Author Archives: Nicole Yunger Halpern

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About Nicole Yunger Halpern

I’m a theoretical physicist at the Joint Center for Quantum Information and Computer Science in Maryland. My research group re-envisions 19th-century thermodynamics for the 21st century, using the mathematical toolkit of quantum information theory. We then apply quantum thermodynamics as a lens through which to view the rest of science. I call this research “quantum steampunk,” after the steampunk genre of art and literature that juxtaposes Victorian settings (à la thermodynamics) with futuristic technologies (à la quantum information). For more information, check out my book for the general public, Quantum Steampunk: The Physics of Yesterday’s Tomorrow. I earned my PhD at Caltech under John Preskill’s auspices; one of my life goals is to be the subject of one of his famous (if not Pullitzer-worthy) poems. Follow me on Twitter @nicoleyh11.

Astrobiology meets quantum computation?

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The origin of life appears to share little with quantum computation, apart from the difficulty of achieving it and its potential for clickbait. Yet similar notions of complexity have recently garnered attention in both fields. Each topic’s researchers expect only special systems to generate high values of such complexity, or complexity at high rates: organisms, in one community, and quantum computers (and perhaps black holes), in the other. 

Each community appears fairly unaware of its counterpart. This article is intended to introduce the two. Below, I review assembly theory from origin-of-life studies, followed by quantum complexity. I’ll then compare and contrast the two concepts. Finally, I’ll suggest that origin-of-life scientists can quantize assembly theory using quantum complexity. The idea is a bit crazy, but, well, so what?

Assembly theory in origin-of-life studies

Imagine discovering evidence of extraterrestrial life. How could you tell that you’d found it? You’d have detected a bunch of matter—a bunch of particles, perhaps molecules. What about those particles could evidence life?

This question motivated Sara Imari Walker and Lee Cronin to develop assembly theory. (Most of my assembly-theory knowledge comes from Sara, about whom I wrote this blog post years ago and with whom I share a mentor.) Assembly theory governs physical objects, from proteins to self-driving cars. 

Imagine assembling a protein from its constituent atoms. First, you’d bind two atoms together. Then, you might bind another two atoms together. Eventually, you’d bind two pairs together. Your sequence of steps would form an algorithm for assembling the protein. Many algorithms can generate the same protein. One algorithm has the least number of steps. That number is called the protein’s assembly number.

Different natural processes tend to create objects that have different assembly numbers. Stars form low-assembly-number objects by fusing two hydrogen atoms together into helium. Similarly, random processes have high probabilities of forming low-assembly-number objects. For example, geological upheavals can bring a shard of iron near a lodestone. The iron will stick to the magnetized stone, forming a two-component object.

My laptop has an enormous assembly number. Why can such an object exist? Because of information, Sara and Lee emphasize. Human beings amassed information about materials science, Boolean logic, the principles of engineering, and more. That information—which exists only because organisms exists—helped engender my laptop.

If any object has a high enough assembly number, Sara and Lee posit, that object evidences life. Absent life, natural processes have too low a probability of randomly throwing together molecules into the shape of a computer. How high is “high enough”? Approximately fifteen, experiments by Lee’s group suggest. (Why do those experiments point to the number fifteen? Sara’s group is working on a theory for predicting the number.)

In summary, assembly number quantifies complexity in origin-of-life studies, according to Sara and Lee. The researchers propose that only living beings create high-assembly-number objects.

Quantum complexity in quantum computation

Quantum complexity defines a stage in the equilibration of many-particle quantum systems. Consider a clump of N quantum particles isolated from its environment. The clump will be in a pure quantum state | \psi(0) \rangle at a time t = 0. The particles will interact, evolving the clump’s state as a function  | \psi(t) \rangle. 

Quantum many-body equilibration is more complicated than the equilibration undergone by your afternoon pick-me-up as it cools.

The interactions will equilibrate the clump internally. One stage of equilibration centers on local observables O. They’ll come to have expectation values \langle \psi(t) | O | \psi(t) \rangle approximately equal to thermal expectation values {\rm Tr} ( O \, \rho_{\rm th} ), for a thermal state \rho_{\rm th} of the clump. During another stage of equilibration, the particles correlate through many-body entanglement. 

The longest known stage centers on the quantum complexity of | \psi(t) \rangle. The quantum complexity is the minimal number of basic operations needed to prepare | \psi(t) \rangle from a simple initial state. We can define “basic operations” in many ways. Examples include quantum logic gates that act on two particles. Another example is an evolution for one time step under a Hamiltonian that couples together at most k particles, for some k independent of N. Similarly, we can define “a simple initial state” in many ways. We could count as simple only the N-fold tensor product | 0 \rangle^{\otimes N} of our favorite single-particle state | 0 \rangle. Or we could call any N-fold tensor product simple, or any state that contains at-most-two-body entanglement, and so on. These choices don’t affect the quantum complexity’s qualitative behavior, according to string theorists Adam Brown and Lenny Susskind.

How quickly can the quantum complexity of | \psi(t) \rangle grow? Fast growth stems from many-body interactions, long-range interactions, and random coherent evolutions. (Random unitary circuits exemplify random coherent evolutions: each gate is chosen according to the Haar measure, which we can view roughly as uniformly random.) At most, quantum complexity can grow linearly in time. Random unitary circuits achieve this rate. Black holes may; they scramble information quickly. The greatest possible complexity of any N-particle state scales exponentially in N, according to a counting argument. 

A highly complex state | \psi(t) \rangle looks simple from one perspective and complicated from another. Human scientists can easily measure only local observables O. Such observables’ expectation values \langle \psi(t) | O | \psi(t) \rangle  tend to look thermal in highly complex states, \langle \psi(t) | O | \psi(t) \rangle \approx {\rm Tr} ( O \, \rho_{\rm th} ), as implied above. The thermal state has the greatest von Neumann entropy, - {\rm Tr} ( \rho \log \rho), of any quantum state \rho that obeys the same linear constraints as | \psi(t) \rangle (such as having the same energy expectation value). Probed through simple, local observables O, highly complex states look highly entropic—highly random—similarly to a flipped coin.

Yet complex states differ from flipped coins significantly, as revealed by subtler analyses. An example underlies the quantum-supremacy experiment published by Google’s quantum-computing group in 2018. Experimentalists initialized 53 qubits (quantum two-level systems) in a tensor product. The state underwent many gates, which prepared a highly complex state. Then, the experimentalists measured the z-component \sigma_z of each qubit’s spin, randomly obtaining a -1 or a 1. One trial yielded a 53-bit string. The experimentalists repeated this process many times, using the same gates in each trial. From all the trials’ bit strings, the group inferred the probability p(s) of obtaining a given string s in the next trial. The distribution \{ p(s) \} resembles the uniformly random distribution…but differs from it subtly, as revealed by a cross-entropy analysis. Classical computers can’t easily generate \{ p(s) \}; hence the Google group’s claiming to have achieved quantum supremacy/advantage. Quantum complexity differs from simple randomness, that difference is difficult to detect, and the difference can evidence quantum computers’ power.

A fridge that holds one of Google’s quantum computers.

