Taming wave functions with neural networks

Note from Nicole Yunger Halpern: One sunny Saturday this spring, I heard Sam Greydanus present about his undergraduate thesis. Sam was about to graduate from Dartmouth with a major in physics. He had worked with quantum-computation theorist Professor James Whitfield. The presentation — about applying neural networks to quantum computation — so intrigued me that I asked him to share his research on Quantum Frontiers. Sam generously agreed; this is his story.

Wave functions in the wild


The wave function, \psi , is a mixed blessing. At first, it causes unsuspecting undergrads (me) some angst via the Schrodinger’s cat paradox. This angst morphs into full-fledged panic when they encounter concepts such as nonlocality and Bell’s theorem (which, by the way, is surprisingly hard to verify experimentally). The real trouble with \psi , though, is that it grows exponentially with the number of entangled particles in a system. We couldn’t even hope to write the wavefunction of 100 entangled particles, much less perform computations on it…but there’s a lot to gain from doing just that.

The thing is, we (a couple of luckless physicists) love \psi . Manipulating wave functions can give us ultra-precise timekeeping, secure encryption, and polynomial-time factoring of integers (read: break RSA). Harnessing quantum effects can also produce better machine learning, better physics simulations, and even quantum teleportation.

Taming the beast

Though \psi grows exponentially with the number of particles in a system, most physical wave functions can be described with a lot less information. Two algorithms for doing this are the Density Matrix Renormalization Group (DMRG) and Quantum Monte Carlo (QMC).


Density Matrix Renormalization Group (DMRG). Imagine we want to learn about trees, but studying a full-grown, 50-foot tall tree in the lab is too unwieldy. One idea is to keep the tree small, like a bonsai tree. DMRG is an algorithm which, like a bonsai gardener, prunes the wave function while preserving its most important components. It produces a compressed version of the wave function called a Matrix Product State (MPS). One issue with DMRG is that it doesn’t extend particularly well to 2D and 3D systems.

Screen Shot 2017-07-29 at 12.01.23 AM

Quantum Monte Carlo (QMC). Another way to study the concept of “tree” in a lab (bear with me on this metaphor) would be to study a bunch of leaf, seed, and bark samples. Quantum Monte Carlo algorithms do this with wave functions, taking “samples” of a wave function (pure states) and using the properties and frequencies of these samples to build a picture of the wave function as a whole. The difficulty with QMC is that it treats the wave function as a black box. We might ask, “how does flipping the spin of the third electron affect the total energy?” and QMC wouldn’t have much of a physical answer.

Brains \gg Brawn

Neural Quantum States (NQS). Some state spaces are far too large for even Monte Carlo to sample adequately. Suppose now we’re studying a forest full of different species of trees. If one type of tree vastly outnumbers the others, choosing samples from random trees isn’t an efficient way to map biodiversity. Somehow, we need to make the sampling process “smarter”. Last year, Google DeepMind used a technique called deep reinforcement learning to do just that – and achieved fame for defeating the world champion human Go player. A recent Science paper by Carleo and Troyer (2017) used the same technique to make QMC “smarter” and effectively compress wave functions with neural networks. This approach, called “Neural Quantum States (NQS)”, produced several state-of-the-art results.


The general idea of my thesis.

My thesis. My undergraduate thesis centered upon much the same idea. In fact, I had to abandon some of my initial work after reading the NQS paper. I then focused on using machine learning techniques to obtain MPS coefficients. Like Carleo and Troyer, I used neural networks to approximate  \psi . Unlike Carleo and Troyer, I trained my model to output a set of Matrix Product State coefficients which have physical meaning (MPS coefficients always correspond to a certain state and site, e.g. “spin up, electron number 3”).

Cool – but does it work?

Yes – for small systems. In my thesis, I considered a toy system of 4 spin-\frac{1}{2} particles interacting via the Heisenberg Hamiltonian. Solving this system is not difficult so I was able to focus on fitting the two disparate parts – machine learning and Matrix Product States – together.

Success! My model solved for ground states with arbitrary precision. Even more interestingly, I used it to automatically obtain MPS coefficients. Shown below, for example, is a visualization of my model’s coefficients for the GHZ state, compared with coefficients taken from the literature.

Screen Shot 2017-07-28 at 11.46.45 PM

A visual comparison of a 4-site Matrix Product State for the GHZ state a) listed in the literature b) obtained from my neural network model. Colored squares correspond to real-valued elements of 2×2 matrices.

