Archive for the ‘Complexity’ Category

NP-complete Problems and Physics: A 2019 View

Sunday, June 2nd, 2019

If I want to get back to blogging on a regular basis, given the negative amount of time that I now have for such things, I’ll need to get better at dispensing with pun-filled titles, jokey opening statements, etc. etc., and resigning myself to a less witty, more workmanline blog.

So in that spirit: a few weeks ago I gave a talk at the Fields Institute in Toronto, at a symposium to celebrate Stephen Cook and the 50th anniversary (or actually more like 48th anniversary) of the discovery of NP-completeness. Thanks so much to the organizers for making this symposium happen.

You can watch the video of my talk here (or read the PowerPoint slides here). The talk, on whether NP-complete problems can be efficiently solved in the physical universe, covers much the same ground as my 2005 survey article on the same theme (not to mention dozens of earlier talks), but this is an updated version and I’m happier with it than I was with most past iterations.

As I explain at the beginning of the talk, I wasn’t going to fly to Toronto at all, due to severe teaching and family constraints—but my wife Dana uncharacteristically urged me to go (“don’t worry, I’ll watch the kids!”). Why? Because in her view, it was the risks that Steve Cook took 50 years ago, as an untenured assistant professor at Berkeley, that gave birth to the field of computational complexity that Dana and I both now work in.

Anyway, be sure to check out the other talks as well—they’re by an assortment of random nobodies like Richard Karp, Avi Wigderson, Leslie Valiant, Michael Sipser, Alexander Razborov, Cynthia Dwork, and Jack Edmonds. I found the talk by Edmonds particularly eye-opening: he explains how he thought about (the objects that we now call) P and NP∩coNP when he first defined them in the early 60s, and how it was similar to and different from the way we think about them today.

Another memorable moment came when Edmonds interrupted Sipser’s talk—about the history of P vs. NP—to deliver a booming diatribe about how what really matters is not mathematical proof, but just how quickly you can solve problems in the real world. Edmonds added that, from a practical standpoint, P≠NP is “true today but might become false in the future.” In response, Sipser asked “what does a mathematician like me care about the real world?,” to roars of approval from the audience. I might’ve picked a different tack—about how for every practical person I meet for whom it’s blindingly obvious that “in real life, P≠NP,” I meet another for whom it’s equally obvious that “in real life, P=NP” (for all the usual reasons: because SAT solvers work so well in practice, because physical systems so easily relax as their ground states, etc). No wonder it took 25+ years of smart people thinking about operations research and combinatorial optimization before the P vs. NP question was even explicitly posed.


Unrelated Announcement: The Texas Advanced Computing Center (TACC), a leading supercomputing facility in North Austin that’s part of the University of Texas, is seeking to hire a Research Scientist focused on quantum computing. Such a person would be a full participant in our Quantum Information Center at UT Austin, with plenty of opportunities for collaboration. Check out their posting!

Not yet retired from research

Friday, April 19th, 2019

Last night, two papers appeared on the quantum physics arXiv that my coauthors and I have been working on for more than a year, and that I’m pretty happy about.

The first paper, with Guy Rothblum, is Gentle Measurement of Quantum States and Differential Privacy (85 pages, to appear in STOC’2019). This is Guy’s first paper that has anything to do with quantum, and also my first paper that has anything to do with privacy. (What do I care about privacy? I just share everything on this blog…) The paper has its origin when I gave a talk at the Weizmann Institute about “shadow tomography” (a task where you have to measure quantum states very carefully to avoid destroying them), and Guy was in the audience, and he got all excited that the techniques sounded just like what they use to ensure privacy in data-mining, and I figured it was just some wacky coincidence and brushed him off, but he persisted, and it turned out that he was 100% right, and our two fields were often studying the same problems from different angles and we could prove it. Anyway, here’s the abstract:

In differential privacy (DP), we want to query a database about n users, in a way that “leaks at most ε about any individual user,” even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure n quantum states, in a way that “damages the states by at most α,” even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is α-gentle for small α is also O(α)-DP, and any product measurement that is ε-DP is also O(ε√n)-gentle.

