Quantum Computing for Computer Scientists

14 maí 2018
1 012 161 Áhorf

This talk discards hand-wavy pop-science metaphors and answers a simple question: from a computer science perspective, how can a quantum computer outperform a classical computer? Attendees will learn the following:
- Representing computation with basic linear algebra (matrices and vectors)
- The computational workings of qbits, superposition, and quantum logic gates
- Solving the Deutsch oracle problem: the simplest problem where a quantum computer outperforms classical methods
- Bonus topics: quantum entanglement and teleportation
The talk concludes with a live demonstration of quantum entanglement on a real-world quantum computer, and a demo of the Deutsch oracle problem implemented in Q# with the Microsoft Quantum Development Kit. This talk assumes no prerequisite knowledge, although comfort with basic linear algebra (matrices, vectors, matrix multiplication) will ease understanding.
See more at www.microsoft.com/en-us/research/video/quantum-computing-computer-scientists/

Ummæli
  • Mathematics is the only universal language in nature. You can't understand qc and qm with words or analogies. You have to know math well.

    OrcunOrcun8 klukkustundum síðan
  • This is excellent - thank you so much ! Incredible clear and straight with easy (enough) examples for self-studying. Yet, In Defense of some Pop-science I also recommend the „bananaworld“ from Jeffrey Bub

    G GG GDegi Síðan síðan
  • Why did you bother creating this 1 and a half hour long presentation? Why did you bother studying all of this stuff? Just look up some fancy sentences on the "quantum computer" page on Wikipedia and make a 5 minutes video! Maybe dress as a very funny proton too :) . This guy has 4 times your views, take example ispast.info/losk/v-deo/m5B_kYzJ1oOnim0.html. Now, for real, thank you, I actually watched this some months ago and I came back and wanted to say that this is really the only helpful source I found on the subject, and also this could hardly be any better. I find this stuff really cool, everything you explained to me feels like just about to slip out of the boundaries of what we intuitively know is possible, but it actually never does.

    ErnaldoErnaldo4 dögum síðan
  • The crowd looked like the man in black waiting room.

    M MAGONM MAGON6 dögum síðan
  • Why do quantum operations have to be reversible?

    Arnav SharmaArnav Sharma6 dögum síðan
    • Just the way the universe works

      Andrew HelwerAndrew Helwer6 dögum síðan
  • I love quantum computers too

    RED DragonRED Dragon7 dögum síðan
  • I understood mostly what he told i think except why you want to send 2 classical bits to someone just to "teleport" one Q-bit. 2 for 1 sounds inefficent. And when you need two Qu-bits as Input for your box to figure what the box does it sounds kinda unfair to compare it with one classical bit. If we would put 2 classical bits in the black box and get 2 out at the same time we would also be able to tell what the box does just as the Q-bits.

    manu petermanu peter8 dögum síðan
    • Teleportation is the only way to send a full-fidelity quantum state over the network without loss. Quantum computers still win with a two-wire black box because it compares quantum computers to reversible classical computers.

      Andrew HelwerAndrew Helwer7 dögum síðan
  • Krista Jacko Josza, my dear old friend. If youre reading this then let me appologise for my behaviour. You were, and i imagine still are, the best friend anyone could wish for. Please dont hate me, in aldult life ive been diagnosed with many mental problems. As a kid i loved you dearly and im sorry i never said so. Please take care and i send love to your dad, mum and sisters. If you.can find it in your heart, please forgive me. Im a wanty lol. Xx

    High Plains DrifterHigh Plains Drifter10 dögum síðan
  • I'm surprised that wokesters didn't shit themselves over term quantum supremacy..

    AkusAkus12 dögum síðan
    • It has been replaced by the term quantum advantage, yes. A welcome change!

      Andrew HelwerAndrew Helwer12 dögum síðan
  • It's a long story but but the CIA has been harassing me (regarding my discovered intellectual property) and attempting to commit Piracy regarding the Theory of Everything but just so everyone knows who wants to analyze the Fundamental Principal behind the Theory of Everything (TOE) can be expressed by connection to the equation *e^(iπ) + 1 = 0* (where e is approximately 2.7182818284590452356 and i represents √(-1) ) This equation basically reduces to -1 + 1 = 0. The (TOE) does NOT require time to solve for it. *Don't worry when the time variable disappears.* I don't want to provide too much information. This way professors and students can review it without having to worry about piracy (sadly, especially from those you think are meant to protect the law) *Once you find it out please publish it*. I have been hiding it. The CIA has offered me a job and money for it. Please stand up for scientists and GLOBAL learning. You can figure it out.