Comparison and contrast

Assembly number and quantum complexity resemble each other as follows:

  1. Each function quantifies the fewest basic operations needed to prepare something.
  2. Only special systems (organisms) can generate high assembly numbers, according to Sara and Lee. Similarly, only special systems (such as quantum computers and perhaps black holes) can generate high complexity quickly, quantum physicists expect.
  3. Assembly number may distinguish products of life from products of abiotic systems. Similarly, quantum complexity helps distinguish quantum computers’ computational power from classical computers’.
  4. High-assembly-number objects are highly structured (think of my laptop). Similarly, high-complexity quantum states are highly structured in the sense of having much many-body entanglement.
  5. Organisms generate high assembly numbers, using information. Similarly, using information, organisms have created quantum computers, which can generate quantum complexity quickly.

Assembly number and quantum complexity differ as follows:

  1. Classical objects have assembly numbers, whereas quantum states have quantum complexities.
  2. In the absence of life, random natural processes have low probabilities of producing high-assembly-number objects. That is, randomness appears to keep assembly numbers low. In contrast, randomness can help quantum complexity grow quickly.
  3. Highly complex quantum states look very random, according to simple, local probes. High-assembly-number objects do not.
  4. Only organisms generate high assembly numbers, according to Sara and Lee. In contrast, abiotic black holes may generate quantum complexity quickly.

Another feature shared by assembly-number studies and quantum computation merits its own paragraph: the importance of robustness. Suppose that multiple copies of a high-assembly-number (or moderate-assembly-number) object exist. Not only does my laptop exist, for example, but so do many other laptops. To Sara, such multiplicity signals the existence of some stable mechanism for creating that object. The multiplicity may provide extra evidence for life (including life that’s discovered manufacturing), as opposed to an unlikely sequence of random forces. Similarly, quantum computing—the preparation of highly complex states—requires stability. Decoherence threatens quantum states, necessitating quantum error correction. Quantum error correction differs from Sara’s stable production mechanism, but both evidence the importance of robustness to their respective fields.

A modest proposal

One can generalize assembly number to quantum states, using quantum complexity. Imagine finding a clump of atoms while searching for extraterrestrial life. The atoms need not have formed molecules, so the clump can have a low classical assembly number. However, the clump can be in a highly complex quantum state. We could detect the state’s complexity only (as far as I know) using many copies of the state, so imagine finding many clumps of atoms. Preparing highly complex quantum states requires special conditions, such as a quantum computer. The clump might therefore evidence organisms who’ve discovered quantum physics. Using quantum complexity, one might extend the assembly number to identify quantum states that may evidence life. However, quantum complexity, or a high rate of complexity generation, alone may not evidence life—for example, if achievable by black holes. Fortunately, a black hole seems unlikely to generate many identical copies of a highly complex quantum state. So we seem to have a low probability of mistakenly attributing a highly complex quantum state, sourced by a black hole, to organisms (atop our low probability of detecting any complex quantum state prepared by anyone other than us).

Would I expect a quantum assembly number to greatly improve humanity’s search for extraterrestrial life? I’m no astrobiology expert (NASA videos notwithstanding), but I’d expect probably not. Still, astrobiology requires chemistry, which requires quantum physics. Quantum complexity seems likely to find applications in the assembly-number sphere. Besides, doesn’t juxtaposing the search for extraterrestrial life and the understanding of life’s origins with quantum computing sound like fun? And a sense of fun distinguishes certain living beings from inanimate matter about as straightforwardly as assembly number does.

With thanks to Jim Al-Khalili, Paul Davies, the From Physics to Life collaboration, and UCLA for hosting me at the workshop that spurred this article.

May I have this dance?

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This July, I came upon a museum called the Haus der Musik in one of Vienna’s former palaces. The museum contains a room dedicated to Johann Strauss II, king of the waltz. The room, dimly lit, resembles a twilit gazebo. I could almost believe that a hidden orchestra was playing the rendition of “The Blue Danube” that filled the room. Glass cases displayed dance cards and accessories that dancers would bring to a nineteenth-century ball.

A ball. Who hasn’t read about one in a novel or seen one in a film? A throng of youngsters and their chaperones, rustling in silk. The glint of candles, the vigor of movement, the thrill of interaction, the anxiety of establishing one’s place in society.

Victoria and Albert at a ball in the film The Young Victoria

Another throng gathered a short walk from the Haus der Musik this summer. The Vienna University of Technology hosted the conference Quantum Thermodynamics (QTD) in the heart of the city. Don’t tell the other annual conferences, but QTD is my favorite. It spotlights the breed of quantum thermodynamics that’s surged throughout the past decade—the breed saturated with quantum information theory. I began attending QTD as a PhD student, and the conference shifts from city to city from year to year. I reveled in returning in person for the first time since the pandemic began.

Yet this QTD felt different. First, instead of being a PhD student, I brought a PhD student of my own. Second, granted, I enjoyed catching up with colleagues-cum-friends as much as ever. I especially relished seeing the “classmates” who belonged to my academic generation. Yet we were now congratulating each other on having founded research groups, and we were commiserating about the workload of primary investigators. 

Third, I found myself a panelist in the annual discussion traditionally called “Quo vadis, quantum thermodynamics?” The panel presented bird’s-eye views on quantum thermodynamics, analyzing trends and opining on the direction our field was taking (or should take).1 Fourth, at the end of the conference, almost the last sentence spoken into any microphone was “See you in Maryland next year.” Colleagues and I will host QTD 2024.


One of my dearest quantum-thermodynamic “classmates,” Nelly Ng, participated in the panel discussion, too. We met as students (see these two blog posts), and she’s now an assistant professor at Nanyang Technological University. Photo credit: Jakub Czartowski.

The day after QTD ended, I boarded an Austrian Airlines flight. Waltzes composed by Strauss played over the loudspeakers. They flipped a switch in my mind: I’d come of age, I thought. I’d attended QTD 2017 as a debutante, presenting my first invited talk at the conference series. I’d danced through QTD 2018 in Santa Barbara, as well as the online iterations held during the pandemic. I’d reveled in the vigor of scientific argumentation, the thrill of learning, the glint of slides shining on projector screens (not really). Now, I was beginning to shoulder responsibilities like a ballgown-wearing chaperone.

As I came of age, so did QTD. The conference series budded around the time I started grad school and embarked upon quantum-thermodynamics research. In 2017, approximately 80 participants attended QTD. This year, 250 people registered to attend in person, and others attended online. Two hundred fifty! Quantum thermodynamics scarcely existed as a field of research fifteen years ago.

I’ve heard that organizers of another annual conference, Quantum Information Processing (QIP), reacted similarly to a 250-person registration list some years ago. Aram Harrow, a professor and quantum information theorist at MIT, has shared stories about co-organizing the first QIPs. As a PhD student, he’d sat in his advisor’s office, taking notes, while the local quantum-information theorists chose submissions to highlight. Nowadays, a small army of reviewers and subreviewers processes the hordes of submissions. And, from what I heard about this year’s attendance, you almost might as well navigate a Disney theme park on a holiday as the QIP crowd. 