Limitations. The careful reader might point out that, according to the schema of my model (above), I still have to write out the full wave function. To scale my model up, I instead trained it variationally over a subspace of the Hamiltonian (just as the authors of the NQS paper did). Results are decent for larger (10-20 particle) systems, but the training itself is still unstable. I’ll finish ironing out the details soon, so keep an eye on arXiv* :).

Outside the ivory tower


A quantum computer developed by Joint Quantum Institute, U. Maryland.

Quantum computing is a field that’s poised to take on commercial relevance. Taming the wave function is one of the big hurdles we need to clear before this happens. Hopefully my findings will have a small role to play in making this happen.

On a more personal note, thank you for reading about my work. As a recent undergrad, I’m still new to research and I’d love to hear constructive comments or criticisms. If you found this post interesting, check out my research blog.

*arXiv is an online library for electronic preprints of scientific papers

The sign problem(s)

The thirteen-month-old had mastered the word “dada” by the time I met her. Her parents were teaching her to communicate other concepts through sign language. Picture her, dark-haired and bibbed, in a high chair. Banana and mango slices litter the tray in front of her. More fruit litters the floor in front of the tray. The baby lifts her arms and flaps her hands.

Dada looks up from scrubbing the floor.

“Look,” he calls to Mummy, “she’s using sign language! All done.” He performs the gesture that his daughter seems to have aped: He raises his hands and rotates his forearms about his ulnas, axes perpendicular to the floor. “All done!”

The baby looks down, seizes another morsel, and stuffs it into her mouth.

“Never mind,” Dada amends. “You’re not done, are you?”

His daughter had a sign(-language) problem.


So does Dada, MIT professor Aram Harrow. Aram studies quantum information theory. His interests range from complexity to matrices, from resource theories to entropies. He’s blogged for The Quantum Pontiff, and he studies—including with IQIM postdoc Elizabeth Crossonthe quantum sign problem.

Imagine calculating properties of a chunk of fermionic quantum matter. The chunk consists of sites, each inhabited by one particle or by none. Translate as “no site can house more than one particle” the jargon “the particles are fermions.”

The chunk can have certain amounts of energy. Each amount E_j corresponds to some particle configuration indexed by j: If the system has some amount E_1 of energy, particles occupy certain sites and might not occupy others. If the system has a different amount E_2 \neq E_1 of energy, particles occupy different sites. A Hamiltonian, a mathematical object denoted by H, encodes the energies E_j and the configurations. We represent H with a matrix, a square grid of numbers.

Suppose that the chunk has a temperature T = \frac{ 1 }{ k_{\rm B} \beta }. We could calculate the system’s heat capacity, the energy required to raise the chunk’s temperature by one Kelvin. We could calculate the free energy, how much work the chunk could perform in powering a motor or lifting a weight. To calculate those properties, we calculate the system’s partition function, Z.

How? We would list the configurations j. With each configuration, we would associate the weight e^{ - \beta E_j }. We would sum the weights: Z = e^{ - \beta E_1 }  +  e^{ - \beta E_2}  +  \ldots  =  \sum_j e^{ - \beta E_j}.

Easier—like feeding a 13-month-old—said than done. Let N denote the number of qubits in the chunk. If N is large, the number of configurations is gigantic. Our computers can’t process so many configurations. This inability underlies quantum computing’s promise of speeding up certain calculations.

We don’t have quantum computers, and we can’t calculate Z. Can we  approximate Z?

Yes, if H “lacks the sign problem.” The math that models our system models also a classical system. If our system has D dimensions, the classical system has D+1 dimensions. Suppose, for example, that our sites form a line. The classical system forms a square.

We replace the weights e^{ - \beta E_j } with different weights—numbers formed from a matrix that represents H. If H lacks the sign problem, the new weights are nonnegative and behave like probabilities. Many mathematical tools suit probabilities. Aram and Elizabeth apply such tools to Z, here and here, as do many other researchers.

We call Hamiltonians that lack the sign problem “stoquastic,” which I think fanquastic.Stay tuned for a blog post about stoquasticity by Elizabeth.

What if H has the sign problem? The new weights can assume negative and nonreal values. The weights behave unlike probabilities; we can’t apply those tools. We find ourselves knee-deep in banana and mango chunks.

