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I attended a liberal-arts college, and I reveled in the curriculum’s breadth. My coursework included art history, psychology, biology, economics, computer science, German literature, archaeology, and chemistry. My major sat halfway between the physics major and the create-your-own major; the requirements consisted mostly of physics but included math, philosophy, and history. By the end of college, I’d determined to dive into physics. So I undertook a physics research assistantship, enlisted in a Master’s program and then a PhD program, and became a theoretical physicist. I’m now building a physics research group that spans a government institute and the University of Maryland. One might think that I became a physicist despite my art history and archaeology. 

My liberal-arts education did mortify me a little as I pursued my Master’s degree. Most of my peers had focused on physics, mathematics, and computer science while I’d been reading Aristotle. They seemed to breeze through coursework that I clawed my way through. I still sigh wistfully over math courses, such as complex analysis, that I’ve never taken. Meanwhile, a debate about the liberal arts has been raging across the nation. Debt is weighing down recent graduates, and high-school students are loading up on STEMM courses. Colleges are cutting liberal-arts departments, and educational organizations are broadcasting the value of liberal-arts educations.

I’m not an expert in public policy or school systems; I’m a physicist. As a physicist, I’m grateful for my liberal-arts education. It’s enhanced my physics research in at least five ways.

(1) I learned to seek out, and take advantage of, context. Early in my first German-literature course, I’d just completed my first reading assignment. My professor told my class to fetch out our books and open them to the beginning. A few rustles later, we held our books open to page one of the main text. 

No, no, said my professor. Open your books to the beginning. Did anyone even look at the title page?

We hadn’t, we admitted. We’d missed a wealth of information, as the book contained a reproduction of an old title page. Publishers, fonts, and advertisement styles have varied across the centuries and the globe. They, together with printing and reprinting dates, tell stories about the book’s origin, popularity, role in society, and purposes. Furthermore, a frontispiece is worth a thousand words, all related before the main text begins. When my class turned to the main text, much later in the lecture, we saw it in a new light. Context deepens and broadens our understanding.

When I read a physics paper, I start at the beginning—the true beginning. I note the publication date, the authors, their institutions and countries, and the journal. X’s lab performed the experiment reported on? X was the world’s expert in Y back then but nursed a bias against Z, a bias later proved to be unjustified. So I should aim to learn from the paper about Y but should take statements about Z with a grain of salt. Seeking and processing context improves my use of physics papers, thanks to a German-literature course.

(2) I learned argumentation. Doing physics involves building, analyzing, criticizing, and repairing arguments. I argue that mathematical X models physical system Y accurately, that an experiment I’ve proposed is feasible with today’s technology, and that observation Z supports a conjecture of mine. Physicists also prove mathematical statements deductively. I received proof-writing lessons in a math course, halfway through college. One of the most competent teachers I’ve ever encountered taught the course. But I learned about classes of arguments and about properties of arguments in a philosophy course, Informal Logic. 

There, I learned to distinguish deduction from inference and an argument’s validity and soundness from an argument’s strength and cogency. I learned strategies for proving arguments and learned fallacies to criticize. I came to respect the difference between “any” and “every,” which I see interchanged in many physics papers. This philosophical framework helps me formulate, process, dissect, criticize, and correct physics arguments. 

For instance, I often parse long, dense, technical proofs of mathematical statements. First, I identify whether the proof strategy is reductio ad absurdum, proof by counterexample, or another strategy. Upon identifying the overarching structure, I can fill my understanding with details. Additionally, I check proofs by students, and I respond to criticisms of my papers by journal referees. I could say, upon reading an argument, “Something feels a bit off, and it’s sort of like the thing that felt a bit off in that paper I read last Tuesday.” But I’d rather group the argument I’m given together with arguments I know how to tackle. I’d rather be able to say, “They’re straw-manning my argument” or “That argument begs the question.” Doing so, I put my finger on the problem and take a step toward solving it.

(3) I learned to analyze materials to bits, then extract meaning from the analysis. English and German courses trained me to wring from literature every drop of meaning that I could discover. I used to write one to three pages about a few-line quotation. The analysis would proceed from diction and punctuation to literary devices; allusions; characters’ relationships with each other, themselves, and nature; and the quotation’s role in the monograph. Everything from minutia to grand themes required scrutiny, according to the dissection technique I trained in. Every pincer probe lifted another skein of skin or drew aside another tendon, offering deeper insights into the literary work. I learned to find the skeins to lift, lift them in the right direction, pinpoint the insights revealed, and integrate the insights into a coherent takeaway.

This training has helped me assess and interpret mathematics. Physicists pick a physical system to study, model the system with equations, and solve the equations. The next two steps are intertwined: evaluating whether one solved the equations correctly and translating the solution into the physical system’s behavior. These two steps necessitate a dissection of everything from minutia to grand themes: Why should this exponent be 4/5, rather than any other number? Should I have expected this energy to depend on that length in this way? Is the physical material aging quickly or resisting change? These questions’ answers inform more-important questions: Who cares? Do my observations shed light worth anyone’s time, or did I waste a week solving equations no one should care about?

To answer all these questions, I draw on my literary training: I dissect content, pinpoint insights, and extract meaning. Having performed this analysis in literature courses facilitates an arguably deeper analysis than my physics training did: In literature courses, I had to organize my thoughts and articulate them in essays. This process revealed holes in my argumentation, as well as connections that I’d overlooked. In contrast, a couple of lines in my physics homework earned full marks. The critical analysis of literature has deepened my assessment of solutions’ correctness, physical interpretation of mathematics, and extraction of meaning from solutions. 

(4) I learned what makes a physicist a physicist. In college, I had a friend who was studying applied mathematics and economics. Over dinner, he described a problem he’d encountered in his studies. I replied, almost without thinking, “From a physics perspective, I’d approach the problem like this.” I described my view, which my friend said he wouldn’t have thought of. I hadn’t thought of myself, and of the tools I was obtaining in the physics department, the way I did after our conversation. 

Physics involves a unique toolkit,1 set of goals, and philosophy. Physicists identify problems, model them, solve them, and analyze the results in certain ways. Students see examples of these techniques in lectures and practice these techniques for homework. But, as a student, I rarely heard articulations of the general principles that underlay the examples scattered across my courses like a handful of marbles across a kitchen floor. Example principles include, if you don’t understand an abstract idea, construct a simple example. Once you’ve finished a calculation, check whether your answer makes sense in the most extreme scenarios possible. After solving an equation, interpret the solution in terms of physical systems—of how particles and waves move and interact. 

I was learning these techniques, in college, without realizing that I was learning them. I became conscious of the techniques by comparing the approach natural to me with the approach taken in another discipline. Becoming conscious of my toolkit enabled me to wield it more effectively; one can best fry eggs when aware that one owns a spatula. The other disciplines at my liberal-arts college served as a foil for physics. Seeing other disciplines, I saw what makes physics physics—and improved my ability to apply my physics toolkit.

