## From shtetl to Forum

January 18th, 2020

Today I’m headed to the 50th World Economic Forum in Davos, where on Tuesday I’ll participate in a panel discussion on “The Quantum Potential” with Jeremy O’Brien of the quantum computing startup PsiQuantum, and will also host an ask-me-anything session about quantum computational supremacy and Google’s claim to have achieved it.

I’m well aware that this will be unlike any other conference I’ve ever attended: STOC or FOCS it ain’t. As one example, also speaking on Tuesday—although not conflicting with my QC sessions—will be a real-estate swindler and reality-TV star who’s somehow (alas) the current President of the United States. Yes, even while his impeachment trial in the Senate gets underway. Also speaking on Tuesday, a mere hour and a half after him, will be TIME’s Person of the Year, 17-year-old climate activist Greta Thunberg.

In short, this Davos is shaping up to be an epic showdown between two diametrically opposed visions for the future of life on Earth. And your humble blogger will be right there in the middle of it, to … uhh … explain how quantum computers can sample probability distributions that are classically intractable unless the polynomial hierarchy collapses to the third level. I feel appropriately sheepish.

Since the experience will be so unusual for me, I’m planning to “live-blog Davos”: I’ll be updating this post, all week, with any strange new things that I see or learn. As a sign of my devotion to you, my loyal readers, I’ll even clothespin my nose and attend Trump’s speech so I can write about it.

And Greta: on the off chance that you happen to read Shtetl-Optimized, let me treat you to a vegan lunch or dinner! I’d like to try to persuade you of just how essential nuclear power will be to a carbon-free future. Oh, and if it’s not too much trouble, I’d also like a selfie with you for this blog. (Alas, a friend pointed out to me that it would probably be easier to meet Trump: unlike Greta, he won’t be swarmed with thousands of fans!)

Anyway, check back here throughout the week for updates. And if you’re in Davos and would like to meet, please shoot me an email. And please use the comment section to give me your advice, suggestions, well-wishes, requests, or important messages for me to fail to deliver to the “Davoisie” who run the world.

## An alternative argument for why women leave STEM: Guest post by Karen Morenz

January 16th, 2020

Scott’s preface: Imagine that every time you turned your blog over to a certain topic, you got denounced on Twitter and Reddit as a privileged douchebro, entitled STEMlord, counterrevolutionary bourgeoisie, etc. etc. The sane response would simply be to quit blogging about that topic. But there’s also an insane (or masochistic?) response: the response that says, “but if everyone like me stopped talking, we’d cede the field by default to the loudest, angriest voices on all sides—thereby giving those voices exactly what they wanted. To hell with that!”

A few weeks ago, while I was being attacked for sharing Steven Pinker’s guest post about NIPS vs. NeurIPS, I received a beautiful message of support from a PhD student in physical chemistry and quantum computing named Karen Morenz. Besides her strong words of encouragement, Karen wanted to share with me an essay she had written on Medium about why too many women leave STEM.

Karen’s essay, I found, marshaled data, logic, and her own experience in support of an insight that strikes me as true and important and underappreciated—one that dovetails with what I’ve heard from many other women in STEM fields, including my wife Dana. So I asked Karen for permission to reprint her essay on this blog, and she graciously agreed.

Briefly: anyone with a brain and a soul wants there to be many more women in STEM. Karen outlines a realistic way to achieve this shared goal. Crucially, Karen’s way is not about shaming male STEM nerds for their deep-seated misogyny, their arrogant mansplaining, or their gross, creepy, predatory sexual desires. Yes, you can go the shaming route (God knows it’s being tried). If you do, you’ll probably snare many guys who really do deserve to be shamed as creeps or misogynists, along with many more who don’t. Yet for all your efforts, Karen predicts, you’ll no more solve the original problem of too few women in STEM, than arresting the kulaks solved the problem of lifting the masses out of poverty.

For you still won’t have made a dent in the real issue: namely that, the way we’ve set things up, pursuing an academic STEM career demands fanatical devotion, to the exclusion of nearly everything else in life, between the ages of roughly 18 and 35. And as long as that’s true, Karen says, the majority of talented women are going to look at academic STEM, in light of all the other great options available to them, and say “no thanks.” Solving this problem might look like more money for maternity leave and childcare. It might also look like re-imagining the academic career trajectory itself, to make it easier to rejoin it after five or ten years away. Way back in 2006, I tried to make this point in a blog post called Nerdify the world, and the women will follow. I’m grateful to Karen for making it more cogently than I did.

Without further ado, here’s Karen’s essay. –SA

### Is it really just sexism? An alternative argument for why women leave STEM

by Karen Morenz

Everyone knows that you’re not supposed to start your argument with ‘everyone knows,’ but in this case, I think we ought to make an exception:

Everyone knows that STEM (Science, Technology, Engineering and Mathematics) has a problem retaining women (see, for example Jean, Payne, and Thompson 2015). We pour money into attracting girls and women to STEM fields. We pour money into recruiting women, training women, and addressing sexism, both overt and subconscious. In 2011, the United States spent nearly $3 billion tax dollars on STEM education, of which roughly one third was spent supporting and encouraging underrepresented groups to enter STEM (including women). And yet, women are still leaving at alarming rates. Alarming? Isn’t that a little, I don’t know, alarmist? Well, let’s look at some stats. A recent report by the National Science Foundation (2011) found that women received 20.3% of the bachelor’s degrees and 18.6% of the PhD degrees in physics in 2008. In chemistry, women earned 49.95% of the bachelor’s degrees but only 36.1% of the doctoral degrees. By comparison, in biology women received 59.8% of the bachelor’s degrees and 50.6% of the doctoral degrees. A recent article in Chemical and Engineering News showed a chart based on a survey of life sciences workers by Liftstream and MassBio demonstrating how women are vastly underrepresented in science leadership despite earning degrees at similar rates, which I’ve copied below. The story is the same in academia, as you can see on the second chart — from comparable or even larger number of women at the student level, we move towards a significantly larger proportion of men at the more and more advanced stages of an academic career. Although 74% of women in STEM report “loving their work,” half (56%, in fact) leave over the course of their career — largely at the “mid-level” point, when the loss of their talent is most costly as they have just completed training and begun to contribute maximally to the work force. A study by Dr. Flaherty found that women who obtain faculty position in astronomy spent on average 1 year less than their male counterparts between completing their PhD and obtaining their position — but he concluded that this is because women leave the field at a rate 3 to 4 times greater than men, and in particular, if they do not obtain a faculty position quickly, will simply move to another career. So, women and men are hired at about the same rate during the early years of their post docs, but women stop applying to academic positions and drop out of the field as time goes on, pulling down the average time to hiring for women. There are many more studies to this effect. At this point, the assertion that women leave STEM at an alarming rate after obtaining PhDs is nothing short of an established fact. In fact, it’s actually a problem across all academic disciplines, as you can see in this matching chart showing the same phenomenon in humanities, social sciences, and education. The phenomenon has been affectionately dubbed the “leaky pipeline.” But hang on a second, maybe there just aren’t enough women qualified for the top levels of STEM? Maybe it’ll all get better in a few years if we just wait around doing nothing? Nope, sorry. This study says that 41% of highly qualified STEM people are female. And also, it’s clear from the previous charts and stats that a significantly larger number of women are getting PhDs than going on the be professors, in comparison to their male counterparts. Dr. Laurie Glimcher, when she started her professorship at Harvard University in the early 1980s, remembers seeing very few women in leadership positions. “I thought, ‘Oh, this is really going to change dramatically,’ ” she says. But 30 years later, “it’s not where I expected it to be.” Her experiences are similar to those of other leading female faculty. So what gives? Why are all the STEM women leaving? It is widely believed that sexism is the leading problem. A quick google search of “sexism in STEM” will turn up a veritable cornucopia of articles to that effect. And indeed, around 60% of women report experiencing some form of sexism in the last year (Robnett 2016). So, that’s clearly not good. And yet, if you ask leading women researchers like Nobel Laureate in Physics 2018, Professor Donna Strickland, or Canada Research Chair in Advanced Functional Materials (Chemistry), Professor Eugenia Kumacheva, they say that sexism was not a barrier in their careers. Moreover, extensive research has shown that sexism has overall decreased since Professors Strickland and Kumacheva (for example) were starting their careers. Even more interestingly, Dr. Rachael Robnett showed that more mathematical fields such as Physics have a greater problem with sexism than less mathematical fields, such as Chemistry, a finding which rings true with the subjective experience of many women I know in Chemistry and Physics. However, as we saw above, women leave the field of Chemistry in greater proportions following their BSc than they leave Physics. On top of that, although 22% of women report experiencing sexual harassment at work, the proportion is the same among STEM and non-STEM careers, and yet women leave STEM careers at a much higher rate than non-STEM careers. So,it seems that sexism can not fully explain why women with STEM PhDs are leaving STEM. At the point when women have earned a PhD, for the most part they have already survived the worst of the sexism. They’ve already proven themselves to be generally thick-skinned and, as anyone with a PhD can attest, very stubborn in the face of overwhelming difficulties. Sexism is frustrating, and it can limit advancement, but it doesn’t fully explain why we have so many women obtaining PhDs in STEM, and then leaving. In fact, at least in the U of T chemistry department, faculty hires are directly proportional to the applicant pool —although the exact number of applicants are not made public, from public information we can see that approximately one in four interview invitees are women, and approximately one in four hires are women. Our hiring committees have received bias training, and it seems that it has been largely successful. That’s not to say that we’re done, but it’s time to start looking elsewhere to explain why there are so few women sticking around. So why don’t more women apply? Well, one truly brilliant researcher had the groundbreaking idea of asking women why they left the field. When you ask women why they left, the number one reason they cite is balancing work/life responsibilities — which as far as I can tell is a euphemism for family concerns. The research is in on this. Women who stay in academia expect to marry later, and delay or completely forego having children, and if they do have children, plan to have fewer than their non-STEM counterparts (Sassler et al 2016Owens 2012). Men in STEM have no such difference compared to their non-STEM counterparts; they marry and have children about the same ages and rates as their non-STEM counterparts (Sassler et al 2016). Women leave STEM in droves in their early to mid thirties (Funk and Parker 2018) — the time when women’s fertility begins to decrease, and risks of childbirth complications begin to skyrocket for both mother and child. Men don’t see an effect on their fertility until their mid forties. Of the 56% of women who leave STEM, 50% wind up self-employed or using their training in a not for profit or government, 30% leave to a non-STEM more ‘family friendly’ career, and 20% leave to be stay-at-home moms (Ashcraft and Blithe 2002). Meanwhile, institutions with better childcare and maternity leave policies have twice(!) the number of female faculty in STEM (Troeger 2018). In analogy to the affectionately named “leaky pipeline,” the challenge of balancing motherhood and career has been titled the “maternal wall.” To understand the so-called maternal wall better, let’s take a quick look at the sketch of a typical academic career. For the sake of this exercise, let’s all pretend to be me. I’m a talented 25 year old PhD candidate studying Physical Chemistry — I use laser spectroscopy to try to understand atypical energy transfer processes in innovative materials that I hope will one day be used to make vastly more efficient solar panels. I got my BSc in Chemistry and Mathematics at the age of 22, and have published 4 scientific papers in two different fields already (Astrophysics and Environmental Chemistry). I’ve got a big scholarship, and a lot of people supporting me to give me the best shot at an academic career — a career I dearly want. But, I also want a family — maybe two or three kids. Here’s what I can expect if I pursue an academic career: With any luck, 2–3 years from now I’ll graduate with a PhD, at the age of 27. Academics are expected to travel a lot, and to move a lot, especially in their 20s and early 30s — all of the key childbearing years. I’m planning to go on exchange next year, and then the year after that I’ll need to work hard to wrap up research, write a thesis, and travel to several conferences to showcase my work. After I finish my PhD, I’ll need to undertake one or two post doctoral fellowships, lasting one or two years each, probably in completely different places. During that time, I’ll start to apply for professorships. In order to do this, I’ll travel around to conferences to advertise my work and to meet important leaders in my field, and then, if I am invited for interviews, I’ll travel around to different universities for two or three days at a time to undertake these interviews. This usually occurs in a person’s early 30s — our helpful astronomy guy, Dr. Flaherty, found the average time to hiring was 5 years, so let’s say I’m 32 at this point. If offered a position, I’ll spend the next year or two renovating and building a lab, buying equipment, recruiting talented graduate students, and designing and teaching courses. People work really, really hard during this time and have essentially no leisure time. Now I’m 34. Within usually 5 years I’ll need to apply for tenure. This means that by the time I’m 36, I’ll need to be making significant contributions in my field, and then in the final year before applying for tenure, I will once more need to travel to many conferences to promote my work, in order to secure tenure — if I fail to do so, my position at the university would probably be terminated. Although many universities offer a “tenure extension” in cases where an assistant professor has had a child, this does not solve all of the problems. Taking a year off during that critical 5 or 6 year period often means that the research “goes bad” — students flounder, projects that were promising get “scooped” by competitors at other institutions, and sometimes, in biology and chemistry especially, experiments literally go bad. You wind up needing to rebuild much more than just a year’s worth of effort. At no point during this time do I appear stable enough, career-wise, to take even six months off to be pregnant and care for a newborn. Hypothetical future-me is travelling around, or even moving, conducting and promoting my own independent research and training students. As you’re likely aware, very pregnant people and newborns don’t travel well. And academia has a very individualistic and meritocratic culture. Starting at the graduate level, huge emphasis is based on independent research, and independent contributions, rather than valuing team efforts. This feature of academia is both a blessing and a curse. The individualistic culture means that people have the independence and the freedom to pursue whatever research interests them — in fact this is the main draw for me personally. But it also means that there is often no one to fall back on when you need extra support, and because of biological constraints, this winds up impacting women more than men. At this point, I need to make sure that you’re aware of some basics of female reproductive biology. According to Wikipedia, the unquestionable source of all reliable knowledge, at age 25, my risk of conceiving a baby with chromosomal abnormalities (including Down’s Syndrome) is 1 in about 1400. By 35, that risk more than quadruples to 1 in 340. At 30, I have a 75% chance of a successful birth in one year, but by 35 it has dropped to 66%, and by 40 it’s down to 44%. Meanwhile, 87 to 94% of women report at least 1 health problem immediately after birth, and 1.5% of mothers have a severe health problem, while 31% have long-term persistent health problems as a result of pregnancy (defined as lasting more than six months after delivery). Furthermore, mothers over the age of 35 are at higher risk for pregnancy complications like preterm delivery, hypertension, superimposed preeclampsia, severe preeclampsia (Cavazos-Rehg et al 2016). Because of factors like these, pregnancies in women over 35 are known as “geriatric pregnancies” due to the drastically increased risk of complications. This tight timeline for births is often called the “biological clock” — if women want a family, they basically need to start before 35. Now, that’s not to say it’s impossible to have a child later on, and in fact some studies show that it has positive impacts on the child’s mental health. But it is riskier. So, women with a PhD in STEM know that they have the capability to make interesting contributions to STEM, and to make plenty of money doing it. They usually marry someone who also has or expects to make a high salary as well. But this isn’t the only consideration. Such highly educated women are usually aware of the biological clock and the risks associated with pregnancy, and are confident in their understanding of statistical risks. The Irish say, “The common challenge facing young women is achieving a satisfactory work-life balance, especially when children are small. From a career perspective, this period of parenthood (which after all is relatively short compared to an entire working life) tends to coincide exactly with the critical point at which an individual’s career may or may not take off. […] All the evidence shows that it is at this point that women either drop out of the workforce altogether, switch to part-time working or move to more family-friendly jobs, which may be less demanding and which do not always utilise their full skillset.” And in the Netherlands, “The research project in Tilburg also showed that women academics have more often no children or fewer children than women outside academia.” Meanwhile in Italy “On a personal level, the data show that for a significant number of women there is a trade-off between family and work: a large share of female economists in Italy do not live with a partner and do not have children” Most jobs available to women with STEM PhDs offer greater stability and a larger salary earlier in the career. Moreover, most non-academic careers have less emphasis on independent research, meaning that employees usually work within the scope of a larger team, and so if a person has to take some time off, there are others who can help cover their workload. By and large, women leave to go to a career where they will be stable, well funded, and well supported, even if it doesn’t fulfill their passion for STEM — or they leave to be stay-at-home moms or self-employed. I would presume that if we made academia a more feasible place for a woman with a family to work, we could keep almost all of those 20% of leavers who leave to just stay at home, almost all of the 30% who leave to self-employment, and all of those 30% who leave to more family friendly careers (after all, if academia were made to be as family friendly as other careers, there would be no incentive to leave). Of course, there is nothing wrong with being a stay at home parent — it’s an admirable choice and contributes greatly to our society. One estimate valued the equivalent salary benefit of stay-at-home parenthood at about$160,000/year. Moreover, children with a stay-at-home parent show long term benefits such as better school performance — something that most academic women would want for their children. But a lot of people only choose it out of necessity — about half of stay-at-home moms would prefer to be working (Ciciolla, Curlee, & Luthar 2017). When the reality is that your salary is barely more than the cost of daycare, then a lot of people wind up giving up and staying home with their kids rather than paying for daycare. In a heterosexual couple it will usually be the woman that winds up staying home since she is the one who needs to do things like breast feed anyways. And so we lose these women from the workforce.

And yet, somehow, during this informal research adventure of mine, most scholars and policy makers seem to be advising that we try to encourage young girls to be interested in STEM, and to address sexism in the workplace, with the implication that this will fix the high attrition rate in STEM women. But from what I’ve found, the stats don’t back up sexism as the main reason women leave. There is sexism, and that is a problem, and women do leave STEM because of it — but it’s a problem that we’re already dealing with pretty successfully, and it’s not why the majority of women who have already obtained STEM PhDs opt to leave the field. The whole family planning thing is huge and for some reason, almost totally swept under the rug — mostly because we’re too shy to talk about it, I think.

In fact, I think that the plethora of articles suggesting that the problem is sexism actually contribute to our unwillingness to talk about the family planning problem, because it reinforces the perception that that men in power will not hire a woman for fear that she’ll get pregnant and take time off. Why would anyone talk about how they want to have a family when they keep hearing that even the mere suggestion of such a thing will limit their chances of being hired? I personally know women who have avoided bringing up the topic with colleagues or supervisors for fear of professional repercussions. So we spend all this time and energy talking about how sexism is really bad, and very little time trying to address the family planning challenge, because, I guess, as the stats show, if women are serious enough about science then they just give up on the family (except for the really, really exceptional ones who can handle the stresses of both simultaneously).

To be very clear, I’m not saying that sexism is not a problem. What I am saying is that, thanks to the sustained efforts of a large number of people over a long period of time, we’ve reduced the sexism problem to the point where, at least at the graduate level, it is no longer the largest major barrier to women’s advancement in STEM. Hurray! That does not mean that we should stop paying attention to the issue of sexism, but does mean that it’s time to start paying more attention to other issues, like how to properly support women who want to raise a family while also maintaining a career in STEM.

So what can we do to better support STEM women who want families?

A couple of solutions have been tentatively tested. From a study mentioned above, it’s clear that providing free and conveniently located childcare makes a colossal difference to women’s choices of whether or not to stay in STEM, alongside extended and paid maternity leave. Another popular and successful strategy was implemented by a leading woman in STEM, Laurie Glimcher, a past Harvard Professor in Immunology and now CEO of Dana-Farber Cancer Institute. While working at NIH, Dr. Glimcher designed a program to provide primary caregivers (usually women) with an assistant or lab technician to help manage their laboratories while they cared for children. Now, at Dana-Farber Cancer Institute, she has created a similar program to pay for a technician or postdoctoral researcher for assistant professors. In the academic setting, Dr. Glimcher’s strategies are key for helping to alleviate the challenges associated with the individualistic culture of academia without compromising women’s research and leadership potential.

For me personally, I’m in the ideal situation for an academic woman. I graduated my BSc with high honours in four years, and with many awards. I’ve already had success in research and have published several peer reviewed papers. I’ve faced some mild sexism from peers and a couple of TAs, but nothing that’s seriously held me back. My supervisors have all been extremely supportive and feminist, and all of the people that I work with on a daily basis are equally wonderful. Despite all of this support, I’m looking at the timelines of an academic career, and the time constraints of female reproduction, and honestly, I don’t see how I can feasible expect to stay in academia and have the family life I want. And since I’m in the privileged position of being surrounded by supportive and feminist colleagues, I can say it: I’m considering leaving academia, if something doesn’t change, because even though I love it, I don’t see how it can fit in to my family plans.

But wait! All of these interventions are really expensive. Money doesn’t just grow on trees, you know!

It doesn’t in general, but in this case it kind of does — well, actually, we already grew it. We spend billions of dollars training women in STEM. By not making full use of their skills, if we look at only the american economy, we are wasting about $1.5 billion USD per year in economic benefits they would have produced if they stayed in STEM. So here’s a business proposal: let’s spend half of that on better family support and scientific assistants for primary caregivers, and keep the other half in profit. Heck, let’s spend 99% —$1.485 billion (in the states alone) on better support. That should put a dent in the support bill, and I’d sure pick up \$15 million if I saw it lying around. Wouldn’t you?

