# The paper that begged for a theme song

A year ago, the “I’m a little teapot” song kept playing in my head.

I was finishing a collaboration with David Limmer, a theoretical chemist at the University of California Berkeley. David studies quantum and classical systems far from equilibrium, including how these systems exchange energy and information with their environments. Example systems include photoisomers.

A photoisomer is a molecular switch. These switches appear across nature and technologies. We have photoisomers in our eyes, and experimentalists have used photoisomers to boost solar-fuel storage. A photoisomer has two functional groups, or collections of bonded atoms, attached to a central axis.

Your average-Joe photoisomer spends much of its life in equilibrium, exchanging heat with room-temperature surroundings. The molecule has the shape above, called the cis configuration. Imagine shining a laser or sunlight on the photoisomer. The molecule can absorb a photon, or particle of light, gaining energy. The energized switch has the opportunity to switch: One chemical group can rotate downward. The molecule will occupy its trans configuration.

The molecule now has more energy than it had while equilibrium, albeit less energy than it had right after absorbing the photon. The molecule can remain in this condition for a decent amount of time. (Experts: The molecule occupies a metastable state.) That is, the molecule can store sunlight. For that reason, experimentalists at Harvard and MIT attached photoisomers to graphene nanotubules, improving the nanotubules’ storage of solar fuel.

With what probability does a photoisomer switch upon absorbing a photon? This question has resisted easy answering, because photoisomers prove difficult to model: They’re small, quantum, and far from equilibrium. People have progressed by making assumptions, but such assumptions can lack justifications or violate physical principles. David wanted to derive a simple, general bound—of the sort in which thermodynamicists specialize—on a photoisomer’s switching probability.

He had a hunch as to how he could derive such a bound. I’ve blogged, many times, about thermodynamic resource theories. Thermodynamic resource theories are simple models, developed in quantum information theory, for exchanges of heat, particles, information, and more. These models involve few assumptions: the conservation of energy, quantum theory, and, to some extent, the existence of a large environment (Markovianity). With such a model, David suspected, he might derive his bound.

I knew nothing about photoisomers when I met David, but I knew about thermodynamic resource theories. I’d contributed to their development, to the theorems that have piled up in the resource-theory corner of quantum information theory. Then, the corner had given me claustrophobia. Those theorems felt so formal, abstract, and idealized. Formal, abstract theory has drawn me ever since I started studying physics in college. But did resource theories model physical reality? Could they impact science beyond our corner of quantum information theory? Did resource theories matter?

I called for connecting thermodynamic resource theories to physical reality four years ago, in a paper that begins with an embarrassing story about me. Resource theorists began designing experiments whose results should agree with our theorems. Theorists also tried to improve the accuracy with which resource theories model experimentalists’ limitations. See David’s and my paper for a list of these achievements. They delighted me, as a step toward the broadening of resource theories’ usefulness.

Like any first step, this step pointed toward opportunities. Experiments designed to test our theorems essentially test quantum mechanics. Scientists have tested quantum mechanics for decades; we needn’t test it much more. Such experimental proposals can push experimentalists to hone their abilities, but I hoped that the community could accomplish more. We should be able to apply resource theories to answer questions cultivated in other fields, such as condensed matter and chemistry. We should be useful to scientists outside our corner of quantum information.

David’s idea lit me up like photons on a solar-fuel-storage device. He taught me about photoisomers, I taught him about resource theories, and we derived his bound. Our proof relies on the “second laws of thermodynamics.” These abstract resource-theory results generalize the second law of thermodynamics, which helps us understand why time flows in only one direction. We checked our bound against numerical simulations (experts: of Lindbladian evolution). Our bound is fairly tight if the photoisomer has a low probability of absorbing a photon, as in the Harvard-MIT experiment.

Experts: We also quantified the photoisomer’s coherences relative to the energy eigenbasis. Coherences can’t boost the switching probability, we concluded. But, en route to this conclusion, we found that the molecule is a natural realization of a quantum clock. Our quantum-clock modeling extends to general dissipative Landau-Zener transitions, prevalent across condensed matter and chemistry.

As I worked on our paper one day, a jingle unfolded in my head. I recognized the tune first: “I’m a little teapot.” I hadn’t sung that much since kindergarten, I realized. Lyrics suggested themselves:

I’m a little isomer
with two hands.
Here is my cis pose;
here is my trans.

Stand me in the sunlight;
watch me spin.
I’ll keep solar
energy in!

The song lodged itself in my head for weeks. But if you have to pay an earworm to collaborate with David, do.