(5) I learned to draw connections between diverse ideas. Senior year of high school, my courses extended from physics to English literature. One might expect such a curriculum to feel higgledy-piggledy, but I found threads that ran through all my courses. For instance, I practiced public speaking in Reasoning, Research, and Rhetoric. Because I studied rhetoric, my philosophy teacher turned to me for input when introducing the triumvirate “thesis, antithesis, synthesis.”2 The philosophy curriculum included the feminist essay “If Men Could Menstruate,” which complemented the feminist book Wide Sargasso Sea in my English-literature course. In English literature, I learned that Baldassare Castiglione codified how Renaissance noblemen should behave, in The Book of the Courtier. The author’s name was the answer to the first question on my AP Modern European History exam. My history course covered Isaac Newton and Gottfried Wilhelm Leibniz, who invented calculus during the 17th century. I leveraged their discoveries in my calculus course, which I applied in my physics course. My physics teacher hoped that his students would solve the world’s energy problems—perhaps averting the global thermonuclear war that graced every debate in my rhetoric course (“If you don’t accept my team’s policy, then X will happen, leading to Y, leading to Z, which will cause a global thermonuclear war”). 

Threads linked everything across my liberal-arts education; every discipline featured an idea that paralleled an idea in another discipline. Finding those parallels grew into a game for me, a game that challenged my creativity. Cultivating that creativity paid off when I began doing physics research. Much of my research has resulted from finding, in one field, a concept that resembles a concept in another field. I smash the ideas together to gain insight into each discipline from the other discipline’s perspective. For example, during my PhD studies, I found a thread connecting the physics of DNA strands to the physics of black holes. That thread initiated a research program of mine that’s yielded nine papers, garnered 19 collaborators, and spawned two experiments. Studying diverse subjects trained me to draw creative connections, which underlie much physics research.

I haven’t detailed all the benefits that a liberal-arts education can accrue to a physics career. For instance, the liberal arts enhance one’s communication skills, key to collaborating on research and to conveying one’s research. Without conveying one’s research adroitly, one likely won’t impact a field much. Also, a liberal-arts education can help one connect with researchers from across the globe on a personal level.3 Personal connections enhance science, which scientists—humans—undertake.

As I began building my research group, I sought advice from an MIT professor who’d attended MIT as an undergraduate. He advised me to seek students who have unusual backgrounds, including liberal-arts educations. Don’t get me wrong; I respect and cherish the colleagues and friends of mine who attended MIT, Caltech, and other tech schools as undergraduates. Still, I wouldn’t trade my German literature and economics. The liberal arts have enriched my physics research no less than they’ve enriched the rest of my life.

1A toolkit that overlaps partially with other disciplines’ toolkits, as explained in (3).

2I didn’t help much. When asked to guess the last concept in the triumvirate, I tried “debate.”

3I once met a Ukrainian physicist who referred to Ilya Muromets in a conversation. Ilya Muromets is a bogatyr, a knight featured in Slavic epics set in the Middle Ages. I happened to have taken a Slavic-folklore course the previous year. So I responded with a reference to Muromets’s pals, Dobrynya Nikitich and Alyosha Popovich. The physicist and I hit it off, and he taught me much about condensed matter over the following months.

I’m publishing a book! Quantum Steampunk: The Physics of Yesterday’s Tomorrow is hitting bookstores next spring, and you can preorder it now.

As Quantum Frontiers regulars know, steampunk is a genre of literature, art and film. Steampunkers fuse 19th-century settings (such as Victorian England, the Wild West, and Meiji Japan) with futuristic technologies (such as dirigibles, time machines, and automata). So does my field of research, a combination of thermodynamics, quantum physics, and information processing. 

Thermodynamics, the study of energy, developed during the Industrial Revolution. The field grew from practical concerns (How efficiently can engines pump water out of mines?) but wound up addressing fundamental questions (Why does time flow in only one direction?). Thermodynamics needs re-envisioning for 21st-century science, which spotlights quantum systems—electrons, protons, and other basic particles. Early thermodynamicists couldn’t even agree that atoms existed, let alone dream that quantum systems could process information in ways impossible for nonquantum systems. Over the past few decades, we’ve learned that quantum technologies can outperform their everyday counterparts in solving certain computational problems, in securing information, and in transmitting information. The study of quantum systems’ information-processing power forms a mathematical and conceptual toolkit, quantum information science. My colleagues and I leverage this toolkit to reconceptualize thermodynamics. As we combine a 19th-century framework (thermodynamics) with advanced technology (quantum information), I call our field quantum steampunk.

Glimpses of quantum steampunk have surfaced on this blog throughout the past eight years. The book is another animal, a 15-chapter closeup of the field. The book sets the stage with introductions to information processing, quantum physics, and thermodynamics. Then, we watch these three perspectives meld into one coherent whole. We tour the landscape of quantum thermodynamics—the different viewpoints and discoveries championed by different communities. These viewpoints, we find, offer a new lens onto the rest of science, including chemistry, black holes, and materials physics. Finally, we peer through a brass telescope to where quantum steampunk is headed next. Throughout the book, the science interleaves with anecdotes, history, and the story of one woman’s (my) journey into physics—and with snippets from a quantum-steampunk novel that I’ve dreamed up.

On this blog, different parts of my posts are intended for different audiences. Each post contains something for everyone, but not everyone will understand all of each post. In contrast, the book targets the general educated layperson. One of my editors majored in English, and another majored in biology, so the physics should come across clearly to everyone (and if it doesn’t, blame my editors). But the book will appeal to physicists, too. Reviewer Jay Lawrence, a professor emeritus of Dartmouth College’s physics department, wrote, “Presenting this vision [of quantum thermodynamics] in a manner accessible to laypeople discovering new interests, Quantum Steampunk will also appeal to specialists and aspiring specialists.” This book is for you.

Strange to say, I began writing Quantum Steampunk under a year ago. I was surprised to receive an email from Tiffany Gasbarrini, a senior acquisitions editor at Johns Hopkins University Press, in April 2020. Tiffany had read the article I’d written about quantum steampunk for Scientific American. She wanted to expand the press’s offerings for the general public. Would I be interested in writing a book proposal? she asked.

Not having expected such an invitation, I poked around. The press’s roster included books that caught my eye, by thinkers I wanted to meet. From Wikipedia, I learned that Johns Hopkins University Press is “the oldest continuously running university press in the United States.” Senior colleagues of mine gave the thumbs-up. So I let my imagination run.

I developed a table of contents while ruminating on long walks, which I’d begun taking at the start of the pandemic. In late July, I submitted my book proposal. As the summer ended, I began writing the manuscript.