By demonstrating that we will support women in STEM who choose to have a family, we will encourage more women with PhDs to apply for the academic positions that they are eminently qualified for. Our institutions will benefit from the wider applicant pool, and our whole society will benefit from having the skills of these highly trained and intelligent women put to use innovating new solutions to our modern day challenges.

## MIP*=RE

January 14th, 2020

Another Update (Jan. 16): Yet another reason to be excited about this result—one that somehow hadn’t occurred to me—is that, as far as I know, it’s the first-ever fully convincing example of a non-relativizing computability result. See this comment for more.

Update: If you’re interested in the above topic, then you should probably stop reading this post right now, and switch to this better post by Thomas Vidick, one of the authors of the new breakthrough. (Or this by Boaz Barak or this by Lance Fortnow or this by Ken Regan.) (For background, also see Thomas Vidick’s excellent piece for the AMS Notices.)

Still here? Alright, alright…

Here’s the paper, which weighs in at 165 pages. The authors are Zhengfeng Ji, Anand Natarajan, my former postdoc Thomas Vidick, John Wright (who will be joining the CS faculty here at UT Austin this fall), and my wife Dana’s former student Henry Yuen. Rather than pretending that I can provide intelligent commentary on this opus in the space of a day, I’ll basically just open my comment section to discussion and quote the abstract:

We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages. Our proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS 2018) by integrating recent developments from (Natarajan and Wright, FOCS 2019) and combining them with the recursive compression framework of (Fitzsimons et al., STOC 2019).
An immediate byproduct of our result is that there is an efficient reduction from the Halting Problem to the problem of deciding whether a two-player nonlocal game has entangled value 1 or at most 1/2. Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson’s problem: we show, by providing an explicit example, that the closure Cqa of the set of quantum tensor product correlations is strictly included in the set Cqc of quantum commuting correlations. Following work of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our results provide a refutation of Connes’ embedding conjecture from the theory of von Neumann algebras.

To say it differently (in response to a commenter’s request), some of the major implications are as follows.

(1) There is a protocol by which two entangled provers can convince a polynomial-time verifier of the answer to any computable problem whatsoever (!!), or indeed that a given Turing machine halts.

(2) There is a two-prover game, analogous to the Bell/CHSH game, for which Alice and Bob can do markedly better with a literally infinite amount of entanglement than they can with any finite amount of entanglement.

(3) There is no algorithm even to approximate the entangled value of a two-prover game (i.e., the probability that Alice and Bob win the game, if they use the best possible strategy and as much entanglement as they like). Instead, this problem is equivalent to the halting problem.

(4) There are types of correlations between Alice and Bob that can be produced using infinite entanglement, but that can’t even be approximated using any finite amount of entanglement.

(5) The Connes embedding conjecture, a central conjecture from the theory of operator algebras dating back to the 1970s, is false.

Note that all of these implications—including the ones for pure math and the foundations of quantum physics—were obtained using tools that originated in theoretical computer science, specifically the study of interactive proof systems.

I can remember when the class MIP* was first defined and studied, back around 2003, and people made the point that we didn’t know any reasonable upper bound on the class’s power—not NEXP, not NEEEEXP, not even the set of all computable languages. Back then, the joke was how far our proof techniques were from what was self-evidently the truth. I don’t remember a single person who seriously contemplated that two entangled provers could convince a polynomial-time verifier than an arbitrary Turing machine halts.

Still, ever since Natarajan and Wright’s NEEXP in MIP* breakthrough last year, all of us in quantum computing theory knew that MIP*=RE was a live possibility—and all through the summer and fall, I heard many hints that such a breakthrough was imminent.

It’s worth pointing out that, with only classical correlations between the provers, MIP gives “merely” the power of NEXP (Nondeterministic Exponential Time), while with arbitrary non-signalling correlations between the provers, the so-called MIPns gives the power of EXP (Deterministic Exponential Time). So it’s particularly striking that quantum entanglement, which is “intermediate” between classical correlations and arbitrary non-signalling correlations, yields such wildly greater computational power than either of those two.

The usual proviso applies: when I’ve blogged excitedly about preprints with amazing new results, most have stood, but at least two ended up being retracted. Still, assuming this one stands (as I’m guessing it will), I regard it as easily one of the biggest complexity-theoretic (and indeed computability-theoretic!) surprises so far in this century. Huge congratulations to the authors on what looks to be a historic achievement.

In unrelated news, for anyone for whom the 165-page MIP* paper is too heavy going (really??), please enjoy this CNBC video on quantum computing, which features several clips of yours truly speaking in front of a fake UT tower.

In other unrelated news, I’m also excited about this preprint by Avishay Tal, which sets a new record for the largest known separation between quantum query complexity and classical randomized query complexity, making substantial progress toward proving a conjecture by me and Andris Ambainis from 2015. (Not the “Aaronson-Ambainis Conjecture,” a different conjecture.)

## Quantum computing motte-and-baileys

December 28th, 2019

In the wake of two culture-war posts—the first on the term “quantum supremacy,” the second on the acronym “NIPS”—it’s clear that we all need to cool off with something anodyne and uncontroversial. Fortunately, this holiday season, I know just the thing to bring everyone together: groaning about quantum computing hype!

When I was at the Q2B conference in San Jose, I learned about lots of cool stuff that’s happening in the wake of Google’s quantum supremacy announcement. I heard about the 57-qubit superconducting chip that the Google group is now building, following up on its 53-qubit one; and also about their first small-scale experimental demonstration of my certified randomness protocol. I learned about recent progress on costing out the numbers of qubits and gates needed to do fault-tolerant quantum simulations of useful chemical reactions (IIRC, maybe a hundred thousand qubits and a few hours’ worth of gates—scary, but not Shor’s algorithm scary).

I also learned about two claims about quantum algorithms that startups have made, and which are being wrongly interpreted. The basic pattern is one that I’ve come to know well over the years, and which you could call a science version of the motte-and-bailey. (For those not up on nerd blogosphere terminology: in medieval times, the motte was a dank castle to which you’d retreat while under attack; the bailey was the desirable land that you’d farm once the attackers left.)

To wit:

1. Startup makes claims that have both a true boring interpretation (e.g., you can do X with a quantum computer), as well as a false exciting interpretation (e.g., you can do X with a quantum computer, and it would actually make sense to do this, because you’ll get an asymptotic speedup over the best known classical algorithm).
2. Lots of business and government people get all excited, because they assume the false exciting interpretation must be true (or why else would everyone be talking about this?). Some of those people ask me for comment.
3. I look into it, perhaps by asking the folks at the startup. The startup folks clarify that they meant only the true boring interpretation. To be sure, they’re actively exploring the false exciting interpretation—whether some parts of it might be true after all—but they’re certainly not making any claims about it that would merit, say, a harsh post on Shtetl-Optimized.
4. I’m satisfied to have gotten to the bottom of things, and I tell the startup folks to go their merry way.
5. Yet many people continue to seem as excited as if the false exciting interpretation had been shown to be true. They continue asking me questions that presuppose its truth.

Our first instance of this pattern is the recent claim, by Zapata Computing, to have set a world record for integer factoring (1,099,551,473,989 = 1,048,589 × 1,048,601) with a quantum computer, by running a QAOA/variational algorithm on IBM’s superconducting device. Gosh! That sure sounds a lot better than the 21 that’s been factored with Shor’s algorithm, doesn’t it?

I read the Zapata paper that this is based on, entitled “Variational Quantum Factoring,” and I don’t believe that a single word in it is false. My issue is something the paper omits: namely, that once you’ve reduced factoring to a generic optimization problem, you’ve thrown away all the mathematical structure that Shor’s algorithm cleverly exploits, and that makes factoring asymptotically easy for a quantum computer. And hence there’s no reason to expect your quantum algorithm to scale any better than brute-force trial division (or in the most optimistic scenario, trial division enhanced with Grover search). On large numbers, your algorithm will be roundly outperformed even by classical algorithms that do exploit structure, like the Number Field Sieve. Indeed, the quantum computer’s success at factoring the number will have had little or nothing to do with its being quantum at all—a classical optimization algorithm would’ve served as well. And thus, the only reasons to factor a number on a quantum device in this way, would seem to be stuff like calibrating the device.

Admittedly, to people who work in quantum algorithms, everything above is so obvious that it doesn’t need to be said. But I learned at Q2B that there are interested people for whom this is not obvious, and even comes as a revelation. So that’s why I’m saying it.

Again and again over the past twenty years, I’ve seen people reinvent the notion of a “simpler alternative” to Shor’s algorithm: one that cuts out all the difficulty of building a fault-tolerant quantum computer. In every case, the trouble, typically left unstated, has been that these alternatives also cut out the exponential speedup that’s Shor’s algorithm’s raison d’être.

Our second example today of a quantum computing motte-and-bailey is the claim, by Toronto-based quantum computing startup Xanadu, that Gaussian BosonSampling can be used to solve all sorts of graph problems, like graph isomorphism, graph similarity, and densest subgraph. As the co-inventor of BosonSampling, few things would warm my heart more than finding an actual application for that model (besides quantum supremacy experiments and, perhaps, certified random number generation). But I still regard this as an open problem—if by “application,” we mean outperforming what you could’ve done classically.

In papers (see for example here, here, here), members of the Xanadu team have given all sorts of ways to take a graph, and encode it into an instance of Gaussian BosonSampling, in such a way that the output distribution will then reveal features of the graph, like its isomorphism type or its dense subgraphs. The trouble is that so far, I’ve seen no indications that this will actually lead to quantum algorithms that outperform the best classical algorithms, for any graph problems of practical interest.

In the case of Densest Subgraph, the Xanadu folks use the output of a Gaussian BosonSampler to seed (that is, provide an initial guess for) a classical local search algorithm. They say they observe better results this way than if they seed that classical local search algorithm with completely random initial conditions. But of course, the real question is: could we get equally good results by seeding with the output of some classical heuristic? Or by solving Densest Subgraph with a different approach entirely? Given how hard it’s turned out to be just to verify that the outputs of a BosonSampling device come from such a device at all, it would seem astonishing if the answer to these questions wasn’t “yes.”

In the case of Graph Isomorphism, the situation is even clearer. There, the central claim made by the Xanadu folks is that given a graph G, they can use a Gaussian BosonSampling device to sample a probability distribution that encodes G’s isomorphism type. So, isn’t this “promising” for solving GI with a quantum computer? All you’d need to do now is invent some fast classical algorithm that could look at the samples coming from two graphs G and H, and tell you whether the probability distributions were the same.