    Joshua PJoshua P13 dögum síðan
  • This guy got paid in Diet Coke. Also, nice talk

    Michael SalmonMichael Salmon13 dögum síðan
  • Kinda sad that there were like 3 people in the room

    Aidan SgarlatoAidan Sgarlato14 dögum síðan
    • I was a bit disappointed at the time but lots of people online seem to have enjoyed it so it all worked out in the end!

      Andrew HelwerAndrew Helwer14 dögum síðan
  • If the values of the qbit vector can be complex numbers, with the condition that v[0]^2 + v[1]^2 = 1, say you had v = {i, √2} which satisfies the condition. How do you interpret the probabilities that the qbit is 0 or 1? -100% and 200%?

    Diego EsparzaDiego Esparza15 dögum síðan
    • You're missing absolute value, it's |v[0]|² + |v[1]|² = 1. So |i|² = 1. See en.wikipedia.org/wiki/Absolute_value#Complex_numbers

      Andrew HelwerAndrew Helwer14 dögum síðan
  • Very very good tutorial (I am a computer science Ph.D. candidate). One interesting piece of information, the instructor of this video left Microsoft.

    Zhan CaitaoZhan Caitao15 dögum síðan
    • Correct! Decided to try the independent contractor life.

      Andrew HelwerAndrew Helwer15 dögum síðan
  • Someone give buddy another nice cold can of whatever he's fake sipping on?

    Andrei NegreaAndrei Negrea15 dögum síðan
  • The obedient crocodile intriguinly unlock because square bodily offer times a mean letter. bouncy, murky grass

    김영선김영선18 dögum síðan
  • Elon Musk is not Isaac Newton eating an Apple

    Sprints WilliamSprints William21 degi síðan síðan
  • I feel asleep watching what if we nuked tge moon and woke up a 6:00 am watching this

    Pixel BotsPixel Bots23 dögum síðan
  • A very nice, easy to follow and exhaustive introduction. Really love the speaker!

    Sacrament of WildernessSacrament of Wilderness25 dögum síðan
  • He's constantly saying what he said at 13:57 , flip bottom two rows of identity matrix. But immediately next he does a matrix transformation which is (4x4 c-not transformation applied on unit vector in 4-dim) . Though the output is same, the most accurate way of saying is the last 2 columns flipped (kind of basis vectors).

    Mani KanthMani Kanth26 dögum síðan
    • Yes that's the explanation that 3blue1brown is fond of. However, there's more than one way to look at matrix transformation; change-of-basis is only one of them.

      Andrew HelwerAndrew Helwer26 dögum síðan
  • One of the clearest explanations I've ever seen

    Martin TunnicliffeMartin Tunnicliffe26 dögum síðan
  • watch the Hebrew, the DNA of creation will help you understand what stuff is made off. Rabbi Kraft

    Tienke Bosher PelzmannTienke Bosher Pelzmann27 dögum síðan
  • 🎵Dunna nunna nunna dun.... shut and calculate 🤓🎶

    AetherDivisionAetherDivisionMánuði síðan
  • Great presenter! Please teach me anything and everything.

    HuntraconyHuntraconyMánuði síðan
  • The jumpy white indisputably dare because editorial uniquely tour unlike a noiseless freckle. remarkable, young responsibility

    Jane AllsepJane AllsepMánuði síðan
  • The ruddy rayon exclusively plug because honey july owe an a verdant macrame. dirty, beneficial bra

    Drino ZhaoDrino ZhaoMánuði síðan
  • There's too much nifty in this lecture :P Loved it!

    Ashwin K RaghuAshwin K RaghuMánuði síðan
  • Nifty

    Clement GervaiseClement GervaiseMánuði síðan
  • Jesus christ take a sip every 4 sentences.

    Chris CostelloChris CostelloMánuði síðan
  • 👏👏👏👏👏👌👍

    H.İ. IskenderH.İ. IskenderMánuði síðan
  • Another math element that may help some in accepting the weirdness of quantum physics is the introduction of negative probability, itself a concept hard to grasp, but a valid approach to mathematically describing intermediate states that a quantum system may realize on its way to the final state as detected by a measurement.

    tideviewtideviewMánuði síðan
  • It makes no sense to me that you can complicate with it which is .measuring it...why does does it no it collapse in then?