Writing the first draft—73,000 words—took about five months. The process didn’t disrupt life much. I’m used to writing regularly; I’ve written one blog post per month here since 2013, and I wrote two novels during and after college. I simply picked up my pace. At first, I wrote only on weekends. Starting in December 2020, I wrote 1,000 words per day. The process wasn’t easy, but it felt like a morning workout—healthy and productive. That productivity fed into my science, which fed back into the book. One of my current research projects grew from the book’s epilogue. A future project, I expect, will evolve from Chapter 5.

As soon as I finished draft one—last January—Tiffany and I hunted for an illustrator. We were fortunate to find Todd Cahill, a steampunk artist. He transformed the poor sketches that I’d made into works of art.

Early this spring, I edited the manuscript. That edit was to a stroll as the next edit was to the Boston Marathon. Editor Michael Zierler coached me through the marathon. He identified concepts that needed clarification, discrepancies between explanations, and analogies that had run away with me—as well as the visions and turns of phrase that delighted him, to balance the criticism. As Michael and I toiled, 17 of my colleagues were kind enough to provide feedback. They read sections about their areas of expertise, pointed out subtleties, and confirmed facts.

Soon after Michael and I crossed the finished line, copyeditor Susan Matheson took up the baton. She hunted for typos, standardized references, and more. Come June, I was editing again—approving and commenting on her draft. Simultaneously, Tiffany designed the cover, shown above, with more artists. The marketing team reached out, and I began planning this blog post. Scratch that—I’ve been dreaming about this blog post for almost a year. But I forced myself not to spill the beans here till I told the research group I’ve been building. I shared about the book with them two Thursdays ago, and I hope that book critics respond as they did.

Every time I’ve finished a draft, my husband and I have celebrated by ordering takeout sandwiches from our favorite restaurant. Three sandwich meals are down, and we have one to go.

Having dreamed about this blog post for a year, I’m thrilled to bits to share my book with you. It’s available for preordering, and I encourage you to support your local bookstore by purchasing through bookshop.org. The book is available also through Barnes & Noble, Amazon, Waterstones, and the other usual suspects. For press inquiries, or to request a review copy, contact Kathryn Marguy at [email protected].

Over the coming year, I’ll continue sharing about my journey into publishing—the blurbs we’ll garner for the book jacket, the first copies hot off the press, the reviews and interviews. I hope that you’ll don your duster coat and goggles (every steampunker wears goggles), hop into your steam-powered gyrocopter, and join me.

I had a relative to whom my parents referred, when I was little, as “that great-aunt of yours who walked into a glass door at your cousin’s birthday party.” I was a small child in a large family that mostly lived far away; little else distinguished this great-aunt from other relatives, in my experience. She’d intended to walk from my grandmother’s family room to the back patio. A glass door stood in the way, but she didn’t see it. So my great-aunt whammed into the glass; spent part of the party on the couch, nursing a nosebleed; and earned the epithet via which I identified her for years.

After growing up, I came to know this great-aunt as a kind, gentle woman who adored her family and was adored in return. After growing into a physicist, I came to appreciate her as one of my earliest instructors in necessary and sufficient conditions.

My great-aunt’s intended path satisfied one condition necessary for her to reach the patio: Nothing visible obstructed the path. But the path failed to satisfy a sufficient condition: The invisible obstruction—the glass door—had been neither slid nor swung open. Sufficient conditions, my great-aunt taught me, mustn’t be overlooked.

Her lesson underlies a paper I published this month, with coauthors from the Cambridge other than mine—Cambridge, England: David Arvidsson-Shukur and Jacob Chevalier Drori. The paper concerns, rather than pools and patios, quasiprobabilities, which I’ve blogged about many times [1,2,3,4,5,6,7]. Quasiprobabilities are quantum generalizations of probabilities. Probabilities describe everyday, classical phenomena, from Monopoly to March Madness to the weather in Massachusetts (and especially the weather in Massachusetts). Probabilities are real numbers (not dependent on the square-root of -1); they’re at least zero; and they compose in certain ways (the probability of sun or hail equals the probability of sun plus the probability of hail). Also, the probabilities that form a distribution, or a complete set, sum to one (if there’s a 70% chance of rain, there’s a 30% chance of no rain). 

In contrast, quasiprobabilities can be negative and nonreal. We call such values nonclassical, as they’re unavailable to the probabilities that describe classical phenomena. Quasiprobabilities represent quantum states: Imagine some clump of particles in a quantum state described by some quasiprobability distribution. We can imagine measuring the clump however we please. We can calculate the possible outcomes’ probabilities from the quasiprobability distribution.

My favorite quasiprobability is an obscure fellow unbeknownst even to most quantum physicists: the Kirkwood-Dirac distribution. John Kirkwood defined it in 1933, and Paul Dirac defined it independently in 1945. Then, quantum physicists forgot about it for decades. But the quasiprobability has undergone a renaissance over the past few years: Experimentalists have measured it to infer particles’ quantum states in a new way. Also, colleagues and I have generalized the quasiprobability and discovered applications of the generalization across quantum physics, from quantum chaos to metrology (the study of how we can best measure things) to quantum thermodynamics to the foundations of quantum theory.

In some applications, nonclassical quasiprobabilities enable a system to achieve a quantum advantage—to usefully behave in a manner impossible for classical systems. Examples include metrology: Imagine wanting to measure a parameter that characterizes some piece of equipment. You’ll perform many trials of an experiment. In each trial, you’ll prepare a system (for instance, a photon) in some quantum state, send it through the equipment, and measure one or more observables of the system. Say that you follow the protocol described in this blog post. A Kirkwood-Dirac quasiprobability distribution describes the experiment.1 From each trial, you’ll obtain information about the unknown parameter. How much information can you obtain, on average over trials? Potentially more information if some quasiprobabilities are negative than if none are. The quasiprobabilities can be negative only if the state and observables fail to commute with each other. So noncommutation—a hallmark of quantum physics—underlies exceptional metrological results, as shown by Kirkwood-Dirac quasiprobabilities.

Exceptional results are useful, and we might aim to design experiments that achieve them. We can by designing experiments described by nonclassical Kirkwood-Dirac quasiprobabilities. When can the quasiprobabilities become nonclassical? Whenever the relevant quantum state and observables fail to commute, the quantum community used to believe. This belief turns out to mirror the expectation that one could access my grandmother’s back patio from the living room whenever no visible barriers obstructed the path. As a lack of visible barriers was necessary for patio access, noncommutation is necessary for Kirkwood-Dirac nonclassicality. But noncommutation doesn’t suffice, according to my paper with David and Jacob. We identified a sufficient condition, sliding back the metaphorical glass door on Kirkwood-Dirac nonclassicality. The condition depends on simple properties of the system, state, and observables. (Experts: Examples include the Hilbert space’s dimensionality.) We also quantified and upper-bounded the amount of nonclassicality that a Kirkwood-Dirac quasiprobability can contain.

From an engineering perspective, our results can inform the design of experiments intended to achieve certain quantum advantages. From a foundational perspective, the results help illuminate the sources of certain quantum advantages. To achieve certain advantages, noncommutation doesn’t cut the mustard—but we now know a condition that does.

For another take on our paper, check out this news article in Physics Today.  

1Really, a generalized Kirkwood-Dirac quasiprobability. But that phrase contains a horrendous number of syllables, so I’ll elide the “generalized.”

Intelligent beings have the ability to receive, process, store information, and based on the processed information, predict what would happen in the future and act accordingly.

An illustration of receiving, processing, and storing information. Based on the processed information, one can make prediction about the future.
[Credit: Claudia Cheng]

We, as intelligent beings, receive, process, and store classical information. The information comes from vision, hearing, smell, and tactile sensing. The data is encoded as analog classical information through the electrical pulses sending through our nerve fibers. Our brain processes this information classically through neural circuits (at least that is our current understanding, but one should check out this blogpost). We then store this processed classical information in our hippocampus that allows us to retrieve it later to combine it with future information that we obtain. Finally, we use the stored classical information to make predictions about the future (imagine/predict the future outcomes if we perform certain action) and choose the action that would most likely be in our favor.

Such abilities have enabled us to make remarkable accomplishments: soaring in the sky by constructing accurate models of how air flows around objects, or building weak forms of intelligent beings capable of performing basic conversations and play different board games. Instead of receiving/processing/storing classical information, one could imagine some form of quantum intelligence that deals with quantum information instead of classical information. These quantum beings can receive quantum information through quantum sensors built up from tiny photons and atoms. They would then process this quantum information with quantum mechanical evolutions (such as quantum computers), and store the processed qubits in a quantum memory (protected with a surface code or toric code).

A caricature of human intelligence dating long before 1950, artificial intelligence that began in the 50’s, and the emergence of quantum intelligence.
[Credit: Claudia Cheng]

It is natural to wonder what a world of quantum intelligence would be like. While we have never encountered such a strange creature in the real world (yet), the mathematics of quantum mechanics, machine learning, and information theory allow us to peek into what such a fantastic world would be like. The physical world we live in is intrinsically quantum. So one may imagine that a quantum being is capable of making more powerful predictions than a classical being. Maybe he/she/they could better predict events that happened further away, such as tell us how a distant black hole was engulfing another? Or perhaps he/she/they could improve our lives, for example by presenting us with an entirely new approach for capturing energy from sunlight?

One may be skeptical about finding quantum intelligent beings in nature (and rightfully so). But it may not be so absurd to synthesize a weak form of quantum (artificial) intelligence in an experimental lab, or enhance our classical human intelligence with quantum devices to approximate a quantum-mechanical being. Many famous companies, like Google, IBM, Microsoft, and Amazon, as well as many academic labs and startups have been building better quantum machines/computers day by day. By combining the concepts of machine learning on classical computers with these quantum machines, the future of us interacting with some form of quantum (artificial) intelligence may not be so distant.

Before the day comes, could we peek into the world of quantum intelligence? And could one better understand how much more powerful they could be over classical intelligence?

A cartoon depiction of me (Left), Richard Kueng (Middle), and John Preskill (Right).
[Credit: Claudia Cheng]

In a recent publication [1], my advisor John Preskill, my good friend Richard Kueng, and I made some progress toward these questions. We consider a quantum mechanical world where classical beings could obtain classical information by measuring the world (performing POVM measurement). In contrast, quantum beings could retrieve quantum information through quantum sensors and store the data in a quantum memory. We study how much better quantum over classical beings could learn from the physical world to accurately predict the outcomes of unseen events (with the focus on the number of interactions with the physical world instead of computation time). We cast these problems in a rigorous mathematical framework and utilize high-dimensional probability and quantum information theory to understand their respective prediction power. Rigorously, one refers to a classical/quantum being as a classical/quantum model, algorithm, protocol, or procedure. This is because the actions of these classical/quantum beings are the center of the mathematical analysis.

Formally, we consider the task of learning an unknown physical evolution described by a CPTP map that takes in -qubit state and maps to -qubit state. The classical model can select an arbitrary classical input to the CPTP map and measure the output state of the CPTP map with some POVM measurement. The quantum model can access the CPTP map coherently and obtain quantum data from each access, which is equivalent to composing multiple CPTP maps with quantum computations to learn about the CPTP map. The task is to predict a property of the output state , given by , for a new classical input . And the goal is to achieve the task while accessing as few times as possible (i.e., fewer interactions or experiments in the physical world). We denote the number of interactions needed by classical and quantum models as .

In general, quantum models could learn from fewer interactions with the physical world (or experiments in the physical world) than classical models. This is because coherent quantum information can facilitate better information synthesis with information obtained from previous experiments. Nevertheless, in [1], we show that there is a fundamental limit to how much more efficient quantum models can be. In order to achieve a prediction error

where is the hypothesis learned from the classical/quantum model and is an arbitrary distribution over the input space , we found that the speed-up is upper bounded by , where is the number of qubits each experiment provides (the output number of qubits in the CPTP map ), and is the desired prediction error (smaller means we want to predict more accurately).

In contrast, when we want to accurately predict all unseen events, we prove that quantum models could use exponentially fewer experiments than classical models. We give a construction for predicting properties of quantum systems showing that quantum models could substantially outperform classical models. These rigorous results show that quantum intelligence shines when we seek stronger prediction performance.

We have only scratched the surface of what is possible with quantum intelligence. As the future unfolds, I am hopeful that we will discover more that can be done only by quantum intelligence, through mathematical analysis, rigorous numerical studies, and physical experiments.

Further information:

• A classical model that can be used to accurately predict properties of quantum systems is the classical shadow formalism [2] that we proposed a year ago. In many tasks, this model can be shown to be one of the strongest rivals that quantum models have to surpass.
• Even if a quantum model only receives and stores classical data, the ability to process the data using a quantum-mechanical evolution can still be advantageous [3]. However, obtaining large advantage will be harder in this case as the computational power in data can slightly boost classical machines/intelligence [3].
• Another nice paper by Dorit Aharonov, Jordan Cotler, and Xiao-Liang Qi [4] also proved advantages of quantum models over classical one in some classification tasks.

References:

[1] Huang, Hsin-Yuan, Richard Kueng, and John Preskill. “Information-Theoretic Bounds on Quantum Advantage in Machine Learning.” Physical Review Letters 126: 190505 (2021).https://doi.org/10.1103/PhysRevLett.126.190505

[2] Huang, Hsin-Yuan, Richard Kueng, and John Preskill. “Predicting many properties of a quantum system from very few measurements.” Nature Physics 16: 1050-1057 (2020). https://doi.org/10.1038/s41567-020-0932-7

[3] Huang, Hsin-Yuan, et al. “Power of data in quantum machine learning.” Nature communications 12.1 (2021): 1-9. https://doi.org/10.1038/s41467-021-22539-9

[4] Aharonov, Dorit, Jordan Cotler, and Xiao-Liang Qi. “Quantum Algorithmic Measurement.” arXiv preprint arXiv:2101.04634 (2021).

The autumn of my sophomore year of college was mildly hellish. I took the equivalent of three semester-long computer-science and physics courses, atop other classwork; co-led a public-speaking self-help group; and coordinated a celebrity visit to campus. I lived at my desk and in office hours, always declining my flatmates’ invitations to watch The West Wing. 

Hard as I studied, my classmates enjoyed greater facility with the computer-science curriculum. They saw immediately how long an algorithm would run, while I hesitated and then computed the run time step by step. I felt behind. So I protested when my professor said, “You’re good at this.” 

I now see that we were focusing on different facets of learning. I rued my lack of intuition. My classmates had gained intuition by exploring computer science in high school, then slow-cooking their experiences on a mental back burner. Their long-term exposure to the material provided familiarity—the ability to recognize a new problem as belonging to a class they’d seen examples of. I was cooking course material in a mental microwave set on “high,” as a semester’s worth of material was crammed into ten weeks at my college.

My professor wasn’t measuring my intuition. He only saw that I knew how to compute an algorithm’s run time. I’d learned the material required of me—more than I realized, being distracted by what I hadn’t learned that difficult autumn.

We can learn a staggering amount when pushed far from our comfort zones—and not only we humans can. So can simple collections of particles.

Examples include a classical spin glass. A spin glass is a collection of particles that shares some properties with a magnet. Both a magnet and a spin glass consist of tiny mini-magnets called spins. Although I’ve blogged about quantum spins before, I’ll focus on classical spins here. We can imagine a classical spin as a little arrow that points upward or downward.  A bunch of spins can form a material. If the spins tend to point in the same direction, the material may be a magnet of the sort that’s sticking the faded photo of Fluffy to your fridge.

The spins may interact with each other, similarly to how electrons interact with each other. Not entirely similarly, though—electrons push each other away. In contrast, a spin may coax its neighbors into aligning or anti-aligning with it. Suppose that the interactions are random: Any given spin may force one neighbor into alignment, gently ask another neighbor to align, entreat a third neighbor to anti-align, and having nothing to say to neighbors four and five.

The spin glass can interact with the external world in two ways. First, we can stick the spins in a magnetic field, as by placing magnets above and below the glass. If aligned with the field, a spin has negative energy; and, if antialigned, positive energy. We can sculpt the field so that it varies across the spin glass. For instance, spin 1 can experience a strong upward-pointing field, while spin 2 experiences a weak downward-pointing field.

Second, say that the spins occupy a fixed-temperature environment, as I occupy a 74-degree-Fahrenheit living room. The spins can exchange heat with the environment. If releasing heat to the environment, a spin flips from having positive energy to having negative—from antialigning with the field to aligning.

Let’s perform an experiment on the spins. First, we design a magnetic field using random numbers. Whether the field points upward or downward at any given spin is random, as is the strength of the field experienced by each spin. We sculpt three of these random fields and call the trio a drive.

Let’s randomly select a field from the drive and apply it to the spin glass for a while; again, randomly select a field from the drive and apply it; and continue many times. The energy absorbed by the spins from the fields spikes, then declines.

Now, let’s create another drive of three random fields. We’ll randomly pick a field from this drive and apply it; again, randomly pick a field from this drive and apply it; and so on. Again, the energy absorbed by the spins spikes, then tails off.

Here comes the punchline. Let’s return to applying the initial fields. The energy absorbed by the glass will spike—but not as high as before. The glass responds differently to a familiar drive than to a new drive. The spin glass recognizes the original drive—has learned the first fields’ “fingerprint.” This learning happens when the fields push the glass far from equilibrium,1 as I learned when pushed during my mildly hellish autumn.

So spin glasses learn drives that push them far from equilibrium. So do many other simple, classical, many-particle systems: polymers, viscous liquids, crumpled sheets of Mylar, and more. Researchers have predicted such learning and observed it experimentally. 

Scientists have detected many-particle learning by measuring thermodynamic observables. Examples include the energy absorbed by the spin glass—what thermodynamicists call work. But thermodynamics developed during the 1800s, to describe equilibrium systems, not to study learning. 

One study of learning—the study of machine learning—has boomed over the past two decades. As described by the MIT Technology Review, “[m]achine-learning algorithms use statistics to find patterns in massive amounts of data.” Users don’t tell the algorithms how to find those patterns.

It seems natural and fitting to use machine learning to learn about the learning by many-particle systems. That’s what I did with collaborators from the group of Jeremy England, a GlaxoSmithKline physicist who studies complex behaviors of many particle systems. Weishun Zhong, Jacob Gold, Sarah Marzen, Jeremy, and I published our paper last month. 

Using machine learning, we detected and measured many-particle learning more reliably and precisely than thermodynamic measures seem able to. Our technique works on multiple facets of learning, analogous to the intuition and the computational ability I encountered in my computer-science course. We illustrated our technique on a spin glass, but one can apply our approach to other systems, too. I’m exploring such applications with collaborators at the University of Maryland.

The project pushed me far from my equilibrium: I’d never worked with machine learning or many-body learning. But it’s amazing, what we can learn when pushed far from equilibrium. I first encountered this insight sophomore fall of college—and now, we can quantify it better than ever.

1Equilibrium is a quiet, restful state in which the glass’s large-scale properties change little. No net flow of anything—such as heat or particles—enter or leave the system.

Happy National Poetry Month! The United States salutes word and whimsy in April, and Quantum Frontiers is continuing its tradition of celebrating. As a resident of Cambridge, Massachusetts and as a quantum information scientist, I have trouble avoiding the poem “Paul Revere’s Ride.” 

Henry Wadsworth Longfellow wrote the poem, as well as others in the American canon, during the 1800s. Longfellow taught at Harvard in Cambridge, and he lived a few blocks away from the university, in what’s now a national historic site. Across the street from the house, a bust of the poet gazes downward, as though lost in thought, in Longfellow Park. Longfellow wrote one of his most famous poems about an event staged a short drive from—and, arguably, partially in—Cambridge.

The event took place “on the eighteenth of April, in [Seventeen] Seventy-Five,” as related by the narrator of “Paul Revere’s Ride.” Revere was a Boston silversmith and a supporter of the American colonies’ independence from Britain. Revolutionaries anticipated that British troops would set out from Boston sometime during the spring. The British planned to seize revolutionaries’ weapons in the nearby town of Concord and to jail revolutionary leaders in Lexington. The troops departed Boston during the night of April 18th. 

Upon learning of their movements, sexton Robert Newman sent a signal from Boston’s old North Church to Charlestown. Revere and the physician William Dawes rode out from Charlestown to warn the people of Lexington and the surrounding areas. A line of artificial hoof prints, pressed into a sidewalk a few minutes from the Longfellow house, marks part of Dawes’s trail through Cambridge. The initial riders galvanized more riders, who stirred up colonial militias that resisted the troops’ advance. The Battles of Lexington and Concord ensued, initiating the Revolutionary War.

Longfellow took liberties with the facts he purported to relate. But “Paul Revere’s Ride” has blown the dust off history books for generations of schoolchildren. The reader shares Revere’s nervous excitement as he fidgets, awaiting Newman’s signal: 

Now he patted his horse’s side, 
Now gazed on the landscape far and near, 
Then impetuous stamped the earth, 
And turned and tightened his saddle-girth;
But mostly he watched with eager search 
The belfry-tower of the old North Church.

The moment the signal arrives, that excitement bursts its seams, and Revere leaps astride his horse. The reader comes to gallop through with the silversmith the night, the poem’s clip-clop-clip-clop rhythm evoking a horse’s hooves on cobblestones.

Not only does “Paul Revere’s Ride” revitalize history, but it also offers a lesson in information theory. While laying plans, Revere instructs Newman: 

He said to his friend, “If the British march
By land or sea from the town to-night,
Hang a lantern aloft in the belfry-arch
Of the North-Church-tower, as a signal light.

Then comes one of the poem’s most famous lines: “One if by land, and two if by sea.” The British could have left Boston by foot or by boat, and Newman had to communicate which. Specifying one of two options, he related one bit, or one basic unit of information. Newman thereby exemplifies a cornerstone of information theory: the encoding of a bit of information—an abstraction—in a physical system that can be in one of two possible states—a light that shines from one or two lanterns.

Benjamin Schumacher and Michael Westmoreland point out the information-theoretic interpretation of Newman’s action in their quantum-information textbook. I used their textbook in my first quantum-information course, as a senior in college. Before reading the book, I’d never felt that I could explain what information is or how it can be quantified. Information is an abstraction and a Big Idea, like consciousness, life, and piety. But, Schumacher and Westmoreland demonstrated, most readers already grasp the basics of information theory; some readers even studied the basics while memorizing a poem in elementary school. So I doff my hat—or, since we’re discussing the 1700s, my mobcap—to the authors.

Reading poetry enriches us more than we realize. So read a poem this April. You can find Longfellow’s poem here or ride off wherever your fancy takes you.  

I used to catch lizards—brown anoles, as I learned to call them later—as a child. They were colored as their name suggests, were about as long as one of my hands, and resented my attention. But they frequented our back porch, and I had a butterfly net. So I’d catch lizards, with my brother or a friend, and watch them. They had throats that occasionally puffed out, exposing red skin, and tails that detached and wriggled of their own accord, to distract predators.

Some theorists might appreciate butterfly nets, I imagine, for catching experimentalists. Some of us theorists will end a paper or a talk with “…and these predictions are experimentally accessible.” A pause will follow the paper’s release or the talk, in hopes that a reader or an audience member will take up the challenge. Usually, none does, and the writer or speaker retires to the Great Deck Chair of Theory on the Back Patio of Science.

So I was startled when an anole, metaphorically speaking, volunteered a superconducting qubit for an experiment I’d proposed.

The experimentalist is one of the few people I can compare to a reptile without fear that he’ll take umbrage: Kater Murch, an associate professor of physics at Washington University in St. Louis. The most evocative description of Kater that I can offer appeared in an earlier blog post: “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.”

Kater expressed interest in an uncertainty relation I’d proved with theory collaborators. According to some of the most famous uncertainty relations, a quantum particle can’t have a well-defined position and a well-defined momentum simultaneously. Measuring the position disturbs the momentum; any later momentum measurement outputs a completely random, or uncertain, number. We measure uncertainties with entropies: The greater an entropy, the greater our uncertainty. We can cast uncertainty relations in terms of entropies.

I’d proved, with collaborators, an entropic uncertainty relation that describes chaos in many-particle quantum systems. Other collaborators and I had shown that weak measurements, which don’t disturb a quantum system much, characterize chaos. So you can check our uncertainty relation using weak measurements—as well as strong measurements, which do disturb quantum systems much. One can simplify our uncertainty relation—eliminate the chaos from the problem and even eliminate most of the particles. An entropic uncertainty relation for weak and strong measurements results.

Kater specializes in weak measurements, so he resolved to test our uncertainty relation. Physical Review Letters published the paper about our collaboration this month. Quantum measurements can not only create uncertainty, the paper shows, but also reduce it: Kater and his PhD student Jonathan Monroe used light to measure a superconducting qubit, a tiny circuit in which current can flow forever. The qubit had properties analogous to position and momentum (the spin’s z– and x-components). If the atom started with a well-defined “position” (the z-component) and the “momentum” (the x-component) was measured, the outcome was highly random; the total uncertainty about the two measurements was large. But if the atom started with a well-defined “position” (z-component) and another property (the spin’s y-component) was measured before the “momentum” (the x-component) was measured strongly, the total uncertainty was lower. The extra measurement was designed not to disturb the atom much. But the nudge prodded the atom enough, rendering the later “momentum” measurement (the x measurement) more predictable. So not only can quantum measurements create uncertainty, but gentle quantum measurements can also reduce it.

I didn’t learn only physics from our experiment. When I’d catch a lizard, I’d tip it into a tank whose lid contained a magnifying lens, and I’d watch the lizard. I didn’t trap Kater and Jonathan under a magnifying glass, but I did observe their ways. Here’s what I learned about the species experimentalus quanticus.

1) They can run experiments remotely when a pandemic shuts down campus: A year ago, when universities closed and cities locked down, I feared that our project would grind to a halt. But Jonathan twiddled knobs and read dials via his computer, and Kater popped into the lab for the occasional fixer-upper. Jonathan even continued his experiment from another state, upon moving to Texas to join his parents. And here we theorists boast of being able to do our science almost anywhere.

2) They speak with one less layer of abstraction than I: We often discussed, for instance, the thing used to measure the qubit. I’d call the thing “the detector.” Jonathan would call it “the cavity mode,” referring to the light that interacts with the qubit, which sits in a box, or cavity. I’d say “poh-tay-toe”; they’d say “poh-tah-toe”; but I’m glad we didn’t call the whole thing off.

