But before I dive into these new results, let me give a brief history on the topic of Majorana states and their experimental observation. The story starts with the young talented physicist Ettore Majorana, who hypothesized back in 1937 the existence of fermionic particles which were their own antiparticles. These hypothetical particles, now called Majorana fermions, were proposed in the context of elementary particle physics, but never observed. Some 60 years later, in the early 2000s, theoretical work emerged showing that Majorana fermionic states can exist as the quasiparticle excitations in certain low-dimensional superconducting systems (not a real particle as originally proposed, but otherwise having the exact same properties). Since then theorists have proposed half a dozen possible ways to realize Majorana modes using readily available materials such as superconductors, semiconductors, magnets, as well as topological insulators (for curious readers, I recommend manuscripts [1, 2, 3] for an overview of the different proposed methods to realize Majorana states in the lab).

The most fascinating thing about Majorana states is that they belong to the class of anyons, which means that they behave neither as bosons nor as fermions upon exchange. For example, if you have two identical fermionic (or bosonic) states and you exchange their positions, the quantum mechanical function describing the two states will acquire a phase factor of -1 (or +1). Anyons, on the other hand, can have an arbitrary phase factor e^{i}^{φ} upon exchange. For this reason, they are considered to be a starting point for topological quantum computation. If you want to learn more about anyons, check out the video below featuring IQIM’s Gil Refael and Jason Alicea.

Back in 2012, a group in Delft (led by Prof. Leo Kouwenhoven) announced the observation of zero-energy states in a nanoscale device consisting of a semiconductor nanowire coupled to a superconductor. These states behaved very similarly to the Majoranas that were previously predicted to occur in this system. The key word here is ‘similar’, since the behavior of these modes was not fully consistent with the theoretical predictions. Namely, the electrical conductance carried through the observed zero energy states was only about ~5% of the expected perfect transmission value for Majoranas. This part of the data was very puzzling, and immediately cast some doubts throughout the community. The physicists were quickly divided into what I will call enthusiasts (believers that these initial results indeed originated from Majorana states) and skeptics (who were pointing out that effects, other than Majoranas, can result in similarly looking zero energy peaks). And thus a great debate started.

In the coming years, experimentalists tried to observe zero energy features in improved devices, track how these features evolve with external parameters, such as gate voltages, length of the wires, etc., or focus on completely different platforms for hosting Majorana states, such as magnetic flux vortices in topological superconductors and magnetic atomic chains placed on a superconducting surface. However, these results were not enough to convince skeptics that the observed states indeed originated from the Majoranas and not some other yet-to-be-discovered phenomenon. And so, the debate continued. With each generation of the experiments some of the alternative proposed scenarios were ruled out, but the final verification was still missing.

Fast forward to the events of this Fall and the exciting recent results. The manuscript I would like to invite you to read was just posted on ArXiv a couple of weeks ago. The main result is the observation of the perfectly quantized 2e^{2}/h conductance at zero energy, the long sought signature of the Majorana states. This quantization implies that in this latest generation of semiconducting-superconducting devices zero-energy states exhibit perfect electron-hole symmetry and thus allow for perfect Andreev reflection. These remarkable results may finally end the debate and convince most of the skeptics out there.

To fully appreciate these results, it is useful to quickly review the physics of Andreev reflection (Fig. 1c-e) that occurs at the interface between a normal region with a superconductor [4]. As the electron (blue) in the normal region enters a superconductor and pulls an additional electron with it to form a Copper pair, an extra hole (red) is left behind (Fig. 1(c)). You can also think about this process as the transmission through two leads, one connecting the superconductor to the electrons and the other to the holes (Fig. 1d). This allows us to view this problem as a transmission through the double barrier that is generally low. In the presence of a Majorana state, however, there is a resonant level at zero energy which is coupled with the same amplitude with both electrons and holes. This in turn results in the resonant Andreev reflection with a perfect quantization of 2e^{2}/h (Fig. 1e). Note that, even in the configuration without Majorana modes, perfect quantization is possible but highly unlikely as it requires very careful tuning of the barrier potential (the authors did show that their quantization is robust against tuning the voltages on the gates, ruling out this possibility).