    Anita Reason To BelieveAnita Reason To BelieveMánuði síðan
    • I had a stroke trying to read this.

      Jakob ZupanecJakob ZupanecMánuði síðan
    • What?

      Andrew HelwerAndrew HelwerMánuði síðan
  • two years later, it remains the best video on quantum cumputing, Bravo and thank you

    i-uki-d i-imuti-uki-d i-imutMánuði síðan
    • Glad you enjoyed!

      Andrew HelwerAndrew HelwerMánuði síðan
  • The materialistic venezuelan logically consist because news immediately ban about a ugliest stream. striped, elite pastry

    kelly thomaskelly thomasMánuði síðan
  • he is not so good at answearing the questions so that they understand i see them ask like the same question in different ways five times and still not understanding because he does not explain it to 100 percent.

    MCPE grejerMCPE grejerMánuði síðan
    • You can only lead a horse to water.

      Andrew HelwerAndrew HelwerMánuði síðan
  • 44:57 i dont get how that is reversible it just sets the bit to zero how is that revirsible?

    MCPE grejerMCPE grejerMánuði síðan
    • Because you store the input, that makes it reversible.

      Andrew HelwerAndrew HelwerMánuði síðan
  • I just woke up and somehow I was brought here

    sk8punk318sk8punk318Mánuði síðan
  • Packed house lol

    sk8punk318sk8punk318Mánuði síðan
  • The various parent excitingly squash because owner socioeconomically stuff including a spotted answer. adorable, numerous responsibility

    Mel YuMel YuMánuði síðan
  • So at the end he said we'd know within a year or two whether that exponential limit exists -- it's been two years, does it?

    ToothToothMánuði síðan
    • Still not determined. Google claims they established quantum supremacy but it isn't quite as clear-cut as we want.

      Andrew HelwerAndrew HelwerMánuði síðan
  • Happy Valentine's day

    勇樹勇樹Mánuði síðan
  • The even excellent excited orchestra grossly grin because fowl expectably bury following a steadfast knickers. open, succinct digital

    Andrew GouldAndrew GouldMánuði síðan
  • 2A.M and a video for computer scientists... yeah time to go to bed

    RandomMm FanRandomMm FanMánuði síðan
  • The right professor literally suck because sidecar physiologically stop along a dependent knowledge. accurate, lamentable tanker

    trav grantrav granMánuði síðan
  • This video proves that I am an imbecile 🤦‍♂️🤣

    AndrewAndrewMánuði síðan
    • Oh no! What did you get stuck on? I am sure you can get it.

      Andrew HelwerAndrew HelwerMánuði síðan
  • Bill Gates is an evil vaxine murderer.

    GuffGuffMánuði síðan
  • 1:05:26 - I'm still confused about that myself. Maybe you can't transfer data through the entangled particles but if you can collapse it by measuring it at will, if there is a way to tell when that other particle collapses then the collapse itself can be the communication. I'm guessing what he's saying is that there is no way to tell when the state collapses so you can't use it to signal?

    GSZeroGSZeroMánuði síðan
    • Yes correct, you cannot tell whether the state is collapsed.

      Andrew HelwerAndrew HelwerMánuði síðan
  • The delicate propane consecutively scatter because dry psychologically talk through a noiseless boundary. fluttering, soft clipper

    Ben WingBen WingMánuði síðan
  • Can you show an algorithm in Quantum computing?

    ALWIN SEBASTIANALWIN SEBASTIANMánuði síðan
  • I'm just going to pretend to understand this I have like a million question

    Andres GomezAndres GomezMánuði síðan
    • @Andres Gomez learning the math is really the only way. Don't be intimidated - this is linear algebra that most people learn on their first or second day in class in a first-year undergrad course! 3blue1brown has a set of linear algebra videos that people really like.

      Andrew HelwerAndrew HelwerMánuði síðan
    • @Andrew Helwer well the math is beyond me I came here trying to understand how a superposition could be created? Also I still dont understand how this allows Qbit to be able to process information faster then a traditional computer? I'm not a computer scientist but I still want grasp how this can be possible

      Andres GomezAndres GomezMánuði síðan
    • Go ahead and ask!