Except, not really. While the Xanadu paper never says so, if all you want is to sample a distribution that encodes a graph’s isomorphism type, that’s easy to do classically! (I even put this on the final exam for my undergraduate Quantum Information Science course a couple weeks ago.) Here’s how: given as input a graph G, just output G but with its vertices randomly permuted. Indeed, this will even provide a further property, better than anything the BosonSampling approach has been shown to provide (or than it probably does provide): namely, if G and H are not isomorphic, then the two probability distributions will not only be different but will have disjoint supports. Alas, this still leaves us with the problem of distinguishing which distribution a given sample came from, which is as hard as Graph Isomorphism itself. None of these approaches, classical or quantum, seem to lead to any algorithm that’s subexponential time, let alone competitive with the “Babai approach” of thinking really hard about graphs.

All of this stuff falls victim to what I regard as the Fundamental Error of Quantum Algorithms Research: namely, to treat it as “promising” that a quantum algorithm works at all, or works better than some brute-force classical algorithm, without asking yourself whether there are any indications that your approach will ever be able to exploit interference of amplitudes to outperform the best classical algorithm.

Incidentally, I’m not sure exactly why, but in practice, a major red flag that the Fundamental Error is about to be committed is when someone starts talking about “hybrid quantum/classical algorithms.” By this they seem to mean: “outside the domain of traditional quantum algorithms, so don’t judge us by the standards of that domain.” But I liked the way someone at Q2B put it to me: every quantum algorithm is a “hybrid quantum/classical algorithm,” with classical processors used wherever they can be, and qubits used only where they must be.

The other thing people do, when challenged, is to say “well, admittedly we have no rigorous proof of an asymptotic quantum speedup”—thereby brilliantly reframing the whole conversation, to make people like me look like churlish theoreticians insisting on an impossible and perhaps irrelevant standard of rigor, blind to some huge practical quantum speedup that’s about to change the world. The real issue, of course, is not that they haven’t given a proof of a quantum speedup (in either the real world or the black-box world); rather, it’s that they’ve typically given no reasons whatsoever to think that there might be a quantum speedup, compared to the best classical algorithms available.

In the holiday spirit, let me end on a positive note. When I did the Q&A at Q2B—the same one where Sarah Kaiser asked me to comment on the term “quantum supremacy”—one of my answers touched on the most important theoretical open problems about sampling-based quantum supremacy experiments. At the top of the list, I said, was whether there’s some interactive protocol by which a near-term quantum computer can not only exhibit quantum supremacy, but prove it to a polynomial-time-bounded classical skeptic. I mentioned that there was one proposal for how to do this, in the IQP model, due to Bremner and Shepherd, from way back in 2008. I said that their proposal deserved much more attention than it had received, and that trying to break it would be one obvious thing to work on. Little did I know that, literally while I was speaking, a paper was being posted to the arXiv, by Gregory Kahanamoku-Meyer, that claims to break Bremner and Shepherd’s protocol. I haven’t yet studied the paper, but assuming it’s correct, it represents the first clear progress on this problem in years (even though of a negative kind). Cool!!

## NIPS vs. NeurIPS: guest post by Steven Pinker

December 23rd, 2019

Scott’s Update (Dec. 26): Comments on this post are now closed, since I felt that whatever progress could be made, had been, and I wanted to move on to more interesting topics. Thanks so much to everyone who came here to hash things out in good faith—which, as far as I’m concerned, included the majority of the participants on both sides.

If you want to see the position paper that led to the name change movement, see What’s In A Name? The Need to Nip NIPS, by Daniela Witten, Elana Fertig, Anima Anandkumar, and Jeff Dean. I apologize for not linking to this paper in the original post.

To recap what I said many times in this post and the comments: I myself am totally fine with the name NeurIPS. I think several of the arguments for changing the name were good arguments—and I thank some of the commenters on this post for elucidating those arguments without shaming anybody or calling them names. In any case the decision is done, and it belongs to the ML community, not to me and not to Steven Pinker.

The one part that I’m against is the bullying of anyone who disagrees by smearing them as a misogynist. And then, recursively, the smearing as a misogynist of anyone who objected to that bullying, and so on and so on. Most supporters of the name change did not engage in such bullying, but one leader of the movement very conspicuously did, and continues to do it even now (to, I’m told, the consternation even of many of her allies).

Since this post went up, something extremely interesting happened: Steven Pinker and I started getting emails from researchers in the NeurIPS community that said, in various words: “thank you for openly airing perspectives that we could not air, without jeopardizing our careers.” We were told that even women in ML, and even those who agreed with the activists on most points, could no longer voice opposition without risking their hiring or tenure. This put into a slightly different light, I thought, the constant claims of some movement leaders about their own marginalization and powerlessness.

Since I was 7 or 8 years old, the moral lodestar of my life has been my yearning (too often left unfulfilled) to stand up to the world’s bullies. Bullies come in all shapes and sizes: some are gangsters or men who sexually exploit vulnerable women; one, alas, is even the President of the United States. But bullying knows no bounds of ideology or gender. Some bullies resort to whisper networks, or Twitter shaming campaigns, or their power in academic hierarchies, to shut down dissenting voices. With the latter kinds of bully—well, to whatever extent this blog is now in a position to make some difference, I’d feel morally complicit if it didn’t.

As I wrote in the comments: may the 2020s be an era of intellectual freedom, compassion, and understanding for all people regardless of background. –SA

Scott’s prologue:

Happy Christmas and Merry Chanukah!

As a followup to last Thursday’s post about the term “quantum supremacy,” today all of us here at Shtetl-Optimized are humbled to host a guest post by Steven Pinker: the Johnstone Professor of Psychology at Harvard University, and author of The Language Instinct, How the Mind Works, The Blank Slate, Enlightenment Now (which I reviewed here), and other books.

The former NIPS—Neural Information Processing Systems—has been the premier conference for machine learning for 30 years. As many readers might know, last year NIPS changed its name to NeurIPS: ironically, giving greater emphasis to an aspect that I’m told has been de-emphasized at that conference over time. The reason, apparently, was that some male attendees had made puns involving the acronym “NIPS” and nipples.

I confess that the name change took me by surprise, simply because it had never occurred to me to make the NIPS/nipples connection—not when I gave a plenary at NIPS in 2012, and not when my collaborators and I coauthored a NIPS paper. It’s not that I’m averse to puerile humor. It’s just that neither I, nor anyone else I knew, had apparently ever felt the need for a shorthand for “nipples.” Of course, once I did learn about this controversy, it became hard to hear “NIPS” without thinking about it.

Steve’s guest post today consists of his response to this email. (He told me that, after sending it, he received no further responses.)

I don’t have any dog in the NIPS/NeurIPS debate, being at most on the “margin” (har!) of machine learning. And in any case the debate ended a year ago: the name is now NeurIPS and it’s not changing back. Reopening the issue would seem to invite a strong risk of social-media denunciation for no possible gain.

So why am I doing this? Mostly because I thought it was in the interest of humanity to know that, even when Steven Pinker is answering someone’s email, with no expectation that his reply will be made public, he writes the same way he does in his books: with clarity, humor, and an amusing quote from his mom.

But also because—again, without taking a position on the NIPS vs. NeurIPS issue itself—there’s a tactic displayed by Pinker’s detractors that fundamentally grates on me. This is where you pretend to an open mind, but it turns out that you’re open only to the possibility that your opponent might not have read enough reports and studies to “do better”—i.e., that they sinned out of ignorance rather than out of malice. You don’t open your mind even a crack to the possibility that the opponent might have a point.

Without further ado, here’s Steven Pinker’s email:

I appreciate your frank comments. At the same time, I do not agree with them. Please allow me to explain.

If this were a matter of sexual harassment or other hostile behavior toward women, I would of course support strong measures to combat it. Any member of the Symposium who uttered demeaning comments toward or about women certainly deserves censure.

But that is not what is at issue here. It’s an utterly irrelevant matter: the three-decades-old acronym for the Neural Information Processing Symposium, the pleasingly pronounceable NIPS. To state what should be obvious: nip is not a sexual word. As Chair of the Usage Panel of the American Heritage Dictionary, I can support this claim.

(And as my mother wrote to me: “I don’t get it. I thought Nips was a brand of caramel candy.”)  [Indeed, I enjoyed those candies as a kid. –SA] Even if people with an adolescent mindset think of nipples when hearing the sound “nips,” the society should not endorse the idea that the concept of nipples is sexist. Men have nipples too, and women’s nipples evolved as organs of nursing, not sexual gratification. Indeed, many feminists have argued that it’s sexist to conceptualize women’s bodies from the point of view of male sexuality.

If some people make insulting puns that demean women, the society should condemn them for the insults, not concede to their puerility by endorsing their appropriation of an innocent sound. (The Linguistics Society of America and Boston Debate League do not change their names to disavow jejune clichés about cunning linguists and master debaters.) To act as if anything with the remotest connection to sexuality must be censored to protect delicate female sensibilities is insulting to women and reminiscent of prissy Victorian taboos against uncovered piano legs or the phrase “with the naked eye.”

Any harm to the community of computer scientists has been done not by me but by the pressure group and the Symposium’s surrender. As a public figure who hears from a broad range of people outside the academic bubble, I can tell you that this episode has not played well. It’s seen as the latest sign that academia has lost its mind—that it has traded reasoned argument, conceptual rigor, proportionality, and common sense for prudish censoriousness, snowflake sensibility, and virtue signaling. I often hear from intelligent non-leftists, “Why should I be impressed by the scientific consensus on climate change? Everyone knows that academics just fall into line with the politically correct position.” To secure the credibility of the academy, we have to make reasoned distinctions, and stop turning our enterprise into a laughingstock.

To repeat: none of this deprecates the important effort to stamp out harassment and misogyny in science, which I’m well aware of and thoroughly support, but which has nothing to do with the acronym NIPS.

Best,
Steve

## Quantum Dominance, Hegemony, and Superiority

December 19th, 2019

Yay! I’m now a Fellow of the ACM. Along with my fellow new inductee Peter Shor, who I hear is a real up-and-comer in the quantum computing field. I will seek to use this awesome responsibility to steer the ACM along the path of good rather than evil.

Also, last week, I attended the Q2B conference in San Jose, where a central theme was the outlook for practical quantum computing in the wake of the first clear demonstration of quantum computational supremacy. Thanks to the folks at QC Ware for organizing a fun conference (full disclosure: I’m QC Ware’s Chief Scientific Advisor). I’ll have more to say about the actual scientific things discussed at Q2B in future posts.

None of that is why you’re here, though. You’re here because of the battle over “quantum supremacy.”