Going back to the experiments, you may wonder what made this breakthrough possible? It seems to be the combination of various factors, including using epitaxially grown superconductors and more sophisticated fabrication methods. As often happens in experimental physics, this milestone did not come from one ingenious idea, but rather from numerous technical improvements obtained by several generations of hard-working grad students and postdocs.

If you are up for more Majorana reading, you can find two more recent eye-catching manuscripts here and here. Note that the list of interesting recent Majorana papers is a mere selection by the author and not complete by any means. A few months ago, my IQIM colleagues wrote a nice blog entry about topological qubits arriving in 2018. Although this may sound overly optimistic, the recent results suggest that the field is definitely taking off. While there are certainly many challenges to be solved, we may see the next generation of experiments designed to probe control over the Majorana states quite soon. Stay tuned for more!!!!!!

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“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.

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*.

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.

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.

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.

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?

*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. *

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On the day of the interview, I meet Boyd at his office on campus at Caltech. We walk a ways down a sunlit hallway and out to a balcony through two glass doors. There’s a slight breeze in the air, a smell of nearby roses, and the temperature is perfect. It’s a picturesque day in Pasadena. We sit at a table and I ask my first question.

**How many patents do you own?**

I have seven patents. The graphene patent was really hard to get, but we got it. We just got it executed in China, so they are allowed to use it. This is particularly exciting because of all the manufacturing in China. The patent system has changed a bit, so it’s getting harder and harder. You can come up with the idea, but if disparate components have already been patented, then you can’t get the patent for combining them in a unique way. The invention has to provide a result that is unexpected or not obvious, and the patent for growing graphene with a one step process was just that. The one step process refers to cleaning the copper substrate and growing graphene under the same chemistry in a continuous manner. What used to be a two step process can be done in one.

**You don’t have to anneal the substrate to 1000 degrees before growing.**

Exactly. Annealing the copper first and then growing doesn’t allow for a nice continuous process. Removing the annealing step means the graphene is growing in an environment with significantly lower temperatures, which is important for CMOS or computer chip manufacturing.

**Which patents do you hold most dear?**

Usually in the research areas that are really cutting edge. I have three patents in plasmonics, and that was a fun area 10 years ago. It was a new area and we were doing something really exciting. When you patent something, an application may never be realized, sometimes they get used and sometimes they don’t. The graphene patent has already been licensed, so we’ve received quite a bit of traction. As far as commercial success, the graphene has been much more successful than the other ones, but plasmonics were a lot of fun. Water desalinization may be one application, and now there is a whole field of plasmonic chemistry. A company has not yet licensed it, so it may have been too far ahead of its time for application anytime soon.

**When did you realize you wanted to be a scientist?**

I liked Physics in high school, and then I had a great mentor in college, Thomas Wdowiak. Wdowiak showed me how to work in the lab. Science is one of those things where an initial spark of interest drives you into action. I became hooked, because of my love for science, the challenge it offers, and the simple fact I have fun with it. I feel it’s very important to get into the lab and start learning science as early as possible in your education.

**Were you identified as a gifted student?**

I don’t think that’s a good marker. I went to a private school early on, but no, I don’t think I was good at what they were looking for, no I wasn’t. It comes down to what you want to do. If you want to do something and you’re motivated to do it, you’ll find ways to make it happen. If you want to code, you start coding, and that’s how you get good at it. If you want to play music and have a passion for it, at first it may be your parents saying you have to go practice, but in the end it’s the passion that drives everything else.

**Did you like high school?**

I went to high school in Alabama and I had a good Physics teacher. It was not the most academic of places, and if you were into academics the big thing there was to go to medical school. I just hated memorizing things so I didn’t go that route.

**Were AP classes offered at your high school, and if so, were you an AP student?**

Yeah, I did take AP classes. My high school only had AP English and AP Math, but it was just coming onboard at that time. I took AP English because I liked the challenge and I love reading.

**Were you involved in any extracurricular activities in school?**

I earned the rank of Eagle Scout in the Boy Scouts. I also raced bicycles in high school, and I was a several time state champion. I finished high school (in America) and wanted to be a professional cyclist. So, I got involved in the American Field Service (AFS), and did an extra year of high school in Italy as an exchange student where I ended up racing with some of the best cyclists in the world all through Italy. It was a fantastic experience.

