Sense, sensibility, and superconductors

Jonathan Monroe disagreed with his PhD supervisor—with respect. They needed to measure a superconducting qubit, a tiny circuit in which current can flow forever. The qubit emits light, which carries information about the qubit’s state. Jonathan and Kater intensify the light using an amplifier. They’d fabricated many amplifiers, but none had worked. Jonathan suggested changing their strategy—with a politeness to which Emily Post couldn’t have objected. Jonathan’s supervisor, Kater Murch, suggested repeating the protocol they’d performed many times.

“That’s the definition of insanity,” Kater admitted, “but I think experiment needs to involve some of that.”

I watched the exchange via Skype, with more interest than I’d have watched the Oscars with. Someday, I hope, I’ll be able to weigh in on such a debate, despite working as a theorist. Someday, I’ll have partnered with enough experimentalists to develop insight.

I’m partnering with Jonathan and Kater on an experiment that coauthors and I proposed in a paper blogged about here. The experiment centers on an uncertainty relation, an inequality of the sort immortalized by Werner Heisenberg in 1927. Uncertainty relations imply that, if you measure a quantum particle’s position, the particle’s momentum ceases to have a well-defined value. If you measure the momentum, the particle ceases to have a well-defined position. Our uncertainty relation involves weak measurements. Weakly measuring a particle’s position doesn’t disturb the momentum much and vice versa. We can interpret the uncertainty in information-processing terms, because we cast the inequality in terms of entropies. Entropies, described here, are functions that quantify how efficiently we can process information, such as by compressing data. Jonathan and Kater are checking our inequality, and exploring its implications, with a superconducting qubit.

With chip

I had too little experience to side with Jonathan or with Kater. So I watched, and I contemplated how their opinions would sound if expressed about theory. Do I try one strategy again and again, hoping to change my results without changing my approach? 

At the Perimeter Institute for Theoretical Physics, Masters students had to swallow half-a-year of course material in weeks. I questioned whether I’d ever understand some of the material. But some of that material resurfaced during my PhD. Again, I attended lectures about Einstein’s theory of general relativity. Again, I worked problems about observers in free-fall. Again, I calculated covariant derivatives. The material sank in. I decided never to question, again, whether I could understand a concept. I might not understand a concept today, or tomorrow, or next week. But if I dedicate enough time and effort, I chose to believe, I’ll learn.

My decision rested on experience and on classes, taught by educational psychologists, that I’d taken in college. I’d studied how brains change during learning and how breaks enhance the changes. Sense, I thought, underlay my decision—though expecting outcomes to change, while strategies remain static, sounds insane.

Old cover

Does sense underlie Kater’s suggestion, likened to insanity, to keep fabricating amplifiers as before? He’s expressed cynicism many times during our collaboration: Experiment needs to involve some insanity. The experiment probably won’t work for a long time. Plenty more things will likely break. 

Jonathan and I agree with him. Experiments have a reputation for breaking, and Kater has a reputation for knowing experiments. Yet Jonathan—with professionalism and politeness—remains optimistic that other methods will prevail, that we’ll meet our goals early. I hope that Jonathan remains optimistic, and I fancy that Kater hopes, too. He prophesies gloom with a quarter of a smile, and his record speaks against him: A few months ago, I met a theorist who’d collaborated with Kater years before. The theorist marveled at the speed with which Kater had operated. A theorist would propose an experiment, and boom—the proposal would work.

Sea monsters

Perhaps luck smiled upon the implementation. But luck dovetails with the sense that underlies Kater’s opinion: Experiments involve factors that you can’t control. Implement a protocol once, and it might fail because the temperature has risen too high. Implement the protocol again, and it might fail because a truck drove by your building, vibrating the tabletop. Implement the protocol again, and it might fail because you bumped into a knob. Implement the protocol a fourth time, and it might succeed. If you repeat a protocol many times, your environment might change, changing your results.

Sense underlies also Jonathan’s objections to Kater’s opinions. We boost our chances of succeeding if we keep trying. We derive energy to keep trying from creativity and optimism. So rebelling against our PhD supervisors’ sense is sensible. I wondered, watching the Skype conversation, whether Kater the student had objected to prophesies of doom as Jonathan did. Kater exudes the soberness of a tenured professor but the irreverence of a Californian who wears his hair slightly long and who tattooed his wedding band on. Science thrives on the soberness and the irreverence.

Green cover

Who won Jonathan and Kater’s argument? Both, I think. Last week, they reported having fabricated amplifiers that work. The lab followed a protocol similar to their old one, but with more conscientiousness. 

I’m looking forward to watching who wins the debate about how long the rest of the experiment takes. Either way, check out Jonathan’s talk about our experiment if you attend the American Physical Society’s March Meeting. Jonathan will speak on Thursday, March 5, at 12:03, in room 106. Also, keep an eye out for our paper—which will debut once Jonathan coaxes the amplifier into synching with his qubit.

Interaction + Entanglement = Efficient Proofs of Halting

A couple weeks ago my co-authors Zhengfeng Ji (UTS Sydney), Heny Yuen (University of Toronto) and Anand Natarajan and John Wright (both at Caltech’s IQIM, with John soon moving to UT Austin) & I posted a manuscript on the arXiv preprint server entitled

MIP*=RE

The magic of the single-letter formula quickly made its effect, and our posting received some attention on the blogosphere (see links below). Within computer science, complexity theory is at an advantage in its ability to capture powerful statements in few letters: who has not head of P, NP, and, for readers of this blog, BQP and QMA? (In contrast, I am under no illusion that my vague attempt at a more descriptive title has, by the time you reach this line, all but vanished from the reader’s memory.)

Even accounting for this popularity however, it is a safe bet that fewer of our readers have heard of MIP* or RE. Yet we are promised that the above-stated equality has great consequences for physics (“Tsirelson’s problem” in the study of nonlocality) and mathematics (“Connes’ embedding problem” in the theory of von Neumann algebras). How so — how can complexity-theoretic alphabet soup have any consequence for, on the one hand, physical reality, and on the other, abstract mathematics?

The goal of this post and the next one is to help the interested reader grasp the significance of interactive proofs (that lie between the symbols MIP*) and undecidability (that lies behind RE) for quantum mechanics.

The bulk of the present post is an almost identical copy of a post I wrote for my personal blog. To avoid accusations of self-plagiarism, I will substantiate it with a little picture and a story, see below. The post gives a very personal take on the research that led to the aforementioned result. In the next post, my co-author Henry Yuen has offered to give a more scientific introduction to the result and its significance.

