Thanksgiving
I’m thankful to the thousands of readers of this blog. Well, not the few who submit troll comments from multiple pseudonymous handles, but the 99.9% who don’t. I’m thankful that they’ve stayed here even when events (as they do more and more often) send me into a spiral of doomscrolling and just subsisting hour-to-hour—when I’m left literally without words for weeks.
I’m thankful for Thanksgiving itself. As I often try to explain to non-Americans (and to my Israeli-born wife), it’s not primarily about the turkey but rather about the sides: the stuffing, the mashed sweet potatoes with melted marshmallows, the cranberry jello mold. The pumpkin pie is good too.
I’m thankful that we seem to be on the threshold of getting to see the birth of fault-tolerant quantum computing, nearly thirty years after it was first theorized.
I’m thankful that there’s now an explicit construction of pseudorandom unitaries — and that, with further improvement, this would lead to a Razborov-Rudich natural proofs barrier for the quantum circuit complexity of unitaries, explaining for the first time why we don’t have superpolynomial lower bounds for that quantity.
I’m thankful that there’s been recent progress on QMA versus QCMA (that is, quantum versus classical proofs), with a full classical oracle separation now possibly in sight.
I’m thankful that, of the problems I cared about 25 years ago — the maximum gap between classical and quantum query complexities of total Boolean functions, relativized BQP versus the polynomial hierarchy, the collision problem, making quantum computations classically verifiable — there’s now been progress if not a full solution for almost all of them. And yet I’m thankful as well that lots of great problems remain open.
I’m thankful that the presidential election wasn’t all that close (by contemporary US standards, it was a ““landslide,”” 50%-48.4%). Had it been a nail-biter, not only would I fear violence and the total breakdown of our constitutional order, I’d kick myself that I hadn’t done more to change the outcome. As it is, there’s no denying that a plurality of Americans actually chose this, and now they’re going to get it good and hard.
I’m thankful that, while I absolutely do see Trump’s return as a disaster for the country and for civilization, it’s not a 100% unmitigated disaster. The lying chaos monster will occasionally rage for things I support rather than things I oppose. And if he actually plunges the country into another Great Depression through tariffs, mass deportations, and the like, hopefully that will make it easier to repudiate his legacy in 2028.
I’m thankful that, whatever Jews around the world have had to endure over the past year — both the physical attacks and the moral gaslighting that it’s all our fault — we’ve already endured much worse on both fronts, not once but countless times over 3000 years, and this is excellent Bayesian evidence that we’ll survive the latest onslaught as well.
I’m thankful that my family remains together, and healthy. I’m thankful to have an 11-year-old who’s a striking wavy-haired blonde who dances and does gymnastics (how did that happen?) and wants to be an astrophysicist, as well as a 7-year-old who now often beats me in chess and loves to solve systems of two linear equations in two unknowns.
I’m thankful that, compared to what I imagined my life would be as an 11-year-old, my life is probably in the 50th percentile or higher. I haven’t saved the world, but I haven’t flamed out either. Even if I do nothing else from this point, I have a stack of writings and results that I’m proud of. And I fully intend to do something else from this point.
I’m thankful that the still-most-powerful nation on earth, the one where I live, is … well, more aligned with good than any other global superpower in the miserable pageant of human history has been. I’m thankful to live in the first superpower in history that has some error-correction machinery built in, some ability to repudiate its past sins (and hopefully its present sins, in the future). I’m thankful to live in the first superpower that has toleration of Jews and other religious minorities built in as a basic principle, with the possible exception of the Persian Empire under Cyrus.
I’m thankful that all eight of my great-grandparents came to the US in 1905, back when Jewish mass immigration was still allowed. Of course there’s a selection effect here: if they hadn’t made it, I wouldn’t be here to ponder it. Still, it seems appropriate to express gratitude for the fact of existing, whatever metaphysical difficulties might inhere in that act.
I’m thankful that there’s now a ceasefire between Israel and Lebanon that Israel’s government saw fit to agree to. While I fear that this will go the way of all previous ceasefires — Hezbollah “obeys” until it feels ready to strike again, so then Israel invades Lebanon again, then more civilians die, then there’s another ceasefire, rinse and repeat, etc. — the possibility always remains that this time will be the charm, for all people on both sides who want peace.