**Did you have a college in mind for your undergraduate studies? **

No, I didn’t have a school in mind. I had thought about the medical school path, so I considered taking pre-med courses at the local college, University of Alabama at Birmingham (UAB), because they have a good medical school. Then UAB called me and said I earned an academic scholarship. My father advised me that it would be a good idea to go there since it’s paid for. I could take pre-med courses and then go to medical school afterwards if I wanted. Well, I was in an honors program at the university and met an astronomer by the name Thomas Wdowiak. I definitely learned from him how to be a scientist. He also gave me a passion for being a scientist. So, after working with Wdowiak for a while, I decided I didn’t want to go to medical school, I wanted to study Physics. They just named a ridge on Mars after him, Wdowiak Ridge. He was a very smart guy, and a great experimentalist who really grew my interest in science… he was great.

**Did you do research while earning your undergraduate degree? **

Yes, Wdowiak had me in the lab working all the time. We were doing real stuff in the lab. I did a lot of undergraduate research in Astronomy, and the whole point was to get in the lab and work on science. Because I worked with Wdowiak I had one or two papers published by the time I graduated. Wdowiak taught me how to do science. And that’s the thing, you have to want to do science, have a lab or a place to practice, and then start working.

**So, he was professor and experimentalist.**

He was a very hands-on lab guy. I was in the lab breaking things and fixing things. Astronomers are fun to work with. He was an experimental astronomer who taught me, among other things, spectroscopy, vacuum technology, and much about the history of science. In fact, it was Professor Wdowiak who told me about Millikan’s famous “Machine Shop in a Vacuum” experiment that inspired the graphene discovery… it all comes back to Caltech!

**Name another scientist, other than Wdowiak, who has influenced you.**

Richard Feynman also had a big influence on me. I did not know him, but I love his books.

**Were you focused solely on academics in college, or did you have a social life as well?**

I was part of a concert committee that brought bands to the college. We had some great bands like R.E.M. and the Red Hot Chili Peppers play, and I would work as a stagehand and a roadie for the shows.

**So, you weren’t doing keg stands at fraternity parties?**

No, it wasn’t like that. I liked to go out and socialize, but no keg stands. Though, I have had friends that were very successful that did do keg stands.

**What’s your least favorite part of your job?**

You’re always having to raise funds for salaries, equipment, and supplies. It can be difficult, but once you get the funding it is a relief for the moment. As a scientist, your focus isn’t always on just the science.

**What are your responsibilities related to generating revenue for the university?**

I raise funds for my projects via grants. Part of the money goes to Caltech as overhead to pay for the facilities, lab space, and to keep the lights on.

**What do you wish you could do more of in your job?**

Less raising money. I like working in the lab, which is fun. Now that I have worked out the technique to grow graphene, I’m looking for applications. I’m searching for the next impactful thing, and then I’ll figure out the necessary steps that need to be taken to get there.

**Is there an aspect of your job that you believe would surprise people?**

You have to be entrepreneurial, you have to sell your ideas to raise money for these projects. You have to go with what’s hot in research. There are certain things that get funded and things that don’t.

**There may be some things you’re interested in, but other people aren’t, so there’s no funding.**

Yeah, there may not be a need, therefore, no funding. Right now, graphene is a big thing, because there are many applications and problems to be solved. For example, diamonds were huge back in the ‘80’s. But once they solved all the problems, research cooled off and industrial application took over.

**Is there something else you’d really rather be researching, or are the trending ideas right now in line with your interests?**

There is nothing else I’d rather be researching. I’m in a good place right now. We’re trying to commercialize the graphene research. You try to do research projects that are complementary to one another. For example, there’s a project underway, where graphene is being used for hydrogen storage in cars, that really interests me. I do like the graphene work, it’s exciting, we’ll see where that goes.

**What are the two most important personality traits essential to being a good scientist?**

Creativity. You have to think outside the box. Perseverance. I’m always reading and trying to understand something better. Curiosity is, of course, a huge part of it as well. You gotta be obsessive too, I guess. That’s more than two, sorry.

