Machine learning the arXiv

Over the last year or so, the machine learning wave has really been sweeping through the field of condensed matter physics. Machine learning techniques have been applied to condensed matter physics before, but very sparsely and with little recognition. These days, I guess (partially) due to the general machine learning and AI hype, the amount of such studies skyrocketed (I admit to contributing to that..). I’ve been keeping track of this using the arXiv and Twitter (@Evert_v_N), but you should know about this website for getting an overview of the physics & machine learning papers: https://physicsml.github.io/pages/papers.html.

This effort of applying machine learning to physics is a serious attempt at trying to understand how such tools could be useful in a variety of ways. It isn’t very hard to get a neural network to learn ‘something’ from physics data, but it is really hard to find out what – and especially how – the network does that. That’s why toy cases such as the Ising model or the Kosterlitz-Thouless transition have been so important!

When you’re keeping track of machine learning and AI developments, you soon realize that there are examples out there of amazing feats. Being able to generate photo-realistic pictures given just a sentence. e.g. “a brown bird with golden speckles and red wings is sitting on a yellow flower with pointy petals”, is (I think..) pretty cool. I can’t help but wonder if we’ll get to a point where we can ask it to generate “the groundstate of the Heisenberg model on a Kagome lattice of 100×100”…

Another feat I want to mention, and the main motivation for this post, is that of being able to encode words as vectors. That doesn’t immediately seem like a big achievement, but it is once you want to have ‘similar’ words have ‘similar’ vectors. That is, you intuitively understand that Queen and King are very similar, but differ basically only in gender. Can we teach that to a computer (read: neural network) by just having it read some text? Turns out we can. The general encoding of words to vectors is aptly named ‘Word2Vec’, and some of the top algorithms that do that were introduced here (https://arxiv.org/abs/1301.3781) and here (https://arxiv.org/abs/1310.4546). The neat thing is that we can actually do arithmetics with these words encoded as vectors, so that the network learns (with no other input than text!):

  • King – Man + Woman = Queen
  • Paris – France + Italy = Rome

In that spirit, I wondered if we can achieve the same thing with physics jargon. Everyone knows, namely, that “electrons + two dimensions + magnetic field = Landau levels”. But is that clear from condensed matter titles?

Try it yourself

If you decide at this point that the rest of the blog is too long, at least have a look here: everthemore.pythonanywhere.com or skip to the last section. That website demonstrates the main point of this post. If that sparks your curiosity, read on!

This post is mainly for entertainment, and so a small disclaimer is in order: in all of the results below, I am sure things can be improved upon. Consider this a ‘proof of principle’. However, I would be thrilled to see what kind of trained models you can come up with yourself! So for that purpose, all of the code (plus some bonus content!) can be found on this github repository: https://github.com/everthemore/physics2vec.

Harvesting the arXiv

The perfect dataset for our endeavor can be found in the form of the arXiv. I’ve written a small script (see github repository) that harvests the titles of a given section from the arXiv. It also has options for getting the abstracts, but I’ll leave that for a separate investigation. Note that in principle we could also get the source-files of all of these papers, but doing that in bulk requires a payment; and getting them one by one will 1) take forever and 2) probably get us banned.

Collecting all this data of the physics:cond-mat subsection took right about 1.5 hours and resulted in 240737 titles and abstracts (I last ran this script on November 20th, 2017). I’ve filtered them by year and month, and you can see the result in Fig.1 below. Seems like we have some catching up to do in 2017 still (although as the inset shows, we have nothing to fear. November is almost over, but we still have the ‘getting things done before x-mas’ rush coming up!).

numpapers

Figure 1: The number of papers in the cond-mat arXiv section over the years. We’re behind, but the year isn’t over yet! (Data up to Nov 20th 2017)

Analyzing n-grams

After tidying up the titles (removing LaTeX, converting everything to lowercase, etc.), the next thing to do is to train a language model on finding n-grams. N-grams are basically fixed n-word expressions such as ‘cooper pair’ (bigram) or ‘metal insulator transition’ (trigram). This makes it easier to train a Word2Vec encoding, since these phrases are fixed and can be considered a single word. The python module we’ll use for Word2Vec is gensim (https://radimrehurek.com/gensim/), and it conveniently has phrase-detection built-in. The language model it builds reports back to us the n-grams it finds, and assigns them a score indicating how certain it is about them. Notice that this is not the same as how frequently it appears in the dataset. Hence an n-gram can appear fewer times than another, but have a higher certainty because it always appears in the same combination. For example, ‘de-haas-van-alphen’ appears less than, but is more certain than, ‘cooper-pair’, because ‘pair’ does not always come paired (pun intended) with ‘cooper’. I’ve analyzed up to 4-grams in the analysis below.