Mango chunks

Solutions to the sign problem remain elusive. Theorists keep trying to mitigate the problem, though. Aram, Elizabeth, and others are improving calculations of properties of sign-problem-free systems. One scientist-in-the-making has achieved a breakthrough: Aram’s daughter now rotates her hands upon finishing meals and when she wants to leave her car seat or stroller.

One sign problem down; one to go.


With gratitude to Aram’s family for its hospitality and to Elizabeth Crosson for sharing her expertise.

1For experts: A local Hamiltonian is stoquastic relative to the computational basis if each local term is represented, relative to the computational basis, by a matrix whose off-diagonal entries are real and nonpositive.

The world of hackers and secrets

I’m Evgeny Mozgunov, and some of you may remember my earlier posts on Quantum Frontiers. I’ve recently graduated with a PhD after 6 years in the quantum information group at Caltech. As I’m navigating the job market in quantum physics, it was only a matter of time before I got dragged into a race between startups. Those who can promise impressive quantum speedups for practical tasks get a lot of money from venture capitalists. Maybe there’s something about my mind and getting paid: when I’m paid to do something, I suddenly start coming up with solutions that never occurred to me while I was wandering around as a student. And this time, I’ve noticed a possibility of impressing the public with quantum speedups that nobody has ever used before.

Three former members of John Preskill’s group, Gorjan Alagic, Stacey Jeffery and Stephen Jordan, have already proposed this idea (Circuit Obfuscation Using Braids, p.10), but none of the startups seems to have picked it up. You only need a small quantum computer. Imagine you are in the audience. I ask you to come up with a number. Don’t tell it out loud: instead, write it on a secret piece of paper, and take a little time to do a few mathematical operations based on the number. Then announce the result of those operations. Once you are done, people will automatically be split into two categories. Those with access to a small quantum computer (like the one at IBM) will be able to put on a magic hat (the computer…) and recover your number. But the rest of the audience will be left in awe, with no clue as to how this is even possible. There’s nothing they could do to guess your number based only on the result you announced, unless you’re willing to wait for a few days and they have access to the world’s supercomputing powers.

So far I’ve described the general setting of encryption – a cipher is announced, the key to the cipher is destroyed, and only those who can break the code can decipher.  For instance, if RSA encryption is used for the magic show above, indeed only people with a big quantum computer will be able to recover the secret number. To complete my story, I need to describe what the result that you announce (the cipher) looks like:

A sequence of instructions for a small quantum computer that is equivalent to a simple instruction for spitting out your number. However, the announced sequence of instructions is obfuscated, such that you can’t just read off the number from it.

You really need to feed the sequence into a quantum computer, and see what it outputs. Obfuscation is more general than encryption, but here we’re going to use it as a method of encryption.

Alagic et al. taught us how to do something called obfuscation by compiling for a quantum computer: much like when you compile a .c file in your CS class, you can’t really understand the .out file. Of course you can just execute the .out file, but not if it describes a quantum circuit, unless you have access to a quantum computer. The proposed classical compiler turns either a classical or a quantum algorithm into a hard-to-read quantum circuit that looks like braidsBraid_1000.gif. Unfortunately, any obfuscation by compiling scheme has the problem that whoever understands the compiler well enough will be able to actually read the .out file (or notice a pattern in braids reduced to a compact “normal” form), and guess your number without resorting to a quantum computer. Surprisingly, even though Alagic et al.’s scheme doesn’t claim any protection under this attack, it still satisfies one of the theoretical definitions of obfuscation: if two people write two different sets of instructions to perform the same operation, and then each obfuscate their own set of instructions by a restricted set of tricks, then it should be impossible to tell from the end result which program was obtained by whom.


Theoretical obfuscation can be illustrated by these video game Nier cosplayers: when they put on their wig and blindfold, they look like the same person. The character named 2B is an android, whose body is disposable, and whose mind is a set of instructions stored on a server. Other characters try to hack her mind as the story progresses.

Quantum scientists can have their own little world of hackers and secrets, organized in the following way: some researchers present their obfuscated code outputting a secret message, and other researchers become hackers trying to break it. Thanks to another result by Alagic et al, we know that hard-to-break obfuscated circuits secure against classical computers exist. But we don’t know how the obfuscator that produces those worst-case instances reliably looks like, so a bit of crowdsourcing to find it is in order. It’s a free-for-all, where all tools and tricks are allowed. In fact, even you can enter! All you need to know is a universal gate set H,T = R(π/4),CNOT and good old matrix multiplication. Come up with a product of these matrices that multiplies to a bunch of X‘s (X=HT⁴H), but such that only you know on which qubits the X are applied. This code will spit out your secret bitstring on an input of all 0’es. Publish it and wait until some hacker breaks it!