**What does it take for someone to become a scientist?**

You must have the desire to be a scientist, otherwise you’ll go be a stockbroker or something else. It’s more of a passion thing, your personality. You do have to have an aptitude for it though. If you’re getting D’s in math, physics is probably not the place for you. There’s an old joke, the medical student in physics class asks the professor, “Why do we have to take physics? We’ll never use it.” The Physics professor answers, “Physics saves lives, because it keeps idiots out of medical school.” If you like science, but you’re not so good at math, then look at less quantitative areas of science where math is not as essential. Computational physics and experimental physics will require you to be very good at math. It takes a different temperament, a different set of skills. Same curiosity, same drive and intelligence, but different temperament.

**Do you ever doubt your own abilities? Do you have insecurities about not being smart enough?**

Sure, but there’s always going to be someone out there smarter. Although, you really don’t want to ask yourself these types of questions. If you do, you’re looking down the wrong end of the telescope. Everyone has their doubts, but you need to listen to the feedback from the universe. If you’re doing something for a long time and not getting results, then that’s telling you something. Like I said, you must have a passion for what you’re doing. If people are in doubt they should read biographies of scientists and explore their mindset to discover if science seems to be a good fit for them. For a lot of people, it’s not the most fun job, it’s not the most social job, and certainly not the most glamorous type of job. Some people need more social interaction, researchers are usually a little more introverted. Again, it really depends on the person’s temperament. There are some very brilliant people in business, and it’s definitely not the case that only the brilliant people in a society go into science. It doesn’t mean you can’t be doing amazing things just because you’re not in a scientific field. If you like science and building things, then follow that path. It’s also important not to force yourself to study something you don’t enjoy.

**Scientists are often thought to work with giant math problems that are far above the intellectual capabilities of mere mortals. Have you ever been in a particular situation where the lack of a solution to a math problem was impeding progress in the lab? If so, what was the problem and did you discover the solution?**

I’m attempting to model the process of graphene growth, so I’m facing this situation right now. That’s why I have this book here. I’m trying to adapt Professor Dave Goodwin’s Cantera reactor modeling code to model the reaction kinetics in graphene (Goodwin originally developed and wrote the modeling software called Cantera). Dave was a big pioneer in diamond and he died almost 5 years ago here in Pasadena. He developed a reaction modeling code for diamond, and I’m trying to apply that to graphene. So, yeah, it’s a big math problem that I’ve been spending weeks on trying to figure out. It’s not that I’m worried about the algebra or the coding, it’s trying to figure things out conceptually.

**Do you love your job?**

I do, I’ve done it for awhile, it’s fun, and I really enjoy it. When it works, it’s great. Discovering stuff is fun and possesses a great sense of satisfaction. But it’s not always that way, it can be very frustrating. Like any good love affair, it has its peaks and valleys. Sometimes you hate it, but that’s part of the relationship, it’s like… aaarrgghh!!

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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.*

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,

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

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

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.*

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If your story is generic, it probably survived without alterations. Tolstoy’s War and Peace, on the other hand, might turn into a version of Game of Thrones. Passing along complex stories seems to be more difficult than passing on easy ones, and it also becomes more prone to errors the more friends join your circle—which makes intuitive sense.

Let’s add maths to this game, because why not. Take a difficult calculation that follows a certain procedure, such as long division of two integer numbers.

Now you perform one step of the division and pass the piece of paper on to your left. Your friend there is honest and trusts you: she doesn’t check what you did, but happily performs the next step in the division. Once she’s done, she passes the piece of paper on to her left, and so on. By the time the paper reaches you again, you hopefully have the result of the calculation, given you have enough friends to divide your favorite numbers, and given that everyone performed their steps accurately.

I’m not sure if Feynman thought about telephone when he, in 1986, proposed a method of embedding computation into eigenstates (e.g. the ground state) of a Hamiltonian, but the fact remains that the similarity is striking. Remember that writing down a Hamiltonian is a way of describing a quantum-mechanical system, for instance how the constituents of a multi-body system are coupled with each other. The ground state of such a Hamiltonian describes the lowest energy state that a system assumes when it is cooled down as far as possible. Before we dive into how the Hamiltonian looks, let’s try to understand how, in Feynman’s construction, a game of telephone can be represented as a quantum state of a physical system.

In this picture, represents a snapshot of the story or calculation at time t—in the division example, this would be the current divisor and remainder terms; so e.g. the snapshot represents the initial dividend and divisor, and the person next to you is thinking of , one step into the calculation. The label in front of the tensor sign is like a tag that you put on files on your computer, and uniquely associates the snapshot with the t-th time step. We say that the story snapshot is *entangled* with its label.

This is also an example of *quantum superposition*: all the are distinct states (the time labels, if not the story snapshots, are all unique), and by adding these states up we put them into superposition. So if we were to measure the time label, we would obtain one of the snapshots uniformly at random—it’s as if you had a cloth bag full of cards, and you blindly pick one. One side of the card will have the time label on it, while the other side contains the story snapshot. But don’t be fooled—you cannot access all story snapshots by successive measurements! Quantum states collapse; whatever measurement outcome you have dictates what the quantum state will look like after the measurement. In our example, this means that we burn the cloth bag after you pick your card; in this sense, the quantum state behaves differently than a simple juxtaposition of scraps of paper.

Nonetheless, this is the reason why we call such a quantum state a *history state*: it preserves the history of the computation, where every step that is performed is appropriately tagged. If we manage to compare all pairs of successively-labeled snapshots (without measuring them!), one can verify that the end result does, in fact, stem from a valid computation—and not just a random guess. In the division example, this would correspond to checking that each of your friends performs a correct division step.

So history states are clearly useful. But how do you design a Hamiltonian with a history state as the ground state? Is it even possible? The answer is yes, and it all boils down to verifying that two successive snapshots and are related to each other in the correct manner, e.g. that your friend on seat t+1 performs a valid division step from the snapshot prepared by the person on seat t. In fancy physics speak (aka Bra-Ket notation), we can for example write

The actual Hamiltonian will then be a sum of such terms, and one can verify that its ground state is indeed the one representing the history state we introduced above.

I’m glossing over a few details here: there is a minus sign in front of this term, and we have to add its Hermitian conjugate (flip the labels and snapshots around). But this is not essential for the argument, so let’s not go there for now. However, you’re totally right with one thing: it wouldn’t make sense to write down all snapshots themselves into the Hamiltonian! After all, if we had to calculate every snapshot transition like in advance, there would be no use to this construction. So instead, we can write

Perfect. We now have a Hamiltonian which, in its ground state, can encode the history of a computation, and if we replace the transition operator with another desired transition operator (a unitary matrix), we can perform any computation we want (more precisely, any computation that can be written as a unitary matrix; this includes anything your laptop can do). However, this is only half of the story, since we need to have a way of reading out the final answer. So let’s step back for a moment, and go back to the telephone game.

Your friends playing telephone make mistakes.

Ok, let’s assume we give them a little incentive: offer $1 to the person on your right in case the result is an even number. Will he cheat? With so much at stake?

In fact, maybe your friend is not only greedy but also dishonest: he wants to hide the fact that he miscalculates on purpose, and sometimes tells his friend on his right to make a mistake instead (maybe giving him a share of the money). So for a few of your friends close to the person at the end of the chain, there is a real incentive to cheat!

We already discussed how to write down a Hamiltonian that verifies valid computational steps. But can we do the same thing as bribing your friends to procure a certain outcome? Can we give an energy bonus to certain outcomes of the computation?

In fact, we can. Alexei Kitaev proposed adding a term to Feynman’s Hamiltonian which raises the energy of an unwanted outcome, relative to a desirable outcome. How? Again in fancy physics language,

What this term does is that it takes the history state and yields a negative energy contribution (signaled by the minus sign in front) if the last snapshot is an even number. If it isn’t, no bonus is felt; this would correspond to you keeping the dollar you promised to your friend. This simply means that in case the computation has a desirable outcome—i.e. an even number—the Hamiltonian allows a *lower* energy ground state than for any other output. Et voilà, we can distinguish between different outputs of the computation.

The true picture is, of course, a tad more complicated; generally, we give penalty terms to unwanted states instead of bonus terms to desirable ones. The reason for this is somewhat subtle, but can potentially be explained with an analogy: humans fear loss much more than they value gains of the same magnitude. Quantum systems behave in a completely opposite manner: the promise of a bonus at the end of the computation is such a great incentive that most of the weight of the history state will flock to the bonus term (for the physicists: the system now has a bound state, meaning that the wave function is localized around a specific site, and drops off exponentially quickly away from it). This makes it difficult to verify the computation far away from the bonus term.