I can tell you’re curious by now to find out what some of the most certain n-grams in cond-mat are (again, these are not necessarily the most frequent), so here are some interesting findings:

  • The most certain n-grams are all surname combo’s, Affleck-Kennedy-Lieb-Tasaki being the number 1. Kugel-Khomskii is the most certain 2-name combo and Einstein-Podolksi-Rosen the most certain 3-name combo.
  • The first certain non-name n-gram is a ‘quartz tuning fork’, followed by a ‘superconducting coplanar waveguide resonator’. Who knew.
  • The bigram ‘phys. rev.’ and trigram ‘phys. rev. lett.’ are relatively high up in the confidence lists. These seem to come from the “Comment on […]”-titles on the arXiv.
  • I learned that there is such a thing as a Lefschetz thimble. I also learned that those things are called thimbles in English (we (in Holland) call them ‘finger-hats’!).

In terms of frequency however, which is probably more of interest to us, the most dominant n-grams are Two-dimensional, Quantum dot, Phase transition, Magnetic field, One dimensional and Bose-Einstein (in descending order). It seems 2D is still more popular than 1D, and all in all the top n-grams do a good job at ‘defining’ condensed matter physics. I’ll refer you to the github repository code if you want to see a full list! You’ll find there a piece of code that produces wordclouds from the dominant words and n-grams too, such as this one:

caltechwordcloud.png

For fun though, before we finally get to the Word2Vec encoding, I’ve also kept track of all of these as a function of year, so that we can now turn to finding out which bigrams have been gaining the most popularity. The table below shows the top 5 n-grams for the period 2010 – 2016 (not including 2017) and for the period 2015 – Nov 20th 2017.

2010-2016

2015 – November 20th 2017

Spin liquids  Topological phases & transitions
 Weyl semimetals  Spin chains
 Topological phases & transitions  Machine learning
 Surface states  Transition metal dichalcogenides
 Transition metal dichalcogenides  Thermal transport
 Many-body localization  Open quantum systems

Actually, the real number 5 in the left column was ‘Topological insulators’, but given number 3 I skipped it. Also, this top 5 includes a number 6 (!), which I just could not leave off given that everyone seems to have been working on MBL. If we really want to be early adopters though, taking only the last 1.8 years (2015 – now, Nov 20th 2017)  in the right column of the table shows some interesting newcomers. Surprisingly, many-body localization is not even in the top 20 anymore. Suffice it to say, if you have been working on anything topology-related, you have nothing to worry about. Machine learning is indeed gaining lots of attention, but we’ve yet to see if it doesn’t go the MBL-route (I certainly don’t hope so!). Quantum computing does not seem to be on the cond-mat radar, but I’m certain we would find that high up in the quant-ph arXiv section.

CondMat2Vec

Alright, finally time to use some actual neural networks for machine learning. As I started this post, what we’re about to do is try to train a network to encode/decode words into vectors, while simultaneously making sure that similar words (by meaning!) have similar vectors. Now that we have the n-grams, we want the Word2Vec algorithm to treat these as words by themselves (they are, after all, fixed combinations).

In the Word2Vec algorithm, we get to decide the length of the vectors that encode words ourselves. Larger vectors means more freedom in encoding words, but also makes it harder to learn similarity. In addition, we get to choose a window-size, indicating how many words the algorithm will look ahead to analyze relations between words. Both of these parameters are free for you to play with if you have a look at the source code repository. For the website everthemore.pythonanywhere.com, I’ve uploaded a size 100 with window-size 10 model, which I found to produce sensible results. Sensible here means “based on my expectations”, such as the previous example of “2D + electrons + magnetic field = Landau levels”. Let’s ask our network some questions.

First, as a simple check, let’s see what our encoding thinks some jargon is similar to:

  • Superconductor ~ Superconducting, Cuprate superconductor, Superconductivity, Layered superconductor, Unconventional superconductor, Superconducting gap, Cuprate, Weyl semimetal, …
  • Majorana ~ Majorana fermion, Majorana mode, Non-abelian, Zero-energy, braiding, topologically protected, …

It seems we could start to cluster words based on this. But the real test comes now, in the form of arithmetics. According to our network (I am listing the top two choices in some cases; the encoder outputs a list of similar vectors, ordered by similarity):

  • Majorana + Braiding = Non-Abelian
  • Electron + Hole = Exciton, Carrier
  • Spin + Magnetic field = Magnetization, Antiferromagnetic
  • Particle + Charge = Electron, Charged particle

And, sure enough:

  • 2D + electrons + magnetic field = Landau level, Magnetoresistance oscillation

The above is just a small sample of the things I’ve tried. See the link in the try it yourself section above if you want to have a go. Not all of the examples work nicely. For example, neither lattice + wave nor lattice + excitation nor lattice + force seem to result in anything related to the word ‘phonon’. I would guess that increasing the window size will help remedy this problem. Even better probably would be to include abstracts!