Will QTD continue to grow like QIP? Would such growth strengthen or fracture the community? Perhaps we’ll discuss those questions at a “Quo vadis?” session in Maryland next year. But I, at least, hope to continue always to grow—and to dance.2


Ludwig Boltzmann, a granddaddy of thermodynamics, worked in Vienna. I’ve waited for years to make a pilgrimage.

1My opinion: Now that quantum thermodynamics has showered us with fundamental insights, we should apply it in practical applications. How? Collaborators and I suggest one path here.

2I confess to having danced the waltz step (gleaned during my 14 years of ballet training) around that Strauss room in the Haus der Musik. I didn’t waltz around the conference auditorium, though.

The Book of Mark, Chapter 2

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Late in the summer of 2021, I visited a physics paradise in a physical paradise: the Kavli Institute for Theoretical Physics (KITP). The KITP sits at the edge of the University of California, Santa Barbara like a bougainvillea bush at the edge of a yard. I was eating lunch outside the KITP one afternoon, across the street from the beach. PhD student Arman Babakhani, whom a colleague had just introduced me to, had joined me.

The KITP’s Kohn Hall

What physics was I working on nowadays? Arman wanted to know.

Thermodynamic exchanges. 

The world consists of physical systems exchanging quantities with other systems. When a rose blooms outside the Santa Barbara mission, it exchanges pollen with the surrounding air. The total amount of pollen across the rose-and-air whole remains constant, so we call the amount a conserved quantity. Quantum physicists usually analyze conservation of particles, energy, and magnetization. But quantum systems can conserve quantities that participate in uncertainty relations. Such quantities are called incompatible, because you can’t measure them simultaneously. The x-, y-, and z-components of a qubit’s spin are incompatible.

The Santa Barbara mission…
…and its roses

Exchanging and conserving incompatible quantities, systems can violate thermodynamic expectations. If one system is much larger than the other, we expect the smaller system to thermalize; yet incompatibility invalidates derivations of the thermal state’s form. Incompatibility reduces the thermodynamic entropy produced by exchanges. And incompatibility can raise the average amount entanglement in the pair of systems—the total system.

If the total system conserves incompatible quantities, what happens to the eigenstate thermalization hypothesis (ETH)? Last month’s blog post overviewed the ETH, a framework for understanding how quantum many-particle systems thermalize internally. That post labeled Mark Srednicki, a professor at the KITP, a high priest of the ETH. I want, I told Arman, to ask Mark what happens when you combine the ETH with incompatible conserved quantities.

I’ll do it, Arman said.

Soon after, I found myself in the fishbowl. High up in the KITP, a room filled with cushy seats overlooks the ocean. The circular windows lend the room its nickname. Arrayed on the armchairs and couches were Mark, Arman, Mark’s PhD student Fernando Iniguez, and Mark’s recent PhD student Chaitanya Murthy. The conversation went like this:

Mark was frustrated about not being able to answer the question. I was delighted to have stumped him. Over the next several weeks, the group continued meeting, and we emailed out notes for everyone to criticize. I particulary enjoyed watching Mark and Chaitanya interact. They’d grown so intellectually close throughout Chaitanya’s PhD studies, they reminded me of an old married couple. One of them had to express only half an idea for the other to realize what he’d meant and to continue the thread. Neither had any qualms with challenging the other, yet they trusted each other’s judgment.1

In vintage KITP fashion, we’d nearly completed a project by the time Chaitanya and I left Santa Barbara. Physical Review Letters published our paper this year, and I’m as proud of it as a gardener of the first buds from her garden. Here’s what we found.

Southern California spoiled me for roses.

Incompatible conserved quantities conflict with the ETH and the ETH’s prediction of internal thermalization. Why? For three reasons. First, when inferring thermalization from the ETH, we assume that the Hamiltonian lacks degeneracies (that no energy equals any other). But incompatible conserved quantities force degeneracies on the Hamiltonian.2 

Second, when inferring from the ETH that the system thermalizes, we assume that the system begins in a microcanonical subspace. That’s an eigenspace shared by the conserved quantities (other than the Hamiltonian)—usually, an eigenspace of the total particle number or the total spin’s z-component. But, if incompatible, the conserved quantities share no eigenbasis, so they might not share eigenspaces, so microcanonical subspaces won’t exist in abundance.

Third, let’s focus on a system of N qubits. Say that the Hamiltonian conserves the total spin components S_x, S_y, and S_z. The Hamiltonian obeys the Wigner–Eckart theorem, which sounds more complicated than it is. Suppose that the qubits begin in a state | s_\alpha, \, m \rangle labeled by a spin quantum number s_\alpha and a magnetic spin quantum number m. Let a particle hit the qubits, acting on them with an operator \mathcal{O} . With what probability (amplitude) do the qubits end up with quantum numbers s_{\alpha'} and m'? The answer is \langle s_{\alpha'}, \, m' | \mathcal{O} | s_\alpha, \, m \rangle. The Wigner–Eckart theorem dictates this probability amplitude’s form. 

| s_\alpha, \, m \rangle and | s_{\alpha'}, \, m' \rangle are Hamiltonian eigenstates, thanks to the conservation law. The ETH is an ansatz for the form of \langle s_{\alpha'}, \, m' | \mathcal{O} | s_\alpha, \, m \rangle—of the elements of matrices that represent operators \mathcal{O} relative to the energy eigenbasis. The ETH butts heads with the Wigner–Eckart theorem, which also predicts the matrix element’s form.

The Wigner–Eckart theorem wins, being a theorem—a proved claim. The ETH is, as the H in the acronym relates, only a hypothesis.

If conserved quantities are incompatible, we have to kiss the ETH and its thermalization predictions goodbye. But must we set ourselves adrift entirely? Can we cling to no buoy from physics’s best toolkit for quantum many-body thermalization?

No, and yes, respectively. Our clan proposed a non-Abelian ETH for Hamiltonians that conserve incompatible quantities—or, equivalently, that have non-Abelian symmetries. The non-Abelian ETH depends on s_\alpha and on Clebsch–Gordan coefficients—conversion factors between total-spin eigenstates | s_\alpha, \, m \rangle and product states | s_1, \, m_1 \rangle \otimes | s_2, \, m_2 \rangle.

Using the non-Abelian ETH, we proved that many systems thermalize internally, despite conserving incompatible quantities. Yet the incompatibility complicates the proof enormously, extending it from half a page to several pages. Also, under certain conditions, incompatible quantities may alter thermalization. According to the conventional ETH, time-averaged expectation values \overline{ \langle \mathcal{O} \rangle }_t come to equal thermal expectation values \langle \mathcal{O} \rangle_{\rm th} to within O( N^{-1} ) corrections, as I explained last month. The correction can grow polynomially larger in the system size, to O( N^{-1/2} ), if conserved quantities are incompatible. Our conclusion holds under an assumption that we argue is physically reasonable.

So incompatible conserved quantities do alter the ETH, yet another thermodynamic expectation. Physicist Jae Dong Noh began checking the non-Abelian ETH numerically, and more testing is underway. And I’m looking forward to returning to the KITP this fall. Tales do say that paradise is a garden.

View through my office window at the KITP

1Not that married people always trust each other’s judgment.

2The reason is Schur’s lemma, a group-theoretic result. Appendix A of this paper explains the details.

The Book of Mark

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Mark Srednicki doesn’t look like a high priest. He’s a professor of physics at the University of California, Santa Barbara (UCSB); and you’ll sooner find him in khakis than in sacred vestments. Humor suits his round face better than channeling divine wrath would; and I’ve never heard him speak in tongues—although, when an idea excites him, his hands rise to shoulder height of their own accord, as though halfway toward a priestly blessing. Mark belongs less on a ziggurat than in front of a chalkboard. Nevertheless, he called himself a high priest.

Specifically, Mark jokingly called himself a high priest of the eigenstate thermalization hypothesis, a framework for understanding how quantum many-body systems thermalize internally. The eigenstate thermalization hypothesis has an unfortunate number of syllables, so I’ll call it the ETH. The ETH illuminates closed quantum many-body systems, such as a clump of N ultracold atoms. The clump can begin in a pure product state | \psi(0) \rangle, then evolve under a chaotic1 Hamiltonian H. The time-t state | \psi(t) \rangle will remain pure; its von Neumann entropy will always vanish. Yet entropy grows according to the second law of thermodynamics. Breaking the second law amounts almost to a enacting a miracle, according to physicists. Does the clump of atoms deserve consideration for sainthood?

No—although the clump’s state remains pure, a small subsystem’s state does not. A subsystem consists of, for example, a few atoms. They’ll entangle with the other atoms, which serve as an effective environment. The entanglement will mix the few atoms’ state, whose von Neumann entropy will grow.

The ETH predicts this growth. The ETH is an ansatz about H and an operator O—say, an observable of the few-atom subsystem. We can represent O as a matrix relative to the energy eigenbasis. The matrix elements have a certain structure, if O and H satisfy the ETH. Suppose that the operators do and that H lacks degeneracies—that no two energy eigenvalues equal each other. We can prove that O thermalizes: Imagine measuring the expectation value \langle \psi(t) | O | \psi(t) \rangle at each of many instants t. Averaging over instants produces the time-averaged expectation value \overline{ \langle O \rangle_t }. 

Another average is the thermal average—the expectation value of O in the appropriate thermal state. If H conserves just itself,2 the appropriate thermal state is the canonical state, \rho_{\rm can} := e^{-\beta H}/ Z. The average energy \langle \psi(0) | H | \psi(0) \rangle defines the inverse temperature \beta, and Z normalizes the state. Hence the thermal average is \langle O \rangle_{\rm th}  :=  {\rm Tr} ( O \rho_{\rm can} ). 

The time average approximately equals the thermal average, according to the ETH: \overline{ \langle O \rangle_t }  =  \langle O \rangle_{\rm th} + O \big( N^{-1} \big). The correction is small in the total number N of atoms. Through the lens of O, the atoms thermalize internally. Local observables tend to satisfy the ETH, and we can easily observe only local observables. We therefore usually observe thermalization, consistently with the second law of thermodynamics.

I agree that Mark Srednicki deserves the title high priest of the ETH. He and Joshua Deutsch independently dreamed up the ETH in 1994 and 1991. Since numericists reexamined it in 2008, studies and applications of the ETH have exploded like a desert religion. Yet Mark had never encountered the question I posed about it in 2021. Next month’s blog post will share the good news about that question.

1Nonintegrable.

2Apart from trivial quantities, such as projectors onto eigenspaces of H.

Let the great world spin

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I first heard the song “Fireflies,” by Owl City, shortly after my junior year of college. During the refrain, singer Adam Young almost whispers, “I’d like to make myself believe / that planet Earth turns slowly.” Goosebumps prickled along my neck. Yes, I thought, I’ve studied Foucault’s pendulum.

Léon Foucault practiced physics in France during the mid-1800s. During one of his best-known experiments, he hung a pendulum from high up in a building. Imagine drawing a wide circle on the floor, around the pendulum’s bob.1

Pendulum bob and encompassing circle, as viewed from above.

Imagine pulling the bob out to a point above the circle, then releasing the pendulum. The bob will swing back and forth, tracing out a straight line across the circle.

You might expect the bob to keep swinging back and forth along that line, and to do nothing more, forever (or until the pendulum has spent all its energy on pushing air molecules out of its way). After all, the only forces acting on the bob seem to be gravity and the tension in the pendulum’s wire. But the line rotates; its two tips trace out the circle.

How long the tips take to trace the circle depends on your latitude. At the North and South Poles, the tips take one day.

Why does the line rotate? Because the pendulum dangles from a building on the Earth’s surface. As the Earth rotates, so does the building, which pushes the pendulum. You’ve experienced such a pushing if you’ve ridden in a car. Suppose that the car is zipping along at a constant speed, in an unchanging direction, on a smooth road. With your eyes closed, you won’t feel like you’re moving. The only forces you can sense are gravity and the car seat’s preventing you from sinking into the ground (analogous to the wire tension that prevents the pendulum bob from crashing into the floor). If the car turns a bend, it pushes you sidewise in your seat. This push is called a centrifugal force. The pendulum feels a centrifugal force because the Earth’s rotation is an acceleration like the car’s. The pendulum also feels another force—a Coriolis force—because it’s not merely sitting, but moving on the rotating Earth.

We can predict the rotation of Foucault’s pendulum by assuming that the Earth rotates, then calculating the centrifugal and Coriolis forces induced, and then calculating how those forces will influence the pendulum’s motion. The pendulum evidences the Earth’s rotation as nothing else had before debuting in 1851. You can imagine the stir created by the pendulum when Foucault demonstrated it at the Observatoire de Paris and at the Panthéon monument. Copycat pendulums popped up across the world. One ended up next to my college’s physics building, as shown in this video. I reveled in understanding that pendulum’s motion, junior year.

Image from the Smithsonian Institution Archives.

My professor alluded to a grander Foucault pendulum in Paris. It hangs in what sounded like a temple to the Enlightenment—beautiful in form, steeped in history, and rich in scientific significance. I’m a romantic about the Enlightenment; I adore the idea of creating the first large-scale organizational system for knowledge. So I hungered to make a pilgrimage to Paris.

I made the pilgrimage this spring. I was attending a quantum-chaos workshop at the Institut Pascal, an interdisciplinary institute in a suburb of Paris. One quiet Saturday morning, I rode a train into the city center. The city houses a former priory—a gorgeous, 11th-century, white-stone affair of the sort for which I envy European cities. For over 200 years, the former priory has housed the Musée des Arts et Métiers, a museum of industry and technology. In the priory’s chapel hangs Foucault’s pendulum.2

A pendulum of Foucault’s own—the one he exhibited at the Panthéon—used to hang in the chapel. That pendulum broke in 2010; but still, the pendulum swinging today is all but a holy relic of scientific history. Foucault’s pendulum! Demonstrating that the Earth rotates! And in a jewel of a setting—flooded with light from stained-glass windows and surrounded by Gothic arches below a painted ceiling. I flitted around the little chapel like a pollen-happy bee for maybe 15 minutes, watching the pendulum swing, looking at other artifacts of Foucault’s, wending my way around the carved columns.