Illustrating the power of this connection, we apply it to the recently studied problem of shadow tomography. Given an unknown d-dimensional quantum state ρ, as well as known two-outcome measurements E1,…,Em, shadow tomography asks us to estimate Pr[Ei accepts ρ], for every i∈[m], by measuring few copies of ρ. Using our connection theorem, together with a quantum analog of the so-called private multiplicative weights algorithm of Hardt and Rothblum, we give a protocol to solve this problem using O((log m)2(log d)2) copies of ρ, compared to Aaronson’s previous bound of ~O((log m)4(log d)). Our protocol has the advantages of being online (that is, the Ei‘s are processed one at a time), gentle, and conceptually simple.

Other applications of our connection include new lower bounds for shadow tomography from lower bounds on DP, and a result on the safe use of estimation algorithms as subroutines inside larger quantum algorithms.

The second paper, with Robin Kothari, UT Austin PhD student William Kretschmer, and Justin Thaler, is Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. Here’s the abstract:

Given only a membership oracle for S, it is well-known that approximate counting takes Θ(√(N/|S|)) quantum queries. But what if a quantum algorithm is also given “QSamples”—i.e., copies of the state |S⟩=Σi∈S|i⟩—or even the ability to apply reflections about |S⟩? Our first main result is that, even then, the algorithm needs either Θ(√(N/|S|)) queries or else Θ(min{|S|1/3,√(N/|S|)}) reflections or samples. We also give matching upper bounds.

We prove the lower bound using a novel generalization of the polynomial method of Beals et al. to Laurent polynomials, which can have negative exponents. We lower-bound Laurent polynomial degree using two methods: a new “explosion argument” that pits the positive- and negative-degree parts of the polynomial against each other, and a new formulation of the dual polynomials method.

Our second main result rules out the possibility of a black-box Quantum Merlin-Arthur (or QMA) protocol for proving that a set is large. More precisely, we show that, even if Arthur can make T quantum queries to the set S⊆[N], and also receives an m-qubit quantum witness from Merlin in support of S being large, we have Tm=Ω(min{|S|,√(N/|S|)}). This resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA.

Note that QMA is “stronger” than the queries+QSamples model in that Merlin’s witness can be anything, rather than just the specific state |S⟩, but also “weaker” in that Merlin’s witness cannot be trusted. Intriguingly, Laurent polynomials also play a crucial role in our QMA lower bound, but in a completely different manner than in the queries+QSamples lower bound. This suggests that the “Laurent polynomial method” might be broadly useful in complexity theory.

I need to get ready for our family’s Seder now, but after that, I’m happy to answer any questions about either of these papers in the comments.

Meantime, the biggest breakthrough in quantum complexity theory of the past month isn’t either of the above: it’s the paper by Anand Natarajan and John Wright showing that MIP*, or multi-prover interactive proof systems with entangled provers, contains NEEXP, or nondeterministic doubly-exponential time (!!). I’ll try to blog about this later, but if you can’t wait, check out this excellent post by Thomas Vidick.

Congratulations!

Friday, April 5th, 2019

Congrats to Geoffrey Hinton, Yann LeCun, and Yoshua Bengio, who won the 2018 Turing Award for their work on deep learning (i.e., what used to be called neural nets). This might be the first Turing Award ever given for something where no one really understands why it works … and it’s years overdue.

Congrats to Avi Wigderson for winning the Knuth Prize. When I was asked to write a supporting nomination letter, my first suggestion was to submit a blank sheet of paper—since for anyone in theoretical computer science, there’s nothing that needs to be said about why Avi should win any awards we have. I hope Avi remains a guiding light of our community for many years to come.

And congrats to Mark Braverman for winning the Alan T. Waterman Award, one that I have some personal fondness for, along with materials scientist Jennifer Dionne. As Sasha Razborov once put it, after he (Sasha), I, and others recoiled from the task of proving the Linial-Nisan Conjecture, that polylog-wise independent distributions are indistinguishable from uniform by AC0 circuits, a “braver man” stepped in to do the job.

Beware of fake FOCS site!

Wednesday, April 3rd, 2019

As most of you in theoretical computer science will know, the submission deadline for the 2019 FOCS conference is this Friday, April 5. The FOCS’2019 program committee chair, my UT Austin colleague David Zuckerman, has asked me to warn everyone that a fake submission site was set up at aconf.org—apparently as a phishing scam—and is one of the first results to come up when you google “FOCS 2019.” Do not submit there! The true URL is focs2019.cs.jhu.edu; accept no substitutes!

Anyway, I’ve been thrashing for several weeks—just barely escaping spaghettification at the Email Event Horizon—but I hope to be back shortly with your regularly scheduled programming.