So the Feynman-Kitaev Hamiltonian consists of three parts: one which checks each step of the computation, one which penalizes invalid outcomes—and obviously we also need to make sure the input of the computation is valid. Why? Well, are you saying you are more honest than your friends?

If there is one thing I’ve learned throughout my PhD it is that we should always ask what use a theory is. So what can we learn from this construction? Almost 20 years ago, Alexei Kitaev used Feynman’s idea to prove that estimating the ground state energy of a physical system with local interactions is hard, even on a quantum computer (for the experts: QMA-hard under the assumption of a promise gap splitting the embedded YES and NO instances). Why is estimating the ground state energy hard? The energy shift induced by the output penalty depends on the outcome of the computation that we embed (e.g. even or odd outcome). And as fun as long division is, there are much more difficult tasks we can write down as a history state Hamiltonian—in fact, it is this very freedom which makes estimating the ground state energy difficult: if we can embed *any* computation we want, estimating the induced energy shift should be *at least* as hard as actually performing the computation on a quantum computer. This has one curious implication: if we don’t expect that we can estimate the ground state energy efficiently, the physical system will take a long time to actually assume its ground state when cooled down, and potentially behave like a spin glass!

Feynman’s history state construction and the QMA-hardness proof of Kitaev were a big part of the research I did for my PhD. I formalized the case where the message is not passed on along a unique path from neighbor to neighbor, but can take an arbitrary path between beginning and end in a more complicated graph; in this way, computation can in some sense be parallelized.

Well, to be honest, the last statement is not entirely true: while there can be parallel tracks of computation from A to B, these tracks have to perform the same computation (albeit in potentially different steps); otherwise the system becomes much more complicated to analyze. The reason why this admittedly quite restricted form of branching might still be an advantage is somewhat subtle: if your computation has a lot of classical if-else cases, but you don’t have enough space on your piece of paper to store all the variables to check the conditions, it might be worth just taking a gamble: pass your message down one branch, in the hope that the condition is met. The only thing that you have to be careful about is that in case the condition *isn’t* met, you don’t produce invalid results. What use is that in physics? If you don’t have to store a lot of information locally, it means you can get away using a much lower local spin dimension for the system you describe.

Such small and physically realistic models have as of late been proposed as actual computational devices (called Hamiltonian quantum computers), where a prepared initial state is *evolved* under such a history state Hamiltonian for a specific time, in contrast to the static property of a history *ground* state we discussed above. Yet whether or not this is something one could actually build in a lab remains an open question.

Last year, Thomas Vidick invited me to visit Caltech, and I worked with IQIM postdoc Elizabeth Crosson to improve the analysis of the energy penalty that is assigned to any history state that cheats the constraints in the Feynman-Kitaev Hamiltonian. We identified some open problems and also proved limitations on the extent of the energetic penalty that these kinds of Hamiltonians can have. This summer I went back to Caltech to further develop these ideas and make progress towards a complete understanding of such “clock” Hamiltonians, which Elizabeth and I are putting together in a follow-up work that should appear soon.

It is striking how such simple idea can have so profound an implication across fields, and remain relevant, even 30 years after its first proposal.

Feynman concludes his 1986 Foundations of Physics paper with the following words.

At any rate, it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms, and quantum behavior holds dominant sway.

For my part, I hope that he was right and that history state constructions will play a part in this future.

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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.

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.

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.

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, . 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 encodes this information. 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* . This entropy is proportional to the apparent horizon’s area: .

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.*

^{1}Nothing 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.

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An emerging sub-field of semiconductor and two-dimensional research is that of Transition metal dichalcogenide (TDMC) monolayers. In particular, a monolayer of Tungsten disulfide, a TDMC, is believed to exhibit interesting semiconductor properties when exposed to circularly polarized light. My role in the Yeh Lab, as a visiting high school Physics Teacher intern, for the Summer of 2017 has been to help research and set up a vacuum chamber to study Tungsten disulfide samples under circularly polarized light.

What makes semiconductors unique is that conductivity can be controlled by doping or changes in temperature. Higher temperatures or doping can bridge the energy gap between the valence and conduction bands; in other words, electrons can start moving from one side of the material to the other. Like graphene, Tungsten disulfide has a hexagonal, symmetric crystal structure. Monolayers of transition metal dichalcogenides in such a honeycomb structure have two valleys of energy**.** One valley can interact with another valley. Circularly polarized light is used to populate one valley versus another. This gives a degree of control over the population of electrons by polarized light.

The Yeh Lab Group prides itself on making in-house the materials and devices needed for research. For example, in order to study high temperature superconductors, the Yeh Group designed and built their own scanning tunneling microscope. When they began researching graphene, instead of buying vast quantities of graphene, they pioneered new ways of fabricating it. This research topic has been no different: Wei-hsiang Lin, a Caltech graduate student, has been busy fabricating Tungsten disulfide samples via chemical vapor deposition (CVD) using Tungsten oxide and sulfur powder.

The first portion of my assignment was spent learning more about vacuum chambers and researching what to order to confine our sample into the chamber. One must determine how the electronic feeds should be attached, how many are necessary, which vacuum pump will be used, how many flanges and gaskets of each size must be purchased in order to prepare the vacuum chamber.There were also a number of flanges and parts already in the lab that needed to be examined for possible use. After triple checking the details the order was set with Kurt J. Lesker. Following a sufficient amount of anti-seize lubricant and numerous nuts, washers, and bolts, we assembled the vacuum chamber that will hold the TDMC sample.

The second part of my assignment was spent researching how to set up the optics for our experiment and ordering the necessary equipment. Once the experiment is up and running we will be using a milliWatt broad spectrum light source that is directed into a monochromator to narrow down the light to specific wavelengths for testing. Ultimately we will be evaluating the giant wavelength range of 300 nm through 1800 nm. Following the monochromator, light will be refocused by a planoconvex lens. Next, light will pass through a linear polarizer and then a circular polarizer (quarter wave plate). Lastly, the light will be refocused by a biconvex lens into the vacuum chamber and onto a 1 mm by 1 mm area of the sample.

Soon, we are excited to verify how tungsten disulfide responds to circularly polarized light. Does our sample resonate at the exact same wavelengths as the first labs found? Why or why not? What other unique properties are observed? How can they be explained? How is the Hall Effect observed? What does this mean for the possible applications of semiconductors? How can the transfer of information from one valley to another be used in advanced electronics for communication? Then, similar exciting experimentation will take place with graphene under circularly polarized light.

I love the sharp contrast of the high-energy, adolescent classroom to the quiet, calm of the lab. I am grateful for getting to learn a different and new-to-me area of Physics during the summer. Yes, I remember studying polarization and semiconductors in high school and as an undergraduate. But it is completely different to set up an experiment from scratch, to be a part of groundbreaking research in these areas. And it is just fun to get to work with your hands and build research equipment at a world leading research university. Sometimes Science teachers can get bogged down with all the paperwork and meetings. I am grateful to have had this fabulous opportunity during the summer to work on applied Science and to be re-energized in my love for Physics. I look forward to meeting my new batch of students in a few short weeks to share my curiosity and joy for learning how the world works with them.

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It all starts with how far are we going to drive in order to see totality. My family and I are currently in Colorado, so we are relatively close to the path of darkness in Wyoming. I thought about trying to book a hotel room. But if you’d like to see the dusk in Lusk, here is what you get:

Let us just say that I became quite acquainted with small-town WY and any-ville NE before giving up. Driving in the same day for 10 hours with my two children, ages 4 and 5, was not an option. So I will have to be content with 90% coverage.

90% coverage sounds like it is good enough… But when you think about the sun and its output, you realize that it won’t actually be very dark. The sun gives out about 1kW of light and heat per square meter. 90% of that still leaves us with 100W per meter squared. Imagine a room lit by a square array of 100W incandescent bulbs at one meter apart from each other. Not so dark. Luckily, we have really dark eclipse glasses.

All things considered, it is a huge coincidence that the moon is just about the right size and distance from the earth to block the sun exactly, .