# Quantum Error Correction with Molecules

In the previous blog post (titled, “On the Coattails of Quantum Supremacy“) we started with Google and ended up with molecules! I also mentioned a recent paper by John Preskill, Jake Covey, and myself (see also this videoed talk) where we assume that, somewhere in the (near?) future, experimentalists will be able to construct quantum superpositions of several orientations of molecules or other rigid bodies. Next, I’d like to cover a few more details on how to construct error-correcting codes for anything from classical bits in your phone to those future quantum computers, molecular or otherwise.

# Classical error correction: the basics

Error correction is concerned with the design of an encoding that allows for protection against noise. Let’s say we want to protect one classical bit, which is in either “0” or “1”. If the bit is say in “0”, and the environment (say, the strong magnetic field from a magnet you forgot was laying next to your hard drive) flipped it to “1” without our knowledge, an error would result (e.g., making your phone think you swiped right!)

Now let’s encode our single logical bit into three physical bits, whose $2^3=8$ possible states are represented by the eight corners of the cube below. Let’s encode the logical bit as “0” —> 000 and “1” —> 111, corresponding to the corners of the cube marked by the black and white ball, respectively. For our (local) noise model, we assume that flips of only one of the three physical bits are more likely to occur than flips of two or three at the same time.

Error correction is, like many Hollywood movies, an origin story. If, say, the first bit flips in our above code, the 000 state is mapped to 100, and 111 is mapped to 011. Since we have assumed that the most likely error is a flip of one of the bits, we know upon observing that 100 must have come from the clean 000, and 011 from 111. Thus, in either case of the logical bit being “0” or “1”, we can recover the information by simply observing which state the majority of the bits are in. The same things happen when the second or third bits flip. In all three cases, the logical “0” state is mapped to one of its three neighboring points (above, in blue) while the logical “1” is mapped to its own three points, which, crucially, are distinct from the neighbors of “0”. The set of points $\{000,100,010,001\}$ that are closer to 000 than to 111 is called a Voronoi tile.

Now, let’s adapt these ideas to molecules. Consider the rotational states of a dumb-bell molecule consisting of two different atoms. (Let’s assume that we have frozen this molecule to the point that the vibration of the inter-atomic bond is limited, essentially creating a fixed distance between the two atoms.) This molecule can orient itself in any direction, and each such orientation can be represented as a point $\mathbf{v}$ on the surface of a sphere. Now let us encode a classical bit using the north and south poles of this sphere (represented in the picture below as a black and a white ball, respectively). The north pole of the sphere corresponds to the molecule being parallel to the z-axis, while the south pole corresponds to the molecule being anti-parallel.

This time, the noise consists of small shifts in the molecule’s orientation. Clearly, if such shifts are small, the molecule just wiggles a bit around the z-axis. Such wiggles still allow us to infer that the molecule is (mostly) parallel and anti-parallel to the axis, as long as they do not rotate the molecule all the way past the equator. Upon such correctable rotations, the logical “0” state — the north pole — is mapped to a point in the northern hemisphere, while logical “1” — the south pole — is mapped to a point in the southern hemisphere. The northern hemisphere forms a Voronoi tile of the logical “0” state (blue in the picture), which, along with the corresponding tile of the logical “1” state (the southern hemisphere), tiles the entire sphere.

Quantum error correction

To upgrade these ideas to the quantum realm, recall that this time we have to protect superpositions. This means that, in addition to shifting our quantum logical state to other states as before, noise can also affect the terms in the superposition itself. Namely, if, say, the superposition is equal — with an amplitude of $+1/\sqrt{2}$ in “0” and $+1/\sqrt{2}$ in “1” — noise can change the relative sign of the superposition and map one of the amplitudes to $-1/\sqrt{2}$. We didn’t have to worry about such sign errors before, because our classical information would always be the definite state of “0” or “1”. Now, there are two effects of noise to worry about, so our task has become twice as hard!

Not to worry though. In order to protect against both sources of noise, all we need to do is effectively stagger the above constructions. Now we will need to design a logical “0” state which is itself a superposition of different points, with each point separated from all of the points that are superimposed to make the logical “1” state.

Diatomic molecules: For the diatomic molecule example, consider superpositions of all four corners of two antipodal tetrahedra for the two respective logical states.

The logical “0” state for the quantum code is now itself a quantum superposition of orientations of our diatomic molecule corresponding to the four black points on the sphere to the left (the sphere to the right is a top-down view). Similarly, the logical “1” quantum state is a superposition of all orientations corresponding to the white points.