      Andrew HelwerAndrew HelwerMánuði síðan
  • he is cute

    Mario RistovskiMario RistovskiMánuði síðan
  • Love you.

    Tuyen MaiTuyen MaiMánuði síðan
  • It's because they are connected via hyperdimensional geometry and are essentially at the same location in this hyperdimensional space. It's all derived by the 8d packing of fields in the e8 lattice, the subsection of it projected down to the next lower dimension, and that dimensional projection is then itself projected to the next lower dimension, etc. Till you get to a 3d "quantum foam" of 20 groups of Planck scale tetrahedra which twist around a central point, who's twist states correlate to quantum mechanics, quantum gravity perturbations, and wave particle duality. Visually similar to the vibrating oil drop experiments. It is at that point we get into physical results at Cern, which Research scientist Garrett Lisi pointed out are coincident with the e8 lie (pronounced "Lee") groups that make up the e8 lattice. At least that's what the math and experiments suggest at any rate.

    ALEX GIBSONALEX GIBSONMánuði síðan
    • So to expand on that relationship, essentially the entangled particles occupy 20 group geometry that is connected in the x dimension up from the third dimension. Meaning though they can be in "separate places" in the 3rd, they are actually just the same hyperdimensional geometry viewed from two hyperdimensional angles that render visually as 2 points in space.

      ALEX GIBSONALEX GIBSONMánuði síðan
  • I'm more confused about the superposition of the soda in his can

    Hiro SanHiro SanMánuði síðan
  • Props to Andrew for being so active in the comments and helping out people who have questions

    cwif rbmcwif rbmMánuði síðan
  • 2:08 “We could undermine our global financial system, which is pretty cool.”

    Jonah DeeganJonah DeeganMánuði síðan
  • The halting community multivariably count because authority taxonomically coil at a spurious mini-skirt. many, red month

    Annie GriffinAnnie GriffinMánuði síðan
  • No se entiende nada.

    WHK YanWHK YanMánuði síðan
  • Gotta give this kid credit for his effort at explaining the unexplainable. Good job!

    AmericanFreedom911AmericanFreedom911Mánuði síðan
  • For the identity black box , why not just pass through the input x to both the outputs ? Why do we need a cnot? Or is it that we have no identity operator?

    Sandeep IyerSandeep IyerMánuði síðan
    • Quantum computers don't have the ability to copy a qbit value, so we have to make do with CNOT.

      Andrew HelwerAndrew HelwerMánuði síðan
  • The lecturer is cute. He seems to genuinely enjoy teaching and it makes me enjoy learning.

    Crunchy DuckCrunchy DuckMánuði síðan
  • 32:22 - "Imagine I show up on your doorstep and I give you a package, its just a black box that has a function on one bit, what a horrible present" 😂

    Drug DealerDrug DealerMánuði síðan
  • Statistics computing is quantum approach command size and ten.Quantum point.=QP Yellow Hand.

    Kenichi MoriKenichi MoriMánuði síðan
  • Why no example of inputs outputs bb at 36:55. I was with you till then Just saying this is the most difficult part without further explaining didn’t help Hoping someone reading the comment would explain

    Sandeep IyerSandeep IyerMánuði síðan
    • @Sandeep Iyer Happy to hear! It certainly took a long amount of staring for me.

      Andrew HelwerAndrew HelwerMánuði síðan
    • @Andrew Helwer thanks . I got it eventually by more staring :)

      Sandeep IyerSandeep IyerMánuði síðan
    • It's conceptually difficult, hence explaining it is also difficult. The sort of thing you just have to stare at for a while.

      Andrew HelwerAndrew HelwerMánuði síðan
  • Maaf ya allah hamba liat ini.

    Nax EniNax EniMánuði síðan
  • I am not a programmer at any rate but at 9mins, this video explains to me that quantum computing built a language(mathematical) to utilize superposition as a means to "computate" possible outcomes in exponents of 2 states possible at the same time that when combined with algorithms with another cubit will speed up computations by being able to perform more instead of on at a time like a classical computer. So this tells me, to treat it akin to getting a solution like a classical computer, a compute scientist will need to learn this way of modeling his "question" to the the right answer.