Before proceeding, it is important to make it clear that the research described in this post and the next has not been refereed or thoroughly vetted by the community. This process will take place over the coming months, and we should wait until it is completed before placing too much weight on the results. As an author, I am proud of my work; yet I am aware that there is due process to be made before the claims can be officialised. As such, these posts only represent my opinion (and Henry’s) and not necessarily that of the wider scientific community.

For more popular introductions to our result, see the blog posts of Scott Aaronson, Dick Lipton, and Gil Kalai and reporting by Davide Castelvecchi for Nature and Emily Conover for Science.

Now for the personal post…and the promised picture. IMG_8074Isn’t it beautiful? The design is courtesy of Tony Metger and Alexandru Gheorghiu, the first a visiting student and the second a postdoctoral scholar at Caltech’s IQIM. While Tony and Andru came up with the idea, the execution is courtesy of the bakery store employee, who graciously implemented the custom design (apparently writing equations on top of cakes is not  common enough to be part of the standard offerings, so they had to go for the custom option). Although it is unclear if the executioner grasped the full depth of the signs they were copying, note how perfect the execution: not a single letter is out of place! Thanks to Tony, Andru, and the anonymous chef for the tasty souvenir.

Now for the story. In an earlier post on my personal research blog, I had reported on the beautiful recent result by Natarajan and Wright showing the astounding power of multi-prover interactive proofs with quantum provers sharing entanglement: in letters, \text{NEEXP} \subseteq \text{MIP}^\star. In the remainder of this post I will describe our follow-up work with Ji, Natarajan, Wright, and Yuen. In this post I will tell the story from a personal point of view, with all the caveats that this implies: the “hard science” will be limited (but there could be a hint as to how “science”, to use a big word, “progresses”, to use an ill-defined one; see also the upcoming post by Henry Yuen for more), the story is far too long, and it might be mostly of interest to me only. It’s a one-sided story, but that has to be. (In particular below I may at times attribute credit in the form “X had this idea”. This is my recollection only, and it is likely to be inaccurate. Certainly I am ignoring a lot of important threads.) I wrote this because I enjoyed recollecting some of the best moments in the story just as much as some the hardest; it is fun to look back and find meanings in ideas that initially appeared disconnected. Think of it as an example of how different lines of work can come together in unexpected ways; a case for open-ended research. It’s also an antidote against despair that I am preparing for myself: whenever I feel I’ve been stuck on a project for far too long, I’ll come back to this post and ask myself if it’s been 14 years yet — if not, then press on.

It likely comes as a surprise to me only that I am no longer fresh out of the cradle. My academic life started in earnest some 14 years ago, when in the Spring of 2006 I completed my Masters thesis in Computer Science under the supervision of Julia Kempe, at Orsay in France. I had met Julia the previous term: her class on quantum computing was, by far, the best-taught and most exciting course in the Masters program I was attending, and she had gotten me instantly hooked. Julia agreed to supervise my thesis, and suggested that I look into some interesting recent result by Stephanie Wehner that linked the study of entanglement and nonlocality in quantum mechanics to complexity-theoretic questions about interactive proof systems (specifically, this was Stephanie’s paper showing that \text{XOR-MIP}^\star \subseteq \text{QIP}(2)).

At the time the topic was very new. It had been initiated the previous year with a beautiful paper by Cleve et al. (that I have recommended to many a student since!) It was a perfect fit for me: the mathematical aspects of complexity theory and quantum computing connected to my undergraduate background, while the relative concreteness of quantum mechanics (it is a physical theory after all) spoke to my desire for real-world connection (not “impact” or even “application” — just “connection”). Once I got myself up to speed in the area (which consisted of three papers: the two I already mentioned, together with a paper by Kobayashi and Matsumoto where they studied interactive proofs with quantum messages), Julia suggested looking into the “entangled-prover” class \text{MIP}^\star introduced in the aforementioned paper by Cleve et al. Nothing was known about this class! Nothing besides the trivial inclusion of single-prover interactive proofs, IP, and the containment in…ALL, the trivial class that contains all languages.

Yet the characterization MIP=NEXP of its classical counterpart by Babai et al. in the 1990s had led to one of the most productive lines of work in complexity of the past few decades, through the PCP theorem and its use from hardness of approximation to efficient cryptographic schemes. Surely, studying \text{MIP}^\star had to be a productive direction? In spite of its well-established connection to classical complexity theory, via the formalism of interactive proofs, this was a real gamble. The study of entanglement from the complexity-theoretic perspective was entirely new, and bound to be fraught with difficulty; very few results were available and the existing lines of works, from the foundations of non-locality to more recent endeavors in device-independent cryptography, provided little other starting point than strong evidence that even the simplest examples came with many unanswered questions. But my mentor was fearless, and far from a novice in terms of defraying new areas, having done pioneering work in areas ranging from quantum random walks to Hamiltonian complexity through adiabatic computation. Surely this would lead to something?

It certainly did. More sleepless nights than papers, clearly, but then the opposite would only indicate dullness. Julia’s question led to far more unexpected consequences than I, or I believe she, could have imagined at the time. I am writing this post to celebrate, in a personal way, the latest step in 15 years of research by dozens of researchers: today my co-authors and I uploaded to the quant-ph arXiv what we consider a complete characterization of the power of entangled-prover interactive proof systems by proving the equality \text{MIP}^\star = \text{RE}, the class of all recursively enumerable languages (a complete problem for RE is the halting problem). Without going too much into the result itself (if you’re interested, look for an upcoming post here that goes into the proof a bit more), and since this is a more personal post, I will continue on with some personal thoughts about the path that got us there.

When Julia & I started working on the question, our main source of inspiration were the results by Cleve et al. showing that the non-local correlations of entanglement had interesting consequences when seen through the lens of interactive proof systems in complexity theory. Since the EPR paper, a lot of work in understanding entanglement had already been accomplished in the Physics community, most notably by Mermin, Peres, Bell, and more recently the works in device-independent quantum cryptography by Acin, Pironio, Scarani and many others, stimulated by Ekert’s proposal for quantum key distribution and Mayers and Yao’s idea for “device-independent cryptography”. By then we certainly knew that “spooky action-at-a-distance” did not entail any faster-than-light communication, and indeed was not really “action-at-a-distance” in the first place but merely “correlation-at-a-distance”. What Cleve et al. recognized is that these “spooky correlations-at-a-distance” were sufficiently special so as to not only give numerically different values in “Bell inequalities”, the tool invented by Bell to evidence non-locality in quantum mechanics, but also have some potentially profound consequences in complexity theory.