Each orientation (black or white point) present in our logical states rotates under fluctuations in the position of the molecule. However, the entire set of orientations for say logical “0” — the tetrahedron — rotates rigidly under such rotations. Therefore, the region from which we can successfully recover after rotations is fully determined by the Voronoi tile of any one of the corners of the tetrahedron. (Above, we plot the tile for the point at the north pole.) This cell is clearly smaller than the one for classical north-south-pole encoding we used before. However, the tetrahedral code now provides some protection against phase errors — the other type of noise that we need to worry about if we are to protect quantum information. This is an example of the trade-off we must make in order to protect against both types of noise; a licensed quantum mechanic has to live with such trade-offs every day.

Oscillators: Another example of a quantum encoding is the GKP encoding in the phase space of the harmonic oscillator. Here, we have at our disposal the entire two-dimensional plane indexing different values of position and momentum. In this case, we can use a checkerboard approach, superimposing all points at the centers of the black squares for the logical “0” state, and similarly all points at the centers of the white squares for the logical “1”. The region depicting correctable momentum and position shifts is then the Voronoi cell of the point at the origin: if a shift takes our central black point to somewhere inside the blue square, we know (most likely) where that point came from! In solid state circles, the blue square is none other than the primitive or unit cell of the lattice consisting of points making up both of the logical states.

Asymmetric molecules (a.k.a. rigid rotors): Now let’s briefly return to molecules. Above, we considered diatomic molecules that had a symmetry axis, i.e., that were left unchanged under rotations about the axis that connects the two atoms. There are of course more general molecules out there, including ones that are completely asymmetric under any possible (proper) 3D rotation (see figure below for an example).

BONUS: There is a subtle mistake relating to the geometry of the rotation group in the labeling of this figure. Let me know if you can find it in the comments!

All of the orientations of the asymmetric molecule, and more generally a rigid body, can no longer be parameterized by the sphere. They can be parameterized by the 3D rotation group $\mathsf{SO}(3)$: each orientation of an asymmetric molecule is labeled by the 3D rotation necessary to obtain said orientation from a reference state. Such rotations, and in turn the orientations themselves, are parameterized by an axis $\mathbf{v}$ (around which to rotate) and an angle $\omega$ (by which one rotates). The rotation group $\mathsf{SO}(3)$ luckily can still be viewed by humans on a sheet of paper. Namely, $\mathsf{SO}(3)$ can be thought of as a ball of radius $\pi$ with opposite points identified. The direction of each vector $\omega\mathbf{v}$ lying inside the ball corresponds to the axis of rotation, while the length corresponds to the angle. This may take some time to digest, but it’s not crucial to the story.

So far we’ve looked at codes defined on cubes of bits, spheres, and phase-space lattices. Turns out that even $\mathsf{SO}(3)$ can house similar encodings! In other words, $\mathsf{SO}(3)$ can also be cut up into different Voronoi tiles, which in turn can be staggered to create logical “0” and “1” states consisting of different molecular orientations. There are many ways to pick such states, corresponding to various subgroups of $\mathsf{SO}(3)$. Below, we sketch two sets of black/white points, along with the Voronoi tile corresponding to the rotations that are corrected by each encoding.

Voronoi tiles of the black point at the center of the ball representing the 3D rotation group, for two different molecular codes. This and the Voronoi cells corresponding to the other points tile together to make up the entire ball. 3D printing all of these tiles would make for cool puzzles!

In closing…

Achieving supremacy was a big first step towards making quantum computing a practical and universal tool. However, the largest obstacles still await, namely handling superposition-poisoning noise coming from the ever-curious environment. As quantum technologies advance, other possible routes for error correction are by encoding qubits in harmonic oscillators and molecules, alongside the “traditional” approach of using arrays of physical qubits. Oscillator and molecular qubits possess their own mechanisms for error correction, and could prove useful (granted that the large high-energy space required for the procedures to work can be accessed and controlled). Even though molecular qubits are not yet mature enough to be used in quantum computers, we have at least outlined a blueprint for how some of the required pieces can be built. We are by no means done however: besides an engineering barrier, we need to further develop how to run robust computations on these exotic spaces.

Author’s note: I’d like to acknowledge Jose Gonzalez for helping me immensely with the writing of this post, as well as for drawing the comic panels in the previous post. The figures above were made possible by Mathematica 12.