Outlook

I could play with this for hours, and I’m sure that by including the abstracts and tweaking the vector size (plus some more parameters I haven’t even mentioned) one could optimize this more. Once we have an optimized model, we could start to cluster the vectors to define research fields, visualizing the relations between n-grams (both suggestions thanks to Thomas Vidick and John Preskill!), and many other things. This post has become rather long already however, and I will leave further investigation to a possible future post. I’d be very happy to incorporate anything cool you find yourselves though, so please let me know!

Gently yoking yin to yang

The architecture at the University of California, Berkeley mystified me. California Hall evokes a Spanish mission. The main library consists of white stone pillared by ionic columns. A sea-green building scintillates in the sunlight like a scarab. The buildings straddle the map of styles.

Architecture.001

So do Berkeley’s quantum scientists, information-theory users, and statistical mechanics.

The chemists rove from abstract quantum information (QI) theory to experiments. Physicists experiment with superconducting qubits, trapped ions, and numerical simulations. Computer scientists invent algorithms for quantum computers to perform.

Few activities light me up more than bouncing from quantum group to info-theory group to stat-mech group, hunting commonalities. I was honored to bounce from group to group at Berkeley this September.

What a trampoline Berkeley has.

The groups fan out across campus and science, but I found compatibility. Including a collaboration that illuminated quantum incompatibility.

Quantum incompatibility originated in studies by Werner Heisenberg. He and colleagues cofounded quantum mechanics during the early 20th century. Measuring one property of a quantum system, Heisenberg intuited, can affect another property.

The most famous example involves position and momentum. Say that I hand you an electron. The electron occupies some quantum state represented by | \Psi \rangle. Suppose that you measure the electron’s position. The measurement outputs one of many possible values x (unless | \Psi \rangle has an unusual form, the form a Dirac delta function).

But we can’t say that the electron occupies any particular point x = x_0 in space. Measurement devices have limited precision. You can measure the position only to within some error \varepsilon: x = x_0 \pm \varepsilon.

Suppose that, immediately afterward, you measure the electron’s momentum. This measurement, too, outputs one of many possible values. What probability q(p) dp does the measurement have of outputting some value p? We can calculate q(p) dp, knowing the mathematical form of | \Psi \rangle and knowing the values of x_0 and \varepsilon.

q(p) is a probability density, which you can think of as a set of probabilities. The density can vary with p. Suppose that q(p) varies little: The probabilities spread evenly across the possible p values. You have no idea which value your momentum measurement will output. Suppose, instead, that q(p) peaks sharply at some value p = p_0. You can likely predict the momentum measurement’s outcome.

The certainty about the momentum measurement trades off with the precision \varepsilon of the position measurement. The smaller the \varepsilon (the more precisely you measured the position), the greater the momentum’s unpredictability. We call position and momentum complementary, or incompatible.

You can’t measure incompatible properties, with high precision, simultaneously. Imagine trying to do so. Upon measuring the momentum, you ascribe a tiny range of momentum values p to the electron. If you measured the momentum again, an instant later, you could likely predict that measurement’s outcome: The second measurement’s q(p) would peak sharply (encode high predictability). But, in the first instant, you measure also the position. Hence, by the discussion above, q(p) would spread out widely. But we just concluded that q(p) would peak sharply. This contradiction illustrates that you can’t measure position and momentum, precisely, at the same time.

But you can simultaneously measure incompatible properties weakly. A weak measurement has an enormous \varepsilon. A weak position measurement barely spreads out q(p). If you want more details, ask a Quantum Frontiers regular; I’ve been harping on weak measurements for months.

Blame Berkeley for my harping this month. Irfan Siddiqi’s and Birgitta Whaley’s groups collaborated on weak measurements of incompatible observables. They tracked how the measured quantum state | \Psi (t) \rangle evolved in time (represented by t).

Irfan’s group manipulates superconducting qubits.1 The qubits sit in the physics building, a white-stone specimen stamped with an egg-and-dart motif. Across the street sit chemists, including members of Birgitta’s group. The experimental physicists and theoretical chemists teamed up to study a quantum lack of teaming up.

Phys. & chem. bldgs

The experiment involved one superconducting qubit. The qubit has properties analogous to position and momentum: A ball, called the Bloch ball, represents the set of states that the qubit can occupy. Imagine an arrow pointing from the sphere’s center to any point in the ball. This Bloch vector represents the qubit’s state. Consider an arrow that points upward from the center to the surface. This arrow represents the qubit state | 0 \rangle. | 0 \rangle is the quantum analog of the possible value 0 of a bit, or unit of information. The analogous downward-pointing arrow represents the qubit state | 1 \rangle, analogous to 1.

Infinitely many axes intersect the sphere. Different axes represent different observables that Irfan’s group can measure. Nonparallel axes represent incompatible observables. For example, the x-axis represents an observable \sigma_x analogous to position. The y-axis represents an observable \sigma_y analogous to momentum.

Tug-of-war

Siddiqi lab, decorated with the trademark for the paper’s tug-of-war between incompatible observables. Photo credit: Leigh Martin, one of the paper’s leading authors.

Irfan’s group stuck their superconducting qubit in a cavity, or box. The cavity contained light that interacted with the qubit. The interactions transferred information from the qubit to the light: The light measured the qubit’s state. The experimentalists controlled the interactions, controlling the axes “along which” the light was measured. The experimentalists weakly measured along two axes simultaneously.

Suppose that the axes coincided—say, at the x-axis \hat{x}. The qubit would collapse to the state | \Psi \rangle = \frac{1}{ \sqrt{2} } ( | 0 \rangle + | 1 \rangle ), represented by the arrow that points along \hat{x} to the sphere’s surface, or to the state | \Psi \rangle = \frac{1}{ \sqrt{2} } ( | 0 \rangle - | 1 \rangle ), represented by the opposite arrow.

0 deg

(Projection of) the Bloch Ball after the measurement. The system can access the colored points. The lighter a point, the greater the late-time state’s weight on the point.

Let \hat{x}' denote an axis near \hat{x}—say, 18° away. Suppose that the group weakly measured along \hat{x} and \hat{x}'. The state would partially collapse. The system would access points in the region straddled by \hat{x} and \hat{x}', as well as points straddled by - \hat{x} and - \hat{x}'.

18 deg

Finally, suppose that the group weakly measured along \hat{x} and \hat{y}. These axes stand in for position and momentum. The state would, loosely speaking, swirl around the Bloch ball.

90 deg

The Berkeley experiment illuminates foundations of quantum theory. Incompatible observables, physics students learn, can’t be measured simultaneously. This experiment blasts our expectations, using weak measurements. But the experiment doesn’t just destroy. It rebuilds the blast zone, by showing how | \Psi (t) \rangle evolves.

“Position” and “momentum” can hang together. So can experimentalists and theorists, physicists and chemists. So, somehow, can the California mission and the ionic columns. Maybe I’ll understand the scarab building when we understand quantum theory.2

With thanks to Birgitta’s group, Irfan’s group, and the rest of Berkeley’s quantum/stat-mech/info-theory community for its hospitality. The Bloch-sphere figures come from http://www.nature.com/articles/nature19762.

1The qubit is the quantum analog of a bit. The bit is the basic unit of information. A bit can be in one of two possible states, which we can label as 0 and 1. Qubits can manifest in many physical systems, including superconducting circuits. Such circuits are tiny quantum circuits through which current can flow, without resistance, forever.

2Soda Hall dazzled but startled me.

Paradise

The word dominates chapter one of Richard Holmes’s book The Age of WonderHolmes writes biographies of Romantic-Era writers: Mary Wollstonecraft, Percy Shelley, and Samuel Taylor Coleridge populate his bibliography. They have cameos in Age. But their scientific counterparts star.

“Their natural-philosopher” counterparts, I should say. The word “scientist” emerged as the Romantic Era closed. Romanticism, a literary and artistic movement, flourished between the 1700s and the 1800s. Romantics championed self-expression, individuality, and emotion over convention and artificiality. Romantics wondered at, and drew inspiration from, the natural world. So, Holmes argues, did Romantic-Era natural philosophers. They explored, searched, and innovated with Wollstonecraft’s, Shelley’s, and Coleridge’s zest.

Age of Wonder

Holmes depicts Wilhelm and Caroline Herschel, a German brother and sister, discovering the planet Uranus. Humphry Davy, an amateur poet from Penzance, inventing a lamp that saved miners’ lives. Michael Faraday, a working-class Londoner, inspired by Davy’s chemistry lectures.

Joseph Banks in paradise.

So Holmes entitled chapter one.

Banks studied natural history as a young English gentleman during the 1760s. He then sailed around the world, a botanist on exploratory expeditions. The second expedition brought Banks aboard the HMS Endeavor. Captain James Cook steered the ship to Brazil, Tahiti, Australia, and New Zealand. Banks brought a few colleagues onboard. They studied the native flora, fauna, skies, and tribes.

Banks, with fellow botanist Daniel Solander, accumulated over 30,000 plant samples. Artist Sydney Parkinson drew the plants during the voyage. Parkinson’s drawings underlay 743 copper engravings that Banks commissioned upon returning to England. Banks planned to publish the engravings as the book Florilegium. He never succeeded. Two institutions executed Banks’s plan more than 200 years later.

Banks’s Florilegium crowns an exhibition at the University of California at Santa Barbara (UCSB). UCSB’s Special Research Collections will host “Botanical Illustrations and Scientific Discovery—Joseph Banks and the Exploration of the South Pacific, 1768–1771” until May 2018. The exhibition features maps of Banks’s journeys, biographical sketches of Banks and Cook, contemporary art inspired by the engravings, and the Florilegium.

online poster

The exhibition spotlights “plants that have subsequently become important ornamental plants on the UCSB campus, throughout Santa Barbara, and beyond.” One sees, roaming Santa Barbara, slivers of Banks’s paradise.

2 bouganvilleas

In Santa Barbara resides the Kavli Institute for Theoretical Physics (KITP). The KITP is hosting a program about the physics of quantum information (QI). QI scientists are congregating from across the world. Everyone visits for a few weeks or months, meeting some participants and missing others (those who have left or will arrive later). Participants attend and present tutorials, explore beyond their areas of expertise, and initiate research collaborations.

A conference capstoned the program, one week this October. Several speakers had founded subfields of physics: quantum error correction (how to fix errors that dog quantum computers), quantum computational complexity (how quickly quantum computers can solve hard problems), topological quantum computation, AdS/CFT (a parallel between certain gravitational systems and certain quantum systems), and more. Swaths of science exist because of these thinkers.

KITP

One evening that week, I visited the Joseph Banks exhibition.

Joseph Banks in paradise.

I’d thought that, by “paradise,” Holmes had meant “physical attractions”: lush flowers, vibrant colors, fresh fish, and warm sand. Another meaning occurred to me, after the conference talks, as I stood before a glass case in the library.

Joseph Banks, disembarking from the Endeavour, didn’t disembark onto just an island. He disembarked onto terra incognita. Never had he or his colleagues seen the blossoms, seed pods, or sprouts before him. Swaths of science awaited. What could the natural philosopher have craved more?

QI scientists of a certain age reminisce about the 1990s, the cowboy days of QI. When impactful theorems, protocols, and experiments abounded. When they dangled, like ripe fruit, just above your head. All you had to do was look up, reach out, and prove a pineapple.

Cowboy

Typical 1990s quantum-information scientist

That generation left mine few simple theorems to prove. But QI hasn’t suffered extinction. Its frontiers have advanced into other fields of science. Researchers are gaining insight into thermodynamics, quantum gravity, condensed matter, and chemistry from QI. The KITP conference highlighted connections with quantum gravity.

…in paradise.

What could a natural philosopher crave more?

Contemporary

Artwork commissioned by the UCSB library: “Sprawling Neobiotic Chimera (After Banks’ Florilegium),” by Rose Briccetti

Most KITP talks are recorded and released online. You can access talks from the conference here. My talk, about quantum chaos and thermalization, appears here. 

With gratitude to the KITP, and to the program organizers and the conference organizers, for the opportunity to participate. 

Standing back at Stanford

T-shirt 1

This T-shirt came to mind last September. I was standing in front of a large silver-colored table littered with wires, cylinders, and tubes. Greg Bentsen was pointing at components and explaining their functions. He works in Monika Schleier-Smith’s lab, as a PhD student, at Stanford.

Monika’s group manipulates rubidium atoms. A few thousand atoms sit in one of the cylinders. That cylinder contains another cylinder, an optical cavity, that contains the atoms. A mirror caps each of the cavity’s ends. Light in the cavity bounces off the mirrors.

Light bounces off your bathroom mirror similarly. But we can describe your bathroom’s light accurately with Maxwellian electrodynamics, a theory developed during the 1800s. We describe the cavity’s light with quantum electrodynamics (QED). Hence we call the lab’s set-up cavity QED.

The light interacts with the atoms, entangling with them. The entanglement imprints information about the atoms on the light. Suppose that light escaped from the cavity. Greg and friends could measure the light, then infer about the atoms’ quantum state.

A little light leaks through the mirrors, though most light bounces off. From leaked light, you can infer about the ensemble of atoms. You can’t infer about individual atoms. For example, consider an atom’s electrons. Each electron has a quantum property called a spin. We sometimes imagine the spin as an arrow that points upward or downward. Together, the electrons’ spins form the atom’s joint spin. You can tell, from leaked light, whether one atom’s spin points upward. But you can’t tell which atom’s spin points upward. You can’t see the atoms for the ensemble.

Monika’s team can. They’ve cut a hole in their cylinder. Light escapes the cavity through the hole. The light from the hole’s left-hand edge carries information about the leftmost atom, and so on. The team develops a photograph of the line of atoms. Imagine holding a photograph of a line of people. You can point to one person, and say, “Aha! She’s the xkcd fan.” Similarly, Greg and friends can point to one atom in their photograph and say, “Aha! That atom has an upward-pointing spin.” Monika’s team is developing single-site imaging.

Solvay

Aha! She’s the xkcd fan.

Monika’s team plans to image atoms in such detail, they won’t need for light to leak through the mirrors. Light leakage creates problems, including by entangling the atoms with the world outside the cavity. Suppose you had to diminish the amount of light that leaks from a rubidium cavity. How should you proceed?