Here’s mine, can anyone see what’s my secret bitstring?

Obfuscated circuit.png

One can run it on a 5 qubit quantum computer in less than 1ms. But if you try to multiply the corresponding 32×32 matrices on your laptop, it takes more than 1ms. Quantum speedup right there. Of course I didn’t prove that there’s no better way of finding out my secret than multiplying matrices. In fact, had I used only even powers of the matrix T in the picture above, then there is a classical algorithm available in open source (Aaronson, Gottesman) that recovers the number without having to multiply large matrices.

I’m in luck: startups and venture capitalists never cared about theoretical proofs, it only has to work until it fails. I think they should give millions to me instead of D-wave. Seriously, there’s plenty of applications for practical obfuscation, besides magic shows. One can set up a social network where posts are gibberish except for those who have a quantum computer (that would be a good conspiracy theory some years from now). One can verify when a private company claims to sell a small quantum computer.

I’d like to end on a more general note: small quantum computers are already faster than classical hardware at multiplying certain kinds of matrices. This has already been proven for a restricted class of quantum computers and a task called boson sampling. If there’s a competition in matrix multiplication somewhere in the world, we can already win.

The power of information

Sara Imari Walker studies ants. Her entomologist colleague Gabriele Valentini cultivates ant swarms. Gabriele coaxes a swarm from its nest, hides the nest, and offers two alternative nests. Gabriele observe the ants’ responses, then analyzes their data with Sara.

Sara doesn’t usually study ants. She trained in physics, information theory, and astrobiology. (Astrobiology is the study of life; life’s origins; and conditions amenable to life, on Earth and anywhere else life may exist.) Sara analyzes how information reaches, propagates through, and manifests in the swarm.

Some ants inspect one nest; some, the other. Few ants encounter both choices. Yet most of the ants choose simultaneously. (How does Gabriele know when an ant chooses? Decided ants carry other ants toward the chosen nest. Undecided ants don’t.)

Gabriele and Sara plotted each ant’s status (decided or undecided) at each instant. All the ants’ lines start in the “undecided” region, high up in the graph. Most lines drop to the “decided” region together. Physicists call such dramatic, large-scale changes in many-particle systems “phase transitions.” The swarm transitions from the “undecided” phase to the “decided,” as moisture transitions from vapor to downpour.

Sara presentation

Sara versus the ants

Look from afar, and you’ll see evidence of a hive mind: The lines clump and slump together. Look more closely, and you’ll find lags between ants’ decisions. Gabriele and Sara grouped the ants according to their behaviors. Sara explained the grouping at a workshop this spring.

The green lines, she said, are undecided ants.

My stomach dropped like Gabriele and Sara’s ant lines.

People call data “cold” and “hard.” Critics lambast scientists for not appealing to emotions. Politicians weave anecdotes into their numbers, to convince audiences to care.

But when Sara spoke, I looked at her green lines and thought, “That’s me.”

I’ve blogged about my indecisiveness. Postdoc Ning Bao and I formulated a quantum voting scheme in which voters can superpose—form quantum combinations of—options. Usually, when John Preskill polls our research group, I abstain from voting. Politics, and questions like “Does building a quantum computer require only engineering or also science?”,1 have many facets. I want to view such questions from many angles, to pace around the questions as around a sculpture, to hear other onlookers, to test my impressions on them, and to cogitate before choosing.2 However many perspectives I’ve gathered, I’m missing others worth seeing. I commiserated with the green-line ants.


I first met Sara in the building behind the statue. Sara earned her PhD in Dartmouth College’s physics department, with Professor Marcelo Gleiser.

Sara presented about ants at a workshop hosted by the Beyond Center for Fundamental Concepts in Science at Arizona State University (ASU). The organizers, Paul Davies of Beyond and Andrew Briggs of Oxford, entitled the workshop “The Power of Information.” Participants represented information theory, thermodynamics and statistical mechanics, biology, and philosophy.

Paul and Andrew posed questions to guide us: What status does information have? Is information “a real thing” “out there in the world”? Or is information only a mental construct? What roles can information play in causation?