# On the Coattails of Quantum Supremacy

Most readers have by now heard that Google has “achieved” quantum “supremacy”. Notice the only word not in quotes is “quantum”, because unlike previous proposals that have also made some waves, quantumness is mostly not under review here. (Well, neither really are the other two words, but that story has already been covered quite eloquently by John, Scott, and Toby.) The Google team has managed to engineer a device that, although noisy, can do the right thing a large-enough fraction of the time for people to be able to “quantify its quantumness”.

However, the Google device, while less so than previous incarnations, is still noisy. Future devices like it will continue to be noisy. Noise is what makes quantum computers so darn difficult to build; it is what destroys the fragile quantum superpositions that we are trying so hard to protect (remember, unlike a classical computer, we are not protecting things we actually observe, but their superposition).

Protecting quantum information is like taking your home-schooled date (who has lived their entire life in a bunker) to the prom for the first time. It is a fun and necessary part of a healthy relationship to spend time in public, but the price you pay is the possibility that your date will hit it off with someone else. This will leave you abandoned, dancing alone to Taylor Swift’s “You Belong With Me” while crying into your (spiked?) punch.

The high school sweetheart/would-be dance partner in the above provocative example is the quantum superposition — the resource we need for a working quantum computer. You want it all to yourself, but your adversary — the environment — wants it too. No matter how much you try to protect it, you’ll have to observe it eventually (after all, you want to know the answer to your computation). And when you do (take your date out onto the crowded dance floor), you run the risk of the environment collapsing the information before you do, leaving you with nothing.

Protecting quantum information is also like (modern!) medicine. The fussy patient is the quantum information, stored in delicate superposition, while quantumists are the doctors aiming to prevent the patient from getting sick (or “corrupted”). If our patient incurs say “quasiparticle poisoning”, we first diagnose the patient’s syndromes, and, based on this diagnosis, apply procedures like “lattice surgery” and “state injection” to help our patient successfully recover.

Error correction with qubits

Error correction sounds hard, and it should! Not to fear: plenty of very smart people have thought hard about this problem, and have come up with a plan — to redundantly encode the quantum superposition in a way that allows protection from errors caused by noise. Such quantum error-correction is an expansion of the techniques we currently use to protect classical bits in your phone and computer, but now the aim is to protect, not the definitive bit states 0 or 1, but their quantum superpositions. Things are even harder now, as the protection machinery has to do its magic without disturbing the superposition itself (after all, we want our quantum calculation to run to its conclusion and hack your bank).

For example, consider a qubit — the fundamental quantum unit represented by two shelves (which, e.g., could be the ground and excited states of an atom, the absence or presence of a photon in a box, or the zeroth and first quanta of a really cold LC circuit). This qubit can be in any quantum superposition of the two shelves, described by 2 probability amplitudes, one corresponding to each shelf. Observing this qubit will collapse its state onto either one of the shelves, changing the values of the 2 amplitudes. Since the resource we use for our computation is precisely this superposition, we definitely do not want to observe this qubit during our computation. However, we are not the only ones looking: the environment (other people at the prom: the trapping potential of our atom, the jiggling atoms of our metal box, nearby circuit elements) is also observing this system, thereby potentially manipulating the stored quantum state without our knowledge and ruining our computation.

Now consider 50 such qubits. Such a space allows for a superposition with $2^{50}$ different amplitudes (instead of just $2^1$ for the case of a single qubit). We are once again plagued by noise coming from the environment. But what if we now, less ambitiously, want to store only one qubit’s worth of information in this 50-qubit system? Now there is room to play with! A clever choice of how to do this (a.k.a. the encoding) helps protect from the bad environment.

The entire prospect of building a bona-fide quantum computer rests on this extra overhead or quantum redundancy of using a larger system to encode a smaller one. It sounds daunting at first: if we need 50 physical qubits for each robust logical qubit, then we’d need “I-love-you-3000” physical qubits for 60 logical ones? Yes, this is a fact we all have to live with. But granted we can scale up our devices to that many qubits, there is no fundamental obstacle that prevents us from then using error correction to make next-level computers.

To what extent do we need to protect our quantum superposition from the environment? It would be too ambitious to protect it from a meteor shower. Or a power outage (although that would be quite useful here in California). So what then can we protect against?

Our working answer is local noise — noise that affects only a few qubits that are located near each other in the device. We can never be truly certain if this type of noise is all that our quantum computers will encounter. However, our belief that this is the noise we should focus on is grounded in solid physical principles — that nature respects locality, that affecting things far away from you is harder than making an impact nearby. (So far Google has not reported otherwise, although much more work needs to be done to verify this intuition.)

