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 cord: 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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*Dada* looks up from scrubbing the floor.

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

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

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

His daughter had a sign(-language) problem.

So does *Dada,* MIT professor Aram Harrow. Aram studies quantum information theory. His interests range from complexity to matrices, from resource theories to entropies. He’s blogged for *The Quantum Pontiff,* and he studies—including with IQIM postdoc Elizabeth Crosson—*the 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 corresponds to some particle configuration indexed by : If the system has some amount of energy, particles occupy certain sites and might not occupy others. If the system has a different amount of energy, particles occupy different sites. A *Hamiltonian*, a mathematical object denoted by encodes the energies and the configurations. We represent with a matrix, a square grid of numbers.

Suppose that the chunk has a temperature . 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*, .

How? We would list the configurations . With each configuration, we would associate the weight . We would sum the weights: .

Easier—like feeding a 13-month-old—said than done. Let denote the number of qubits in the chunk. If 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 . Can we approximate ?

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

We replace the weights with different weights—numbers formed from a matrix that represents . If 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 , here and here, as do many other researchers.

We call Hamiltonians that lack the sign problem “stoquastic,” which I think fanquastic.^{1 }Stay tuned for a blog post about stoquasticity by Elizabeth.

What if 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.

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

One sign problem down; one to go.

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

^{1}For 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.

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So you can imagine my joy of finally seeing a fully anti-entropic superhero appearing on my facebook account (physics enthusiasts out there – the project is seeking support on Kickstarter):

Apart from the plug for Assa Auerbach’s project (which, for full disclosure, I have just supported), I would like to use this as an excuse to share my lessons about entropy. With the same level of seriousness. Here they are, in order of increasing entropy.

**1. Cost of entropy.** Entropy is always marketed as a very palpable thing. Disorder. In class, however, it is calculated via an enumeration of the ‘microscopic states of the system’. For an atomic gas I know how to calculate the entropy (throw me at the blackboard in the middle of the night, no problem. Bosons or Fermions – anytime!) But how can the concept be applied to our practical existence? I have a proposal:

*Quantify entropy by the cost (in $’s) of cleaning up the mess!*

Examples can be found at all scales. For anything household-related, we should use the constant. =$25/hour for my housekeeper. You break a glass – it takes about 10 minutes to clean. That puts the entropy of the wreckage at $4.17. Having a birthday party takes about 2 hours to clean up: $50 entropy.

Another insight which my combined experience as professor and parent has produced:

**2. Conjecture:** Babies are maximally efficient topological entropy machines. If you raised a 1 year-old you know exactly what I mean. You can at least guess why maximum efficiency. But why topological? A baby sauntering through the house leaves a string of destruction behind itself. The baby is a mess-creation string-operator! If you start lagging behind, doom will emerge – hence the maximum efficiency. By the way, the only strategy viable is to undo the damage as it happens. But this blog post is about entropy, not about parenting.

In fact, this allows us to establish a conversion of entropy measured in units, to its, clearly more natural, measure in dollar units. A baby eats about 1000kCal/day=4200kJ/day. To fully deal with the consequences, we need a housekeeper to visit about once a week. 4200kJ/day times 7 days=29400 kJoules. These are consumed at T=300K. So an entropy of S=Q/T~100J/K, which is also S~ in dimensionless units, converts to S~$120, which is the cost of our weekly housekeeper visit. This gives a value of $ per entropy of a two-level system. Quite a reasonable bang for the buck, don’t you think?

**3. My conjecture (2) fails.** The second law of thermodynamics is an inequality. Entropy Q/T. Why does the conjecture fail? Babies are not ‘maximal’. Consider presidents. Consider the mess that the government can make. It is at the scale of trillions per year. $ . Using the rigorous conversion rule established above, this corresponds to two-level systems. Which happens to quite precisely match the combined number of electrons present in the human bodies of all our military personnel. But the mess, however, is created by very few individuals.

Given the large amounts of taxpayer money we dish out to deal with entropy in the world, Auerbach’s book is bound to make a big impact. In fact, maybe Max the demon would one day be nominated for the national medal of freedom, or at least be inducted into the National Academy of Sciences.

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

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

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

A sequence of instructions for a

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

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

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

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

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

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

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

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

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