Four updates

Tuesday, February 12th, 2019

A few weeks ago, I was at QIP’2019 in Boulder, CO. This week I was at SQuInT’2019 in Albuquerque, NM. There were lots of amazing talks—feel free to ask in the comments section.

There’s an interview with me at the website “GigaOm,” conducted by Byron Reese and entitled Quantum Computing: Capabilities and Limits. I didn’t proofread the transcript and it has some errors in it, but hopefully the meaning comes through. In other interview news, if you were interested in my podcast with Adam Ford in Melbourne but don’t like YouTube, Adam has helpfully prepared transcripts of the two longest segments: The Ghost in the Quantum Turing Machine and The Winding Road to Quantum Supremacy.

The New York Times ran an article entitled The Hard Part of Computer Science? Getting Into Class, about the surge in computer science majors all over the US, and the shortage of professors to teach them. The article’s go-to example of a university where this is happening is UT Austin, and there’s extensive commentary from my department chair, Don Fussell.

The STOC’2019 accepted papers list is finally out. Lots of cool stuff!

The NP genie

Tuesday, December 11th, 2018

Hi from the Q2B conference!

Every nerd has surely considered the scenario where an all-knowing genie—or an enlightened guru, or a superintelligent AI, or God—appears and offers to answer any question of your choice.  (Possibly subject to restrictions on the length or complexity of the question, to prevent glomming together every imaginable question.)  What do you ask?

(Standard joke: “What question should I ask, oh wise master, and what is its answer?”  “The question you should ask me is the one you just asked, and its answer is the one I am giving.”)

The other day, it occurred to me that theoretical computer science offers a systematic way to generate interesting variations on the genie scenario, which have been contemplated less—variations where the genie is no longer omniscient, but “merely” more scient than any entity that humankind has ever seen.  One simple example, which I gather is often discussed in the AI-risk and rationality communities, is an oracle for the halting problem: what computer program can you write, such that knowing whether it halts would provide the most useful information to civilization?  Can you solve global warming with such an oracle?  Cure cancer?

But there are many other examples.  Here’s one: suppose what pops out of your lamp is a genie for NP questions.  Here I don’t mean NP in the technical sense (that would just be a pared-down version of the halting genie discussed above), but in the human sense.  The genie can only answer questions by pointing you to ordinary evidence that, once you know where to find it, makes the answer to the question clear to every competent person who examines the evidence, with no further need to trust the genie.  Or, of course, the genie could fail to provide such evidence, which itself provides the valuable information that there’s no such evidence out there.

More-or-less equivalently (because of binary search), the genie could do what my parents used to do when my brother and I searched the house for Hanukkah presents, and give us “hotter” or “colder” hints as we searched for the evidence ourselves.

To make things concrete, let’s assume that the NP genie will only provide answers of 1000 characters or fewer, in plain English text with no fancy encodings.  Here are the candidates for NP questions that I came up with after about 20 seconds of contemplation:

  • Which pieces of physics beyond the Standard Model and general relativity can be experimentally confirmed with the technology of 2018? What are the experiments we need to do?
  • What’s the current location of the Ark of the Covenant, or its remains, if any still exist?  (Similar: where can we dig to find physical records, if any exist, pertaining to the Exodus from Egypt, or to Jesus of Nazareth?)
  • What’s a sketch of a resolution of P vs. NP, from which experts would stand a good chance of filling in the details?  (Similar for other any famous unsolved math problem.)
  • Where, if anywhere, can we point radio telescopes to get irrefutable evidence for the existence of extraterrestrial life?
  • What happened to Malaysia Flight 370, and where are the remains by which it could be verified?  (Similar for Amelia Earhart.)
  • Where, if anywhere, can we find intact DNA of non-avian dinosaurs?

Which NP questions would you ask the genie?  And what other complexity-theoretic genies would be interesting to consider?  (I thought briefly about a ⊕P genie, but I’m guessing that the yearning to know whether the number of sand grains in the Sahara is even or odd is limited.)


Update: I just read Lenny Susskind’s Y Combinator interview, and found it delightful—pure Lenny, and covering tons of ground that should interest anyone who reads this blog.

Ten updates

Wednesday, November 7th, 2018

If you like quantum, complexity, etc., then please read to the end! I’ve gotten a bunch of emails lately of the form “why haven’t you ever blogged about such-and-such?,” when it turned out that I damn well did blog about it; it was just somewhere down in a multi-item post.