The harmonic oscillator

In what other ways can we embed our two-shelf qubit into a larger space? Instead of scaling up using many physical qubits, we can utilize a fact that we have so far swept under the rug: in any physical system, our two shelves are already part of an entire bookcase! Atoms have more than one excited state, there can be more than one photon in a box, and there can be more than one quantum in a cold LC circuit. Why don’t we use some of that higher-energy space for our redundant encoding?

The noise in our bookcase will certainly be different, since the structure of the space, and therefore the notion of locality, is different. How to cope with this? The good news is that such a space — the space of the harmonic oscillator — also has a(t least one) natural notion of locality!

Whatever the incarnation, the oscillator has associated with it a position and momentum (different jargon for these quantities may be used, depending on the context, but you can just think of a child on a swing, just quantized). Anyone who knows the joke about Heisenberg getting pulled over, will know that these two quantities cannot be set simultaneously.

Nevertheless, local errors can be thought of as small shifts in position or momentum, while nonlocal errors are ones that suddenly shift our bewildered swinging quantized child from one side of the swing to the other.

Armed with a local noise model, we can extend our know-how from multi-qubit land to the oscillator. One of the first such oscillator codes were developed by Gottesman, Kitaev, and Preskill (GKP). Proposed in 2001, GKP encodings posed a difficult engineering challenge: some believed that GKP states could never be realized, that they “did not exist”. In the past few years however, GKP states have been realized nearly simultaneously in two experimental platforms. (Food for thought for the non-believers!)

Parallel to GKP codes, another promising oscillator encoding using cat states is also being developed. This encoding has historically been far easier to create experimentally. It is so far the only experimental procedure achieving the break-even point, at which the actively protected logical information has the same lifetime as the system’s best unprotected degree of freedom.

Can we mix and match all of these different systems? Why yes! While Google is currently trying to build the surface code out of qubits, using oscillators (instead of qubits) for the surface code and encoding said oscillators either in GKP (see related IBM post) [1,2,3] or cat [4,5] codes is something people are seriously considering. There is even more overhead, but the extra information one gets from the correction procedure might make for a more fault-tolerant machine. With all of these different options being explored, it’s an exciting time to be into quantum!

Molecules?

It turns out there are still other systems we can consider, although because they are sufficiently more “out there” at the moment, I should first say “bear with me!” as I explain. Forget about atoms, photons in a box, and really cold LC circuits. Instead, consider a rigid 3-dimensional object whose center of mass has been pinned in such a way that the object can rotate any way it wants. Now, “quantize” it! In other words, consider the possibility of having quantum superpositions of different orientations of this object. Just like superpositions of a dead and alive cat, of a photon and no photon, the object can be in quantum superposition of oriented up, sideways, and down, for example. Superpositions of all possible orientations then make up our new configuration space (read: playground), and we are lucky that it too inherits many of the properties we know and love from its multi-qubit and oscillator cousins.

Examples of rigid bodies include airplanes (which can roll, pitch and yaw, even while “fixed” on a particular trajectory vector) and robot arms (which can rotate about multiple joints). Given that we’re not quantizing those (yet?), what rigid body should we have in mind as a serious candidate? Well, in parallel to the impressive engineering successes of the multi-qubit and oscillator paradigms, physicists and chemists have made substantial progress in trapping and cooling molecules. If a trapped molecule is cold enough, it’s vibrational and electronic states can be neglected, and its rotational states form exactly the rigid body we are interested in. Such rotational states, as far as we can tell, are not in the realm of Avengers-style science fiction.

The idea to use molecules for quantum computing dates all the way back to a 2001 paper by Dave DeMille, but in a recent paper by Jacob Covey, John Preskill, and myself, we propose a framework of how to utilize the large space of molecular orientations to protect against (you guessed it!) a type of local noise. In the second part of the story, called “Quantum Error Correction with Molecules“, I will cover a particular concept that is not only useful for a proper error-correcting code (classical and quantum), but also one that is quite fun to try and understand. The concept is based on a certain kind of tiling, called Voronoi tiles or Thiessen polygons, which can be used to tile anything from your bathroom floor to the space of molecular orientations. Stay tuned!

# Bas|ket>ball: A Game for Young Students Learning Quantum Computing

It is no secret that quantum computing has recently become one of the trendiest topics within the physics community, gaining financial support and good press at an ever increasing pace. The new technology not only promises huge advances in information processing, but it also – in theory – has the potential to crack the encryption that currently protects sensitive information inside governments and businesses around the world. Consequently, quantum research has extended beyond academic groups and has entered the technical industry, creating new job opportunities for both experimentalists and theorists. However, in order for this technology to become a reality, we need qualified engineers and scientists that can fill these positions.