Tell the mirrors,

T-shirt 2

You should lengthen the cavity. Why? Imagine a photon, a particle of light, in the cavity. It zooms down the cavity’s length, hits a mirror, bounces off, retreats up the cavity’s length, hits the other mirror, and bounces off. The photon repeats this process until a mirror hit fails to generate a bounce. The mirror transmits the photon to the exterior; the photon leaks out. How can you reduce leaks? By preventing photons from hitting mirrors so often, by forcing the photons to zoom longer, by lengthening the cavity, by shifting the mirrors outward.

So Greg hinted, beside that silver-colored table in Monika’s lab. The hint struck a chord: I recognized the impulse to

T-shirt 3

The impulse had led me to Stanford.

Weeks earlier, I’d written my first paper about quantum chaos and information scrambling. I’d sat and read and calculated and read and sat and emailed and written. I needed to stand up, leave my cavity, and image my work from other perspectives.

Stanford physicists had written quantum-chaos papers I admired. So I visited, presented about my work, and talked. Patrick Hayden introduced me to a result that might help me apply my result to another problem. His group helped me simplify a mathematical expression. Monika reflected that a measurement scheme I’d proposed sounded not unreasonable for cavity QED.

And Greg led me to recognize the principle behind my visit: Sometimes, you have to

T-shirt 4

to move forward.

With gratitude to Greg, Monika, Patrick, and the rest of Monika’s and Patrick’s groups for their time, consideration, explanations, and feedback. With thanks to Patrick and Stanford’s Institute for Theoretical Physics for their hospitality.

Decoding (the allure of) the apparent horizon

I took 32 hours to unravel why Netta Engelhardt’s talk had struck me.

We were participating in Quantum Information in Quantum Gravity III, a workshop hosted by the University of British Columbia (UBC) in Vancouver. Netta studies quantum gravity as a Princeton postdoc. She discussed a feature of black holes—an apparent horizon—I’d not heard of. After hearing of it, I had to grasp it. I peppered Netta with questions three times in the following day. I didn’t understand why, for 32 hours.

After 26 hours, I understood apparent horizons like so.

Imagine standing beside a glass sphere, an empty round shell. Imagine light radiating from a point source in the sphere’s center. Think of the point source as a minuscule flash light. Light rays spill from the point source.

Which paths do the rays follow through space? They fan outward from the sphere’s center, hit the glass, and fan out more. Imagine turning your back to the sphere and looking outward. Light rays diverge as they pass you.

At least, rays diverge in flat space-time. We live in nearly flat space-time. We wouldn’t if we neighbored a supermassive object, like a black hole. Mass curves space-time, as described by Einstein’s theory of general relativity.

Sphere 2

Imagine standing beside the sphere near a black hole. Let the sphere have roughly the black hole’s diameter—around 10 kilometers, according to astrophysical observations. You can’t see much of the sphere. So—imagine—you recruit your high-school-physics classmates. You array yourselves around the sphere, planning to observe light and compare observations. Imagine turning your back to the sphere. Light rays would converge, or flow toward each other. You’d know yourself to be far from Kansas.

Picture you, your classmates, and the sphere falling into the black hole. When would everyone agree that the rays switch from diverging to converging? Sometime after you passed the event horizon, the point of no return.1 Before you reached the singularity, the black hole’s center, where space-time warps infinitely. The rays would switch when you reached an in-between region, the apparent horizon.

Imagine pausing at the apparent horizon with your sphere, facing away from the sphere. Light rays would neither diverge nor converge; they’d point straight. Continue toward the singularity, and the rays would converge. Reverse away from the singularity, and the rays would diverge.

Rose garden 2

UBC near twilight

Rays diverged from the horizon beyond UBC at twilight. Twilight suits UBC as marble suits the Parthenon; and UBC’s twilight suits musing. You can reflect while gazing on reflections in glass buildings, or reflections in a pool by a rose garden. Your mind can roam as you roam paths lined by elms, oaks, and willows. I wandered while wondering why the sphere intrigued me.

Science thrives on instrumentation. Galileo improved the telescope, which unveiled Jupiter’s moons. Alexander von Humboldt measured temperatures and pressures with thermometers and barometers, charting South America during the 1700s. The Large Hadron Collider revealed the Higgs particle’s mass in 2012.

The sphere reminded me of a thermometer. As thermometers register temperature, so does the sphere register space-time curvature. Not that you’d need a sphere to distinguish a black hole from Kansas. Nor do you need a thermometer to distinguish Vancouver from a Brazilian jungle. But thermometers quantify the distinction. A sphere would sharpen your observations’ precision.