1. Like many of you, I watched the US midterm election results with (mostly…) disappointment and dismay.  I think that history will judge us harshly for not totally and unequivocally rebuking everything Trump stands for and every politician associated with him.  But that’s not what I wanted to blog about today.

2. There was a breakthrough in communication complexity by Arkadev Chattopadhyay, Nikhil Mande, and Suhail Sherif: the first exponential separation between randomized communication complexity and log approximate rank for a total Boolean function f.  This falsifies the longstanding conjecture that these measures are polynomially related (though it doesn’t resolve the original log rank conjecture).  For those of you keeping score at home, the quantum communication complexity of f is sandwiched in between randomized CC and log approximate rank.  So, at least one of the following must now be true: either randomized CC is exponentially separated from quantum CC, or else quantum CC is exponentially separated from log approximate rank.  My money’s on the latter.

3. Ewin Tang, who achieved fame with a quantum-inspired classical algorithm for recommendation systems (which I blogged about in July), is now back with quantum-inspired classical algorithms for principal component analysis and supervised clustering.  Well, with the announcements of such algorithms; details of the analysis are to come later.

4. A bunch of people asked me about the paper by Sergey Bravyi, David Gosset, and Robert Koenig, Quantum advantage with shallow circuits.  tl;dr: it’s great!  And it was deservedly a highlight of the QIP conference back in January!  That’s why it confused me when everyone started asking about it a couple weeks ago.  The resolution is that the paper was just recently published in Science magazine, which led to popular coverage like this, which in turn led to people asking me whether this result unconditionally proves P≠BQP (that is, quantum computers can solve more problems in polynomial time than classical computers), and if not why not.  The answer is no: the paper proves an unconditional separation, but one that’s a long long way from P≠BQP, or anything else that would entail solving the central open problems of complexity theory like P vs. PSPACE.  Basically, it shows there are problems solvable in constant time with a quantum computer that aren’t solvable in constant time classically, for suitable meanings of “problem” (namely, a relation problem) and “in constant time” (namely, NC0 circuits, in which each output bit depends on only a constant number of input bits).  I understand that a stronger separation has since been achieved, between quantum NC0 and classical AC0, in work that’s not yet on the arXiv.  The problems in question, however, are all easy to solve in P, or even in classical logarithmic time, given a polynomial number of parallel processors.

5. A bunch of people also asked me about the paper by Xun Gao and Luming Duan, Efficient classical simulation of noisy quantum computation.  This paper tries to formalize something that many of us have suspected/feared for years, namely that random quantum circuits (the whole thing is specific to random circuits) can tolerate only a tiny amount of noise and decoherence before they become efficiently simulable classically.  If true, this has obvious implications for the sampling-based quantum supremacy experiments that Google and others are planning for the next few years: namely, that all the engineering effort they’ve already been investing anyway to push down the noise rate, will actually be necessary!  However, correspondence with the authors revealed that there’s a significant gap in the analysis as it currently stands: namely, the current proof applies only to closed quantum systems, which would (for example) rule out all the techniques that people eventually hope to use to achieve quantum fault-tolerance—all of which are based on constantly measuring subsets of the qubits, doing essentially error-free classical computation on the measurement outcomes, throwing away noisy qubits, and pumping in fresh qubits.  Xun and Duan say that they’re currently working on an extension to open systems; in my personal view, such an extension seems essential for this interesting result to have the informal interpretation that the authors want.

6. My friend Bram Cohen asked me to announce that his company, Chia, has launched a competition for best implementation of its Verifiable Delay Functions (VDFs), with real money rewards.  You can find the details at this Github page.

7. The second Q2B (“Quantum 2 Business”) conference, organized by QC Ware Corp., will be held this coming December 10-12, at the Computer History Museum in Mountain View.  There will be two keynote addresses, one by John Preskill and the other by me.  I hope I’ll get a chance to meet some of you there!

8. Longtime colleague and friend-of-the-blog Ashwin Nayak asked me to announce that the 2019 Conference on Computational Complexity, to be held July 18-20 in exciting New Brunswick, NJ, is now accepting submissions.  I hope to be there!

9. OK, what the hell: the 21st annual, and nearly impossible to capitalize correctly, SQuInT (Southwest Quantum Information and Technology) workshop will be held February 2019 in Albuquerque, NM.  UT Austin is now a node of the SQuInT network, and I’ll hopefully be attending along with a posse of students and postdocs.  The deadline for abstract submission is coming up soon: Monday November 12!