Increasing the number of individuals with an interest in this field starts with educating our youth. While it does not take particularly advanced mathematics to explain the basics of quantum computing, there are still involved topics such as quantum superposition, unitary evolution, and projective measurement that can be difficult to conceptualize. In order to explain these topics at a middle and high school level to encourage more students to enter this area, we decided to design an educational game called Bas|ket>ball, which allows students to directly engage with quantum computing concepts outside of the classroom while being physically active.

After playing the game with students in our local Quantum Information High (QIHigh) Program at Stevens Institute of Technology, we realized that the game is a fun learning tool worth sharing with the broader physics community. Here, we describe a non-gender specific activity that can be used to effectively teach the basics of quantum computing at a high school level.

Quantum Basketball is something that helps you understand a very confusing topic, especially for a ninth grader! In the QI-High Program at Stevens, I was approached with a challenge of learning about quantum computing, and while I was hesitant at first, my mentors made the topic so much more understandable by relating it to a sport that I love!

Grace Conlin, Freshman Student from High Tech High School

## The Rules of Bas|ket>ball

The game can have up to 10 student players and only requires one basketball. Each player acts as a quantum bit (qubit) in a quantum register and is initialized to the |0> position. During each turn, a player will perform one of the allowed quantum gates depending on their position on the court. A diagram of the court positions is displayed at the bottom.

There are four options of quantum gates from which players can choose to move around the court:

1. X Gate – This single qubit gate will take a player from the |0> to the |1> position, and vice versa.
2. Hadamard Gate – This single qubit gate will take a player from the |0> to the (|0> + |1>) / $\sqrt{2}$ position and the |1> to the (|0> – |1>) / $\sqrt{2}$ position, and vice versa.
3. Control-Not Gate – This two-qubit gate allows one player to control another only if they are in the |1> position, or in superposition between |0> and |1>. The player in the |1> position can move a player back and forth between the |0> and |1> positions. The player in the superposition can choose to entangle with a player in the |0> position.
4. Z Measurement – The player takes a shot. The player measures a 1 if he/she makes the shot and measures a 0 if he/she misses. Once the player shoots, he/she has to return back to the |0> position no matter what was measured.

The first player to measure ten 1’s (make ten shots) wins! In order to make the game more interesting, the following additional rules are put in place:

1. Each player has one SWAP gate that can be utilized per game to switch positions with any other player, including players that are entangled. This is an effective way to replace yourself with someone in an entangled state.
2. Up to five players can be entangled at any given time. The only way to break the entanglement is to make a Z Measurement by taking a shot. If one of the entangled players makes a shot, each player entangled with that player receives a point value equal to the number of individuals they are entangled with (including themselves). If the player misses, the entanglement is broken and no points are awarded. Either way, all players go back to the |0> position.

## Example Bas|ket>ball Match

For example, let’s say that we have three student players. Each will start at the red marker, behind the basketball hoop. One by one, each student will choose from the list of gate operations above. If the first player chooses an X-gate, he/she will physically move to the blue marker and be in a better position to make a measurement (take a shot) during the next turn. If the second player chooses to perform a Hadamard gate, he/she will move to the green marker. Each of the students will continue to move around the court and score points by making measurements until 10 points are reached.

However, things can get more interesting when players start to form alliances by entangling. If player 1 is at the green marker and player 3 is at the red marker, then player 1 can perform a C-Not gate on player 3 to become entangled with them. Now if either player takes a shot and scores a point, both players will be awarded 2 points (1 x the number of players entangled).

We believe that simple games such as this, along with Quantum TiqTaqToe and Quantum Chess, will attract more young students to pursue degrees in physics and computer science, and eventually specialize in quantum information and computing fields. Not only is this important for the overall progression of the field, but also to encourage more diversity and inclusion in STEM.

# The quantum steampunker by Massachusetts Bay

Every spring, a portal opens between Waltham, Massachusetts and another universe.

The other universe has a Watch City dual to Waltham, known for its watch factories. The cities throw a festival to which explorers, inventors, and tourists flock. Top hats, goggles, leather vests, bustles, and lace-up boots dot the crowds. You can find pet octopodes, human-machine hybrids, and devices for bending space and time. Steam powers everything.