In particular, examples such as the “Magic Square game” demonstrated that enough correlation could be gained from entanglement so as to defeat basic proof systems whose soundness relied only on the absence of communication between the provers, an assumption that until then had been wrongly equated with the assumption that any computation performed by the provers could be modeled entirely locally. I think that the fallacy of this implicit assumption came as a surprise to complexity theorists, who may still not have entirely internalized it. Yet the perfect quantum strategy for the Magic Square game provides a very concrete “counter-example” to the soundness of the “clause-vs-variable” game for 3SAT. Indeed this game, a reformulation by Aravind and Cleve-Mermin of a Bell Inequality discovered by Mermin and Peres in 1990, can be easily re-framed as a 3SAT system of equations that is not satisfiable, and yet is such that the associated two-player clause-vs-variable game has a perfect quantum strategy. It is this observation, made in the paper by Cleve et al., that gave the first strong hint that the use of entanglement in interactive proof systems could make many classical results in the area go awry.

By importing the study of non-locality into complexity theory Cleve et al. immediately brought it into the realm of asymptotic analysis. Complexity theorists don’t study fixed objects, they study families of objects that tend to have a uniform underlying structure and whose interesting properties manifest themselves “in the limit”. As a result of this new perspective focus shifted from the study of single games or correlations to infinite families thereof. Some of the early successes of this translation include the “unbounded violations” that arose from translating asymptotic separations in communication complexity to the language of Bell inequalities and correlations (e.g. this paper). These early successes attracted the attention of some physicists working in foundations as well as some mathematical physicists, leading to a productive exploration that combined tools from quantum information, functional analysis and complexity theory.

The initial observations made by Cleve et al. had pointed to \text{MIP}^\star as a possibly interesting complexity class to study. Rather amazingly, nothing was known about it! They had shown that under strong restrictions on the verifier’s predicate (it should be an XOR of two answer bits), a collapse took place: by the work of Hastad, XOR-MIP equals NEXP, but \text{MIP}^\star is included in EXP. This seemed very fortuitous (the inclusion is proved via a connection with semidefinite programming that seems tied to the structure of XOR-MIP protocols): could entanglement induce a collapse of the entire, unrestricted class? We thought (at this point mostly Julia thought, because I had no clue) that this ought not to be the case, and so we set ourselves to show that the equality \text{MIP}^\star=\text{NEXP}, that would directly parallel Babai et al.’s characterization MIP=NEXP, holds. We tried to show this by introducing techniques to “immunize” games against entanglement: modify an interactive proof system so that its structure makes it “resistant” to the kind of “nonlocal powers” that can be used to defeat the clause-vs-variable game (witness the Magic Square). This was partially successful, and led to one of the papers I am most proud of — I am proud of it because I think it introduced elementary techniques (such as the use of the Cauchy-Schwarz inequality — inside joke — more seriously, basic things such as “prover-switching”, “commutation tests”, etc.) that are now routine manipulations in the area. The paper was a hard sell! It’s good to remember the first rejections we received. They were not unjustified: the main point of criticism was that we were only able to establish a hardness result for exponentially small completeness-soundness gap. A result for such a small gap in the classical setting follows directly from a very elementary analysis based on the Cook-Levin theorem. So then why did we have to write so many pages (and so many applications of Cauchy-Schwarz!) to arrive at basically the same result (with a ^\star)?

Eventually we got lucky and the paper was accepted to a conference. But the real problem, of establishing any non-trivial lower bound on the class \text{MIP}^\star with constant (or, in the absence of any parallel repetition theorem, inverse-polynomial) completeness-soundness gap, remained. By that time I had transitioned from a Masters student in France to a graduate student in Berkeley, and the problem (pre-)occupied me during some of the most difficult years of my Ph.D. I fully remember spending my first year entirely thinking about this (oh and sure, that systems class I had to pass to satisfy the Berkeley requirements), and then my second year — yet, getting nowhere. (I checked the arXiv to make sure I’m not making this up: two full years, no posts.) I am forever grateful to my fellow student Anindya De for having taken me out of the cycle of torture by knocking on my door with one of the most interesting questions I have studied, that led me into quantum cryptography and quickly resulted in an enjoyable paper. It was good to feel productive again! (Though the paper had fun reactions as well: after putting it on the arXiv we quickly heard from experts in the area that we had solved an irrelevant problem, and that we better learn about information theory — which we did, eventually leading to another paper, etc.) The project had distracted me and I set interactive proofs aside; clearly, I was stuck.

About a year later I visited IQC in Waterloo. I don’t remember in what context the visit took place. What I do remember is a meeting in the office of Tsuyoshi Ito, at the time a postdoctoral scholar at IQC. Tsuyoshi asked me to explain our result with Julia. He then asked a very pointed question: the bedrock for the classical analysis of interactive proof systems is the “linearity test” of Blum-Luby-Rubinfeld (BLR). Is there any sense in which we could devise a quantum version of that test?

What a question! This was great. At first it seemed fruitless: in what sense could one argue that quantum provers apply a “linear function”? Sure, quantum mechanics is linear, but that is besides the point. The linearity is a property of the prover’s answers as a function of their question. So what to make of the quantum state, the inherent randomness, etc.?

It took us a few months to figure it out. Once we got there however, the answer was relatively simple — the prover should be making a question-independent measurement that returns a linear function that it applies to its question in order to obtain the answer returned to the verifier — and it opened the path to our subsequent paper showing that the inclusion of NEXP in \text{MIP}^\star indeed holds. Tsuyoshi’s question about linearity testing had allowed us to make the connection with PCP techniques; from there to MIP=NEXP there was only one step to make, which is to analyze multi-linearity testing. That step was suggested by my Ph.D. advisor, Umesh Vazirani, who was well aware of the many pathways towards the classical PCP theorem, since the theorem had been obtained in great part by his former student Sanjeev Arora. It took a lot of technical work, yet conceptually a single question from my co-author had sufficed to take me out of a 3-year slumber.

This was in 2012, and I thought we were done. For some reason the converse inclusion, of \text{MIP}^\star in NEXP, seemed to resist our efforts, but surely it couldn’t resist much longer. Navascues et al. had introduced a hierarchy of semidefinite programs that seemed to give the right answer (technically they could only show convergence to a relaxation, the commuting value, but that seemed like a technicality; in particular, the values coincide when restricted to finite-dimensional strategies, which is all we computer scientists cared about). There were no convergence bounds on the hierarchy, yet at the same time commutative SDP hierarchies were being used to obtain very strong results in combinatorial optimization, and it seemed like it would only be a matter of time before someone came up with an analysis of the quantum case. (I had been trying to solve a related “dimension reduction problem” with Oded Regev for years, and we were making no progress; yet it seemed someone ought to!)