10. I went to morning Shabbat services in Austin this past weekend, exactly one week after the tragedy in Pittsburgh.  There was massively increased security, with armed guards interrogating us, Israeli-style, about why we had no membership sticker on our car, whether we knew the name of the rabbi, etc.  Attendance was maybe a factor of three higher than usual.  About thirty priests, ministers, and Christian seminary students, and one Muslim, came up to the bima to say a prayer of solidarity with Jews.  The mayor of Austin, Steve Adler, was also on hand to give a speech.  Then the rabbi read a letter to the synagogue by Sen. Ted Cruz denouncing antisemitism (well, parts of it; he said the letter was really long).  There were murmurs of disapproval from the crowd when Cruz’s name was mentioned, but then everyone quieted down and listened.  The thing is, the US and large parts of the world are now so far outside the norms of liberal democracy, in territory so terrifyingly uncharted since the end of World War II, that shooting up synagogues is bad is actually something that it’s better than not for powerful people to affirm explicitly.  Anyway, while I’m neither a believer nor much of a synagogue-goer, I found the show of unity moving.

CS and quantum information at UT Austin: come join us!

Wednesday, September 19th, 2018

Merry Yom Kippur!

This is my annual post where I tell you about opportunities available at UT Austin, which has long been a safe space for CS research, and which we hope will rapidly become (or return to its historical role as…) a safe space for quantum computing and information.

If you’re interested in faculty positions in computer science at UT, I have some great news: we plan to do a lot of hiring this year!  Because of the sheer volume of interviews we’ll be doing, we’d like to start our recruiting season already in the fall.  So we’re extending an unusual invitation: if you already have your materials ready, we encourage you to apply for faculty positions right now.  If you’re chosen for an interview, we could schedule it for the next few months.

We’ll be looking for great candidates across all parts of CS, but one particular interest is hiring another quantum computing theorist in CS (i.e., besides me), most likely a junior person.  While not everyone who reads this blog is a plausible candidate, and not every plausible candidate reads this blog, the intersection is surely non-negligible!  So again: we encourage you to apply right now, so we can start scheduling interviews already.

I’m also on the lookout for postdocs, mainly in theoretical quantum computing and information.  (I, and others in the theory group, are also collectively interested in postdocs in classical computational complexity.)  If you’re interested in doing a postdoc with me starting in Fall 2019, the procedure, like in previous years, is this:

  • Email me introducing yourself (if I don’t already know you), and include your CV and up to three representative papers.  Do this even if you already emailed me before.
  • Arrange for two recommendation letters to be emailed to me.

We’ll set a deadline for this of December 15.

Finally, if you’re interested in pursuing a PhD in CS at UT, please apply here!  The deadline, again, is December 15.  Just like every year, I’m on the lookout for superb, complexity-loving, quantum- or quantum-curious, lower-bound-hungry students of every background, and if you specify that you want to work with me, I’ll be sure to see your application.  Emailing me won’t help: everything is done through the application process.

As we like to say down here in Texas, hook ’em Hadamards!  (Well OK, no, we don’t especially like to say that.  It’s just a slogan that I found amusing a few years ago.)

My Tomassoni-Chisesi Prize talk

Saturday, September 15th, 2018

Update (Sep. 21) Video of Philip Kim’s and my talks is now available! (But not streaming, just a giant mp4 that you can download.)


On Thursday, I had the incredible honor of accepting the 2018 Tomassoni-Chisesi Prize in Physics at Università “La Sapienza” in Rome—“incredible” mostly because I’m of course not a physicist.  (I kept worrying they’d revoke the award when they realized I could barely solve the wave equation.)  This is not the first time quantum information was recognized; the prize has previously gone to Serge Haroche and Alain Aspect.  This year, for the first time, there was both an under-40 and an over-40 award; the latter went to Philip Kim, a quantum materials researcher at Harvard who I had the privilege to meet on this trip (he’s the taller one below).

I’m unbelievably grateful, not only to the committee, and its chair Giorgio Parisi (whose seminal work on phase transitions and satisfiability I’d long known, but who I met for the first time on this trip), but to Fabio Sciarrino, Paolo Mataloni, Fernanda Lupinacci, and everyone else who graciously hosted me and helped make my hastily-planned visit to Europe a success.