3) Experiments take longer than expected—even if you expect them to take longer than estimated: Kater and I hatched the plan for this project during June 2018. The experiment would take a few months, Kater estimated. It terminated last summer.

4) How they explain their data: Usually in terms of decoherence, the qubit’s leaking of quantum information into its environment. For instance, to check that the setup worked properly, Jonathan ran a simple test that ended with a measurement. (Experts: He prepared a eigenstate, performed a Hadamard gate, and measured .) The measurement should have had a 50% chance of yielding and a 50% chance of yield . But the outcome dominated the trials. Why? Decoherence pushed the qubit toward toward . (Amplitude damping dominated the noise.)

5) Seeing one’s theoretical proposal turn into an experiment feels satisfying: Due to point (3), among other considerations, experiments aren’t cheap. The lab’s willingness to invest in the idea I’d developed with other theorists was heartening. Furthermore, the experiment pushed us to uncover more theory—for example, how tight the uncertainty bound could grow.

After getting to know an anole, I’d release it into our backyard and bid it adieu.1 So has Kater moved on to experimenting with topology, and Jonathan has progressed toward graduation. But more visitors are wriggling in the Butterfly Net of Theory-Experiment Collaboration. Stay tuned.

1Except for the anole I accidentally killed, by keeping it in the tank for too long. But let’s not talk about that.

The craziest challenge I’ve undertaken hasn’t been skydiving; sailing the Amazon on a homemade raft; scaling Mt. Everest; or digging for artifacts atop a hill in a Middle Eastern desert, near midday, during high summer.1 The craziest challenge has been to study the possibility that quantum phenomena affect cognition significantly. 

Most physicists agree that quantum phenomena probably don’t affect cognition significantly. Cognition occurs in biological systems, which have high temperatures, many particles, and watery components. Such conditions quash entanglement (a relationship that quantum particles can share and that can produce correlations stronger than any produceable by classical particles). 

Yet Matthew Fisher, a condensed-matter physicist, proposed a mechanism by which entanglement might enhance coordinated neuron firing. Phosphorus nuclei have spins (quantum properties similar to angular momentum) that might store quantum information for long times when in Posner molecules. These molecules may protect the information from decoherence (leaking quantum information to the environment), via mechanisms that Fisher described.

I can’t check how correct Fisher’s proposal is; I’m not a biochemist. But I’m a quantum information theorist. So I can identify how Posners could process quantum information if Fisher were correct. I undertook this task with my colleague Elizabeth Crosson, during my PhD. 

Experimentalists have begun testing elements of Fisher’s proposal. What if, years down the road, they find that Posners exist in biofluids and protect quantum information for long times? We’ll need to test whether Posners can share entanglement. But detecting entanglement tends to require control finer than you can exert with a stirring rod. How could you check whether a beakerful of particles contains entanglement?

I asked that question of Adam Bene Watts, a PhD student at MIT, and John Wright, then an MIT postdoc and now an assistant professor in Texas. John gave our project its codename. At a meeting one day, he reported that he’d watched the film Avengers: Endgame. Had I seen it? he asked.

No, I replied. The only superhero movie I’d seen recently had been Ant-Man and the Wasp—and that because, according to the film’s scientific advisor, the movie riffed on research of mine. 

Go on, said John.

Spiros Michalakis, the Caltech mathematician in charge of this blog, served as the advisor. The film came out during my PhD; during a meeting of our research group, Spiros advised me to watch the movie. There was something in it “for you,” he said. “And you,” he added, turning to Elizabeth. I obeyed, to hear Laurence Fishburne’s character tell Ant-Man that another character had entangled with the Posner molecules in Ant-Man’s brain.2 

John insisted on calling our research Project Ant-Man.

John and Adam study Bell tests. Bell test sounds like a means of checking whether the collar worn by your cat still jingles. But the test owes its name to John Stewart Bell, a Northern Irish physicist who wrote a groundbreaking paper in 1964. 

Say you’d like to check whether two particles share entanglement. You can run an experiment, described by Bell, on them. The experiment ends with a measurement of the particles. You repeat this experiment in many trials, using identical copies of the particles in subsequent trials. You accumulate many measurement outcomes, whose statistics you calculate. You plug those statistics into a formula concocted by Bell. If the result exceeds some number that Bell calculated, the particles shared entanglement.

We needed a variation on Bell’s test. In our experiment, every trial would involve hordes of particles. The experimentalists—large, clumsy, classical beings that they are—couldn’t measure the particles individually. The experimentalists could record only aggregate properties, such as the intensity of the phosphorescence emitted by a test tube.

Adam, MIT physicist Aram Harrow, and I concocted such a Bell test, with help from John. Physical Review A published our paper this month—as a Letter and an Editor’s Suggestion, I’m delighted to report.

Can you distinguish my coauthor Adam Bene Watts from John Stewart Bell?

For experts: The trick was to make the Bell correlation function nonlinear in the state. We assumed that the particles shared mostly pairwise correlations, though our Bell inequality can accommodate small aberrations. Alas, no one can guarantee that particles share only mostly pairwise correlations. Violating our Bell inequality therefore doesn’t rule out hidden-variables theories. Under reasonable assumptions, though, a not-completely-paranoid experimentalist can check for entanglement using our test. 

One can run our macroscopic Bell test on photons, using present-day technology. But we’re more eager to use the test to characterize lesser-known entities. For instance, we sketched an application to Posner molecules. Detecting entanglement in chemical systems will require more thought, as well as many headaches for experimentalists. But our paper broaches the cask—which I hope to see flow in the next Ant-Man film. Due to debut in 2022, the movie has the subtitle Quantumania. Sounds almost as crazy as studying the possibility that quantum phenomena affect cognition.

1Of those options, I’ve undertaken only the last.

2In case of any confusion: We don’t know that anyone’s brain contains Posner molecules. The movie features speculative fiction.

A college professor of mine proposed a restaurant venture to our class. He taught statistical mechanics, the physics of many-particle systems. Examples range from airplane fuel to ice cubes to primordial soup. Such systems contain 1024 particles each—so many particles that we couldn’t track them all if we tried. We can gather only a little information about the particles, so their actions look random.

So does a drunkard’s walk. Imagine a college student who (outside of the pandemic) has stayed out an hour too late and accepted one too many red plastic cups. He’s arrived halfway down a sidewalk, where he’s clutching a lamppost, en route home. Each step has a 50% chance of carrying him leftward and a 50% chance of carrying him rightward. This scenario repeats itself every Friday. On average, five minutes after arriving at the lamppost, he’s back at the lamppost. But, if we wait for a time , we have a decent chance of finding him a distance away. These characteristic typify a simple random walk.