We paced around these questions as around a Chinese viewing stone. We sat on a bench in front of those questions, stared, debated, and cogitated. We taught each other about ants, artificial atoms, nanoscale machines, and models for information processing.


Chinese viewing stone in Yuyuan Garden in Shanghai

I wonder if I’ll acquire opinions about Paul and Andrew’s questions. Maybe I’ll meander from “undecided” to “decided” over a career. Maybe I’ll phase-transition like Sara’s ants. Maybe I’ll remain near the top of her diagram, a green holdout.

I know little about information’s power. But Sara’s plot revealed one power of information: Information can move us—from homeless to belonging, from ambivalent to decided, from a plot’s top to its bottom, from passive listener to finding yourself in a green curve.


With thanks to Sara Imari Walker, Paul Davies, Andrew Briggs, Katherine Smith, and the Beyond Center for their hospitality and thoughts.


1By “only engineering,” I mean not “merely engineering” pejoratively, but “engineering and no other discipline.”

2I feel compelled to perform these activities before choosing. I try to. Psychological experiments, however, suggest that I might decide before realizing that I’ve decided.

Glass beads and weak-measurement schemes

Richard Feynman fiddled with electronics in a home laboratory, growing up. I fiddled with arts and crafts.1 I glued popsicle sticks, painted plaques, braided yarn, and designed greeting cards. Of the supplies in my family’s crafts box, I adored the beads most. Of the beads, I favored the glass ones.

I would pour them on the carpet, some weekend afternoons. I’d inherited a hodgepodge: The beads’ sizes, colors, shapes, trimmings, and craftsmanship varied. No property divided the beads into families whose members looked like they belonged together. But divide the beads I tried. I might classify them by color, then subdivide classes by shape. The color and shape groupings precluded me from grouping by size. But, by loosening my original classification and combining members from two classes, I might incorporate trimmings into the categorization. I’d push my classification scheme as far as I could. Then, I’d rake the beads together and reorganize them according to different principles.

Why have I pursued theoretical physics? many people ask. I have many answers. They include “Because I adored organizing craft supplies at age eight.” I craft and organize ideas.


I’ve blogged about the out-of-time-ordered correlator (OTOC), a signature of how quantum information spreads throughout a many-particle system. Experimentalists want to measure the OTOC, to learn how information spreads. But measuring the OTOC requires tight control over many quantum particles.

I proposed a scheme for measuring the OTOC, with help from Chapman University physicist Justin Dressel. The scheme involves weak measurements. Weak measurements barely disturb the systems measured. (Most measurements of quantum systems disturb the measured systems. So intuited Werner Heisenberg when formulating his uncertainty principle.)

I had little hope for the weak-measurement scheme’s practicality. Consider the stereotypical experimentalist’s response to a stereotypical experimental proposal by a theorist: Oh, sure, we can implement that—in thirty years. Maybe. If the pace of technological development doubles. I expected to file the weak-measurement proposal in the “unfeasible” category.

But experimentalists started collaring me. The scheme sounds reasonable, they said. How many trials would one have to perform? Did the proposal require ancillas, helper systems used to control the measured system? Must each ancilla influence the whole measured system, or could an ancilla interact with just one particle? How did this proposal compare with alternatives?

I met with a cavity-QED2 experimentalist and a cold-atoms expert. I talked with postdocs over skype, with heads of labs at Caltech, with grad students in Taiwan, and with John Preskill in his office. I questioned an NMR3 experimentalist over lunch and fielded superconducting-qubit4 questions in the sunshine. I read papers, reread papers, and powwowed with Justin.

I wouldn’t have managed half so well without Justin and without Brian Swingle. Brian and coauthors proposed the first OTOC-measurement scheme. He reached out after finding my first OTOC paper.

According to that paper, the OTOC is a moment of a quasiprobability.5 How does that quasiprobability look, we wondered? How does it behave? What properties does it have? Our answers appear in a paper we released with Justin this month. We calculate the quasiprobability in two examples, prove properties of the quasiprobability, and argue that the OTOC motivates generalizations of quasiprobability theory. We also enhance the weak-measurement scheme and analyze it.

Amidst that analysis, in a 10 x 6 table, we classify glass beads.


We inventoried our experimental conversations and distilled them. We culled measurement-scheme features analogous to bead size, color, and shape. Each property labels a row in the table. Each measurement scheme labels a column. Each scheme has, I learned, gold flecks and dents, hues and mottling, an angle at which it catches the light.