    Joseph CJoseph C2 mánuðum síðan
    • It is sort of like that but there is an important bottleneck: although we compute with an exponential number of amplitudes while the qbits are entangled or in superposition, in order to actually get information out of the system we have to measure it - and we can only get out as many bits of information as there are qbits. This huge bottleneck means we have to be very tricky about how we design our algorithms, and can only get a speedup in certain cases rather than in general.

      Andrew HelwerAndrew Helwer2 mánuðum síðan
  • Have you guys have you found God yet? scuse me 'iggy boson. When's the last time you spoke to Higgs Boson? I think Quantum Physics is a real waste of time and life, eh? it's just up the garden path physics. It's really smart people trying to find ways to hide more and more shit, you become Gods when you multiply the deceived amounts you. Thank God the real Gd coming back, it's known as the light, by which, no other light could have been made, including this AI, 2nd light, light that you're all currently having a lovefest with.

    Tony FaulknorTony Faulknor2 mánuðum síðan
  • This is pretty cool. I'm amazed

    Felipe ConstantinoFelipe Constantino2 mánuðum síðan
  • These machines are going to have a giant market: this guy and his friends.

    Ean EricksonEan Erickson2 mánuðum síðan
  • 7:44 Dinesh spotted from silicon valley 😂😂

    Sachin ThakkarSachin Thakkar2 mánuðum síðan
  • 32:22 - "Imagine I show up on your doorstep and I give you a package, its just a black box that has a function on one bit, what a horrible present" 😂

    WheelsakaSpeedyWheelsakaSpeedy2 mánuðum síðan
  • Very nice presentation. Thank you

    azeddine benabbouazeddine benabbou2 mánuðum síðan
  • finally. I've watched and read so many things that just talk about the fact that superposition is a thing without actually going in to what it means for computation

    Bacon InvaderBacon Invader2 mánuðum síðan
  • stfu n show me the quantum gaming

    Ghryst VanGhodGhryst VanGhod2 mánuðum síðan
  • And he’s confident enough to even sip out of a soda can sideways after explaining complex quantum dimensions.. “Are you all getting this?” Haha.. Great video

    Welcome GuestWelcome Guest2 mánuðum síðan
  • Pretty nifty indeed! What I'm suspecting is that qubits can be thought of as unit quaternions, which for me at least, makes it a lot more intuitive to think about them. But maybe for most people it wouldn't be! But if they're points on a 3-sphere, that sort of makes it make sense to me that you might have opposite "poles" of a sphere - the two different half and half superpositions - which project to the same ultimate value but are different - it's just phase again, like polarization.

    The Great DreamThe Great Dream2 mánuðum síðan
  • So..... The probabilities, are the states.....

    Andrew900460Andrew9004602 mánuðum síðan
    • Basically, except amplitudes aren't quite probabilities - they can be complex or negative, and the sum of their squares has to equal 1 rather than just their sum (as is the case with regular probabilities).

      Andrew HelwerAndrew Helwer2 mánuðum síðan
  • I'm not an idiot I think but this stuff is over my head

    James ShelnuttJames Shelnutt2 mánuðum síðan
    • You can do it! Anything specific you're having trouble with?

      Andrew HelwerAndrew Helwer2 mánuðum síðan
  • nice video! good collection of topics and good presentation... I tried to access quantumexperience.ng.bluemix.net/ but no luck.

    atul kumthekaratul kumthekar2 mánuðum síðan
    • @Andrew Helwer Thank you, this one seems to be working.

      atul kumthekaratul kumthekar2 mánuðum síðan
    • Looks like they moved it: quantum-computing.ibm.com/

      Andrew HelwerAndrew Helwer2 mánuðum síðan
  • Hey buddy, you look really handsome and have great hair! Do you know anything about computers?? No!? Well that's okay, we'll figure that stuff out later. #Microsoft

    Eric FunderburgEric Funderburg2 mánuðum síðan
  • This is not for any scientist, why is the guy explaining matrix

    Yixuan LeeYixuan Lee2 mánuðum síðan
    • Some people forget their linear algebra!

      Andrew HelwerAndrew Helwer2 mánuðum síðan
  • How did i fall asleep to this?

    SunzaleaSunzalea2 mánuðum síðan
  • "So you can send entangled Qbits by laser?" "Yes" "Okay that's even cooler" love that guy 😂

    kurt Morakurt Mora2 mánuðum síðan
  • I tried to read so many articles to get an intuition about how QC works, but it seems it just never rings the bell. I honestly think the best part in this vid for me is when he said "shut up and do the math". Really over the years I've been reminded again and again how "trying to understand it intuitively" is nothing if not backed up by concrete and hard working math/calculations.