A week ago, my good friend and collaborator Zach Weinersmith, of SMBC Comics, put out a cartoon with a dark-curly-haired scientist named “Dr. Aaronson,” who’s revealed on a hot mic to be an evil “quantum supremacist.” Apparently a rush job, this cartoon is far from Zach’s finest work. For one thing, if the character is supposed to be me, why not draw him as me, and if he isn’t, why call him “Dr. Aaronson”? In any case, I learned from talking to Zach that the cartoon’s timing was purely coincidental: Zach didn’t even realize what a hornet’s-nest he was poking with this.

Ever since John Preskill coined it in 2012, “quantum supremacy” has been an awkward term. Much as I admire John Preskill’s wisdom, brilliance, generosity, and good sense, in physics as in everything else—yeah, “quantum supremacy” is not a term I would’ve coined, and it’s certainly not a hill I’d choose to die on. Once it had gained common currency, though, I sort of took a liking to it, mostly because I realized that I could mine it for dark one-liners in my talks.

The thinking was: even as white supremacy was making its horrific resurgence in the US and around the world, here we were, physicists and computer scientists and mathematicians of varied skin tones and accents and genders, coming together to pursue a different and better kind of supremacy—a small reflection of the better world that we still believed was possible. You might say that we were reclaiming the word “supremacy”—which, after all, just means a state of being supreme—for something non-sexist and non-racist and inclusive and good.

In the world of 2019, alas, perhaps it was inevitable that people wouldn’t leave things there.

My first intimation came a month ago, when Leonie Mueck—someone who I’d gotten to know and like when she was an editor at Nature handling quantum information papers—emailed me about her view that our community should abandon the term “quantum supremacy,” because of its potential to make women and minorities uncomfortable in our field. She advocated using “quantum advantage” instead.

So I sent Leonie back a friendly reply, explaining that, as the father of a math-loving 6-year-old girl, I understood and shared her concerns—but also, that I didn’t know an alternative term that really worked.

See, it’s like this. Preskill meant “quantum supremacy” to refer to a momentous event that seemed likely to arrive in a matter of years: namely, the moment when programmable quantum computers would first outpace the ability of the fastest classical supercomputers on earth, running the fastest algorithms known by humans, to simulate what the quantum computers were doing (at least on special, contrived problems). And … “the historic milestone of quantum advantage”? It just doesn’t sound right. Plus, as many others pointed out, the term “quantum advantage” is already used to refer to … well, quantum advantages, which might fall well short of supremacy.

But one could go further. Suppose we did switch to “quantum advantage.” Couldn’t that term, too, remind vulnerable people about the unfair advantages that some groups have over others? Indeed, while “advantage” is certainly subtler than “supremacy,” couldn’t that make it all the more insidious, and therefore dangerous?

Oblivious though I sometimes am, I realized Leonie would be unhappy if I offered that, because of my wholehearted agreement, I would henceforth never again call it “quantum supremacy,” but only “quantum superiority,” “quantum dominance,” or “quantum hegemony.”

But maybe you now see the problem. What word does the English language provide to describe one thing decisively beating or being better than a different thing for some purpose, and which doesn’t have unsavory connotations?

I’ve heard “quantum ascendancy,” but that makes it sound like we’re a UFO cult—waiting to ascend, like ytterbium ions caught in a laser beam, to a vast quantum computer in the sky.

I’ve heard “quantum inimitability” (that is, inability to imitate using a classical computer), but who can pronounce that?

Yesterday, my brilliant former student Ewin Tang (yes, that one) relayed to me a suggestion by Kevin Tian: “quantum eclipse” (that is, the moment when quantum computers first eclipse classical ones for some task). But would one want to speak of a “quantum eclipse experiment”? And shouldn’t we expect that, the cuter and cleverer the term, the harder it will be to use unironically?

In summary, while someone might think of a term so inspired that it immediately supplants “quantum supremacy” (and while I welcome suggestions), I currently regard it as an open problem.

Anyway, evidently dissatisfied with my response, last week Leonie teamed up with 13 others to publish a letter in Nature, which was originally entitled “Supremacy is for racists—use ‘quantum advantage,'” but whose title I see has now been changed to the less inflammatory “Instead of ‘supremacy’ use ‘quantum advantage.'” Leonie’s co-signatories included four of my good friends and colleagues: Alan Aspuru-Guzik, Helmut Katzgraber, Anne Broadbent, and Chris Granade (the last of whom got started in the field by helping me edit Quantum Computing Since Democritus).

(Update: Leonie pointed me to a longer list of signatories here, at their website called “quantumresponsibility.org.” A few names that might be known to Shtetl-Optimized readers are Andrew White, David Yonge-Mallo, Debbie Leung, Matt Leifer, Matthias Troyer.)

Their letter says:

The community claims that quantum supremacy is a technical term with a specified meaning. However, any technical justification for this descriptor could get swamped as it enters the public arena after the intense media coverage of the past few months.

In our view, ‘supremacy’ has overtones of violence, neocolonialism and racism through its association with ‘white supremacy’. Inherently violent language has crept into other branches of science as well — in human and robotic spaceflight, for example, terms such as ‘conquest’, ‘colonization’ and ‘settlement’ evoke the terra nullius arguments of settler colonialism and must be contextualized against ongoing issues of neocolonialism.

Instead, quantum computing should be an open arena and an inspiration for a new generation of scientists.

When I did an “Ask Me Anything” session, as the closing event at Q2B, Sarah Kaiser asked me to comment on the Nature petition. So I repeated what I’d said in my emailed response to Leonie—running through the problems with each proposed alternative term, talking about the value of reclaiming the word “supremacy,” and mostly just trying to diffuse the tension by getting everyone laughing together. Sarah later tweeted that she was “really disappointed” in my response.

Then the Wall Street Journal got in on the action, with a brief editorial (warning: paywalled) mocking the Nature petition:

There it is, folks: Mankind has hit quantum wokeness. Our species, akin to Schrödinger’s cat, is simultaneously brilliant and brain-dead. We built a quantum computer and then argued about whether the write-up was linguistically racist.

Taken seriously, the renaming game will never end. First put a Sharpie to the Supremacy Clause of the U.S. Constitution, which says federal laws trump state laws. Cancel Matt Damon for his 2004 role in “The Bourne Supremacy.” Make the Air Force give up the term “air supremacy.” Tell lovers of supreme pizza to quit being so chauvinistic about their toppings. Please inform Motown legend Diana Ross that the Supremes are problematic.

The quirks of quantum mechanics, some people argue, are explained by the existence of many universes. How did we get stuck in this one?

Steven Pinker also weighed in, with a linguistically-informed tweetstorm:

This sounds like something from The Onion but actually appeared in Nature … It follows the wokified stigmatization of other innocent words, like “House Master” (now, at Harvard, Residential Dean) and “NIPS” (Neural Information Processing Society, now NeurIPS). It’s a familiar linguistic phenomenon, a lexical version of Gresham’s Law: bad meanings drive good ones out of circulation. Examples: the doomed “niggardly” (no relation to the n-word) and the original senses of “cock,” “ass,” “prick,” “pussy,” and “booty.” Still, the prissy banning of words by academics should be resisted. It dumbs down understanding of language: word meanings are conventions, not spells with magical powers, and all words have multiple senses, which are distinguished in context. Also, it makes academia a laughingstock, tars the innocent, and does nothing to combat actual racism & sexism.

Others had a stronger reaction. Curtis Yarvin, better known as Mencius Moldbug, is one of the founders of “neoreaction” (and a significant influence on Steve Bannon, Michael Anton, and other Trumpists). Regulars might remember that Yarvin argued with me in Shtetl-Optimized‘s comment section, under a post in which I denounced Trump’s travel ban and its effects on my Iranian PhD student. Since then, Yarvin has sent me many emails, which have ranged from long to extremely long, and whose message could be summarized as: “[labored breathing] Abandon your liberal Enlightenment pretensions, young Nerdwalker. Come over the Dark Side.”

After the “supremacy is for racists” letter came out in Nature, though, Yarvin sent me his shortest email ever. It was simply a link to the letter, along with the comment “I knew it would come to this.”

He meant: “What more proof do you need, young Nerdawan, that this performative wokeness is a cancer that will eventually infect everything you value—even totally apolitical research in quantum information? And by extension, that my whole worldview, which warned of this, is fundamentally correct, while your faith in liberal academia is naïve, and will be repaid only with backstabbing?”

In a subsequent email, Yarvin predicted that in two years, the whole community will be saying “quantum advantage” instead of “quantum supremacy,” and in five years I’ll be saying “quantum advantage” too. As Yarvin famously wrote: “Cthulhu may swim slowly. But he only swims left.”

Truthfully, half of me just wants to switch to “quantum advantage” right now and be done with it. As I said, I know some of the signatories of the Nature letter to be smart and reasonable and kind. They don’t wish to rid the planet of everyone like me. They’re not Amanda Marcottes or Arthur Chus. Furthermore, there’s little I despise more than a meaty scientific debate devolving into a pointless semantic one, with brilliant friend after brilliant friend getting sucked into the vortex (“you too?”). I’m strongly in the Pinkerian camp, which holds that words are just arbitrary designators, devoid of the totemic power to dictate thoughts. So if friends and colleagues—even just a few of them—tell me that they find some word I use to be offensive, why not just be a mensch, apologize for any unintended hurt, switch words midsentence, and continue discussing the matter at hand?

But then the other half of me wonders: once we’ve ceded an open-ended veto over technical terms that remind anyone of anything bad, where does it stop? How do we ever certify a word as kosher? At what point do we all get to stop arguing and laugh together?

To make this worry concrete, look back at Sarah Kaiser’s Twitter thread—the one where she expresses disappointment in me. Below her tweet, someone remarks that, besides “quantum supremacy,” the word “ancilla” (as in ancilla qubit, a qubit used for intermediate computation or other auxiliary purposes) is problematic as well. Here’s Sarah’s response:

I agree, but I wanted to start by focusing on the obvious one, Its harder for them to object to just one to start with, then once they admit the logic, we can expand the list

(What would Curtis Yarvin say about that?)

You’re probably now wondering: what’s wrong with “ancilla”? Apparently, in ancient Rome, an “ancilla” was a female slave, and indeed that’s the Latin root of the English adjective “ancillary” (as in “providing support to”). I confess that I hadn’t known that—had you? Admittedly, once you do know, you might never again look at a Controlled-NOT gate—pitilessly flipping an ancilla qubit, subject only to the whims of a nearby control qubit—in quite the same way.

(Ah, but the ancilla can fight back against her controller! And she does—in the Hadamard basis.)

The thing is, if we’re gonna play this game: what about annihilation operators? Won’t those need to be … annihilated from physics?