Random walks crop up across statistical physics. For instance, consider a grain of pollen dropped onto a thin film of water. The water molecules buffet the grain, which random-walks across the film. Robert Brown observed this walk in 1827, so we call it Brownian motion. Or consider a magnet at room temperature. The magnet’s constituents don’t walk across the surface, but they orient themselves according random-walk mathematics. And, in quantum many-particle systems, information can spread via a random walk. 

So, my statistical-mechanics professor said, someone should open a restaurant near MIT. Serve lo mein and Peking duck, and call the restaurant the Random Wok.

This is the professor who, years later, confronted another alumna and me at a snack buffet.

“You know what this is?” he asked, waving a pastry in front of us. We stared for a moment, concluded that the obvious answer wouldn’t suffice, and shook our heads.

“A brownie in motion!”

Not only pollen grains undergo Brownian motion, and not only drunkards undergo random walks. Many people random-walk to their careers, trying out and discarding alternatives en route. We may think that we know our destination, but we collide with a water molecule and change course.

Such is the thrust of Random Walks, a podcast to which I contributed an interview last month. Abhigyan Ray, an undergraduate in Mumbai, created the podcast. Courses, he thought, acquaint us only with the successes in science. Stereotypes cast scientists as lone geniuses working in closed offices and silent labs. He resolved to spotlight the collaborations, the wrong turns, the lessons learned the hard way—the random walks—of science. Interviewees range from a Microsoft researcher to a Harvard computer scientist to a neurobiology professor to a genomicist.

You can find my episode on Instagram, Apple Podcasts, Google Podcasts, and Spotify. We discuss the bridging of disciplines; the usefulness of a liberal-arts education in physics; Quantum Frontiers; and the delights of poking fun at my PhD advisor, fellow blogger and Institute for Quantum Information and Matter director John Preskill. 

Young noblemen used to undertake a “Grand Tour” during the 1600s and 1700s. Many of the tourists hailed from England, though well-to-do compatriots traveled from Scandinavia, Germany, and the United States. The men had just graduated from university—in many cases, Oxford or Cambridge. They’d studied classical history, language, and literature; and now, they’d experience what they’d read. Tourists flocked to Rome, Venice, and Florence, as well as to Paris; optional additions included Naples, Switzerland, Germany, and the Netherlands.

Tutors accompanied the tourists, guiding their charges across Europe. The tutors rounded out the young men’s education, instructing them in art, music, architecture, and continental society. I felt like those tutors, this month and last.1

I was lecturing in a quantum-thermodynamics mini course, with fellow postdoctoral scholar Matteo Lostaglio. Gabriel Landi, a professor of theoretical physics at the University of São Paolo in Brazil, organized the course. It targeted early-stage graduate students, who’d mastered the core of physics and who wished to immerse in quantum thermodynamics. But the enrollment ranged from PhD and Masters students to undergraduates, postdocs, faculty members, and industry employees.

The course toured quantum thermodynamics similarly to how young noblemen toured Europe. I imagine quantum thermodynamics as a landscape—one inked on a parchment map, with blue whorls representing the sea and with a dragon breathing fire in one corner. Quantum thermodynamics encompasses many communities whose perspectives differ and who wield different mathematical and conceptual tools. These communities translate into city-states, principalities, republics, and other settlements on the map. The class couldn’t visit every city, just as Grand Tourists couldn’t. But tourists had a leg up on us in their time budgets: A Grand Tour lasted months or years, whereas we presented nine hour-and-a-half lectures.

Grand Tourists returned home with trinkets, books, paintings, and ancient artifacts. I like to imagine that the tutors, too, acquired souvenirs. Here are four of my favorite takeaways from the course:

1) Most captivating subfield that I waded into for the course: Thermodynamic uncertainty relations. Researchers have derived these inequalities using nonequilibrium statistical mechanics, a field that encompasses molecular motors, nanorobots, and single strands of DNA. Despite the name “uncertainty relations,” classical and quantum systems obey these inequalities.

Imagine a small system interacting with big systems that have different temperatures and different concentrations of particles. Energy and particles hop between the systems, dissipating entropy () and forming currents. The currents change in time, due to the probabilistic nature of statistical mechanics. 

How much does a current vary, relative to its average value, ? We quantify this variation with the relative variance, . Say that you want a low-variance, predictable current. You’ll have to pay a high entropy cost: , wherein denotes Boltzmann’s constant. 

Thermodynamic uncertainty relations govern systems arbitrarily far from equilibrium. We know loads about systems at equilibrium, in which large-scale properties remain approximately constant and no net flows (such as flows of particles) enter or leave the system. We know much about systems close to equilibrium. The regime arbitrarily far from equilibrium is the Wild, Wild West of statistical mechanics. Proving anything about this regime tends to require assumptions and specific models, to say nothing of buckets of work. But thermodynamic uncertainty relations are general, governing classical and quantum systems from molecular motors to quantum dots.

2) Most unexpected question: During lecture one, I suggested readings that introduce quantum thermodynamics. The suggestions included two reviews and the article I wrote for Scientific American about quantum steampunk, my angle on quantum thermodynamics. The next day, a participant requested recommendations of steampunk novels. I’d prepared more for requests for justifications of the steps in my derivations. But I forwarded a suggestion given to me twice: The Difference Engine, by William Gibson and Bruce Sterling.

3) Most insightful observation: My fellow tutor—I mean lecturer—pointed out how quantum thermodynamics doesn’t and does diverge from classical thermodynamics. Quantum systems can’t break the second law of thermodynamics, as classical systems can’t. Quantum engines can’t operate more efficiently than Carnot’s engine. Erasing information costs work, regardless of whether the information-bearing degree of freedom is classical or quantum. So broad results about quantum thermodynamics coincide with broad results about classical thermodynamics. We can find discrepancies by focusing on specific physical systems, such as a spring that can be classical or quantum.

4) Most staggering numbers: Unlike undertaking a Grand Tour, participating in the mini course cost nothing. We invited everyone across the world to join, and 420 participants from 48 countries enrolled. I learned of the final enrollment days before the course began, scrolling through the spreadsheet of participants. Motivated as I had been to double-check my lecture notes, the number spurred my determination like steel on a horse’s flanks.

The Grand Tour gave rise to travelogues and guidebooks read by tourists across the centuries: Mark Twain has entertained readers—partially at his own expense—since 1869 in the memoir The Innocents Abroad. British characters in the 1908 novel A Room with a View diverge in their views of Baedeker’s Handbook to Northern Italy. Our course material, and videos of the lectures, remain online and available to everyone for free. You’re welcome to pack your trunk, fetch your cloak, and join the trip.

1In addition to guiding their wards, tutors kept the young men out of trouble—and one can only imagine what trouble wealthy young men indulged in the year after college. I didn’t share that responsibility.