In Spring 2014 during an open questions session at a workshop at the Simons Institute in Berkeley Dorit Aharonov suggested that I ask the question of the possible inclusion of QMA-EXP, the exponential-sized-proofs analogue of QMA, in \text{MIP}^\star. A stronger result than the inclusion of NEXP (under assumptions), wouldn’t it be a more natural “fully quantum” analogue of MIP=NEXP? Dorit’s suggestion was motivated by research on the “quantum PCP theorem”, that aims to establish similar hardness results in the realm of the local Hamiltonian problem; see e.g. this post for the connection. I had no idea how to approach the question — I also didn’t really believe the answer could be positive — but what can you do, if Dorit asks you something… So I reluctantly went to the board and asked the question. Joe Fitzsimons was in the audience, and he immediately picked it up! Joe had the fantastic ideas of using quantum error-correction, or more specifically secret-sharing, to distribute a quantum proof among the provers. His enthusiasm overcame my skepticism, and we eventually showed the desired inclusion. Maybe \text{MIP}^\star was bigger than \text{NEXP} after all.

Our result, however, had a similar deficiency as the one with Julia, in that the completeness-soundness gap was exponentially small. Obtaining a result with a constant gap took 3 years of couple more years of work and the fantastic energy and insights of a Ph.D. student at MIT, Anand Natarajan. Anand is the first person I know of to have had the courage to dive into the most technical aspects of the analysis of the aforementioned results, while also bringing in the insights of a “true quantum information theorist” that were supported by Anand’s background in Physics and upbringing in the group of Aram Harrow at MIT. (In contrast I think of myself more as a “raw” mathematician; I don’t really understand quantum states other than as positive-semidefinite matrices…not that I understand math either of course; I suppose I’m some kind of a half-baked mish-mash.) Anand had many ideas but one of the most beautiful ones led to what he poetically called the “Pauli braiding test”, a “truly quantum” analogue of the BLR linearity test that amounts to doing two linearity tests in conjugate bases and piecing the results together into a robust test for {n}-qubit entanglement (I wrote about our work on this here).

At approximately the same time, Zhengfeng Ji had another wonderful idea that was in some sense orthogonal to our work. (My interpretation of) Zhengfeng’s idea is that one can see an interactive proof system as a computation (verifier-prover-verifier) and use Kitaev’s circuit-to-Hamiltonian construction to transform the entire computation into a “quantum CSP” (in the same sense that the local Hamiltonian problem is a quantum analogue of classical constraint satisfaction problems (CSP)) that could then itself be verified by a quantum multi-prover interactive proof system…with exponential gains in efficiency! Zhengfeng’s result implied an exponential improvement in complexity compared to the result by Julia and myself, showing inclusion of NEEXP, instead of NEXP, in \text{MIP}^\star. However, Zhengfeng’s technique suffered from the same exponentially small completeness-soundness gap as we had, so that the best lower bound on \text{MIP}^\star per se remained NEXP.

Both works led to follow-ups. With Natarajan we promoted the Pauli braiding test into a “quantum low-degree test” that allowed us to show the inclusion of QMA-EXP into \text{MIP}^\star, with constant gap, thereby finally answering the question posed by Aharonov 4 years after it was asked. (I should also say that by then all results on \text{MIP}^\star started relying on a sequence of parallel repetition results shown by Bavarian, Yuen, and others; I am skipping this part.) In parallel, with Ji, Fitzsimons, and Yuen we showed that Ji’s compression technique could be “iterated” an arbitrary number of times. In fact, by going back to “first principles” and representing verifiers uniformly as Turing machines we realized that the compression technique could be used iteratively to (up to small caveats) give a new proof of the fact (first shown by Slofstra using an embedding theorem for finitely presented group) that the zero-gap version of \text{MIP}^\star contains the halting problem. In particular, the entangled value is uncomputable! This was not the first time that uncomputability crops in to a natural problem in quantum computing (e.g. the spectral gap paper), yet it still surprises when it shows up. Uncomputable! How can anything be uncomputable!

As we were wrapping up our paper Henry Yuen realized that our “iterated compression of interactive proof systems” was likely optimal, in the following sense. Even a mild improvement of the technique, in the form of a slower closing of the completeness-soundness gap through compression, would yield a much stronger result: undecidability of the constant-gap class \text{MIP}^\star. It was already known by work of Navascues et al., Fritz, and others, that such a result would have, if not surprising, certainly consequences that seemed like they would be taking us out of our depth. In particular, undecidability of any language in \text{MIP}^\star would imply a negative resolution to a series of equivalent conjectures in functional analysis, from Tsirelson’s problem to Connes’ Embedding Conjecture through Kirchberg’s QWEP conjecture. While we liked our result, I don’t think that we believed it could resolve any conjecture(s) in functional analysis.

So we moved on. At least I moved on, I did some cryptography for a change. But Anand Natarajan and his co-author John Wright did not stop there. They had the last major insight in this story, which underlies their recent STOC best paper described in the previous post. Briefly, they were able to combine the two lines of work, by Natarajan & myself on low-degree testing and by Ji et al. on compression, to obtain a compression that is specially tailored to the existing \text{MIP}^\star protocol for NEXP and compresses that protocol without reducing its completeness-soundness gap. This then let them show Ji’s result that \text{MIP}^\star contains NEEXP, but this time with constant gap! The result received well-deserved attention. In particular, it is the first in this line of works to not suffer from any caveats (such as a closing gap, or randomized reductions, or some kind of “unfair” tweak on the model that one could attribute the gain in power to), and it implies an unconditional separation between MIP and \text{MIP}^\star.

As they were putting the last touches on their result, suddenly something happened, which is that a path towards a much bigger result opened up. What Natarajan & Wright had achieved is a one-step gapless compression. In our iterated compression paper we had observed that iterated gapless compression would lead to \text{MIP}^\star=\text{RE}, implying negative answers to the aforementioned conjectures. So then?

I suppose it took some more work, but in some way all the ideas had been laid out in the previous 15 years of work in the complexity of quantum interactive proof systems; we just had to put it together. And so a decade after the characterization QIP = PSPACE of single-prover quantum interactive proof systems, we have arrived at a characterization of quantum multiprover interactive proof systems, \text{MIP}^\star = \text{RE}. With one author in common between the two papers: congratulations Zhengfeng!

Even though we just posted a paper, in a sense there is much more left to do. I am hopeful that our complexity-theoretic result will attract enough interest from the mathematicians’ community, and especially operator algebraists, for whom CEP is a central problem, that some of them will be willing to devote time to understanding the result. I also recognize that much effort is needed on our own side to make it accessible in the first place! I don’t doubt that eventually complexity theory will not be needed to obtain the purely mathematical consequences; yet I am hopeful that some of the ideas may eventually find their way into the construction of interesting mathematical objects (such as, who knows, a non-hyperlinear group).

That was a good Masters project…thanks Julia!

On the merits of flatworm reproduction

On my right sat a quantum engineer. She was facing a melanoma specialist who works at a medical school. Leftward of us sat a networks expert, a flatworm enthusiast, and a condensed-matter theorist.

Farther down sat a woman who slices up mouse brains. 

Welcome to “Coherent Spins in Biology,” a conference that took place at the University of California, Los Angeles (UCLA) this past December. Two southern Californians organized the workshop: Clarice Aiello heads UCLA’s Quantum Biology Tech lab. Thorsten Ritz, of the University of California, Irvine, cofounded a branch of quantum biology.

Clarice logo

Quantum biology served as the conference’s backdrop. According to conventional wisdom, quantum phenomena can’t influence biology significantly: Biological systems have high temperatures, many particles, and fluids. Quantum phenomena, such as entanglement (a relationship that quantum particles can share), die quickly under such conditions.

Yet perhaps some survive. Quantum biologists search for biological systems that might use quantum resources. Then, they model and measure the uses and resources. Three settings (at least) have held out promise during the past few decades: avian navigation, photosynthesis, and olfaction. You can read about them in this book, cowritten by a conference participant for the general public. I’ll give you a taste (or a possibly quantum smell?) by sketching the avian-navigation proposal, developed by Thorsten and colleagues.

Bird + flower

Birds migrate southward during the autumn and northward during the spring. How do they know where to fly? At least partially by sensing the Earth’s magnetic field, which leads compass needles to point northward. How do birds sense the field?

Possibly with a protein called “cryptochrome.” A photon (a particle of light) could knock an electron out of part of the protein and into another part. Each part would have one electron that lacked a partner. The electrons would share entanglement. One electron would interact with the Earth’s magnetic field differently than its partner, because its surroundings would differ. (Experts: The electrons would form a radical pair. One electron would neighbor different atoms than the other, so the electron would experience a different local magnetic field. The discrepancy would change the relative phase between the electrons’ spins.) The discrepancy could affect the rate at which the chemical system could undergo certain reactions. Which reactions occur could snowball into large and larger effects, eventually signaling the brain about where the bird should fly.

Angry bird

Quantum mechanics and life rank amongst the universe’s mysteries. How could a young researcher resist the combination? A postdoc warned me away, one lunchtime at the start of my PhD. Quantum biology had enjoyed attention several years earlier, he said, but noise the obscured experimental data. Controversy marred the field.

I ate lunch with that postdoc in 2013. Interest in quantum biology is reviving, as evidenced in the conference. Two reasons suggested themselves: new technologies and new research avenues. For example, Thorsten described the disabling and deletion of genes that code for cryptochrome. Such studies require years’ more work but might illuminate whether cryptochrome affects navigation.

Open door

The keynote speaker, Harvard’s Misha Lukin, illustrated new technologies and new research avenues. Misha’s lab has diamonds that contain quantum defects, which serve as artificial atoms. The defects sense tiny magnetic fields and temperatures. Misha’s group applies these quantum sensors to biology problems.

For example, different cells in an embryo divide at different times. Imagine reversing the order in which the cells divide. Would the reversal harm the organism? You could find out by manipulating the temperatures in different parts of the embryo: Temperature controls the rate at which cells divide.

Misha’s team injected nanoscale diamonds into a worm embryo. (See this paper for a related study.) The diamonds reported the temperature at various points in the worm. This information guided experimentalists who heated the embryo with lasers.

The manipulated embryos grew into fairly normal adults. But their cells, and their descendants’ cells, cycled through the stages of life slowly. This study exemplified, to me, one of the most meaningful opportunities for quantum physicists interested in biology: to develop technologies and analyses that can answer biology questions.

Thermometer

I mentioned, in an earlier blog post, another avenue emerging in quantum biology: Physicist Matthew Fisher proposed a mechanism by which entanglement might enhance coordinated neuron firing. My collaborator Elizabeth Crosson and I analyzed how the molecules in Matthew’s proposal—Posner clusters—could process quantum information. The field of Posner quantum biology had a population of about two, when Elizabeth and I entered, and I wondered whether anyone would join us.

The conference helped resolve my uncertainty. Three speakers (including me) presented work based on Matthew’s; two other participants were tilling the Posner soil; and another speaker mentioned Matthew’s proposal. The other two Posner talks related data from three experiments. The experimentalists haven’t finished their papers, so I won’t share details. But stay tuned.

Posner 2

Posner molecule (image by Swift et al.)

Clarice and Thorsten’s conference reminded me of a conference I’d participated in at the end of my PhD: Last month, I moonlighted as a quantum biologist. In 2017, I moonlighted as a quantum-gravity theorist. Two years earlier, I’d been dreaming about black holes and space-time. At UCLA, I was finishing the first paper I’ve coauthored with biophysicists. What a toolkit quantum information theory and thermodynamics provide, that it can unite such disparate fields. 

The contrast—on top of what I learned at UCLA—filled my mind for weeks. And reminded me of the description of asexual reproduction that we heard from the conference’s flatworm enthusiast. According to Western Michigan University’s Wendy Beane, a flatworm “glues its butt down, pops its head off, and grows a new one. Y’know. As one does.” 

I hope I never flinch from popping my head off and growing a new one—on my quantum-information-thermodynamics spine—whenever new science calls for figuring out.

 

With thanks to Clarice, Thorsten, and UCLA for their invitation and hospitality.

An equation fit for a novel

Archana Kamal was hunting for an apartment in Cambridge, Massachusetts. She was moving MIT, to work as a postdoc in physics. The first apartment she toured had housed John Updike, during his undergraduate career at Harvard. No other apartment could compete; Archana signed the lease.

The apartment occupied the basement of a red-brick building covered in vines. The rooms spanned no more than 350 square feet. Yet her window opened onto the neighbors’ garden, whose leaves she tracked across the seasons. And Archana cohabited with history.

Apartment photos

She’s now studying the universe’s history, as an assistant professor of physics at the University of Massachusetts Lowell. The cosmic microwave background (CMB) pervades the universe. The CMB consists of electromagnetic radiation, or light. Light has particle-like properties and wavelike properties. The wavelike properties include wavelength, the distance between successive peaks. Long-wavelength light includes red light, infrared light, and radio waves. Short-wavelength light includes blue light, ultraviolet light, and X-rays. Light of one wavelength and light of another wavelength are said to belong to different modes.

Wavelength

Does the CMB have nonclassical properties, impossible to predict with classical physics but (perhaps) predictable with quantum theory? The CMB does according to the theory of inflation. According to the theory, during a short time interval after the Big Bang, the universe expanded very quickly: Spacetime stretched. Inflation explains features of our universe, though we don’t know what mechanism would have effected the expansion.

According to inflation, around the Big Bang time, all the light in the universe crowded together. The photons (particles of light) interacted, entangling (developing strong quantum correlations). Spacetime then expanded, and the photons separated. But they might retain entanglement.

Detecting that putative entanglement poses challenges. For instance, the particles that you’d need to measure could produce a signal too weak to observe. Cosmologists have been scratching their heads about how to observe nonclassicality in the CMB. One team—Nishant Agarwal at UMass Lowell and Sarah Shandera at Pennsylvania State University—turned to Archana for help.

A sky full of stars

Archana studies the theory of open quantum systems, quantum systems that interact with their environments. She thinks most about systems such as superconducting qubits, tiny circuits with which labs are building quantum computers. But the visible universe constitutes an open quantum system.

We can see only part of the universe—or, rather, only part of what we believe is the whole universe. Why? We can see only stuff that’s emitted light that has reached us, and light has had only so long to travel. But the visible universe interacts (we believe) with stuff we haven’t seen. For instance, according to the theory of inflation, that rapid expansion stretched some light modes’ wavelengths. Those wavelengths grew longer than the visible universe. We can’t see those modes’ peak-to-peak variations or otherwise observe the modes, often called “frozen.” But the frozen modes act as an environment that exchanges information and energy with the visible universe.

We describe an open quantum system’s evolution with a quantum master equation, which I blogged about four-and-a-half years ago. Archana and collaborators constructed a quantum master equation for the visible universe. The frozen modes, they found, retain memories of the visible universe. (Experts: the bath is non-Markovian.) Next, they need to solve the equation. Then, they’ll try to use their solution to identify quantum observables that could reveal nonclassicality in the CMB.

Frozen modes

Frozen modes

Archana’s project caught my fancy for two reasons. First, when I visited her in October, I was collaborating on a related project. My coauthors and I were concocting a scheme for detecting nonclassical correlations in many-particle systems by measuring large-scale properties. Our paper debuted last month. It might—with thought and a dash of craziness—be applied to detect nonclassicality in the CMB. Archana’s explanation improved my understanding of our scheme’s potential. 

Second, Archana and collaborators formulated a quantum master equation for the visible universe. A quantum master equation for the visible universe. The phrase sounded romantic to me.1 It merited a coauthor who’d seized on an apartment lived in by a Pulitzer Prize-winning novelist. 

Archana’s cosmology and Updike stories reminded me of one reason why I appreciate living in the Boston area: History envelops us here. Last month, while walking to a grocery, I found a sign that marks the building in which the poet e. e. cummings was born. My walking partner then generously tolerated a recitation of cummings’s “anyone lived in a pretty how town.” History enriches our lives—and some of it might contain entanglement.

 

1It might sound like gobbledygook to you, if I’ve botched my explanations of the terminology.

With thanks to Archana and the UMass Lowell Department of Physics and Applied Physics for their hospitality and seminar invitation.

Breaking up the band structure

Note from the editor: During the Summer of 2019, a group of thirteen undergraduate students from Caltech and universities around the world, spent 10 weeks on campus performing research in experimental quantum physics. Below, Aiden Cullo, a student from Binghampton University in New York, shares his experience working in Professor Yeh’s lab. The program, termed QuantumSURF, will run again during the Summer of 2020.

This summer, I worked in Nai-Chang Yeh’s experimental condensed matter lab. The aim of my project was to observe the effects of a magnetic field on our topological insulator (TI) sample, {(BiSb)}_2{Te}_3. The motivation behind this project was to examine more closely the transformation between a topological insulator and a state exhibiting the anomalous hall effect (AHE).

Both states of matter have garnered a good deal of interest in condensed matter research because of their interesting transport properties, among other things. TIs have gained popularity due to their applications in electronics (spintronics), superconductivity, and quantum computation. TIs are peculiar in that they simultaneously have insulating bulk states and conducting surface state. Due to time-reversal symmetry (TRS) and spin-momentum locking, these surface states have a very symmetric hourglass-like gapless energy band structure (Dirac cone).

The focus of our particular study was the effects of “c-plane” magnetization of our TI’s surface state. Theory predicts TRS and spin-momentum locking will be broken, resulting in a gapped spectrum with a single connection between the valence and conduction bands. This gapping has been theorized and shown experimentally in Chromium (Cr)-doped {(BiSb)}_2{Te}_3 and numerous other TIs with similar make-up.

In 2014, Nai-Chang Yeh’s group showed that Cr-doped {Bi}_2{Se}_3 exhibit this gap opening due to the surface state of {Bi}_2{Se}_3 interacting via the proximity effect with a ferromagnet. Our contention is that a similar material, Cr-doped {(BiSb)}_2{Te}_3, exhibits a similar effect, but more homogeneously because of reduced structural strain between atoms. Specifically, at temperatures below the Curie temperature (Tc), we expect to see a gap in the energy band and an overall increase in the gap magnitude. In short, the main goal of my summer project was to observe the gapping of our TI’s energy band.

Overall, my summer project entailed a combination of reading papers/textbooks and hands-on experimental work. It was difficult to understand fully the theory behind my project in such a short amount of time, but even with a cursory knowledge of topological insulators, I was able to provide a meaningful analysis/interpretation of our data.

Additionally, my experiment relied heavily on external factors such as our supplier for liquid helium, argon gas, etc. As a result, our progress was slowed if an order was delayed or not placed far enough in advance. Most of the issues we encountered were not related to the abstract theory of the materials/machinery, but rather problems with less complex mechanisms such as wiring, insulation, and temperature regulation.

While I expected to spend a good deal of time troubleshooting, I severely underestimated the amount of time that would be spent dealing with quotidian problems such as configuring software or etching STM tips. Working on a machine as powerful as an STM was frustrating at times, but also very rewarding as eventually we were able to collect a large amount of data on our samples.

An important (and extremely difficult) part of our analysis of STM data was whether patterns/features in our data set were artifacts or genuine phenomena, or a combination. I was fortunate enough to be surrounded by other researchers that helped me sift through the volumes of data and identify traits of our samples. Reflecting on my SURF, I believe it was a positive experience as it not only taught me a great deal about research, but also, more importantly, closely mimicked the experience of graduate school.

The paper that begged for a theme song

A year ago, the “I’m a little teapot” song kept playing in my head.

I was finishing a collaboration with David Limmer, a theoretical chemist at the University of California Berkeley. David studies quantum and classical systems far from equilibrium, including how these systems exchange energy and information with their environments. Example systems include photoisomers.

A photoisomer is a molecular switch. These switches appear across nature and technologies. We have photoisomers in our eyes, and experimentalists have used photoisomers to boost solar-fuel storage. A photoisomer has two functional groups, or collections of bonded atoms, attached to a central axis. 

Photoisomer

Your average-Joe photoisomer spends much of its life in equilibrium, exchanging heat with room-temperature surroundings. The molecule has the shape above, called the cis configuration. Imagine shining a laser or sunlight on the photoisomer. The molecule can absorb a photon, or particle of light, gaining energy. The energized switch has the opportunity to switch: One chemical group can rotate downward. The molecule will occupy its trans configuration.

Switch

The molecule now has more energy than it had while equilibrium, albeit less energy than it had right after absorbing the photon. The molecule can remain in this condition for a decent amount of time. (Experts: The molecule occupies a metastable state.) That is, the molecule can store sunlight. For that reason, experimentalists at Harvard and MIT attached photoisomers to graphene nanotubules, improving the nanotubules’ storage of solar fuel.

Teapot 1

With what probability does a photoisomer switch upon absorbing a photon? This question has resisted easy answering, because photoisomers prove difficult to model: They’re small, quantum, and far from equilibrium. People have progressed by making assumptions, but such assumptions can lack justifications or violate physical principles. David wanted to derive a simple, general bound—of the sort in which thermodynamicists specialize—on a photoisomer’s switching probability.

He had a hunch as to how he could derive such a bound. I’ve blogged, many times, about thermodynamic resource theories. Thermodynamic resource theories are simple models, developed in quantum information theory, for exchanges of heat, particles, information, and more. These models involve few assumptions: the conservation of energy, quantum theory, and, to some extent, the existence of a large environment (Markovianity). With such a model, David suspected, he might derive his bound.

Teapot 2

I knew nothing about photoisomers when I met David, but I knew about thermodynamic resource theories. I’d contributed to their development, to the theorems that have piled up in the resource-theory corner of quantum information theory. Then, the corner had given me claustrophobia. Those theorems felt so formal, abstract, and idealized. Formal, abstract theory has drawn me ever since I started studying physics in college. But did resource theories model physical reality? Could they impact science beyond our corner of quantum information theory? Did resource theories matter?

I called for connecting thermodynamic resource theories to physical reality four years ago, in a paper that begins with an embarrassing story about me. Resource theorists began designing experiments whose results should agree with our theorems. Theorists also tried to improve the accuracy with which resource theories model experimentalists’ limitations. See David’s and my paper for a list of these achievements. They delighted me, as a step toward the broadening of resource theories’ usefulness. 

Like any first step, this step pointed toward opportunities. Experiments designed to test our theorems essentially test quantum mechanics. Scientists have tested quantum mechanics for decades; we needn’t test it much more. Such experimental proposals can push experimentalists to hone their abilities, but I hoped that the community could accomplish more. We should be able to apply resource theories to answer questions cultivated in other fields, such as condensed matter and chemistry. We should be useful to scientists outside our corner of quantum information.

Teapot 3

David’s idea lit me up like photons on a solar-fuel-storage device. He taught me about photoisomers, I taught him about resource theories, and we derived his bound. Our proof relies on the “second laws of thermodynamics.” These abstract resource-theory results generalize the second law of thermodynamics, which helps us understand why time flows in only one direction. We checked our bound against numerical simulations (experts: of Lindbladian evolution). Our bound is fairly tight if the photoisomer has a low probability of absorbing a photon, as in the Harvard-MIT experiment. 

Experts: We also quantified the photoisomer’s coherences relative to the energy eigenbasis. Coherences can’t boost the switching probability, we concluded. But, en route to this conclusion, we found that the molecule is a natural realization of a quantum clock. Our quantum-clock modeling extends to general dissipative Landau-Zener transitions, prevalent across condensed matter and chemistry.

Teapot 4

As I worked on our paper one day, a jingle unfolded in my head. I recognized the tune first: “I’m a little teapot.” I hadn’t sung that much since kindergarten, I realized. Lyrics suggested themselves: 

I’m a little isomer
with two hands.
Here is my cis pose;
here is my trans.

Stand me in the sunlight;
watch me spin.
I’ll keep solar
energy in!

The song lodged itself in my head for weeks. But if you have to pay an earworm to collaborate with David, do.

Quantum Error Correction with Molecules

In the previous blog post (titled, “On the Coattails of Quantum Supremacy“) we started with Google and ended up with molecules! I also mentioned a recent paper by John Preskill, Jake Covey, and myself (see also this videoed talk) where we assume that, somewhere in the (near?) future, experimentalists will be able to construct quantum superpositions of several orientations of molecules or other rigid bodies. Next, I’d like to cover a few more details on how to construct error-correcting codes for anything from classical bits in your phone to those future quantum computers, molecular or otherwise.

Classical error correction: the basics

Error correction is concerned with the design of an encoding that allows for protection against noise. Let’s say we want to protect one classical bit, which is in either “0” or “1”. If the bit is say in “0”, and the environment (say, the strong magnetic field from a magnet you forgot was laying next to your hard drive) flipped it to “1” without our knowledge, an error would result (e.g., making your phone think you swiped right!)

Now let’s encode our single logical bit into three physical bits, whose 2^3=8 possible states are represented by the eight corners of the cube below. Let’s encode the logical bit as “0” —> 000 and “1” —> 111, corresponding to the corners of the cube marked by the black and white ball, respectively. For our (local) noise model, we assume that flips of only one of the three physical bits are more likely to occur than flips of two or three at the same time.

Error correction is, like many Hollywood movies, an origin story. If, say, the first bit flips in our above code, the 000 state is mapped to 100, and 111 is mapped to 011. Since we have assumed that the most likely error is a flip of one of the bits, we know upon observing that 100 must have come from the clean 000, and 011 from 111. Thus, in either case of the logical bit being “0” or “1”, we can recover the information by simply observing which state the majority of the bits are in. The same things happen when the second or third bits flip. In all three cases, the logical “0” state is mapped to one of its three neighboring points (above, in blue) while the logical “1” is mapped to its own three points, which, crucially, are distinct from the neighbors of “0”. The set of points \{000,100,010,001\} that are closer to 000 than to 111 is called a Voronoi tile.