And what about unitary matrices? Doesn’t their very name negate the multiplicity of perspectives and cultures?

What about Dirac’s oddly-named bra/ket notation, with its limitless potential for puerile jokes, about the “bra” vectors displaying their contents horizontally and so forth? (Did you smile at that, you hateful pig?)

What about daggers? Don’t we need a less violent conjugate tranpose?

Not to beat a dead horse, but once you hunt for examples, you realize that the whole dictionary is shot through with domination and brutality—that you’d have to massacre the English language to take it out. There’s nothing special about math or physics in this respect.

The same half of me also thinks about my friends and colleagues who oppose claims of quantum supremacy, or even the quest for quantum supremacy, on various scientific grounds. I.e., either they don’t think that the Google team achieved what it said, or they think that the task wasn’t hard enough for classical computers, or they think that the entire goal is misguided or irrelevant or uninteresting.

Which is fine—these are precisely the arguments we should be having—except that I’ve personally seen some of my respected colleagues, while arguing for these positions, opportunistically tack on ideological objections to the term “quantum supremacy.” Just to goose up their case, I guess. And I confess that every time they did this, it made me want to keep saying “quantum supremacy” from now till the end of time—solely to deny these colleagues a cheap and unearned “victory,” one they apparently felt they couldn’t obtain on the merits alone. I realize that this is childish and irrational.

Most of all, though, the half of me that I’m talking about thinks about Curtis Yarvin and the Wall Street Journal editorial board, cackling with glee to see their worldview so dramatically confirmed—as theatrical wokeness, that self-parodying modern monstrosity, turns its gaze on (of all things) quantum computing research. More red meat to fire up the base—or at least that sliver of the base nerdy enough to care. And the left, as usual, walks right into the trap, sacrificing its credibility with the outside world to pursue a runaway virtue-signaling spiral.

The same half of me thinks: do we really want to fight racism and sexism? Then let’s work together to assemble a broad coalition that can defeat Trump. And Jair Bolsonaro, and Viktor Orbán, and all the other ghastly manifestations of humanity’s collective lizard-brain. Then, if we’re really fantasizing, we could liberalize the drug laws, and get contraception and loans and education to women in the Third World, and stop the systematic disenfranchisement of black voters, and open up the world’s richer, whiter, and higher-elevation countries to climate refugees, and protect the world’s remaining indigenous lands (those that didn’t burn to the ground this year).

In this context, the trouble with obsessing over terms like “quantum supremacy” is not merely that it diverts attention, while contributing nothing to fighting the world’s actual racism and sexism. The trouble is that the obsessions are actually harmful. For they make academics—along with progressive activists—look silly. They make people think that we must not have meant it when we talked about the existential urgency of climate change and the world’s other crises. They pump oxygen into right-wing echo chambers.

But it’s worse than ridiculous, because of the message that I fear is received by many outside the activists’ bubble. When you say stuff like “[quantum] supremacy is for racists,” what’s heard might be something more like:

“Watch your back, you disgusting supremacist. Yes, you. You claim that you mentor women and minorities, donate to good causes, try hard to confront the demons in your own character? Ha! None of that counts for anything with us. You’ll never be with-it enough to be our ally, so don’t bother trying. We’ll see to it that you’re never safe, not even in the most abstruse and apolitical fields. We’ll comb through your words—even words like ‘ancilla qubit’—looking for any that we can cast as offensive by our opaque and ever-shifting standards. And once we find some, we’ll have it within our power to end your career, and you’ll be reduced to groveling that we don’t. Remember those popular kids who bullied you in second grade, giving you nightmares of social ostracism that persist to this day? We plan to achieve what even those bullies couldn’t: to shame you with the full backing of the modern world’s moral code. See, we’re the good guys of this story. It’s goodness itself that’s branding you as racist scum.”

In short, I claim that the message—not the message intended, of course, by anyone other than a Chu or a Marcotte or a SneerClubber, but the message received—is basically a Trump campaign ad. I claim further that our civilization’s current self-inflicted catastrophe will end—i.e., the believers in science and reason and progress and rule of law will claw their way back to power—when, and only when, a generation of activists emerges that understands these dynamics as well as Barack Obama did.

Wouldn’t it be awesome if, five years from now, I could say to Curtis Yarvin: you were wrong? If I could say to him: my colleagues and I still use the term ‘quantum supremacy’ whenever we care to, and none of us have been cancelled or ostracized for it—so maybe you should revisit your paranoid theories about Cthulhu and the Cathedral and so forth? If I could say: quantum computing researchers now have bigger fish to fry than arguments over words—like moving beyond quantum supremacy to the first useful quantum simulations, as well as the race for scalability and fault-tolerance? And even: progressive activists now have bigger fish to fry too—like retaking actual power all over the world?

Anyway, as I said, that’s how half of me feels. The other half is ready to switch to “quantum advantage” or any other serviceable term and get back to doing science.

December 2nd, 2019
1. Two weeks ago, I blogged about the claim of Nathan Keller and Ohad Klein to have proven the Aaronson-Ambainis Conjecture. Alas, Keller and Klein tell me that they’ve now withdrawn their preprint (though it may take another day for that to show up on the arXiv), because of what looks for now like a fatal flaw, in Lemma 5.3, discovered by Paata Ivanishvili. (My own embarrassment over having missed this flaw is slightly mitigated by most of the experts in discrete Fourier analysis having missed it as well!) Keller and Klein are now working to fix the flaw, and I wholeheartedly wish them success.
2. In unrelated news, I was saddened to read that Virgil Griffith—cryptocurrency researcher, former Integrated Information Theory researcher, and onetime contributor to Shtetl-Optimized—was arrested at LAX for having traveled to North Korea to teach the DPRK about cryptocurrency, against the admonitions of the US State Department. I didn’t know Virgil well, but I did meet him in person at least once, and I liked his essays for this blog about how, after spending years studying IIT under Giulio Tononi himself, he became disillusioned with many aspects of it and evolved to a position not far from mine (though not identical either).
Personally, I despise the North Korean regime for the obvious reasons—I regard it as not merely evil, but cartoonishly so—and I’m mystified by Virgil’s apparently sincere belief that he could bring peace between the North and South by traveling to North Korea to give a lecture about blockchain. Yet, however world-historically naïve he may have been, his intentions appear to have been good. More pointedly—and here I’m asking not in a legal sense but in a human one—if giving aid and comfort to the DPRK is treasonous, then isn’t the current occupant of the Oval Office a million times guiltier of that particular treason (to say nothing of others)? It’s like, what does “treason” even mean anymore? In any case, I hope some plea deal or other arrangement can be worked out that won’t end Virgil’s productive career.

## Guest post by Greg Kuperberg: Archimedes’ other principle and quantum supremacy

November 26th, 2019

Scott’s Introduction: Happy Thanksgiving! Please join me in giving thanks for the beautiful post below, by friend-of-the-blog Greg Kuperberg, which tells a mathematical story that stretches from the 200s BC all the way to Google’s quantum supremacy result last month.

## Archimedes’ other principle and quantum supremacy

by Greg Kuperberg

Note: UC Davis is hiring in CS theory! Scott offered me free ad space if I wrote a guest post, so here we are. The position is in all areas of CS theory, including QC theory although the search is not limited to that.

In this post, I wear the hat of a pure mathematician in a box provided by Archimedes. I thus set aside what everyone else thinks is important about Google’s 53-qubit quantum supremacy experiment, that it is a dramatic milestone in quantum computing technology. That’s only news about the physical world, whose significance pales in comparison to the Platonic world of mathematical objects. In my happy world, I like quantum supremacy as a demonstration of a beautiful coincidence in mathematics that has been known for more than 2000 years in a special case. The single-qubit case was discovered by Archimedes, duh. As Scott mentions in Quantum Computing Since Democritus, Bill Wootters stated the general result in a 1990 paper, but Wootters credits a 1974 paper by the Czech physicist Stanislav Sýkora. I learned of it in the substantially more general context of symplectic geometric that mathematicians developed independently between Sýkora’s prescient paper and Wootters’ more widely known citation. Much as I would like to clobber you with highly abstract mathematics, I will wait for some other time.

Suppose that you pick a pure state $$|\psi\rangle$$ in the Hilbert space $$\mathbb{C}^d$$ of a $$d$$-dimensional qudit, and then make many copies and fully measure each one, so that you sample many times from some distribution $$\mu$$ on the $$d$$ outcomes. You can think of such a distribution $$\mu$$ as a classical randomized state on a digit of size $$d$$. The set of all randomized states on a $$d$$-digit makes a $$(d-1)$$-dimensional simplex $$\Delta^{d-1}$$ in the orthant $$\mathbb{R}_{\ge 0}^d$$. The coincidence is that if $$|\psi\rangle$$ is uniformly random in the unit sphere in $$\mathbb{C}^d$$, then $$\mu$$ is uniformly random in $$\Delta^{d-1}$$. I will call it the Born map, since it expresses the Born rule of quantum mechanics that amplitudes yield probabilities. Here is a diagram of the Born map of a qutrit, except with the laughable simplification of the 5-sphere in $$\mathbb{C}^3$$ drawn as a 2-sphere.

If you pretend to be a bad probability student, then you might not be surprised by this coincidence, because you might suppose that all probability distributions are uniform other than treacherous exceptions to your intuition. However, the principle is certainly not true for a “rebit” (a qubit with real amplitudes) or for higher-dimensional “redits.” With real amplitudes, the probability density goes to infinity at the sides of the simplex $$\Delta^{d-1}$$ and is even more favored at the corners. It also doesn’t work for mixed qudit states; the projection then favors the middle of $$\Delta^{d-1}$$.

### Archimedes’ theorem

The theorem of Archimedes is that a natural projection from the unit sphere to a circumscribing vertical cylinder preserves area. The projection is the second one that you might think of: Project radially from a vertical axis rather than radially in all three directions. Since Archimedes was a remarkably prescient mathematician, he was looking ahead to the Bloch sphere of pure qubit states $$|\psi\rangle\langle\psi|$$ written in density operator form. If you measure $$|\psi\rangle\langle\psi|$$ in the computational basis, you get a randomized bit state $$\mu$$ somewhere on the interval from guaranteed 0 to guaranteed 1.

This transformation from a quantum state to a classical randomized state is a linear projection to a vertical axis. It is the same as Archimedes’ projection, except without the angle information. It doesn’t preserve dimension, but it does preserve measure (area or length, whatever) up to a factor of $$2\pi$$. In particular, it takes a uniformly random $$|\psi\rangle\langle\psi|$$ to a uniformly random $$\mu$$.

Archimedes’ projection is also known as the Lambert cylindrical map of the world. This is the map that squishes Greenland along with the top of North America and Eurasia to give them proportionate area.