Watch City Steampunk Festival

So I learned thanks to Maxim Olshanyi, a professor of physics at the University of Massachusetts Boston. He hosted my colloquium, “Quantum steampunk: Quantum information meets thermodynamics,” earlier this month. Maxim, I discovered, has more steampunk experience than I. He digs up century-old designs for radios, builds the radios, and improves upon the designs. He exhibits his creations at the Watch City Steampunk Festival.

Maxim Olshanyi

I never would have guessed that Maxim moonlights with steampunkers. But his hobby makes sense: Maxim has transformed our understanding of quantum integrability.

Integrability is to thermalization as Watch City is to Waltham. A bowl of baked beans thermalizes when taken outside in Boston in October: Heat dissipates into the air. After half-an-hour, large-scale properties bear little imprint of their initial conditions: The beans could have begun at 112ºF or 99º or 120º. Either way, the beans have cooled.

Integrable systems avoid thermalizing; more of their late-time properties reflect early times. Why? We can understand through an example, an integrable system whose particles don’t interact with each other (whose particles are noninteracting fermions). The dynamics conserve the particles’ momenta. Consider growing the system by adding particles. The number of conserved quantities grows as the system size. The conserved quantities retain memories of the initial conditions.

Imagine preparing an integrable system, analogously to preparing a bowl of baked beans, and letting it sit for a long time. Will the system equilibrate, or settle down to, a state predictable with a simple rule? We might expect not. Obeying the same simple rule would cause different integrable systems to come to resemble each other. Integrable systems seem unlikely to homogenize, since each system retains much information about its initial conditions.

Maxim and collaborators exploded this expectation. Integrable systems do relax to simple equilibrium states, which the physicists called the generalized Gibbs ensemble (GGE). Josiah Willard Gibbs cofounded statistical mechanics during the 1800s. He predicted the state to which nonintegrable systems, like baked beans in autumnal Boston, equilibrate. Gibbs’s theory governs classical systems, like baked beans, as does the GGE theory. But also quantum systems equilibrate to the GGE, and Gibbs’s conclusions translate into quantum theory with few adjustments. So I’ll explain in quantum terms.

Consider quantum baked beans that exchange heat with a temperature-$T$ environment. Let $\hat{H}$ denote the system’s Hamiltonian, which basically represents the beans’ energy. The beans equilibrate to a quantum Gibbs state, $e^{ - \hat{H} / ( k_{\rm B} T ) } / Z$. The $k_{\rm B}$ denotes Boltzmann’s constant, a fundamental constant of nature. The partition function $Z$ enables the quantum state to obey probability theory (normalizes the state).

Maxim and friends modeled their generalized Gibbs ensemble on the Gibbs state. Let $\hat{I}_m$ denote a quantum integrable system’s $m^{\rm th}$ conserved quantity. This system equilibrates to $e^{ - \sum_m \lambda_m \hat{I}_m } / Z_{\rm GGE}$. The $Z_{\rm GGE}$ normalizes the state. The intensive parameters $\lambda_m$’s serve analogously to temperature and depend on the conserved quantities’ values. Maxim and friends predicted this state using information theory formalized by Ed Jaynes. Inventing the GGE, they unlocked a slew of predictions about integrable quantum systems.

A radio built by Maxim. According to him, “The invention was to replace a diode with a diode bridge, in a crystal radio, thus gaining a factor of two in the output power.”

I define quantum steampunk as the intersection of quantum theory, especially quantum information theory, with thermodynamics, and the application of this intersection across science. Maxim has used information theory to cofound a branch of quantum statistical mechanics. Little wonder that he exhibits homemade radios at the Watch City Steampunk Festival. He also holds a license to drive steam engines and used to have my postdoc position. I appreciate having older cousins to look up to. Here’s hoping that I become half the quantum steampunker that I found by Massachusetts Bay.

With thanks to Maxim and the rest of the University of Massachusetts Boston Department of Physics for their hospitality.

The next Watch City Steampunk Festival takes place on May 9, 2020. Contact me if you’d attend a quantum-steampunk meetup!

# Sultana: The Girl Who Refused To Stop Learning

Caltech attracts some truly unique individuals from all across the globe with a passion for figuring things out. But there was one young woman on campus this past summer whose journey towards scientific research was uniquely inspiring.