The department I visited has a storied history: here are the notes that Enrico Fermi left, documenting what he covered each day in his physics class in 1938.  The reason the last squares are blank is that, when Fermi and his Jewish wife left for Stockholm on the occasion of Fermi’s Nobel Prize, they continued directly to the US rather than return to an Italy that had just passed the racial laws.

On my way to Rome, I also gave two talks at a “quantum computing hackathon” in Zurich, called QuID (Quantum Information for Developers).  Thanks so much to Lidia del Rio for arranging that visit, which was fantastic as well.

To accept the Tomassoni-Chisesi prize, I had to give a 40-minute talk summarizing all my research from 2000 to the present—the hardest part being that I had to do it while wearing a suit, and sweating at least half my body weight.  (I also had a cold and a hacking cough.)  I think there will eventually be video of my and Prof. Kim’s talks, but it’s not yet available.  In the meantime, for those who are interested, here are my PowerPoint slides, and here’s the title and abstract:

Three Questions About Quantum Computing
Scott Aaronson (University of Texas at Austin)

I’ll discuss some of my work in quantum computing over the past 18 years, organizing it in terms of three questions.  First, how can we demonstrate, using near-future hardware, that quantum computers can get any genuine speedups at all over classical computers (ideally useful speedups)?  Second, what sorts of problems would be hard even for quantum computers, and can we turn the intractability of those problems to our advantage?  Third, are there physically reasonable models of computation even more powerful than quantum computing, or does quantum computing represent an ultimate limit?

If you’re a regular reader here, most of the content will be stuff you’ve seen before, with the exception of a story or two like the following:

Last night I was talking to my mom about my grandfather, who as it happens came through Rome 73 years ago, as an engineer with the US Army.  Disabling landmines was, ironically, one of the safer ways to be a Jew in Europe at that time.  If you’d told him then that, three-quarters of a century later, his grandson would be back here in Rome to accept an award for research in quantum computational complexity … well, I’m sure he’d have any number of questions about it.  But one thing I clearly remember is that my grandfather was always full of effusive praise for the warmth of the people he met in Italy—how, for example, Italian farmers would share food with the hungry and inadequately-provisioned Allied soldiers, despite supposedly being on the opposing side.  Today, every time I’m in Italy for a conference or a talk, I get to experience that warmth myself, and certainly the food part.

(Awww!  But I meant it.  Italians are super-warm.)

There’s a view that scientists should just pursue the truth and be serenely unaffected by prizes, recognition, and other baubles.  I think that view has a great deal to be said for it.  But thinking it over recently, I struck the following mental bargain: if I’m going to get depressed on a semi-regular basis by people attacking me online—and experience shows that I will—well then, I also get to enjoy whatever’s the opposite of that with a clear conscience.  It’s not arrogance or self-importance; it’s just trying to balance things out a bit!

So again, thanks so much—to the physics department of La Sapienza, but also to my family, friends, mentors, readers, colleagues at UT Austin and around the world, and everyone else who helps make possible whatever it is that I do.

Lecture notes! Intro to Quantum Information Science

Sunday, August 26th, 2018

Someone recently wrote that my blog is “too high on nerd whining content and too low on actual compsci content to be worth checking too regularly.”  While that’s surely one of the mildest criticisms I’ve ever received, I hope that today’s post will help to even things out.

In Spring 2017, I taught a new undergraduate course at UT Austin, entitled Introduction to Quantum Information Science.  There were about 60 students, mostly CS but also with strong representation from physics, math, and electrical engineering.  One student, Ewin Tang, made a previous appearance on this blog.  But today belongs to another student, Paulo Alves, who took it upon himself to make detailed notes of all of my lectures.  Using Paulo’s notes as a starting point, and after a full year of procrastination and delays, I’m now happy to release the full lecture notes for the course.  Among other things, I’ll be using these notes when I teach the course a second time, starting … holy smokes … this Wednesday.

I don’t pretend that these notes break any new ground.  Even if we restrict to undergrad courses only (which rules out, e.g., Preskill’s legendary notes), there are already other great quantum information lecture notes available on the web, such as these from Berkeley (based on a course taught by, among others, my former adviser Umesh Vazirani and committee member Birgitta Whaley), and these from John Watrous in Waterloo.  There are also dozens of books—including Mermin’s, which we used in this course.  The only difference with these notes is that … well, they cover exactly the topics I’d cover, in exactly the order I’d cover them, and with exactly the stupid jokes and stories I’d tell in a given situation.  So if you like my lecturing style, you’ll probably like these, and if not, not (but given that you’re here, there’s hopefully some bias toward the former).

The only prerequisite for these notes is some minimal previous exposure to linear algebra and algorithms.  If you read them all, you might not be ready yet to do research in quantum information—that’s what a grad course is for—but I feel good that you’ll have an honest understanding of what quantum information is all about and where it currently stands.  (In fact, where it already stood by the late 1990s and early 2000s, but with many comments about the theoretical and experimental progress that’s been made since then.)

Also, if you’re one of the people who read Quantum Computing Since Democritus and who was disappointed by the lack of basic quantum algorithms in that book—a function of the book’s origins, as notes of lectures given to graduate students who already knew basic quantum algorithms—then consider these new notes my restitution.  If nothing else, no one can complain about a dearth of basic quantum algorithms here.

I welcome comments, bugfixes, etc.  Thanks so much, not only to Paulo for transcribing the lectures (and making the figures!), but also to Patrick Rall and Corey Ostrove for TA’ing the course, to Tom Wong and Supartha Podder for giving guest lectures, and of course, to all the students for making the course what it was.

  • Lecture 1: Course Intro, Church-Turing Thesis (3 pages)
  • Lecture 2: Probability Theory and QM (5 pages)
  • Lecture 3: Basic Rules of QM (4 pages)
  • Lecture 4: Quantum Gates and Circuits, Zeno Effect, Elitzur-Vaidman Bomb (5 pages)
  • Lecture 5: Coin Problem, Inner Products, Multi-Qubit States, Entanglement (5 pages)
  • Lecture 6: Mixed States (6 pages)
  • Lecture 7: Bloch Sphere, No-Cloning, Wiesner’s Quantum Money (6 pages)
  • Lecture 8: More on Quantum Money, BB84 Quantum Key Distribution (5 pages)
  • Lecture 9: Superdense Coding (2 pages)
  • Lecture 10: Teleportation, Entanglement Swapping, GHZ State, Monogamy (5 pages)
  • Lecture 11: Quantifying Entanglement, Mixed State Entanglement (4 pages)
  • Lecture 12: Interpretation of QM (Copenhagen, Dynamical Collapse, MWI, Decoherence) (10 pages)
  • Lecture 13: Hidden Variables, Bell’s Inequality (5 pages)
  • Lecture 14: Nonlocal Games (7 pages)
  • Lecture 15: Einstein-Certified Randomness (4 pages)
  • Lecture 16: Quantum Computing, Universal Gate Sets (8 pages)
  • Lecture 17: Quantum Query Complexity, Deutsch-Jozsa (8 pages)
  • Lecture 18: Bernstein-Vazirani, Simon (7 pages)
  • Lecture 19: RSA and Shor’s Algorithm (6 pages)
  • Lecture 20: Shor, Quantum Fourier Transform (8 pages)
  • Lecture 21: Continued Fractions, Shor Wrap-Up (4 pages)
  • Lecture 22: Grover (9 pages)
  • Lecture 23: BBBV, Applications of Grover (7 pages)
  • Lecture 24: Collision and Other Applications of Grover (6 pages)
  • Lecture 25: Hamiltonians (10 pages)
  • Lecture 26: Adiabatic Algorithm (10 pages)
  • Lecture 27: Quantum Error Correction (8 pages)
  • Lecture 28: Stabilizer Formalism (9 pages)
  • Lecture 29: Experimental Realizations of QC (9 pages)

And by popular request, here are the 2017 problem sets!

I might post solutions at a later date.

Note: If you’re taking the course in 2018 or a later year, these sets should be considered outdated and for study purposes only.


Notes and Updates (Aug. 27)

Here’s a 184-page combined file. Thanks so much to Robert Rand, Oscar Cunningham, Petter S, and Noon van der Silk for their help with this.

If it wasn’t explicit: these notes are copyright Scott Aaronson 2018, free for personal or academic use, but not for modification or sale.

I’ve freely moved material between lectures so that it wasn’t arbitrarily cut across lecture boundaries. This is one of the reasons why some lectures are much longer than others.

I apologize that some of the displayed equations are ugly. This is because we never found an elegant way to edit equations in Google Docs.

If you finish these notes and are still hankering for more, try my Quantum Complexity Theory or Great Ideas in Theoretical Computer Science lecture notes, or my Barbados lecture notes.  I now have links to all of them on the sidebar on the right.