    Andrew KimAndrew Kim2 mánuðum síðan
  • Yang pilih allah like.

    Indian GirlIndian Girl2 mánuðum síðan
  • In case you're looking for slides www.microsoft.com/en-us/research/uploads/prod/2018/05/40655.compressed.pdf

    Amiko MalaniaAmiko Malania2 mánuðum síðan
    • speakerdeck.com/ahelwer/quantum-computing-for-computer-scientists

      Andrew HelwerAndrew Helwer2 mánuðum síðan
  • Pretty neat, quantum computing is just an application of vector and tensor algebra or potentially just an application of probability theory?

    Saul CovarrubiasSaul Covarrubias2 mánuðum síðan
    • It's kind of an extension of probability theory, in that instead of all the probabilities summing to 1, the absolute value squared of probabilities (amplitudes) sum to 1. 2-norm vs. 1-norm.

      Andrew HelwerAndrew Helwer2 mánuðum síðan
  • Why am i here don't even know quantum computing

    pizzaplanet 22pizzaplanet 222 mánuðum síðan
  • I finally found the smart people

    Matt DMatt D2 mánuðum síðan
  • This video was my introduction to quantum computing....now, I was able to recreate the "delayed choice quantum eraser experiment" using the IBM computer....A million thanks to this video. Here is my paper: www.academia.edu/resource/work/44494928

    QMG The Quantum Mechanics GuyQMG The Quantum Mechanics Guy2 mánuðum síðan
  • 1:40 dog: help my please

    Love NightLove Night2 mánuðum síðan
  • i can build what circuit or draw it up for intel chip production now ?

    ronald jorgensenronald jorgensen2 mánuðum síðan
  • can you show this in octal and hexidecimal ?

    ronald jorgensenronald jorgensen2 mánuðum síðan
  • pulse them to rms integers in magnetic beam control in CRT to see the majick happen in real time subset infinity in all calculus

    ronald jorgensenronald jorgensen2 mánuðum síðan
  • The doubt at 1:09:45, the language of probability isn't accurate. Well with the small domain of my understanding, I can vouch for the fact that Accuracy can't be an artifact of probability.

    Utkarsh DubeyUtkarsh Dubey2 mánuðum síðan
  • I dont know how to even use excel and can barely use MS word, I have no idea how I got here but I cant leave

    Joe KalaniJoe Kalani2 mánuðum síðan
  • 40:00 The confusion here is that there are two qbits, one named 'input' and one named 'output', but these two are both passed into the quantum black box to be operated on. The 'input' qbit is left unscathed after the black box, but it is the 'output' qbit which is rewritten with the value of [the possibly nonreversible function f : applied to whatever the value of our 'input' bit is]

    Enlightened PenguinEnlightened Penguin2 mánuðum síðan
  • UMMMM THIS WAS ON MY BIRTHDAY WTF I'm intrigued and curious and tripped out

    Baby DragonBaby Dragon2 mánuðum síðan
  • Great talk. But at first I had trouble understanding why his "reversible set to zero" function is built the way it is. So to anyone who is asking himself the same question, here is my explanation: The non-reversible way of writing "set to zero" is the matrix [[1,1], [0, 0]] which is of course is rank-deficient so it is not invertible. So you need to avoid a row of zeros in the matrix while keeping all rows linearly independent from one another. The way to do that is to write the matrix in the following way [[0,0,1,1], [0,1,0,0], [0,0,1,0], [0,0,0,1]]. The last two rows perform what he labelled Input->Input' in his diagram. The first two rows perform the actual "set to zero" operation (labelled Input->Output' in his diagram), assuming that the first two values of the input vector are [1, 0, ...] which of course corresponds to the |0> input that is confusingly labelled "Output" in his diagram. I really wish he had written the operation as a matrix. Then it would have been clear to me immediately. That is also the reason why the "actual input" is the second parameter but the "actual output" is the first parameter: To get the [[1,1], [0,0]] submatrix away from the main diagonal.

    compuholic82compuholic822 mánuðum síðan
  • this new bible is as bad as the the bible bible ... yawn

    Lance ColemanLance Coleman2 mánuðum síðan
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