(I forgive Lambert if he didn’t give prior credit to Archimedes. There was no Internet back then to easily find out who did what first.) Here is a calculus-based proof of Archimedes’ theorem: In spherical coordinates, imagine an annular strip on the sphere at a polar angle of $$\theta$$. (The polar angle is the angle from vertical in spherical coordinates, as depicted in red in the Bloch sphere diagram.) The strip has a radius of $$\sin\theta$$, which makes it shorter than its unit radius friend on the cylinder. But it’s also tilted from vertical by an angle of $$\frac{\pi}2-\theta$$, which makes it wider by a factor of $$1/(\sin \theta)$$ than the height of its projection onto the $$z$$ axis. The two factors exactly cancel out, making the area of the strip exactly proportional to the length of its projection onto the $$z$$ axis. This is a coincidence which is special to the 2-sphere in 3 dimensions. As a corollary, we get that the surface area of a unit sphere is $$4\pi$$, the same as an open cylinder with radius 1 and height 2. If you want to step through this in even more detail, Scott mentioned to me an action video which is vastly spiffier than anything that I could ever make.

The projection of the Bloch sphere onto an interval also shows what goes wrong if you try this with a rebit. The pure rebit states — again expressed in density operator form $$|\psi\rangle\langle\psi|$$ are a great circle in the Bloch sphere. If you linearly project a circle onto an interval, then the length of the circle is clearly bunched up at the ends of the interval and the projected measure on the interval is not uniform.

### Sýkora’s generalization

It is a neat coincidence that the Born map of a qubit preserves measure, but a proof that relies on Archimedes’ theorem seems to depend on the special geometry of the Bloch sphere of a qubit. That the higher-dimensional Born map also preserves measure is downright eerie. Scott challenged me to write an intuitive explanation. I remembered two different (but similar) proofs, neither of which is original to me. Scott and I disagree as to which proof is nicer.

As a first step of the first proof, it is easy to show that the Born map $$p = |z|^2$$ for a single amplitude $$z$$ preserves measure as a function from the complex plane $$\mathbb{C}$$ to the ray $$\mathbb{R}_{\ge 0}$$. The region in the complex numbers $$\mathbb{C}$$ where the length of $$z$$ is between $$a$$ and $$b$$, or $$a \le |z| \le b$$, is $$\pi(b^2 – a^2)$$. The corresponding interval for the probability is $$a^2 \le p \le b^2$$, which thus has length $$b^2-a^2$$. That’s all, we’ve proved it! More precisely, the area of any circularly symmetric region in $$\mathbb{C}$$ is $$\pi$$ times the length of its projection onto $$\mathbb{R}_{\ge 0}$$.

The second step is to show the same thing for the Born map from the $$d$$-qudit Hilbert space $$\mathbb{C}^d$$ to the $$d$$-digit orthant $$\mathbb{R}_{\ge 0}^d$$, again without unit normalization. It’s also measure-preserving, up to a factor of $$\pi^d$$ this time, because it’s the same thing in each coordinate separately. To be precise, the volume ratio holds for any region in $$\mathbb{C}^d$$ that is invariant under separately rotating each of the $$d$$ phases. (Because you can approximate any such region with a union of products of thin annuli.)

The third and final step is the paint principle for comparing surface areas. If you paint the hoods of two cars with the same thin layer of paint and you used the same volume of paint for each one, then you can conclude that the two car hoods have nearly same area. In our case, the Born map takes the region $1 \le |z_0|^2 + |z_1|^2 + \cdots + |z_{d-1}|^2 \le 1+\epsilon$ in $$\mathbb{C}^d$$ to the region $1 \le p_0 + p_1 + \cdots + p_{d-1} \le 1+\epsilon$ in the orthant $$\mathbb{R}_{\ge 0}^d$$. The former is the unit sphere $$S^{2d-1}$$ in $$\mathbb{C}^d$$ painted to a thickness of roughly $$\epsilon/2$$. The latter is the probability simplex $$\Delta^{n-1}$$ painted to a thickness of exactly $$\epsilon$$. Taking the limit $$\epsilon \to 0$$, the Born map from $$S^{2d-1}$$ to $$\Delta^{n-1}$$ preserves measure up to a factor of $$2\pi^n$$.

You might wonder “why” this argument works even if you accept that it does work. The key is that the exponent 2 appears in two different ways: as the dimension of the complex numbers, and as the exponent used to set probabilities and define spheres. If we try the same argument with real amplitudes, then the volume between “spheres” of radius $$a$$ and $$b$$ is just $$2(b-a)$$, which does not match the length $$b^2-a^2$$. The Born map for a single real amplitude is the parabola $$p = x^2$$, which clearly distorts length since it is not linear. The higher-dimensional real Born map similarly distorts volumes, whether or not you restrict to unit-length states.

If you’re a bitter-ender who still wants Archimedes’ theorem for real amplitudes, then you might consider the variant formula $$p = |x|$$ to obtain a probability $$p$$ from a “quantum amplitude” $$x$$. Then the “Born” map does preserve measure, but for the trivial reason that $$x = \pm p$$ is not really a quantum amplitude, it is a probability with a vestigial sign. Also the unit “sphere” in $$\mathbb{R}^d$$ is not really a sphere in this theory, it is a hyperoctahedron. The only “unitary” operators that preserve the unit hyperoctahedron are signed permutation matrices. You can only use them for reversible classical computing or symbolic dynamics; they don’t have the strength of true quantum computing or quantum mechanics.

The fact that the Born map preserves measure also yields a bonus calculation of the volume of the unit ball in $$2d$$ real dimensions, if we interpret that as $$d$$ complex dimensions. The ball $|z_0|^2 + |z_1|^2 + \cdots + |z_{d-1}|^2 \le 1$ in $$\mathbb{C}^d$$ is sent to a different simplex $p_0 + p_1 + \cdots + p_{d-1} \le 1$ in $$\mathbb{R}_{\ge 0}^d$$. If we recall that the volume of a $$d$$-dimensional pyramid is $$\frac1d$$ times base times height and calculate by induction on $$d$$, we get that this simplex has volume $$\frac1{d!}$$. Thus, the volume of the $$2d$$-dimensional unit ball is $$\frac{\pi^d}{d!}$$.

You might ask whether the volume of a $$d$$-dimensional unit ball is always $$\frac{\pi^{d/2}}{(d/2)!}$$ for both $$d$$ even and odd. The answer is yes if we interpret factorials using the gamma function formula $$x! = \Gamma(x+1)$$ and look up that $$\frac12! = \Gamma(\frac32) = \frac{\sqrt{\pi}}2$$. The gamma function was discovered by Euler as a solution to the question of defining fractional factorials, but the notation $$\Gamma(x)$$ and the cumbersome shift by 1 is due to Legendre. Although Wikipedia says that no one knows why Legendre defined it this way, I wonder if his goal was to do what the Catholic church later did for itself in 1978: It put a Pole at the origin.

(Scott wanted to censor this joke. In response, I express my loyalty to my nation of birth by quoting the opening of the Polish national anthem: “Poland has not yet died, so long as we still live!” I thought at first that Stanislav Sýkora is Polish since Stanisław and Sikora are both common Polish names, but his name has Czech spelling and he is Czech. Well, the Czechs are cool too.)

Sýkora’s 1974 proof of the generalized Archimedes’ theorem is different from this one. He calculates multivariate moments of the space of unit states $$S^{2d-1} \subseteq \mathbb{C}^d$$, and confirms that they match the moments in the probability simplex $$\Delta^{d-1}$$. There are inevitably various proofs of this result, and I will give another one.

### Another proof, and quantum supremacy

There is a well-known and very useful algorithm to generate a random point on the unit sphere in either $$\mathbb{R}^d$$ or $$\mathbb{C}^d$$, and a similar algorithm to generate a random point in a simplex. In the former algorithm, we make each real coordinate $$x$$ into an independent Gaussian random variable with density proportional to $$e^{-x^2}\;dx$$, and then rescale the result to unit length. Since the exponents combine as $e^{-x_0^2}e^{-x_1^2}\cdots e^{-x_{d-1}^2} = e^{-(x_0^2 + x_1^2 + \cdots + x_{d-1}^2)},$ we learn that the total exponent is spherically symmetric. Therefore after rescaling, the result is a uniformly random point on the unit sphere $$S^{d-1} \subseteq \mathbb{R}^d$$. Similarly, the other algorithm generates a point in the orthant $$\mathbb{R}_{\ge 0}^d$$ by making each real coordinate $$p \ge 0$$ an independent random variable with exponential distribution $$e^{-p}\;dp$$. This time we rescale the vector until its sum is 1. This algorithm likewise produces a uniformly random point in the simplex $$\Delta^{d-1} \subseteq \mathbb{R}_{\ge 0}^d$$ because the total exponent of the product $e^{-p_0}e^{-p_1}\cdots e^{-p_{d-1}} = e^{-(p_0 + p_1 + \cdots + p_{d-1})}$ only depends on the sum of the coordinates. Wootters describes both of these algorithms in his 1990 paper, but instead of relating them to give his own proof of the generalized Archimedes’ theorem, he cites Sýkora.

The gist of the proof is that the Born map takes the Gaussian algorithm to the exponential algorithm. Explicitly, the Gaussian probability density for a single complex amplitude $z = x+iy = re^{i\theta}$ can be converted from Cartesian to polar coordinate as follows: $\frac{e^{-|z|^2}\;dx\;dy}{\pi} = \frac{e^{-r^2}r\;dr\;d\theta}{\pi}.$ I have included the factor of $$r$$ that is naturally present in an area integral in polar coordinates. We will need it, and it is another way to see that the theorem relies on the fact that the complex numbers are two-dimensional. To complete the proof, we substitute $$p = r^2$$ and remember that $$dp = 2r\;dr$$, and then integrate over $$\theta$$ (trivially, since the integrand does not depend on $$\theta$$). The density simplifies to $$e^{-p}\;dp$$, which is exactly the exponential distribution for a real variable $$p \ge 0$$. Since the Born map takes the Gaussian algorithm to the exponential algorithm, and since each algorithm produces a uniformly random point, the Born map must preserve uniform measure. (Scott likes this proof better because it is algorithmic, and because it is probabilistic.)

Now about quantum supremacy. You might think that a random chosen quantum circuit on $$n$$ qubits produces a nearly uniformly random quantum state $$|\psi\rangle$$ in their joint Hilbert space, but it’s not quite not that simple. When $$n=53$$, or otherwise as $$n \to \infty$$, a manageable random circuit is not nearly creative enough to either reach or approximate most of the unit states in the colossal Hilbert space of dimension $$d = 2^n$$. The state $$|\psi\rangle$$ that you get from (say) a polynomial-sized circuit resembles a fully random state in various statistical and computational respects, both proven and conjectured. As a result, if you measure the qubits in the computational basis, you get a randomized state on $$n$$ bits that resembles a uniformly random point in $$\Delta^{2^n-1}$$.