Almost alone. A handful of visitors trickled in and out. They contrasted with my visit, the previous weekend, to the Louvre. There, I’d witnessed a Disney World–esque line of tourists waiting for a glimpse of the Mona Lisa, camera phones held high. Nobody was queueing up in the musée’s chapel. But this was Foucault’s pendulum! Demonstrating that the Earth rotates!

I confess to capitalizing on the lack of visitors to take a photo with Foucault’s pendulum and Foucault’s Pendulum, though.

Shortly before I’d left for Paris, a librarian friend had recommended Umberto Eco’s novel Foucault’s Pendulum. It occupied me during many a train ride to or from the center of Paris.

The rest of the museum could model in an advertisement for steampunk. I found automata, models of the steam engines that triggered the Industrial Revolution, and a phonograph of Thomas Edison’s. The gadgets, many formed from brass and dark wood, contrast with the priory’s light-toned majesty. Yet the priory shares its elegance with the inventions, many of which gleam and curve in decorative flutes. 

The grand finale at the Musée des Arts et Métiers.

I tore myself away from the Musée des Arts et Métiers after several hours. I returned home a week later and heard the song “Fireflies” again not long afterward. The goosebumps returned worse. Thanks to Foucault, I can make myself believe that planet Earth turns.

With thanks to Kristina Lynch for tolerating my many, many, many questions throughout her classical-mechanics course.

This story’s title refers to a translation of Goethe’s Faust. In the translation, the demon Mephistopheles tells the title character, “You let the great world spin and riot; / we’ll nest contented in our quiet” (to within punctuational and other minor errors, as I no longer have the text with me). A prize-winning 2009 novel is called Let the Great World Spin; I’ve long wondered whether Faust inspired its title.

1Why isn’t the bottom of the pendulum called the alice?

2After visiting the musée, I learned that my classical-mechanics professor had been referring to the Foucault pendulum that hangs in the Panthéon, rather than to the pendulum in the musée. The musée still contains the pendulum used by Foucault in 1851, whereas the Panthéon has only a copy, so I’m content. Still, I wouldn’t mind making a pilgrimage to the Panthéon. Let me know if more thermodynamic workshops take place in Paris!

Winners of the Quantum-Steampunk Short-Story Contest

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During the past seven months, I’ve steamed across the Atlantic, sailed in a flying castle, teleported across the globe, and shuttled forward and backward in time. Literarily, not literally—the Quantum-Steampunk Short-Story Contest began welcoming submissions in October 2022. We challenged everybody aged 13 and over to write a steampunk narrative that involves a real or imagined quantum technology. One hundred sixty-seven entries arrived from 29 countries. Professional writers submitted stories, as did 13-year-olds. Tenured physics professors, librarians, English and math teachers, undergraduates, physicians, graduate students, and a United States Senate staffer entered. Thanks to their creativity, I now have a folder full of other worlds.

I’m over the moon (in a steam-powered ship) to announce the winners. David Wakeham received the $1,500 grand prize for the story The Creature of Ashen House. First runner-up Gerard McCaul won $1,000 for Doctor Up and Mister Down, and second runner-up Paulo Barreto won $500 for Eikonal. The People’s Choice Award ($500) went to Cristina Legarda for Pursuit, also nominated by two judges for a “Please Turn This into a Novel” award. Thanks to the 261 of you who voted in the People’s Choice competition!

In addition to traditional awards, we created four idiosyncratic ones, each entailing $250. We recognized Jeff Provine’s Stealing Buttons for its badass steampunk heroine; Matt King’s Three Imperiled Scientists for its wit and (relatedly) its portrayal of academia; Rick Searle’s The Recurrence Machine for its steampunk atmosphere; and Claudia Clarke’s Looking Forward, Looking Back, for its heart-capturing automaton. You can read all the finalist stories here.

Quantum-steampunk graphic by contest entrant Kayla Phan, who used YouChat Imagine

Sending our judges the finalists in March, I felt not only exhilaration (and relief, as whittling down 167 entries entails no little hand wringing), but also anxiety. Would the stories measure up? So I must have glowed when the first judge submitted his evaluations: Speculative-fiction author Ken Liu enthused, “The entries were so fun to read.” Similar reactions followed from across the panel, which featured experts in mathematics, philosophy, creative writing, experimental quantum physics, and history: “I had a very good time reading these stories,” another panelist wrote. “This was fun and some excellent spring break airplane (no dirigibles, I’m afraid) reading,” said another. Many thanks to our judges and short-listing committee for their input. University of Maryland undergraduates Hannah Cho and Jade Leschack led the team of students who narrowed down the candidates. I couldn’t resist treating the committee to a Victorian-inspired thank-you upon announcing the winners.

Thank-yous baked by Panera, not me

Although this year’s contest has ended, quantum-steampunk literature has just shipped out from its berth. Two contest entrants have posted their stories on their own online domains: You can read the mystery by Duke physics professor Ken Brown here and the adventure by quantum-algorithm designer Brian Siegelwax here. All other entrants, please feel free to post your stories and to submit them to other literary contests. Drop me a line, and leave a link in the chat below, when your story is published. I’d love to hear how your journey continues.

Also, stay tuned for v2.0 of the Quantum-Steampunk Short-Story Contest. An organization has expressed interest in a reboot during the 2024–2025 academic year. AI-collaboration category, anyone? Bonus points if you use a quantum neural network. Please email me if you’d like to support the effort!

Quantum-steampunk graphic by contest entrant Necklace Devkota

The opportunity to helm this contest has been a privilege and a dream. Many thanks to our writers, readers, funder (the John Templeton Foundation), staff (especially webmaster Anıl Zenginoğlu), judges, and shortlisting committee. Keep writing, and keep experimenting.

Quantum computing vs. Grubhub

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pon receiving my speaking assignments for the Tucson Festival of Books, I mentally raised my eyebrows. I’d be participating in a panel discussion with Mike Evans, the founder of Grubhub? But I hadn’t created an app that’s a household name. I hadn’t transformed 30 million people’s eating habits. I’m a theoretical physicist; I build universes in my head for a living. I could spend all day trying to prove a theorem and failing, and no stocks would tumble as a result.

Once the wave of incredulity had crested, I noticed that the panel was entitled “The Future of Tech.” Grubhub has transformed technology, I reasoned, and quantum computing is in the process of doing so. Fair enough. 

Besides, my husband pointed out, the food industry requires fridges. Physicists building quantum computers from superconductors need fridges. The latter fridges require temperatures ten million times lower than restaurateurs do, but we still share an interest.

Very well, I thought. Game on.

Tucson hosts the third-largest book festival in the United States. And why shouldn’t it, as the festival takes place in early March, when much of the country is shivering and eyeing Arizona’s T-shirt temperatures with envy? If I had to visit any institution in the winter, I couldn’t object to the festival’s home, the University of Arizona.

The day before the festival, I presented a colloquium at the university, for the Arizona Quantum Alliance. The talk took place in the Wyant College of Optical Sciences, the home of an optical-instruments museum. Many of the instruments date to the 1800s and, built from brass and wood, smack of steampunk. I approved. Outside the optics building, workers were setting up tents to house the festival’s science activities.

The next day—a Saturday—dawned clear and bright. Late in the morning, I met Mike and our panel’s moderator, Bob Griffin, another startup veteran. We sat down at a table in the back of a broad tent, the tent filled up with listeners, and the conversation began.

I relished the conversation as I’d relished an early-morning ramble along the trails by my hotel at the base of the Santa Catalina Mountains. I joined theoretical physics for the love of ideas, and this exchange of ideas offered an intellectual workout. One of Mike’s points resonated with me most: Grubhub didn’t advance technology much. He shifted consumers from ordering pizza via phone call to ordering pizza via computer, then to ordering pizza via apps on phones. Yet these small changes, accumulated across a population and encouraged by a pandemic, changed society. Food-delivery services exploded and helped establish the gig economy (despite Mike’s concerns about worker security). One small step for technology, adopted by tens of millions, can constitute one giant leap for commerce.

To me, Grubhub offered a foil for quantum computing, which offers a giant leap in technology: The physical laws best-suited to describing today’s computers can’t describe quantum computers. Some sources portray this advance as bound to transform all our lives in countless ways. This portrayal strikes some quantum scientists as hype that can endanger quality work. 

Quantum computers will transform cybersecurity, being able to break the safeguards that secure our credit-card information when we order food via Grubhub. Yet most consumers don’t know what safeguards are protecting us. We simply trust that safeguards exist. How they look under the hood will change by the time large-scale quantum computers exist—will metamorphose perhaps as dramatically as did Gregor Samsa before he woke up as an insect. But consumers’ lives might not metamorphose.

Quantum scientists hope and anticipate that quantum computers will enable discoveries in chemistry, materials science, and pharmacology. Molecules are quantum, and many materials exhibit quantum properties. Simulating quantum systems takes classical (everyday) computers copious amounts of time and memory—in some cases, so much that a classical computer the size of the universe would take ages. Quantum computers will be able to simulate quantum subjects naturally. But how these simulations will impact everyday life remains a question.

For example, consider my favorite potential application of quantum computers: fertilizer production, as envisioned by Microsoft’s quantum team. Humanity spends about 3% of the world’s energy on producing fertilizer, using a technique developed in 1909. Bacteria accomplish the same goal far more efficiently. But those bacteria use a molecule—nitrogenase—too complicated for us to understand using classical computers. Being quantum, the molecule invites quantum computation. Quantum computers may crack the molecule’s secrets and transform fertilizer production and energy use. The planet and humanity would benefit. We might reduce famines or avert human-driven natural disasters. But would the quantum computation change my neighbor’s behavior as Grubhub has? I can’t say.

Finally, evidence suggests that quantum computers can assist with optimization problems. Imagine a company that needs to transport supplies to various places at various times. How can the company optimize this process—implement it most efficiently? Quantum computers seem likely to be able to help. The evidence isn’t watertight, however, and quantum computers might not solve optimization problems exactly. If the evidence winds up correct, industries will benefit. But would this advance change Jane Doe’s everyday habits? Or will she only receive pizza deliveries a few minutes more quickly?

Don’t get me wrong; quantum technology has transformed our lives. It’s enabled the most accurate, most precise clocks in the world, which form the infrastructure behind GPS. Quantum physics has awed us, enabling the detection of gravitational waves—ripples, predicted by Einstein, in spacetime. But large-scale quantum computers—the holy grail of quantum technology—don’t suit all problems, such as totting up the miles I traveled en route to Tucson; and consumers might not notice quantum computers’ transformation of cybersecurity. I expect quantum computing to change the world, but let’s think twice about whether quantum computing will change everyone’s life like a blockbuster app.

I’ve no idea how many people have made this pun about Mike’s work, but the panel discussion left me with food for thought. He earned his undergraduate degree at MIT, by the way; so scientifically inclined Quantum Frontiers readers might enjoy his memoir, Hangry. It conveys a strong voice and dishes on data and diligence through stories. (For the best predictor of whether you’ll enjoy a burrito, ignore the starred reviews. Check how many people have reordered the burrito.)

The festival made my week. After the panel, I signed books; participated in a discussion about why “The Future Is Quantum!” with law professor Jane Bambauer; and narrowly missed a talk by Lois Lowry, a Newbury Award winner who wrote novels that I read as a child. (The auditorium filled up before I reached the door, but I’m glad that it did; Lois Lowry deserves a packed house and then some.) I learned—as I’d wondered—that yes, there’s something magical to being an author at a book festival. And I learned about how the future of tech depends on more than tech.

Memories of things past

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My best friend—who’s held the title of best friend since kindergarten—calls me the keeper of her childhood memories. I recall which toys we played with, the first time I visited her house,1 and which beverages our classmates drank during snack time in kindergarten.2 She wouldn’t be surprised to learn that the first workshop I’ve co-organized centered on memory.

Memory—and the loss of memory—stars in thermodynamics. As an example, take what my husband will probably do this evening: bake tomorrow’s breakfast. I don’t know whether he’ll bake fruit-and-oat cookies, banana muffins, pear muffins, or pumpkin muffins. Whichever he chooses, his baking will create a scent. That scent will waft across the apartment, seep into air vents, and escape into the corridor—will disperse into the environment. By tomorrow evening, nobody will be able to tell by sniffing what my husband will have baked. 

That is, the kitchen’s environment lacks a memory. This lack contributes to our experience of time’s arrow: We sense that time passes partially by smelling less and less of breakfast. Physicists call memoryless systems and processes Markovian.

A snippet from Quantum Steampunk: The Physics of Yesterday’s Tomorrow

Our kitchen’s environment is Markovian because it’s large and particles churn through it randomly. But not all environments share these characteristics. Metaphorically speaking, a dispersed memory of breakfast may recollect, return to a kitchen, and influence the following week’s baking. For instance, imagine an atom in a quantum computer, rather than a kitchen in an apartment. A few other atoms may form our atom’s environment. Quantum information may leak from our atom into that environment, swish around in the environment for a time, and then return to haunt our atom. We’d call the atom’s evolution and environment non-Markovian.

I had the good fortune to co-organize a workshop about non-Markovianity—about memory—this February. The workshop took place at the Banff International Research Station, abbreviated BIRS, which you pronounce like the plural of what you say when shivering outdoors in Canada. BIRS operates in the Banff Centre for Arts and Creativity, high in the Rocky Mountains. The Banff Centre could accompany a dictionary entry for pristine, to my mind. The air feels crisp, the trees on nearby peaks stand out against the snow like evergreen fringes on white velvet, and the buildings balance a rustic-mountain-lodge style with the avant-garde. 

The workshop balanced styles, too, but skewed toward the theoretical and abstract. We learned about why the world behaves classically in our everyday experiences; about information-theoretic measures of the distances between quantum states; and how to simulate, on quantum computers, chemical systems that interact with environments. One talk, though, brought our theory back down to (the snow-dusted) Earth.

Gabriela Schlau-Cohen runs a chemistry lab at MIT. She wants to understand how plants transport energy. Energy arrives at a plant from the sun in the form of light. The light hits a pigment-and-protein complex. If the plant is lucky, the light transforms into a particle-like packet of energy called an exciton. The exciton traverses the receptor complex, then other complexes. Eventually, the exciton finds a spot where it can enable processes such as leaf growth. 

A high fraction of the impinging photons—85%—transform into excitons. How do plants convert and transport energy as efficiently as they do?

Gabriela’s group aims to find out—not by testing natural light-harvesting complexes, but by building complexes themselves. The experimentalists mimic the complex’s protein using DNA. You can fold DNA into almost any shape you want, by choosing the DNA’s base pairs (basic units) adroitly and by using “staples” formed from more DNA scraps. The sculpted molecules are called DNA origami.

Gabriela’s group engineers different DNA structures, analogous to complexes’ proteins, to have different properties. For instance, the experimentalists engineer rigid structures and flexible structures. Then, the group assesses how energy moves through each structure. Each structure forms an environment that influences excitons’ behaviors, similarly to how a memory-containing environment influences an atom.

Courtesy of Gabriela Schlau-Cohen

The Banff environment influenced me, stirring up memories like powder displaced by a skier on the slopes above us. I first participated in a BIRS workshop as a PhD student, and then I returned as a postdoc. Now, I was co-organizing a workshop to which I brought a PhD student of my own. Time flows, as we’re reminded while walking down the mountain from the Banff Centre into town: A cemetery borders part of the path. Time flows, but we belong to that thermodynamically remarkable class of systems that retain memories…memories and a few other treasures that resist change, such as friendships held since kindergarten.

1Plushy versions of Simba and Nala from The Lion King. I remain grateful to her for letting me play at being Nala.

2I’d request milk, another kid would request apple juice, and everyone else would request orange juice.

A (quantum) complex legacy: Part deux

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I didn’t fancy the research suggestion emailed by my PhD advisor.

A 2016 email from John Preskill led to my publishing a paper about quantum complexity in 2022, as I explained in last month’s blog post. But I didn’t explain what I thought of his email upon receiving it.

It didn’t float my boat. (Hence my not publishing on it until 2022.)

The suggestion contained ingredients that ordinarily would have caulked any cruise ship of mine: thermodynamics, black-hole-inspired quantum information, and the concept of resources. John had forwarded a paper drafted by Stanford physicists Adam Brown and Lenny Susskind. They act as grand dukes of the community sussing out what happens to information swallowed by black holes. 

From Rare-Gallery

We’re not sure how black holes work. However, physicists often model a black hole with a clump of particles squeezed close together and so forced to interact with each other strongly. The interactions entangle the particles. The clump’s quantum state—let’s call it | \psi(t) \rangle—grows not only complicated with time (t), but also complex in a technical sense: Imagine taking a fresh clump of particles and preparing it in the state | \psi(t) \rangle via a sequence of basic operations, such as quantum gates performable with a quantum computer. The number of basic operations needed is called the complexity of | \psi(t) \rangle. A black hole’s state has a complexity believed to grow in time—and grow and grow and grow—until plateauing. 

This growth echoes the second law of thermodynamics, which helps us understand why time flows in only one direction. According to the second law, every closed, isolated system’s entropy grows until plateauing.1 Adam and Lenny drew parallels between the second law and complexity’s growth.

The less complex a quantum state is, the better it can serve as a resource in quantum computations. Recall, as we did last month, performing calculations in math class. You needed clean scratch paper on which to write the calculations. So does a quantum computer. “Scratch paper,” to a quantum computer, consists of qubits—basic units of quantum information, realized in, for example, atoms or ions. The scratch paper is “clean” if the qubits are in a simple, unentangled quantum state—a low-complexity state. A state’s greatest possible complexity, minus the actual complexity, we can call the state’s uncomplexity. Uncomplexity—a quantum state’s blankness—serves as a resource in quantum computation.

Manny Knill and Ray Laflamme realized this point in 1998, while quantifying the “power of one clean qubit.” Lenny arrived at a similar conclusion while reasoning about black holes and firewalls. For an introduction to firewalls, see this blog post by John. Suppose that someone—let’s call her Audrey—falls into a black hole. If it contains a firewall, she’ll burn up. But suppose that someone tosses a qubit into the black hole before Audrey falls. The qubit kicks the firewall farther away from the event horizon, so Audrey will remain safe for longer. Also, the qubit increases the uncomplexity of the black hole’s quantum state. Uncomplexity serves as a resource also to Audrey.

A resource is something that’s scarce, valuable, and useful for accomplishing tasks. Different things qualify as resources in different settings. For instance, imagine wanting to communicate quantum information to a friend securely. Entanglement will serve as a resource. How can we quantify and manipulate entanglement? How much entanglement do we need to perform a given communicational or computational task? Quantum scientists answer such questions with a resource theory, a simple information-theoretic model. Theorists have defined resource theories for entanglement, randomness, and more. In many a blog post, I’ve eulogized resource theories for thermodynamic settings. Can anyone define, Adam and Lenny asked, a resource theory for quantum uncomplexity?

Resource thinking pervades our world.

By late 2016, I was a quantum thermodynamicist, I was a resource theorist, and I’d just debuted my first black-hole–inspired quantum information theory. Moreover, I’d coauthored a review about the already-extant resource theory that looked closest to what Adam and Lenny sought. Hence John’s email, I expect. Yet that debut had uncovered reams of questions—questions that, as a budding physicist heady with the discovery of discovery, I could own. Why would I answer a question of someone else’s instead?

So I thanked John, read the paper draft, and pondered it for a few days. Then, I built a research program around my questions and waited for someone else to answer Adam and Lenny.

Three and a half years later, I was still waiting. The notion of uncomplexity as a resource had enchanted the black-hole-information community, so I was preparing a resource-theory talk for a quantum-complexity workshop. The preparations set wheels churning in my mind, and inspiration struck during a long walk.2

After watching my workshop talk, Philippe Faist reached out about collaborating. Philippe is a coauthor, a friend, and a fellow quantum thermodynamicist and resource theorist. Caltech’s influence had sucked him, too, into the black-hole community. We Zoomed throughout the pandemic’s first spring, widening our circle to include Teja Kothakonda, Jonas Haferkamp, and Jens Eisert of Freie University Berlin. Then, Anthony Munson joined from my nascent group in Maryland. Physical Review A published our paper, “Resource theory of quantum uncomplexity,” in January.

The next four paragraphs, I’ve geared toward experts. An agent in the resource theory manipulates a set of n qubits. The agent can attempt to perform any gate U on any two qubits. Noise corrupts every real-world gate implementation, though. Hence the agent effects a gate chosen randomly from near U. Such fuzzy gates are free. The agent can’t append or discard any system for free: Appending even a maximally mixed qubit increases the state’s uncomplexity, as Knill and Laflamme showed. 

Fuzzy gates’ randomness prevents the agent from mapping complex states to uncomplex states for free (with any considerable probability). Complexity only grows or remains constant under fuzzy operations, under appropriate conditions. This growth echoes the second law of thermodynamics. 

We also defined operational tasks—uncomplexity extraction and expenditure analogous to work extraction and expenditure. Then, we bounded the efficiencies with which the agent can perform these tasks. The efficiencies depend on a complexity entropy that we defined—and that’ll star in part trois of this blog-post series.

Now, I want to know what purposes the resource theory of uncomplexity can serve. Can we recast black-hole problems in terms of the resource theory, then leverage resource-theory results to solve the black-hole problem? What about problems in condensed matter? Can our resource theory, which quantifies the difficulty of preparing quantum states, merge with the resource theory of magic, which quantifies that difficulty differently?

Unofficial mascot for fuzzy operations

I don’t regret having declined my PhD advisor’s recommendation six years ago. Doing so led me to explore probability theory and measurement theory, collaborate with two experimental labs, and write ten papers with 21 coauthors whom I esteem. But I take my hat off to Adam and Lenny for their question. And I remain grateful to the advisor who kept my goals and interests in mind while checking his email. I hope to serve Anthony and his fellow advisees as well.

1…en route to obtaining a marriage license. My husband and I married four months after the pandemic throttled government activities. Hours before the relevant office’s calendar filled up, I scored an appointment to obtain our license. Regarding the metro as off-limits, my then-fiancé and I walked from Cambridge, Massachusetts to downtown Boston for our appointment. I thank him for enduring my requests to stop so that I could write notes.

2At least, in the thermodynamic limit—if the system is infinitely large. If the system is finite-size, its entropy grows on average.

A (quantum) complex legacy

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Early in the fourth year of my PhD, I received a most John-ish email from John Preskill, my PhD advisor. The title read, “thermodynamics of complexity,” and the message was concise the way that the Amazon River is damp: “Might be an interesting subject for you.” 

Below the signature, I found a paper draft by Stanford physicists Adam Brown and Lenny Susskind. Adam is a Brit with an accent and a wit to match his Oxford degree. Lenny, known to the public for his books and lectures, is a New Yorker with an accent that reminds me of my grandfather. Before the physicists posted their paper online, Lenny sought feedback from John, who forwarded me the email.

The paper concerned a confluence of ideas that you’ve probably encountered in the media: string theory, black holes, and quantum information. String theory offers hope for unifying two physical theories: relativity, which describes large systems such as our universe, and quantum theory, which describes small systems such as atoms. A certain type of gravitational system and a certain type of quantum system participate in a duality, or equivalence, known since the 1990s. Our universe isn’t such a gravitational system, but never mind; the duality may still offer a toehold on a theory of quantum gravity. Properties of the gravitational system parallel properties of the quantum system and vice versa. Or so it seemed.

The gravitational system can have two black holes linked by a wormhole. The wormhole’s volume can grow linearly in time for a time exponentially long in the black holes’ entropy. Afterward, the volume hits a ceiling and approximately ceases changing. Which property of the quantum system does the wormhole’s volume parallel?

Envision the quantum system as many particles wedged close together, so that they interact with each other strongly. Initially uncorrelated particles will entangle with each other quickly. A quantum system has properties, such as average particle density, that experimentalists can measure relatively easily. Does such a measurable property—an observable of a small patch of the system—parallel the wormhole volume? No; such observables cease changing much sooner than the wormhole volume does. The same conclusion applies to the entanglement amongst the particles.

What about a more sophisticated property of the particles’ quantum state? Researchers proposed that the state’s complexity parallels the wormhole’s volume. To grasp complexity, imagine a quantum computer performing a computation. When performing computations in math class, you needed blank scratch paper on which to write your calculations. A quantum computer needs the quantum equivalent of blank scratch paper: qubits (basic units of quantum information, realized, for example, as atoms) in a simple, unentangled, “clean” state. The computer performs a sequence of basic operations—quantum logic gates—on the qubits. These operations resemble addition and subtraction but can entangle the qubits. What’s the minimal number of basic operations needed to prepare a desired quantum state (or to “uncompute” a given state to the blank state)? The state’s quantum complexity.1 

Quantum complexity has loomed large over multiple fields of physics recently: quantum computing, condensed matter, and quantum gravity. The latter, we established, entails a duality between a gravitational system and a quantum system. The quantum system begins in a simple quantum state that grows complicated as the particles interact. The state’s complexity parallels the volume of a wormhole in the gravitational system, according to a conjecture.2 

The conjecture would hold more water if the quantum state’s complexity grew similarly to the wormhole’s volume: linearly in time, for a time exponentially large in the quantum system’s size. Does the complexity grow so? The expectation that it does became the linear-growth conjecture.

Evidence supported the conjecture. For instance, quantum information theorists modeled the quantum particles as interacting randomly, as though undergoing a quantum circuit filled with random quantum gates. Leveraging probability theory,3 the researchers proved that the state’s complexity grows linearly at short times. Also, the complexity grows linearly for long times if each particle can store a great deal of quantum information. But what if the particles are qubits, the smallest and most ubiquitous unit of quantum information? The question lingered for years.

Jonas Haferkamp, a PhD student in Berlin, dreamed up an answer to an important version of the question.4 I had the good fortune to help formalize that answer with him and members of his research group: master’s student Teja Kothakonda, postdoc Philippe Faist, and supervisor Jens Eisert. Our paper, published in Nature Physics last year, marked step one in a research adventure catalyzed by John Preskill’s email 4.5 years earlier.

Imagine, again, qubits undergoing a circuit filled with random quantum gates. That circuit has some architecture, or arrangement of gates. Slotting different gates into the architecture effects different transformations5 on the qubits. Consider the set of all transformations implementable with one architecture. This set has some size, which we defined and analyzed.

What happens to the set’s size if you add more gates to the circuit—let the particles interact for longer? We can bound the size’s growth using the mathematical toolkits of algebraic geometry and differential topology. Upon bounding the size’s growth, we can bound the state’s complexity. The complexity, we concluded, grows linearly in time for a time exponentially long in the number of qubits.

Our result lends weight to the complexity-equals-volume hypothesis. The result also introduces algebraic geometry and differential topology into complexity as helpful mathematical toolkits. Finally, the set size that we bounded emerged as a useful concept that may elucidate circuit analyses and machine learning.

John didn’t have machine learning in mind when forwarding me an email in 2017. He didn’t even have in mind proving the linear-growth conjecture. The proof enables step two of the research adventure catalyzed by that email: thermodynamics of quantum complexity, as the email’s title stated. I’ll cover that thermodynamics in its own blog post. The simplest of messages can spin a complex legacy.

The links provided above scarcely scratch the surface of the quantum-complexity literature; for a more complete list, see our paper’s bibliography. For a seminar about the linear-growth paper, see this video hosted by Nima Lashkari’s research group.

1The term complexity has multiple meanings; forget the rest for the purposes of this article.

2According to another conjecture, the quantum state’s complexity parallels a certain space-time region’s action. (An action, in physics, isn’t a motion or a deed or something that Hamlet keeps avoiding. An action is a mathematical object that determines how a system can and can’t change in time.) The first two conjectures snowballed into a paper entitled “Does complexity equal anything?” Whatever it parallels, complexity plays an important role in the gravitational–quantum duality. 

3Experts: Such as unitary t-designs.

4Experts: Our work concerns quantum circuits, rather than evolutions under fixed Hamiltonians. Also, our work concerns exact circuit complexity, the minimal number of gates needed to prepare a state exactly. A natural but tricky extension eluded us: approximate circuit complexity, the minimal number of gates needed to approximate the state.

5Experts: Unitary operators.