On a more personal note, another coincidence of a lesser cosmic meaning is that my wife, Jocelyn Holland, a professor of comparative literature at UCSB and Caltech, has also done research on eclipses. She has recently published an essay that shows how, for nineteenth-century observers, and astronomers in particular, the unique darkness associated with the eclipse during totality shook their subjective experience of time. Readers might want to share their own personal experiences at the end of this blog so that we can see how a twenty-first century perspective compares.

As for Jocelyn’s paper, here is a redacted ‘poetry for scientists’ excerpt from it.

Eclipses are well-known objects of scientific study but it is just as true that, throughout history, they have been perceived as the most supernatural of events, permitting superstition and fear to intrude. As a result, eclipses have frequently been used across cultures, in particular, by the community of scientists and scholars, as an index of “enlightenment.” Astronomers in the nineteenth century – an epoch that witnessed several mathematical advances in the calculation of solar and lunar eclipses, as exemplified in the work of Friedrich Bessel – looked back at prior centuries with scorn, mocking the irrational fears of times past. The German astronomer Ludwig August Busch, in text published shortly before a total eclipse in 1851, points out with some smugness that scarcely 200 years before then, in Germany, “the majority of the population threw itself upon its knees in desperation during a total eclipse,” and that the composure with which the next eclipse will be greeted is “the most certain proof how only science is able to conquer prejudices and superstition which prior centuries have gone through.”

Two solar eclipses were witnessed by Europeans in the mid-nineteenth century, on July 8th, 1842 and July 28th, 1851, when the first photographic image of an eclipse was made by Julius Berkowski (see below).

What Berkowski’s daguerreotype cannot convey, however, is a particular perception shared by both professional astronomers and amateur observers of these eclipses: that the darkness of the eclipse’s totality is unlike any darkness they had experienced before. As it turns out, this perception posed a challenge to their self-proclaimed enlightenment.

There was already a historical record in place describing the strange darkness of a total eclipse. As another nineteenth-century astronomer, Jacob Lehmann, phrased it, “How is it now to be explained, namely what several observers report during the eclipse of 1706, that the darkness at the time of the total occultation of the sun compares neither to night nor to dusk, but rather is of a particular kind. What is this particular kind?” The strange darkness of the eclipse presents a problem that one can state quite simply in temporal terms: it corresponds to no prior experience of natural light or time of day.

It might strike us as odd that August Ludwig Busch, the same astronomer who derided the superstition of prior generations, writes the following with reference to eclipses past, and in anticipation of the eclipse of 1851:

You will all remember the inexplicable melancholic frame of mind which one already experiences during large if not even total eclipses, when all objects appear in a dull, unusual light, there lies namely in the sight of great plains and far-spread drifts, upon which trees and rocks, although still illuminated by sunlight, still seem to cast no shadow, such a thing which causes mourning, that one is involuntarily overcome by horror. This feeling should occur more intensely in people when, during the total eclipse, a very peculiar darkness arrives which can be named neither night nor dusk.

August Ludwig Busch.

One can say that the perceived relationship between the quality of light and time of day is based on expectations that are so innate as to be taken as infallible until experience teaches otherwise. It is natural for us to use the available light in the sky as the basis for a measure of time when no time-keeping piece is on hand. The cyclical predictability of a steady increase and decrease in available light during the course of the day, however, in addition to all the nuances of how the midday light differs from dawn and twilight, is less than helpful in the rare event of an eclipse. The quality of light does not correspond to any experience of lived time. As a consequence, not only August Ludwig Busch, but also numerous other observers, attributed it to death, as if for lack of an alternative.

For all their claims of rationality, nineteenth-century observers were troubled by this darkness that conformed to no experienced time of day. It signaled to them, among other things, that time and light are out of joint. In short, as natural and as it may be, a full solar eclipse has, historically, posed a real challenge: not to the predictability of mechanical time-keeping, but rather to a very human experience of time.

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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.

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 . 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

** (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.

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.

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 that is exponentially small in the separation distance between Majorana modes divided by the superconducting coherence length . 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, , vanishes exponentially while the dephasing time becomes exponentially long. That is,

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:

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 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” . 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:

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.

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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.

The wave function, , 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 , 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 . 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.

Though 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.

*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.

*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.

*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 . 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”).

*Yes – for small systems. *In my thesis, I considered a toy system of 4 spin- 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.

*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* :).

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*

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