Now, let’s adapt these ideas to molecules. Consider the rotational states of a dumb-bell molecule consisting of two different atoms. (Let’s assume that we have frozen this molecule to the point that the vibration of the inter-atomic bond is limited, essentially creating a fixed distance between the two atoms.) This molecule can orient itself in any direction, and each such orientation can be represented as a point \mathbf{v} on the surface of a sphere. Now let us encode a classical bit using the north and south poles of this sphere (represented in the picture below as a black and a white ball, respectively). The north pole of the sphere corresponds to the molecule being parallel to the z-axis, while the south pole corresponds to the molecule being anti-parallel.

This time, the noise consists of small shifts in the molecule’s orientation. Clearly, if such shifts are small, the molecule just wiggles a bit around the z-axis. Such wiggles still allow us to infer that the molecule is (mostly) parallel and anti-parallel to the axis, as long as they do not rotate the molecule all the way past the equator. Upon such correctable rotations, the logical “0” state — the north pole — is mapped to a point in the northern hemisphere, while logical “1” — the south pole — is mapped to a point in the southern hemisphere. The northern hemisphere forms a Voronoi tile of the logical “0” state (blue in the picture), which, along with the corresponding tile of the logical “1” state (the southern hemisphere), tiles the entire sphere.

Quantum error correction

To upgrade these ideas to the quantum realm, recall that this time we have to protect superpositions. This means that, in addition to shifting our quantum logical state to other states as before, noise can also affect the terms in the superposition itself. Namely, if, say, the superposition is equal — with an amplitude of +1/\sqrt{2} in “0” and +1/\sqrt{2} in “1” — noise can change the relative sign of the superposition and map one of the amplitudes to -1/\sqrt{2}. We didn’t have to worry about such sign errors before, because our classical information would always be the definite state of “0” or “1”. Now, there are two effects of noise to worry about, so our task has become twice as hard!

Not to worry though. In order to protect against both sources of noise, all we need to do is effectively stagger the above constructions. Now we will need to design a logical “0” state which is itself a superposition of different points, with each point separated from all of the points that are superimposed to make the logical “1” state.

Diatomic molecules: For the diatomic molecule example, consider superpositions of all four corners of two antipodal tetrahedra for the two respective logical states.

blog_tet

The logical “0” state for the quantum code is now itself a quantum superposition of orientations of our diatomic molecule corresponding to the four black points on the sphere to the left (the sphere to the right is a top-down view). Similarly, the logical “1” quantum state is a superposition of all orientations corresponding to the white points.

Each orientation (black or white point) present in our logical states rotates under fluctuations in the position of the molecule. However, the entire set of orientations for say logical “0” — the tetrahedron — rotates rigidly under such rotations. Therefore, the region from which we can successfully recover after rotations is fully determined by the Voronoi tile of any one of the corners of the tetrahedron. (Above, we plot the tile for the point at the north pole.) This cell is clearly smaller than the one for classical north-south-pole encoding we used before. However, the tetrahedral code now provides some protection against phase errors — the other type of noise that we need to worry about if we are to protect quantum information. This is an example of the trade-off we must make in order to protect against both types of noise; a licensed quantum mechanic has to live with such trade-offs every day.

Oscillators: Another example of a quantum encoding is the GKP encoding in the phase space of the harmonic oscillator. Here, we have at our disposal the entire two-dimensional plane indexing different values of position and momentum. In this case, we can use a checkerboard approach, superimposing all points at the centers of the black squares for the logical “0” state, and similarly all points at the centers of the white squares for the logical “1”. The region depicting correctable momentum and position shifts is then the Voronoi cell of the point at the origin: if a shift takes our central black point to somewhere inside the blue square, we know (most likely) where that point came from! In solid state circles, the blue square is none other than the primitive or unit cell of the lattice consisting of points making up both of the logical states.

Asymmetric molecules (a.k.a. rigid rotors): Now let’s briefly return to molecules. Above, we considered diatomic molecules that had a symmetry axis, i.e., that were left unchanged under rotations about the axis that connects the two atoms. There are of course more general molecules out there, including ones that are completely asymmetric under any possible (proper) 3D rotation (see figure below for an example).

mol-f0 - blog

BONUS: There is a subtle mistake relating to the geometry of the rotation group in the labeling of this figure. Let me know if you can find it in the comments!

All of the orientations of the asymmetric molecule, and more generally a rigid body, can no longer be parameterized by the sphere. They can be parameterized by the 3D rotation group \mathsf{SO}(3): each orientation of an asymmetric molecule is labeled by the 3D rotation necessary to obtain said orientation from a reference state. Such rotations, and in turn the orientations themselves, are parameterized by an axis \mathbf{v} (around which to rotate) and an angle \omega (by which one rotates). The rotation group \mathsf{SO}(3) luckily can still be viewed by humans on a sheet of paper. Namely, \mathsf{SO}(3) can be thought of as a ball of radius \pi with opposite points identified. The direction of each vector \omega\mathbf{v} lying inside the ball corresponds to the axis of rotation, while the length corresponds to the angle. This may take some time to digest, but it’s not crucial to the story.

So far we’ve looked at codes defined on cubes of bits, spheres, and phase-space lattices. Turns out that even \mathsf{SO}(3) can house similar encodings! In other words, \mathsf{SO}(3) can also be cut up into different Voronoi tiles, which in turn can be staggered to create logical “0” and “1” states consisting of different molecular orientations. There are many ways to pick such states, corresponding to various subgroups of \mathsf{SO}(3). Below, we sketch two sets of black/white points, along with the Voronoi tile corresponding to the rotations that are corrected by each encoding.

Voronoi tiles of the black point at the center of the ball representing the 3D rotation group, for two different molecular codes. This and the Voronoi cells corresponding to the other points tile together to make up the entire ball. 3D printing all of these tiles would make for cool puzzles!

In closing…

Achieving supremacy was a big first step towards making quantum computing a practical and universal tool. However, the largest obstacles still await, namely handling superposition-poisoning noise coming from the ever-curious environment. As quantum technologies advance, other possible routes for error correction are by encoding qubits in harmonic oscillators and molecules, alongside the “traditional” approach of using arrays of physical qubits. Oscillator and molecular qubits possess their own mechanisms for error correction, and could prove useful (granted that the large high-energy space required for the procedures to work can be accessed and controlled). Even though molecular qubits are not yet mature enough to be used in quantum computers, we have at least outlined a blueprint for how some of the required pieces can be built. We are by no means done however: besides an engineering barrier, we need to further develop how to run robust computations on these exotic spaces.

Author’s note: I’d like to acknowledge Jose Gonzalez for helping me immensely with the writing of this post, as well as for drawing the comic panels in the previous post. The figures above were made possible by Mathematica 12.