I’m thankful that our laws of physics are so constructed that G, c, and ℏ, three constants that are relatively easy to measure, can be combined to tell us the fundamental units of length and time, even though those units — the Planck time, 10-43 seconds, and the Planck length, 10-33 centimeters — are themselves below the reach of any foreseeable technology, and to atoms as atoms are to the solar system.
I’m thankful that, almost thirty years after I could have and should have, I’ve now finally learned the proof of the irrationality of π.
I’m thankful that, if I could go back in time to my 14-year-old self, I could tell him firstly, that female heterosexual attraction to men is a real phenomenon in the world, and secondly, that it would sometimes fixate on him (the future him, that is) in particular.
I’m thankful for red grapefruit, golden mangos, seedless watermelons, young coconuts (meat and water), mangosteen, figs, dates, and even prunes. Basically, fruit is awesome, the more so after whatever selective breeding and genetic engineering humans have done to it.
I’m thankful for Futurama, and for the ability to stream every episode of it in order, as Dana, the kids, and I have been doing together all fall. I’m thankful that both of my kids love it as much as I do—in which case, how far from my values and worldview could they possibly be? Even if civilization is destroyed, it will have created 100 episodes of something this far out on the Pareto frontier of lowbrow humor, serious intellectual content, and emotional depth for a future civilization to discover. In short: “good news, everyone!”
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Comment #1 November 28th, 2024 at 1:00 pm
Happy Thanksgiving 🙏❤️🦃❗️
Comment #2 November 28th, 2024 at 1:12 pm
Happy Thanksgiving! I’m glad to see that you’re handling the aftermath of the election okay.
I’m sure plenty of people have asked you this and I don’t want to be annoying, but are you planning to write a blog post about Google’s AlphaQubit? I read their announcement but was not sure how to situate it in the context of the general challenges around quantum error correction that you’ve written about in the past.
Comment #3 November 28th, 2024 at 2:49 pm
Thank *you* for this blog post, Scott, and for the blog in general.
Comment #4 November 28th, 2024 at 3:12 pm
Taymon A. Beal #2: Here’s what I wrote to a journalist who asked me:
I think it’s tremendously exciting!
It’s been clear for a while that decoding and correcting the errors quickly enough, in a fault-tolerant quantum computation, was going to push classical computing to the limit also. It’s also become clear that for just about anything classical computers do involving optimization or uncertainty, you can now throw ML at it and they might do it better. So using neural nets to decode the errors in a QC is a natural enough idea — but what’s notable here is that they already get state-of-the-art performance this way. I could easily imagine that this is just how people will handle decoding going forward.
Comment #5 November 28th, 2024 at 6:21 pm
“I haven’t saved the world, but I haven’t flamed out either.” S.A.
Me too..
Happy Thanksgiving Brother.. Cheers from a land ruled by muslim dictator!
Comment #6 November 28th, 2024 at 6:50 pm
Happy Thanksgiving, Scott. 🙂
Comment #7 November 28th, 2024 at 9:01 pm
Good to see you’re back at it. By the way, Cambridge University Press has just released the online version of Steven Weinberg: A Life in Physics. Enjoy.
Comment #8 November 28th, 2024 at 9:58 pm
I’m going to be rereading this a bunch of times in the near future. Always very much appreciate reading your thoughts.
Comment #9 November 28th, 2024 at 10:00 pm
From a passive reader the highest gratitude to you Scott for writing this blog. Intellectually honest, with moral clarity, captivating, of course educating, and somehow heartwarming, too!
Comment #10 November 29th, 2024 at 12:01 am
Hi Scott,
Happy Thanksgiving!
Thank you for this post 🙂 and for your blog too!!
Comment #11 November 29th, 2024 at 8:41 am
Glad to get you back for another new great post!
Happy Thanksgiving!
Comment #12 November 29th, 2024 at 10:20 am
Thanks for writing this, Scott.
My older granddaughter is also 11, and also interested in a wide range of physical and intellectual pursuits. A (generally) happy and active pre-teen, given the resources she needs to thrive, is a wonder. My younger granddaughter is also 7, and is also working to figure out what sort of person she’ll be. (Having a physicist and a librarian for parents has, I suspect, been useful to them in this process, though of course there are many roads through childhood.)
I hope we’ll be able to hand them all a viable future, and whatever the troubles of the present, I think that’s worth striving for.
Comment #13 November 29th, 2024 at 10:47 am
You can get the Queen of Fruits (mangosteen) in Austin?
Now that’s something to be thankful for! I’ve never seen it in the US.
Comment #14 November 29th, 2024 at 11:01 am
Happy Thanksgiving, Scott!
What proof of the irrationality of pi did you learn? I’m aware of one proof with crazy integrals and another with crazy continued fractions.
Comment #15 November 29th, 2024 at 11:46 am
> Even if I do nothing else from this point, I have a stack of writings and results that I’m proud of. And I fully intend to do something else from this point.
I am thankful for you willing to live your life out loud for the rest of us, despite the enormous psychic cost.
I am thankful that you didn’t give in to despair, but continue to find reasons to be thankful.
Thank you.
Love, Ernie
Comment #16 November 29th, 2024 at 12:02 pm
One of the many things I am thankful for is this blog.
Comment #17 November 29th, 2024 at 12:12 pm
thank you Scott, for this and the blog generally.
“I haven’t saved the world, but I haven’t flamed out either.”
is where I am too..
When, in disgrace with fortune and men’s eyes
then I remember I could have done much worse, and fight on..
Comment #18 November 29th, 2024 at 12:32 pm
Thanks so much for the kind words, everyone!!
Comment #19 November 29th, 2024 at 12:35 pm
Jimmy Koppel #13: No, I’ve never seen it fresh here, but I love the freeze-dried mangosteen at Trader Joe’s. I like it even more now that TJ’s has stood down a massive pressure campaign to stop selling Israeli products, telling the protesters to take a hike.
Comment #20 November 29th, 2024 at 4:30 pm
Very nice post and considering the entirety of human history we have so much to be thankful for and too much of it simply the unnoticed way of things.
Comment #21 November 29th, 2024 at 5:29 pm
Martin Mertens #14: Greg Kuperberg has given me permission to share the following beautiful argument of his, which is a modification of an argument of Niven, which in turn is a modification of the original argument of Hermite.
Let
In = integral from 0 to π of xn (π-x)n / n! (sin x) dx
Here are some of its values:
I5 = 60π4 – 6720π2 + 60480 ~ 0.80
I6 = -2π6 + 1680π4 – 151200π2 + 1330560 ~ 0.31
I7 = -112π6 + 50400π4 – 3991680π2 + 34594560 ~ 0.10
I8 = 2π8 – 5040π6 + 1663200π4 – 121080960π2 + 1037836800 ~ 0.030
On the one hand, In = pn(π) is an integer polynomial in π of degree at most n, in fact π2.
You can prove this with iterated integration by parts, which eats the n! denominator before producing any non-zero terms.
On the other hand, the values start to fall rapidly. The integrand is positive and unimodal, and the value in the middle is (π/2)2n/n!. Thus, In converges to zero at a superexponential rate, more precisely at a factorial rate.
It follows that π (moreover π2) cannot be a rational number a/b, because if pn(x) is any sequence of integer polynomials of degree n with pn(a/b) > 0, then pn(a/b) ≥ 1/bn. I.e., if pn(a/b) converges to zero from above, then the rate of convergence is at most exponential.
Comment #22 November 29th, 2024 at 8:34 pm
Now Demis is taking AlphaQubit to next level and predicting QC is irrelevant compared to ML ?
Comment #23 November 29th, 2024 at 9:21 pm
Prasanna #22:
No. AlphaQubit is a decoder. It is a technology that enables quantum computing.
Demis’ comments are that classical techniques could have more applications to modeling quantum systems. This is known – things like DFT have been around for decades. ML approaches are an interesting somewhat new approach which could expand the scope of these classical techniques.
Comment #24 November 30th, 2024 at 3:58 am
Happy Thanksgiving, Scott ; I was surprised you didn’t mention ChatGPT in your list. Somehow, lots of people I respect seem to actively resist it, from dismissing it as a word corrector on steroids to predicting it will never be able to do things it usually does in the next version. Meanwhile, I am almost every day surprised by some of its productions, from plausible explanations of some discrepancies in a minor book to its capacity to find the least upper bound of two elements in a Boolean ring (a+b-ab) without knowing it is one (i.e. without cheating)
Comment #25 November 30th, 2024 at 8:22 am
What further improvement on pseudorandom unitaries would lead to the natural proofs barrier for unitary circuit complexity?
Comment #26 November 30th, 2024 at 10:48 am
Hi Scott,
“as well as a 7-year-old who … loves to solve systems of two linear equations in two unknowns” – cool!!
Of course Daniel is your and Dana’s son, but still, would you mind sharing with us the pedagogical tricks and principles that you follow with him (and Lily too)?
(Hey, if there are enough things to say about this then how about a separate post in itself, please? 🙂 – we don’t get to see how theoretical scientists teach their kids usually)
Comment #27 November 30th, 2024 at 2:56 pm
Tachyonic message from 2025/11/27:
I am thankful that Trump has ended the Ukrainian war.
Comment #28 November 30th, 2024 at 6:28 pm
Ashley #26: I haven’t been doing anything the slightest bit systematic with Daniel, even if I probably should’ve been. Mostly he just asks for a math problem when he’s bored (and doesn’t have his iPad for Roblox or Minecraft). So then I give him one—e.g., what’s the prime factorization of 98? A rectangle has area 50 and perimeter 30; what are the sides? X+Y=10 and X-Y=2; what are X and Y? And then if he solves it quickly I give him a harder one, etc.
Comment #29 November 30th, 2024 at 7:00 pm
Greg Rosenthal #25: Well, the current result gives a 2n×2n unitary that’s pseudorandom against a roughly 2n/2-time adversary—so, one who doesn’t even have time to look at the whole matrix. To get a Razborov-Rudich barrier, you’d need to improve that to get security against a 2p(n)-time adversary, for any polynomial p (which in turn would require arbitrary polynomial-size seeds).
Note: I learned all this when Fermi Ma came here to UT and gave a talk about the recent breakthroughs on this! But any errors are mine of course.
Comment #30 December 1st, 2024 at 3:01 am
I was waiting for a new post-election post from you, happy it’s finally here and that it’s full of positivity.
I’m not American, but I’ll jump on the Thanksgiving bandwagon to say I’m thankful for this blog, and for you Scott. I’m sure you face plenty of trolls, but I’m sure I speak for many when I say that this blog is very important for many of us. It certainly is for me.
Scott #28:
As for pedagogical methods, how did this giving a math problem thing even get started? Do you remember at what age? I can’t really imagine my 8-year-old being too happy with me giving him school-like problems, because he doesn’t love school.
Also, I assume the level of problems they are solving is higher than the current level in their class, doesn’t that present a problem?
Comment #31 December 1st, 2024 at 7:34 am
Happy Thanksgiving! I’m thankful for you and your blog, which for me has been a constant source of sanity and intellectual enrichment.
Comment #32 December 1st, 2024 at 8:55 am
I’m pretty sure the proof presented in comment 21 is the proof in Bourbaki (you can also see the wikipedia page: https://en.wikipedia.org/wiki/Proof_that_π_is_irrational#Bourbaki's_proof)
Comment #33 December 1st, 2024 at 10:30 am
Anonymous #32: Thanks!! Yes, Bourbaki’s proof and Greg’s are essentially the same (the only difference being whether you multiply the integral by bn to get an integer).
One thing I’m still wondering about: is there a calculus-free proof of the irrationality of π? Or at least, a proof that makes it more transparent what property of π is being exploited, rather than burying it in an integration by parts involving the sine function happening to work a certain way and sine having a period involving π?
Comment #34 December 1st, 2024 at 12:47 pm
The great Mathologer explains the Hermite-Niven-Zhou-… proof of pi’s irrationality in this terrific video: https://www.youtube.com/watch?v=jGZtVl4XfGo
Comment #35 December 1st, 2024 at 1:03 pm
Well, Mathologer earlier gives this more ‘motivated’ proof, which is Hermite’s original:
The short form is:
. tan(x) has a nice continued fraction form (following from power series for sin and cos)
. by continued-fraction-type reasoning (kinda of the sort used to show e is irrational), tan(u/v) is irrational for any (nonzero) rational u/v
. but tan(pi/4) = 1.
It’s reasonable to take the last fact as the *definition* of pi. (I mean, if you want to define pi in terms of area/circumference of a circle and be formal about it, you need to introduce integrals.)
So this proof has no calculus per se, just the definition of pi in terms of the trig functions (which are defined via power series).
Comment #36 December 1st, 2024 at 1:25 pm
Scott #33:
“One thing I’m still wondering about: is there a calculus-free proof of the irrationality of π? Or at least, a proof that makes it more transparent what property of π is being exploited, rather than burying it in an integration by parts involving the sine function happening to work a certain way and sine having a period involving π?”
There isn’t a great rigorous definition of π that doesn’t involve at least some calculus. One might hope for something like the standard proof of the irrationality of e where one just needs to agree that the series converges and manipulate it accordingly. However, that sort of proof relies generally on a pretty fast converging series or one with fast growth of the denominators, and π doesn’t seem to have any obvious series representation that grows fast enough for that to work.
Comment #37 December 1st, 2024 at 1:56 pm
Joshua Zelinsky #36: Thanks! Given that there’s a geometric proof of the irrationality of √2, and given how much is done with π in Euclidean geometry with no calculus, I had hoped that there might be a geometric proof of the irrationality of π, but maybe that’s too much to ask!
Comment #38 December 1st, 2024 at 2:27 pm
Ryan O’Donnell: Thanks for the links—I’ll watch both!
Comment #39 December 1st, 2024 at 6:34 pm
Is there any known and natural irrational number whose irrationality can’t be simply proved by looking at its continued fractions expansion and observing that it is not finite? (there are tricky cases, such as Apéry’s constant.) Any number that can be expressed by a hypergeometric function (and that’s a huge category!) would automatically fall into that framework. Important exceptions include such constants as the Euler–Mascheroni constant, whose irrationality remains open.
Comment #40 December 1st, 2024 at 9:32 pm
Mahdi #39: Not sure I understand. Even once you’ve proven that a given infinite continued fraction expansion is correct, there’s still the problem of proving that it’s not “secretly” rational (as they can be).
In the case of Lambert’s original proof of the irrationality of tan(r) when r is a nonzero rational (and as a byproduct, the irrationality of π), this latter part (as I just learned today!) requires a special argument involving the magnitudes of the terms in the continued fraction.
Comment #41 December 2nd, 2024 at 8:08 am
you wrote ” it’s not primarily about the turkey but rather about the sides: the stuffing, the mashed sweet potatoes with melted marshmallows, the cranberry jello mold.” i agree but i would also include the turkey tail (note: you can buy turkey tail in supermarkets but it is always sold after being smoked and i dislike smoked food)
Comment #42 December 2nd, 2024 at 11:43 am
Scott #40: That’s what I mean. A simple continued fraction is finite iff the number is irrational. Constants such as Pi don’t have a well-understood simple continued fraction so one has to look at continued fractions of Gauss type (that works for all hypergeometric functions) and then argue that the rationality requiring finiteness still holds for those. That can be tricky at times but the general recipe is still: Look at some continued fraction and use the fact that it doesn’t terminate. Is there a nontrivial known irrational number whose irrationality proof can’t be cast in this framework?
Comment #43 December 2nd, 2024 at 2:03 pm
@Mahdi #40
“Is there a nontrivial known irrational number whose irrationality proof can’t be cast in this framework?”
Seems like the answer is yes. Take your favorite(computable ordering of Turing machines, with t(n) being the nth Turing machine, and let h(n) be the function that is zero when t(n) halts on the blank tape and one when it does not. Then consider the number c, where c is given by digits 0.h(1)h(2)h(3)h(4)h(5). It is not too hard to see that if c were rational then one would have a Turing machine that could solve the halting problem. So c must be irrational. (I think a variant of this argument is originally due to Chaitin.)
I’m not sure this example cannot be cast in that framework, but it seems like it would be very artificial to put it in that framework.
So, are there specific constants which arise not from computability or decidability, but arise naturally where we cannot put them in this framework? Then also the answer still seems to be yes. Here are two examples. Consider the constant x= 0.0110101… where the nth digit is 1 if and only if n is prime, and the digit is zero otherwise. It is not too hard to show that this constant is irrational simply by thinking about the digit distribution. Similar remarks would apply to replacing primes with whatever your favorite zero density infinite set of natural numbers. A slightly trickier is to look at the number which in base 10 is 0.23571113177192329… . Showing this is irrational has a similar flavor but is more involved.
So, is there an example of a constant which is arising from “natural” objects and isn’t being defined by pointing to its digits or the like, where a proof cannot be cast in that form? I’m not sure, but some of them at least don’t seem to obviously fit in that framework. There’s an incomplete list of ways to prove irrationality here https://mathoverflow.net/questions/435236/compilation-of-strategies-to-show-that-some-constant-is-irrational and for some of the answers given there, it isn’t obvious how to convert the proofs into that framework.
Comment #44 December 2nd, 2024 at 2:25 pm
Scott #21: Wow, thank you and thanks Greg Kuperberg! That’s an amazing sequence of integrals. It certainly wasn’t obvious to me that I(n) is always an integer polynomial in pi^2, but then I worked out the recurrence relation
I(0) = 2
I(1) = 4
I(n + 2) = (4n + 6) I(n+1) – pi^2 I(n)
Comment #45 December 2nd, 2024 at 3:42 pm
Joshua Zelinsky #43: Thanks. These two examples are what I consider “trivial,” as the numbers are constructed by a (non-periodic) binary expansion, from which deducing irrationality is trivial. But the MathOverflow post of compilations of techniques is interesting, even if the numbers investigated under each technique seem known to be irrational via a continued fractions argument too.
Comment #46 December 3rd, 2024 at 7:35 am
I’m thankful that Trump is incompetent. This will likely protect the world from his evil.
Comment #47 December 3rd, 2024 at 9:12 am
Happy late Thanksgiving, nice to hear from you.
In the forthcoming years, while Nyarlathotep rules, it is good to know sanity and wisdom still exists in America.
Comment #48 December 3rd, 2024 at 9:40 am
I heard an interesting point about AIs.
It will be interesting to see whether advanced “thinking” AIs will ever be able to “swear”, because for now this ability appears to be a fundamental human trait coming from a frustration to not be able to express something in language, because not all our thinking can be reduced in terms of language – for example our inability to fully express things that can’t be reduced to concepts, like consciousness/subjective experience.
But it’s possible that once AIs truly interface with the “real” world, they may also experience an inability to map patterns in streams of sensory data into a consistent framework of concepts.
Comment #49 December 3rd, 2024 at 9:44 am
@Mahdi #45,
If you want another example which likely cannot be put into the generalized continue fraction framework, one can play with decidability in a similar way. For example, given a computable listing of Turing machines, let a(1), a(2), a(3)… be the sequence of which machines on the list halt on the blank tape. Then consider S as the sum of (-1)^n / a(n) from n=1 to infinity. This series converges by the alternating series test. If S was rational then we could solve the Halting problem. So this example still uses decidability tricks but doesn’t involve any base trick.
A possible more precise version of your question might be the following (but this may be too narrow): Is every irrational number which is a period https://en.wikipedia.org/wiki/Period_(algebraic_geometry) provably a period using continued fraction techniques? My guess is that this is not true either, but this at least is starting to look like a mathematically precise statement.
Comment #50 December 3rd, 2024 at 7:35 pm
> I’m thankful that the presidential election wasn’t all that close (by contemporary US standards, it was a ““landslide,”” 50%-48.4%).
It’s actually the second smallest margin of the century, ahead of only 2000’s hanging chads.
Comment #51 December 4th, 2024 at 9:03 am
Ori Vandewalle #50:
Could you please expound on that? A link to the data please.
Comment #52 December 4th, 2024 at 6:36 pm
DR #51:
Start here and work your way back?
https://en.wikipedia.org/wiki/2024_United_States_presidential_election
For 2020 and earlier, the stats listed on Wikipedia come from the FEC. https://www.fec.gov/resources/cms-content/documents/federalelections [year] .pdf seems to do the trick (2-digit year for 2000 and earlier).
Comment #53 December 5th, 2024 at 11:56 am
“I’m thankful that, almost thirty years after I could have and should have, I’ve now finally learned the proof of the irrationality of π.”
Love this one!
And love the comments section full of nerdiness following it.
Comment #54 December 5th, 2024 at 3:09 pm
π being irrational could probably also be derived from the fact that an angle such as π/3 can’t be trisected (no finite sequence of ruler-and-compass constructions can compose an angle of π/9 or 20 degrees, which is the trisection of the angle π/3 or 60 degrees.)?
Comment #55 December 10th, 2024 at 8:14 am
I’m thankful that, after seeing headlines about a new quantum chip “Willow” from Google, I don’t have to try to read through the misunderstandings of journalists, but can keep checking back here, and eventually there will be a meaningful explanation of the chip’s real capabilities/significance that I can trust 🙂
Comment #56 December 10th, 2024 at 12:55 pm
yay I see the new post, thanks!!
Comment #57 December 30th, 2024 at 12:47 pm
I was just reading your Willow and Thanksgiving blog posts. What delightful human beings you and your family are Scott, I’m thankful for people like you in the world. Also appreciate your academic work, and being a voice of reason as a US citizen.