I’ve kept most of the glass beads that fascinated me at age eight. Some of the beads have dispersed to necklaces, picture frames, and eyeglass leashes. I moved the remnants, a few years ago, to a compartmentalized box. Doesn’t it resemble the table?


That’s why I work at the IQIM.


1I fiddled in a home laboratory, too, in a garage. But I lived across the street from that garage. I lived two rooms from an arts-and-crafts box.

2Cavity QED consists of light interacting with atoms in a box.

3Lots of nuclei manipulated with magnetic fields. “NMR” stands for “nuclear magnetic resonance.” MRI machines, used to scan brains, rely on NMR.

4Superconducting circuits are tiny, cold quantum circuits.

5A quasiprobability resembles a probability but behaves more oddly: Probabilities range between zero and one; quasiprobabilities can dip below zero. Think of a moment as like an average.

With thanks to all who questioned me; to all who answered questions of mine; to my wonderful coauthors; and to my parents, who stocked the crafts box.

Local operations and Chinese communications

The workshop spotlighted entanglement. It began in Shanghai, paused as participants hopped the Taiwan Strait, and resumed in Taipei. We discussed quantum operations and chaos, thermodynamics and field theory.1 I planned to return from Taipei to Shanghai to Los Angeles.

Quantum thermodynamicist Nelly Ng and I drove to the Taipei airport early. News from Air China curtailed our self-congratulations: China’s military was running an operation near Shanghai. Commercial planes couldn’t land. I’d miss my flight to LA.


Two quantum thermodynamicists in Shanghai

An operation?

Quantum information theorists use a mindset called operationalism. We envision experimentalists in separate labs. Call the experimentalists Alice, Bob, and Eve (ABE). We tell stories about ABE to formulate and analyze problems. Which quantum states do ABE prepare? How do ABE evolve, or manipulate, the states? Which measurements do ABE perform? Do they communicate about the measurements’ outcomes?

Operationalism concretizes ideas. The outlook checks us from drifting into philosophy and into abstractions difficult to apply physics tools to.2 Operationalism infuses our language, our framing of problems, and our mathematical proofs.

Experimentalists can perform some operations more easily than others. Suppose that Alice controls the magnets, lasers, and photodetectors in her lab; Bob controls the equipment in his; and Eve controls the equipment in hers. Each experimentalist can perform local operations (LO). Suppose that Alice, Bob, and Eve can talk on the phone and send emails. They exchange classical communications (CC).

You can’t generate entanglement using LOCC. Entanglement consists of strong correlations that quantum systems can share and that classical systems can’t. A quantum system in Alice’s lab can hold more information about a quantum system of Bob’s than any classical system could. We must create and control entanglement to operate quantum computers. Creating and controlling entanglement poses challenges. Hence quantum information scientists often model easy-to-perform operations with LOCC.

Suppose that some experimentalist Charlie loans entangled quantum systems to Alice, Bob, and Eve. How efficiently can ABE compute some quantity, exchange quantum messages, or perform other information-processing tasks, using that entanglement? Such questions underlie quantum information theory.


Taipei’s night market. Or Caltech’s neighborhood?

Local operations.

Nelly and I performed those, trying to finagle me to LA. I inquired at Air China’s check-in desk in English. Nelly inquired in Mandarin. An employee smiled sadly at each of us.

We branched out into classical communications. I called Expedia (“No, I do not want to fly to Manila”), United Airlines (“No flights for two days?”), my credit-card company, Air China’s American reservations office, Air China’s Chinese reservations office, and Air China’s Taipei reservations office. I called AT&T to ascertain why I couldn’t reach Air China (“Yes, please connect me to the airline. Could you tell me the number first? I’ll need to dial it after you connect me and the call is then dropped”).

As I called, Nelly emailed. She alerted Bob, aka Janet (Ling-Yan) Hung, who hosted half the workshop at Fudan University in Shanghai. Nelly emailed Eve, aka Feng-Li Lin, who hosted half the workshop at National Taiwan University in Taipei. Janet twiddled the magnets in her lab (investigated travel funding), and Feng-Li cooled a refrigerator in his.

ABE can process information only so efficiently, using LOCC. The time crept from 1:00 PM to 3:30.


Nelly Ng uses classical communications.

What could we have accomplished with quantum communication? Using LOCC, Alice can manipulate quantum states (like an electron’s orientation) in her lab. She can send nonquantum messages (like “My flight is delayed”) to Bob. She can’t send quantum information (like an electron’s orientation).