If you choose $$d$$ probabilities, and if each one is an independent exponential random variable, then the law of large numbers says that the sum (which you use for rescaling) is close to $$d$$ when $$d$$ is large. When $$d$$ is really big like $$2^{53}$$, a histogram of the probabilities of the bit strings of a supremacy experiment looks like an exponential curve $$f(p) \propto e^{-pd}$$. In a sense, the statistical distribution of the bit strings is almost the same almost every time, independent of which random quantum circuit you choose to generate them. The catch is that the position of any given bit string does depend on the circuit and is highly scrambled. I picture it in my mind like this:

It is thought to be computationally intractable to calculate where each bit string lands on this exponential curve, or even where just one of them does. (The exponential curve is attenuated by noise in the actual experiment, but it’s the same principle.) That is one reason that random quantum circuits are supreme.

## The Aaronson-Ambainis Conjecture (2008-2019)

November 17th, 2019

Update: Sadly, Nathan Keller and Ohad Klein tell me that they’ve retracted their preprint, because of what currently looks like a fatal flaw in Lemma 5.3, uncovered by Paata Ivanishvili. I wish them the best of luck in fixing the problem. In the meantime, I’m leaving up this post for “historical” reasons.

Around 1999, one of the first things I ever did in quantum computing theory was to work on a problem that Fortnow and Rogers suggested in a paper: is it possible to separate P from BQP relative to a random oracle? (That is, without first needing to separate P from PSPACE or whatever in the real world?) Or to the contrary: suppose that a quantum algorithm Q makes T queries to a Boolean input string X. Is there then a classical simulation algorithm that makes poly(T) queries to X, and that approximates Q’s acceptance probability for most values of X? Such a classical simulation, were it possible, would still be consistent with the existence of quantum algorithms like Simon’s and Shor’s, which are able to achieve exponential (and even greater) speedups in the black-box setting. It would simply demonstrate the importance, for Simon’s and Shor’s algorithms, of global structure that makes the string X extremely non-random: for example, encoding a periodic function (in the case of Shor’s algorithm), or encoding a function that hides a secret string s (in the case of Simon’s). It would underscore that superpolynomial quantum speedups depend on structure.

I never managed to solve this problem. Around 2008, though, I noticed that a solution would follow from a perhaps-not-obviously-related conjecture, about influences in low-degree polynomials. Namely, let p:Rn→R be a degree-d real polynomial in n variables, and suppose p(x)∈[0,1] for all x∈{0,1}n. Define the variance of p to be
Var(p):=Ex,y[|p(x)-p(y)|],
and define the influence of the ith variable to be
Infi(p):=Ex[|p(x)-p(xi)|].
Here the expectations are over strings in {0,1}n, and xi means x with its ith bit flipped between 0 and 1. Then the conjecture is this: there must be some variable i such that Infi(p) ≥ poly(Var(p)/d) (in other words, that “explains” a non-negligible fraction of the variance of the entire polynomial).

Why would this conjecture imply the statement about quantum algorithms? Basically, because of the seminal result of Beals et al. from 1998: that if a quantum algorithm makes T queries to a Boolean input X, then its acceptance probability can be written as a real polynomial over the bits of X, of degree at most 2T. Given that result, if you wanted to classically simulate a quantum algorithm Q on most inputs—and if you only cared about query complexity, not computation time—you’d simply need to do the following:
(1) Find the polynomial p that represents Q’s acceptance probability.
(2) Find a variable i that explains at least a 1/poly(T) fraction of the total remaining variance in p, and query that i.
(3) Keep repeating step (2), until p has been restricted to a polynomial with not much variance left—i.e., to nearly a constant function p(X)=c. Whenever that happens, halt and output the constant c.
The key is that by hypothesis, this algorithm will halt, with high probability over X, after only poly(T) steps.

Anyway, around the same time, Andris Ambainis had a major break on a different problem that I’d told him about: namely, whether randomized and quantum query complexities are polynomially related for all partial functions with permutation symmetry (like the collision and the element distinctness functions). Andris and I decided to write up the two directions jointly. The result was our 2011 paper entitled The Need for Structure in Quantum Speedups.

Of the two contributions in the “Need for Structure” paper, the one about random oracles and influences in low-degree polynomials was clearly the weaker and less satisfying one. As the reviewers pointed out, that part of the paper didn’t solve anything: it just reduced one unsolved problem to a new, slightly different problem that was also unsolved. Nevertheless, that part of the paper acquired a life of its own over the ensuing decade, as the world’s experts in analysis of Boolean functions and polynomials began referring to the “Aaronson-Ambainis Conjecture.” Ryan O’Donnell, Guy Kindler, and many others had a stab. I even got Terry Tao to spend an hour or two on the problem when I visited UCLA.

Now, at long last, Nathan Keller and Ohad Klein say they’ve proven the Aaronson-Ambainis Conjecture, in a preprint whose title is a riff on ours: “Quantum Speedups Need Structure.”

Their paper hasn’t yet been peer-reviewed, and I haven’t yet carefully studied it, but I could and should: at 19 pages, it looks very approachable and clear, if not as radically short as (say) Huang’s proof of the Sensitivity Conjecture. Keller and Klein’s argument subsumes all the earlier results that I knew would need to be subsumed, and involves all the concepts (like a real analogue of block sensitivity) that I knew would need to be involved somehow.

My plan had been as follows:
(1) Read their paper in detail. Understand every step of their proof.
(2) Write a blog post that reflects my detailed understanding.

Unfortunately, this plan did not sufficiently grapple with the fact that I now have two kids. It got snagged for a week at step (1). So I’m now executing an alternative plan, which is to jump immediately to the blog post.

Assuming Keller and Klein’s result holds up—as I expect it will—by combining it with the observations in my and Andris’s paper, one immediately gets an explanation for why no one has managed to separate P from BQP relative to a random oracle, but only relative to non-random oracles. This complements the work of Kahn, Saks, and Smyth, who around 2000 gave a precisely analogous explanation for the difficulty of separating P from NP∩coNP relative to a random oracle.

Unfortunately, the polynomial blowup is quite enormous: from a quantum algorithm making T queries, Keller and Klein apparently get a classical algorithm making O(T18) queries. But such things can almost always be massively improved.

Feel free to use the comments to ask any questions about this result or its broader context. I’ll either do my best to answer from the limited amount I know, or else I’ll pass the questions along to Nathan and Ohad themselves. Maybe, at some point, I’ll even be forced to understand the new proof.

Update (Nov. 20): Tonight I finally did what I should’ve done two weeks ago, and worked through the paper from start to finish. Modulo some facts about noise operators, hypercontractivity, etc. that I took on faith, I now have a reasonable (albeit imperfect) understanding of the proof. It’s great!

In case it’s helpful to anybody, here’s my one-paragraph summary of how it works. First, you hit your bounded degree-d function f with a random restriction to attenuate its higher-degree Fourier coefficients (reminiscent of Linial-Mansour-Nisan).  Next, in that attenuated function, you find a small “coalition” of influential variables—by which we mean, a set of variables for which there’s some assignment that substantially biases f.  You keep iterating—finding influential coalitions in subfunctions on n/4, n/8, etc. variables. All the while, you keep track of the norm of the vector of all the block-sensitivities of all the inputs (the authors don’t clearly explain this in the intro, but they reveal it near the end). Every time you find another influential coalition, that norm goes down by a little, but by approximation theory, it can only go down O(d2) times until it hits rock bottom and your function is nearly constant. By the end, you’ll have approximated f itself by a decision tree of depth poly(d, 1/ε, log(n)).  Finally, you get rid of the log(n) term by using the fact that f essentially depended on at most exp(O(d)) variables anyway.

Anyway, I’m not sure how helpful it is to write more: the paper itself is about 95% as clear as it could possibly be, and even where it isn’t, you’d probably need to read it first (and, uh, know something about influences, block sensitivity, random restrictions, etc.) before any further clarifying remarks would be of use. But happy to discuss more in the comments, if anyone else is reading it.

## Annual recruitment post

November 12th, 2019

Just like I did last year, and the year before, I’m putting up a post to let y’all know about opportunities in our growing Quantum Information Center at UT Austin.

I’m proud to report that we’re building something pretty good here. This fall Shyam Shankar joined our Electrical and Computer Engineering (ECE) faculty to do experimental superconducting qubits, while (as I blogged in the summer) the quantum complexity theorist John Wright will join me on the CS faculty in Fall 2020. Meanwhile, Drew Potter, an expert on topological qubits, rejoined our physics faculty after a brief leave. Our weekly quantum information group meeting now regularly attracts around 30 participants—from the turnout, you wouldn’t know it’s not MIT or Caltech or Waterloo. My own group now has five postdocs and six PhD students—as well as some amazing undergrads striving to meet the bar set by Ewin Tang. Course offerings in quantum information currently include Brian La Cour’s Freshman Research Initiative, my own undergrad Intro to Quantum Information Science honors class, and graduate classes on quantum complexity theory, experimental realizations of QC, and topological matter (with more to come). We’ll also be starting an undergraduate Quantum Information Science concentration next fall.

(1) If you’re interested in pursuing a PhD focused on quantum computing and information (and/or classical theoretical computer science) at UT Austin: please apply! If you want to work with me or John Wright on quantum algorithms and complexity, apply to CS (I can also supervise physics students in rare cases). Also apply to CS, of course, if you want to work with our other CS theory faculty: David Zuckerman, Dana Moshkovitz, Adam Klivans, Anna Gal, Eric Price, Brent Waters, Vijaya Ramachandran, or Greg Plaxton. If you want to work with Drew Potter on nonabelian anyons or suchlike, or with Allan MacDonald, Linda Reichl, Elaine Li, or others on many-body quantum theory, apply to physics. If you want to work with Shyam Shankar on superconducting qubits, apply to ECE. Note that the deadline for CS and physics is December 1, while the deadline for ECE is December 15.

You don’t need to ask me whether I’m on the lookout for great students: I always am! If you say on your application that you want to work with me, I’ll be sure to see it. Emailing individual faculty members is not how it works and won’t help. Admissions are extremely competitive, so I strongly encourage you to apply broadly to maximize your options.

(2) If you’re interested in a postdoc in my group, I’ll have approximately two openings starting in Fall 2020. To apply, just send me an email by January 1, 2020 with the following info: