of him by the end. | | I'm pretty sure they all know/read each other - related | communities/ideas. | | PBS also has a (now discontinued) YouTube show called | infinite series that did a decent overview of the | algorithm and showed examples of a lot of the stuff | described here. | [deleted] | [deleted] | ithinkso wrote: | > "I don't give a fuck anymore" | | This article is from 2007 | benreesman wrote: | Haha stop upvoting my wrong comment people! | | Apparently Scott never gave a fuck. :) Dope. | mikotodomo wrote: | Wow. That such high end technical content is hosted on Wordpress | really shows how battle tested and reliable it is. | jefftk wrote: | It does? I think mostly it shows that WordPress is a popular | blogging platform. | | Also note that this was published in 2007, 4 years after | WordPress was originally released, which probably makes Scott a | bit of an early adopter. | Ma8ee wrote: | It's static text. What is hard about it? | jstanley wrote: | > if you think about quantum computing in terms of "parallel | universes" (and whether you do or don't is up to you) | | But if you _do_ think of it that way, why not just pick a random | number using some quantum process, classically test whether or | not it 's a divisor of the number you're trying to factor, and | kill yourself if it's not? In every universe where you survive | (which, for sake of argument, are the only ones you care about) | you find a factor on the first try. | smegsicle wrote: | note that this is what CERN is doing to 'manipulate the | timeline'- | | the entire planet is obliterated except in cases pursuant to | their interests, with none the wiser since in those cases, they | 'didn't do anything' | buu700 wrote: | Sounds like quantum bogosort: | https://en.wikipedia.org/wiki/Bogosort#Quantum_bogosort | | _The algorithm generates a random permutation of its input | using a quantum source of entropy, checks if the list is | sorted, and, if it is not, destroys the universe. Assuming that | the many-worlds interpretation holds, the use of this algorithm | will result in at least one surviving universe where the input | was successfully sorted in O(n) time._ | Strilanc wrote: | Isn't the multiverse where one version of you solves the | problem and the other versions _don 't die_ strictly better | than the version where they commit suicide? It's not like their | experiences and memories get merged into the surviving copy. | They're just gone, instead of not gone. | layer8 wrote: | For the winner (survivor) it doesn't matter whether their | "clones" in the other branches kill themselves, so why plan | that in the first place? Also, the losers will probably prefer | to not kill themselves, despite not getting lucky with their | random number. So, the end result is the same as for any | regular random guess. | moffkalast wrote: | Besides, infinite universes doesn't mean they'll collectively | give you a mutually exclusive and exhaustive set like | everyone is for some reason assuming. You could just as well | get the same wrong number in all of them and kill yourself | off permanently in the entire multiverse. | | That's the nature of true randomness, if you set up a dice | throw a million times they could just as well all come up as | six. Incredibly unlikely, but possible. | espadrine wrote: | > _why not just pick a random number using some quantum | process, classically test whether or not it 's a divisor of the | number you're trying to factor, and kill yourself if it's not?_ | | All universes affect the probability of something occurring in | each universe. That is how each universe contributes a bit of | the Fourier offset which gives off the period in pretty much | all universes. | | Because finding the right divisor is so unlikely, you run a | high risk of killing yourself in all universes. As Scott | emphasizes, quantum computers don't run possibilities in | parallel! | quickthrower2 wrote: | That is quite the prisoners dilemma! | YetAnotherNick wrote: | The thing is that even in that sense collapse will select a | random universe. So you will get a universe with killed output | or whatever you said it. | | Also quantum parallel universe is not what you expect parallel | universe to be. Universes interact with each other. | cwillu wrote: | Then you're talking about the complexity class PostBQP, rather | than the class BQP. | | Postselection is ridiculously powerful, as you've noted. | kspinka wrote: | This is thought provoking, but suffers from a fundamental flaw. | Let's play this out...now you're alive and have Satoshi's keys, | but all of the would-be bidders and pumpers in your universe | have killed themselves, value goes to zero. | | I think you may have accidentally discovered the Quantum Bag | Holder Paradox. | __ryan__ wrote: | In every universe where you survive (which, for sake of | argument, are the only ones you care about) you find a factor | on the first try. | | What's more likely, randomly guessing a prime factor of a giant | number, or miraculously surviving and being permanently | incapacitated? Or being interrupted, saved by modern medicine, | bitten by a black widow immediately after getting it right... | | You have to think that the odds really aren't in your favor | there. | shagie wrote: | Quantum immortality is a neat thing ( https://en.wikipedia.or | g/wiki/Quantum_suicide_and_immortalit... ). I'd suggest the | short story All the Myriad Ways by Larry Niven ( | https://erenow.net/common/the-best-alternate-history- | stories... ) and the exceedingly long story Nangurz ol Arny | Fgrcurafba (rot13 because it kind of is a spoiler). | Groxx wrote: | As a middleground, at around 42 pages long, there's Divided | by Infinity: https://www.tor.com/2010/08/05/divided-by- | infinity/ | JadeNB wrote: | When discussing quantum immortality and sci-fi, one must | also mention Greg Egan's early _Quarantine_ | (https://en.wikipedia.org/wiki/Quarantine_(Egan_novel)). | ctoth wrote: | Also when discussing Quantum Immortality and Sci Fi, one | must certainly mention Divided By Infinity. | | https://www.tor.com/2010/08/05/divided-by-infinity/ | cwillu wrote: | The universe likes to play with its food. | | (What, did you think this logic is only valid when you're | trying to gain godlike powers?) | einpoklum wrote: | I believe that you would enjoy this film: | | https://www.imdb.com/title/tt0482571/ | walnutclosefarm wrote: | Congratulations to Scott Aaronson for the must lucide explanation | of Shor's algorithm for the merely mathematically literate, ever! | georgehm wrote: | > And once we knew (p-1)(q-1), we could then use some more little | tricks to recover p and q, the prime factors we wanted. | | I am curious to know about these tricks to recover p, q. Does | anyone know? | YetAnotherNick wrote: | You have pq(original number) and (p-1)(q-1). You could get (p + | q) = pq+1-(p-1)(q-1). Then we know p and q satisfies | x^2-(p+q)x+pq=0. You could solve this quadratic equation using | quadratic formula to get p and q. | eliben wrote: | I like how there's a comment by Peter Shor commending Scott on | the article :-) Knowing Scott's blog, this is likely legit | ryan-duve wrote: | If the part about the tack on the board under the clocks wasn't | clear, and you're a visual learner, this video from 3Blue1Brown | about the [classic] Fourier Transform might help clarify the | relationship to periodicity: | | https://www.youtube.com/watch?v=spUNpyF58BY | omoikane wrote: | Near the bit that describes "repeated squaring", there is this | formula (note the "http"): http://www.scottaaronson.com/cgi- | bin/mimetex.cgi?x^r%20=%203... | | Firefox doesn't want to load that due to mixed content (https | page loading http), and also doesn't want to follow the 301 | redirect to the https version. Chrome appears to follow the 301 | redirect (or maybe lazy-images.js does something different) such | that the page renders properly. ___________________________________________________________________ (page generated 2022-07-31 23:00 UTC)