Alice and Bob can ape quantum communication, given entanglement. Suppose that Charlie strongly correlates two electrons. Suppose that Charlie gives Alice one electron and gives Bob the other. Alice can send one qubit–one unit of quantum information–to Bob. We call that sending quantum teleportation.

Suppose that air-traffic control had loaned entanglement to Janet, Feng-Li, and me. Could we have finagled me to LA quickly?

Quantum teleportation differs from human teleportation.



We didn’t need teleportation. Feng-Li arranged for me to visit Taiwan’s National Center for Theoretical Sciences (NCTS) for two days. Air China agreed to return me to Shanghai afterward. United would fly me to LA, thanks to help from Janet. Nelly rescued my luggage from leaving on the wrong flight.

Would I rather have teleported? I would have avoided a bushel of stress. But I wouldn’t have learned from Janet about Chinese science funding, wouldn’t have heard Feng-Li’s views about gravitational waves, wouldn’t have glimpsed Taiwanese countryside flitting past the train we rode to the NCTS.

According to some metrics, classical resources outperform quantum.


At Taiwan’s National Center for Theoretical Sciences

The workshop organizers have generously released videos of the lectures. My lecture about quantum chaos and fluctuation relations appears here and here. More talks appear here.

With gratitude to Janet Hung, Feng-Li Lin, and Nelly Ng; to Fudan University, National Taiwan University, and Taiwan’s National Center for Theoretical Sciences for their hospitality; and to Xiao Yu for administrative support.

Glossary and other clarifications:

1Field theory describes subatomic particles and light.

2Physics and philosophy enrich each other. But I haven’t trained in philosophy. I benefit from differentiating physics problems that I’ve equipped to solve from philosophy problems that I haven’t.


My brother and I played the video game Sonic the Hedgehog on a Sega Dreamcast. The hero has spiky electric-blue fur and can run at the speed of sound.1 One of us, then the other, would battle monsters. Monster number one oozes onto a dark city street as an aquamarine puddle. The puddle spreads, then surges upward to form limbs and claws.2 The limbs splatter when Sonic attacks: Aqua globs rain onto the street.


The monster’s master, Dr. Eggman, has ginger mustachios and a body redolent of his name. He scoffs as the heroes congratulate themselves.

“Fools!” he cries, the pauses in his speech heightening the drama. “[That monster is] CHAOS…the GOD…of DE-STRUC-TION!” His cackle could put a Disney villain to shame.

Dr. Eggman’s outburst comes to mind when anyone asks what topic I’m working on.

“Chaos! And the flow of time, quantum theory, and the loss of information.”


Alexei Kitaev, a Caltech physicist, hooked me on chaos. I TAed his spring-2016 course. The registrar calls the course Ph 219c: Quantum Computation. I call the course Topics that Interest Alexei Kitaev.

“What do you plan to cover?” I asked at the end of winter term.

Topological quantum computation, Alexei replied. How you simulate Hamiltonians with quantum circuits. Or maybe…well, he was thinking of discussing black holes, information, and chaos.

If I’d had a tail, it would have wagged.

“What would you say about black holes?” I asked.


Sonic’s best friend, Tails the fox.

I fwumped down on the couch in Alexei’s office, and Alexei walked to his whiteboard. Scientists first noticed chaos in classical systems. Consider a double pendulum—a pendulum that hangs from the bottom of a pendulum that hangs from, say, a clock face. Imagine pulling the bottom pendulum far to one side, then releasing. The double pendulum will swing, bend, and loop-the-loop like a trapeze artist. Imagine freezing the trapeze artist after an amount t of time.

What if you pulled another double pendulum a hair’s breadth less far? You could let the pendulum swing, wait for a time t, and freeze this pendulum. This pendulum would probably lie far from its brother. This pendulum would probably have been moving with a different speed than its brother, in a different direction, just before the freeze. The double pendulum’s motion changes loads if the initial conditions change slightly. This sensitivity to initial conditions characterizes classical chaos.

A mathematical object F(t) reflects quantum systems’ sensitivities to initial conditions. [Experts: F(t) can evolve as an exponential governed by a Lyapunov-type exponent: \sim 1 - ({\rm const.})e^{\lambda_{\rm L} t}.] F(t) encodes a hypothetical process that snakes back and forth through time. This snaking earned F(t) the name “the out-of-time-ordered correlator” (OTOC). The snaking prevents experimentalists from measuring quantum systems’ OTOCs easily. But experimentalists are trying, because F(t) reveals how quantum information spreads via entanglement. Such entanglement distinguishes black holes, cold atoms, and specially prepared light from everyday, classical systems.

Alexei illustrated, on his whiteboard, the sensitivity to initial conditions.

“In case you’re taking votes about what to cover this spring,” I said, “I vote for chaos.”

We covered chaos. A guest attended one lecture: Beni Yoshida, a former IQIM postdoc. Beni and colleagues had devised quantum error-correcting codes for black holes.3 Beni’s foray into black-hole physics had led him to F(t). He’d written an OTOC paper that Alexei presented about. Beni presented about a follow-up paper. If I’d had another tail, it would have wagged.


Sonic’s friend has two tails.

Alexei’s course ended. My research shifted to many-body localization (MBL), a quantum phenomenon that stymies the spread of information. OTOC talk burbled beyond my office door.

At the end of the summer, IQIM postdoc Yichen Huang posted on Facebook, “In the past week, five papers (one of which is ours) appeared . . . studying out-of-time-ordered correlators in many-body localized systems.”

I looked down at the MBL calculation I was performing. I looked at my computer screen. I set down my pencil.


I marched to John Preskill’s office.


The bosses. Of different sorts, of course.

The OTOC kept flaring on my radar, I reported. Maybe the time had come for me to try contributing to the discussion. What might I contribute? What would be interesting?

We kicked around ideas.

“Well,” John ventured, “you’re interested in fluctuation relations, right?”

Something clicked like the “power” button on a video-game console.

Fluctuation relations are equations derived in nonequilibrium statistical mechanics. They describe systems driven far from equilibrium, like a DNA strand whose ends you’ve yanked apart. Experimentalists use fluctuation theorems to infer a difficult-to-measure quantity, a difference \Delta F between free energies. Fluctuation relations imply the Second Law of Thermodynamics. The Second Law relates to the flow of time and the loss of information.

Time…loss of information…Fluctuation relations smelled like the OTOC. The two had to join together.


I spent the next four days sitting, writing, obsessed. I’d read a paper, three years earlier, that casts a fluctuation relation in terms of a correlator. I unearthed the paper and redid the proof. Could I deform the proof until the paper’s correlator became the out-of-time-ordered correlator?

Apparently. I presented my argument to my research group. John encouraged me to clarify a point: I’d defined a mathematical object A, a probability amplitude. Did A have physical significance? Could anyone measure it? I consulted measurement experts. One identified A as a quasiprobability, a quantum generalization of a probability, used to model light in quantum optics. With the experts’ assistance, I devised two schemes for measuring the quasiprobability.

The result is a fluctuation-like relation that contains the OTOC. The OTOC, the theorem reveals, is a combination of quasiprobabilities. Experimentalists can measure quasiprobabilities with weak measurements, gentle probings that barely disturb the probed system. The theorem suggests two experimental protocols for inferring the difficult-to-measure OTOC, just as fluctuation relations suggest protocols for inferring the difficult-to-measure \Delta F. Just as fluctuation relations cast \Delta F in terms of a characteristic function of a probability distribution, this relation casts F(t) in terms of a characteristic function of a (summed) quasiprobability distribution. Quasiprobabilities reflect entanglement, as the OTOC does.


Collaborators and I are extending this work theoretically and experimentally. How does the quasiprobability look? How does it behave? What mathematical properties does it have? The OTOC is motivating questions not only about our quasiprobability, but also about quasiprobability and weak measurements. We’re pushing toward measuring the OTOC quasiprobability with superconducting qubits or cold atoms.

Chaos has evolved from an enemy to a curiosity, from a god of destruction to an inspiration. I no longer play the electric-blue hedgehog. But I remain electrified.


1I hadn’t started studying physics, ok?

2Don’t ask me how the liquid’s surface tension rises enough to maintain the limbs’ shapes.

3Black holes obey quantum mechanics. Quantum systems can solve certain problems more quickly than ordinary (classical) computers. Computers make mistakes. We fix mistakes using error-correcting codes. The codes required by quantum computers differ from the codes required by ordinary computers. Systems that contain black holes, we can regard as performing quantum computations. Black-hole systems’ mistakes admit of correction via the code constructed by Beni & co.