A sphere and a light source—free of supercolliders, superconductors, and superfridges. The instrument boasts not only profundity, but also simplicity.

von Humboldt.001

Alexander von Humboldt

Netta proved a profound theorem about apparent horizons, with coauthor Aron Wall. Jakob Bekenstein and Stephen Hawking had studied event horizons during the 1970s. An event horizon’s area, Bekenstein and Hawking showed, is proportional to the black hole’s thermodynamic entropy. Netta and Aron proved a proportionality between another area and another entropy.

They calculated an apparent horizon’s area, A. The math that represents their black hole represents also a quantum system, by a duality called AdS/CFT. The quantum system can occupy any of several states. Different states encode different information about the black hole. Consider the information needed to describe, fully and only, the region outside the apparent horizon. Some quantum state \rho encodes this information. \rho encodes no information about the region behind the apparent horizon, closer to the black hole. How would you quantify this lack of information? With the von Neumann entropy S(\rho). This entropy is proportional to the apparent horizon’s area: S( \rho )  \propto  A.

Netta and Aron entitled their paper “Decoding the apparent horizon.” Decoding the apparent horizon’s allure took me 32 hours and took me to an edge of campus. But I didn’t mind. Edges and horizons suited my visit as twilight suits UBC. Where can we learn, if not at edges, as where quantum information meets other fields?

 

With gratitude to Mark van Raamsdonk and UBC for hosting Quantum Information in Quantum Gravity III; to Mark, the other organizers, and the “It from Qubit” Simons Foundation collaboration for the opportunity to participate; and to Netta Engelhardt for sharing her expertise.

1Nothing that draws closer to a black hole than the event horizon can turn around and leave, according to general relativity. The black hole’s gravity pulls too strongly. Quantum mechanics implies that information leaves, though, in Hawking radiation.

Topological qubits: Arriving in 2018?

Editor‘s note: This post was prepared jointly by Ryan Mishmash and Jason Alicea.

Physicists appear to be on the verge of demonstrating proof-of-principle “usefulness” of small quantum computers.  Preskill’s notion of quantum supremacy spotlights a particularly enticing goal: use a quantum device to perform some computation—any computation in fact—that falls beyond the reach of the world’s best classical computers.  Efforts along these lines are being vigorously pursued along many fronts, from academia to large corporations to startups.  IBM’s publicly accessible 16-qubit superconducting device, Google’s pursuit of a 7×7 superconducting qubit array, and the recent synthesis of a 51-qubit quantum simulator using rubidium atoms are a few of many notable highlights.  While the number of qubits obtainable within such “conventional” approaches has steadily risen, synthesizing the first “topological qubit” remains an outstanding goal.  That ceiling may soon crumble however—vaulting topological qubits into a fascinating new chapter in the quest for scalable quantum hardware.

Why topological quantum computing?

As quantum computing progresses from minimalist quantum supremacy demonstrations to attacking real-world problems, hardware demands will naturally steepen.  In, say, a superconducting-qubit architecture, a major source of overhead arises from quantum error correction needed to combat decoherence.  Quantum-error-correction schemes such as the popular surface-code approach encode a single fault-tolerant logical qubit in many physical qubits, perhaps thousands.  The number of physical qubits required for practical applications can thus rapidly balloon.

The dream of topological quantum computing (introduced by Kitaev) is to construct hardware inherently immune to decoherence, thereby mitigating the need for active error correction.  In essence, one seeks physical qubits that by themselves function as good logical qubits.  This lofty objective requires stabilizing exotic phases of matter that harbor emergent particles known as “non-Abelian anyons”.  Crucially, nucleating non-Abelian anyons generates an exponentially large set of ground states that cannot be distinguished from each other by any local measurement.  Topological qubits encode information in those ground states, yielding two key virtues:

(1) Insensitivity to local noise.  For reference, consider a conventional qubit encoded in some two-level system, with the 0 and 1 states split by an energy \hbar \omega.  Local noise sources—e.g., random electric and magnetic fields—cause that splitting to fluctuate stochastically in time, dephasing the qubit.  In practice one can engender immunity against certain environmental perturbations.  One famous example is the transmon qubit (see “Charge-insensitive qubit design derived from the Cooper pair box” by Koch et al.) used extensively at IBM, Google, and elsewhere.  The transmon is a superconducting qubit that cleverly suppresses the effects of charge noise by operating in a regime where Josephson couplings are sizable compared to charging energies.  Transmons remain susceptible, however, to other sources of randomness such as flux noise and critical-current noise.  By contrast, topological qubits embed quantum information in global properties of the system, building in immunity against all local noise sources.  Topological qubits thus realize “perfect” quantum memory.

(2) Perfect gates via braiding.  By exploiting the remarkable phenomenon of non-Abelian statistics, topological qubits further enjoy “perfect” quantum gates: Moving non-Abelian anyons around one another reshuffles the system among the ground states—thereby processing the qubits—in exquisitely precise ways that depend only on coarse properties of the exchange.

Disclaimer: Adjectives like “perfect” should come with the qualifier “up to exponentially small corrections”, a point that we revisit below.

Experimental status

The catch is that systems supporting non-Abelian anyons are not easily found in nature.  One promising topological-qubit implementation exploits exotic 1D superconductors whose ends host “Majorana modes”—novel zero-energy degrees of freedom that underlie non-Abelian-anyon physics.  In 2010, two groups (Lutchyn et al. and Oreg et al.) proposed a laboratory realization that combines semiconducting nanowires, conventional superconductors, and modest magnetic fields.

Since then, the materials-science progress on nanowire-superconductor hybrids has been remarkable.  Researchers can now grow extremely clean, versatile devices featuring various manipulation and readout bells and whistles.  These fabrication advances paved the way for experiments that have reported increasingly detailed Majorana characteristics: tunneling signatures including recent reports of long-sought quantized response, evolution of Majorana modes with system size, mapping out of the phase diagram as a function of external parameters, etc.  Alternate explanations are still being debated though.  Perhaps the most likely culprit are conventional localized fermionic levels (“Andreev bound states”) that can imitate Majorana signatures under certain conditions; see in particular Liu et al.  Still, the collective experimental effort on this problem over the last 5+ years has provided mounting evidence for the existence of Majorana modes.  Revealing their prized quantum-information properties poses a logical next step.

Validating a topological qubit

Ideally one would like to verify both hallmarks of topological qubits noted above—“perfect” insensitivity to local noise and “perfect” gates via braiding.  We will focus on the former property, which can be probed in simpler device architectures.  Intuitively, noise insensitivity should imply long qubit coherence times.  But how do you pinpoint the topological origin of long coherence times, and in any case what exactly qualifies as “long”?

Here is one way to sharply address these questions (for more details, see our work in Aasen et al.).  As alluded to in our disclaimer above, logical 0 and 1 topological-qubit states aren’t exactly degenerate.  In nanowire devices they’re split by an energy \hbar \omega that is exponentially small in the separation distance L between Majorana modes divided by the superconducting coherence length \xi.  Correspondingly, the qubit states are not quite locally indistinguishable either, and hence not perfectly immune to local noise.  Now imagine pulling apart Majorana modes to go from a relatively poor to a perfect topological qubit.  During this process two things transpire in tandem: The topological qubit’s oscillation frequency, \omega, vanishes exponentially while the dephasing time T_2 becomes exponentially long.  That is,

scaling

This scaling relation could in fact be used as a practical definition of a topologically protected quantum memory.  Importantly, mimicking this property in any non-topological qubit would require some form of divine intervention.  For example, even if one fine-tuned conventional 0 and 1 qubit states (e.g., resulting from the Andreev bound states mentioned above) to be exactly degenerate, local noise could still readily produce dephasing.

As discussed in Aasen et al., this topological-qubit scaling relation can be tested experimentally via Ramsey-like protocols in a setup that might look something like the following:

Aasen

This device contains two adjacent Majorana wires (orange rectangles) with couplings controlled by local gates (“valves” represented by black switches).  Incidentally, the design was inspired by a gate-controlled variation of the transmon pioneered in Larsen et al. and de Lange et al.  In fact, if only charge noise was present, we wouldn’t stand to gain much in the way of coherence times: both the transmon and topological qubit would yield exponentially long T_2 times.  But once again, other noise sources can efficiently dephase the transmon, whereas a topological qubit enjoys exponential protection from all sources of local noise.  Mathematically, this distinction occurs because the splitting for transmon qubit states is exponentially flat only with respect to variations in a “gate offset” n_g.  For the topological qubit, the splitting is exponentially flat with respect to variations in all external parameters (e.g., magnetic field, chemical potential, etc.), so long as Majorana modes still survive.  (By “exponentially flat” we mean constant up to exponentially small deviations.)  Plotting the energies of the qubit states in the two respective cases versus external parameters, the situation can be summarized as follows:

energies

Outlook: Toward “topological quantum ascendancy”

These qubit-validation experiments constitute a small stepping stone toward building a universal topological quantum computer.  Explicitly demonstrating exponentially protected quantum information as discussed above would, nevertheless, go a long way toward establishing practical utility of Majorana-based topological qubits.  One might even view this goal as single-qubit-level “topological quantum ascendancy”.  Completion of this milestone would further set the stage for implementing “perfect” quantum gates, which requires similar capabilities albeit in more complex devices.  Researchers at Microsoft and elsewhere have their sights set on bringing a prototype topological qubit to life in the very near future.  It is not unreasonable to anticipate that 2018 will mark the debut of the topological qubit.  We could of course be off target.  There is, after all, still plenty of time in 2017 to prove us wrong.

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.

Banana

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.

Mess

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.