# What Clocks have to do with Quantum Computation

Have you ever played the game “telephone”? You might remember it from your nursery days, blissfully oblivious to the fact that quantum mechanics governs your existence, and not yet wondering why Fox canceled Firefly. For everyone who forgot, here is the gist of the game: sit in a circle with your friends. Now you think of a story (prompt: a spherical weapon that can destroy planets). Once you have the story laid out in your head, tell it to your neighbor on your left. She takes the story and tells it to her friend on her left. It is important to master the art of whispering for this game: you don’t want to be overheard when the story is passed on. After one round, the friend on your right tells you what he heard from his friend on his right. Does the story match your masterpiece?

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.

## So what does this have to do with physics or quantum computation?

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, $| \psi_t \rangle$ 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 $| \psi_1 \rangle$ represents the initial dividend and divisor, and the person next to you is thinking of $| \psi_2 \rangle$, one step into the calculation. The label $|t\rangle$ in front of the tensor sign $\otimes$ is like a tag that you put on files on your computer, and uniquely associates the snapshot $| \psi_t \rangle$ 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 $|t\rangle\otimes|\psi_t\rangle$ 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 $| \psi_t \rangle$ and $| \psi_{t+1} \rangle$ 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 $| \psi_2 \rangle \langle \psi_1 |$ 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 $\mathbf U_\text{DIVISION}$ 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.

## Can you motivate your friends to cheat?

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!

## Can we motivate spins 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 $| \psi_T \rangle$ 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?

## Physical Implications of History State Hamiltonians

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 $1/\text{poly}$ 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.

# Decoding (the allure of) the apparent horizon

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

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

After 26 hours, I understood apparent horizons like so.

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

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

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

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.

UBC near twilight

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

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

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

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

Alexander von Humboldt

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

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

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

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

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

# Topological qubits: Arriving in 2018?

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

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

# Why topological quantum computing?

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

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

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

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

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

# Experimental status

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

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

# Validating a topological qubit

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

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

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 $T_2$ times.  But once again, other noise sources can efficiently dephase the transmon, whereas a topological qubit enjoys exponential protection from all sources of local noise.  Mathematically, this distinction occurs because the splitting for transmon qubit states is exponentially flat only with respect to variations in a “gate offset” $n_g$.  For the topological qubit, the splitting is exponentially flat with respect to variations in all external parameters (e.g., magnetic field, chemical potential, etc.), so long as Majorana modes still survive.  (By “exponentially flat” we mean constant up to exponentially small deviations.)  Plotting the energies of the qubit states in the two respective cases versus external parameters, the situation can be summarized as follows:

# Outlook: Toward “topological quantum ascendancy”

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

# Taming wave functions with neural networks

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

# Wave functions in the wild

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

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

# Taming the beast

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

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

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

# Brains $\gg$ Brawn

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

The general idea of my thesis.

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

# Cool – but does it work?

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

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

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

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

# Outside the ivory tower

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

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

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

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

# The sign problem(s)

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

Dada looks up from scrubbing the floor.

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

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

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

His daughter had a sign(-language) problem.

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

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

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

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

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

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

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

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

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

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

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

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

One sign problem down; one to go.

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

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

# The power of information

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

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

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

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

Sara versus the ants

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

The green lines, she said, are undecided ants.

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

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

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

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

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

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

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

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

Chinese viewing stone in Yuyuan Garden in Shanghai

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

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

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

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

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

# Glass beads and weak-measurement schemes

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

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

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

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

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

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

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

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

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

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

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

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

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

That’s why I work at the IQIM.

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

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

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

4Superconducting circuits are tiny, cold quantum circuits.

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

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