Sultana spent the summer at Caltech in the SURF program, working on next generation quantum error correction codes under the supervision of Dr. John Preskill. As she wrapped up her summer project, returning to her “normal” undergraduate education in Arizona, I had the honor of helping her document her remarkable journey. This is her story:

Afghanistan

My name is Sultana. I was born in Afghanistan. For years I was discouraged and outright prevented from going to school by the war. It was not safe for me because of the active war and violence in the region, even including suicide bombings. Society was still recovering from the decades long civil war, the persistent influence of a dethroned, theocratically regressive regime and the current non-functioning government. These forces combined to make for a very insecure environment for a woman. It was tacitly accepted that the only place safe for a woman was to remain at home and stay quiet. Another consequence of these circumstances was that the teachers at local schools were all male and encouraged the girls to not come to school and study. What was the point if at the end of the day a woman’s destiny was to stay at home and cook?

For years, I would be up every day at 8am and every waking hour was devoted to housework and preparing the house to host guests, typically older women and my grandmother’s friends. I was destined to be a homemaker and mother. My life had no meaning outside of those roles.

My brothers would come home from school, excited about mathematics and other subjects. For them, it seemed like life was full of infinite possibilities. Meanwhile I had been confined to be behind the insurmountable walls of my family’s compound. All the possibilities for my life had been collapsed, limited to a single identity and purpose.

At fourteen I had had enough. I needed to find a way out of the mindless routine and depressing destiny. And more specifically, I wanted to understand how complex, and clearly powerful, human social systems, such as politics, economics and culture, combined to create overtly negative outcomes like imbalance and oppression. I made the decision to wake up two hours early every day to learn English, before taking on the day’s expected duties.

My grandfather had a saying, “If you know English, then you don’t have to worry about where the food is going to come from.”

He taught himself English and eventually became a professor of literature and humanities. He had even encouraged his five daughters to pursue advanced education. My aunts became medical doctors and chemists (one an engineer, another a teacher). My mother became a lecturer at a university, a profession she would be forced to leave when the Mujaheddin came to power.

I started by studying newspapers and any book I could get my hands on. My hunger for knowledge proved insatiable.

When my father got the internet, the floodgates of information opened. I found and took online courses through sites like Khan Academy and, later, Coursera.

I was intrigued by discussions between my brothers on mathematics. Countless pages of equations and calculations could propagate from a single, simple question; just like how a complex and towering tree can emerge from a single seed.

Khan Academy provided a superbly structured approach to learning mathematics from scratch. Most importantly, mathematics did not rely on a mastery of English as a prerequisite.

Over the next few years I consumed lesson after lesson, expanding my coursework into physics. I would supplement this unorthodox yet structured education with a more self-directed investigation into philosophy through books like Kant’s Critique of Pure Reason. While math and physics helped me develop confidence and ability, ultimately, I was still driven by trying to understand the complexities of human behavior and social systems.

Emily from Iowa

To further develop my hold on English I enrolled in a Skype-based student exchange program and made a critical friend in Emily from Iowa. After only a few conversations, Emily suggested that my English was so good that I should consider taking the SAT and start applying for schools. She soon became a kind of college counselor for me.

Even though my education was stonewalled by an increasingly repressive socio-political establishment, I had the full support of my family. There were no SAT testing locations in Afghanistan. So when it was clear to my family I had the potential to get a college education, my uncle took me across the border into Pakistan, to take the SAT. However, a passport from Afghanistan was required to take the test and, when it was finally granted, it had to be smuggled across the border. Considering that I had no formal education and little time to study for the SAT, I earned a surprisingly competitive score on the exam.

My confidence soared and I convinced my family to make the long trek to the American embassy and apply for a student visa. I was denied in less than sixty seconds! They thought I would end up not studying and becoming an economic burden. I was crushed. And my immaturely formed vision of the world was clearly more idealized than the reality that presented itself and slammed its door in my face. I was even more confused by how the world worked and I immediately became invested in understanding politics.

The New York Times

Emily was constantly working in the background on my behalf, and on the other side of the world, trying to get the word out about my struggle. This became her life’s project, to somehow will me into a position to attend a university. New York Times writer Nicholas Kristoff heard about my story and we conducted an interview over Skype. The story was published in the summer of 2016.

The New York Times opinion piece was published in June. Ironically, I didn’t have much say or influence on the opinion-editorial piece. I felt that the piece was overly provocative.

Even now, because family members still live under the threat of violence, I will not allow myself to be photographed. Suffice to say, I never wanted to stir up trouble, or call attention to myself. Even so, the net results of that article are overwhelmingly positive. I was even offered a scholarship to attend Arizona State University; that was, if I could secure a visa.

I was pessimistic. I had been rejected twice already by what should have been the most logical and straightforward path towards formal education in America. How was this special asylum plea going to result in anything different? But Nicholas Kristoff was absolutely certain I would get it. He gave my case to an immigration lawyer with a relationship to the New York Times. In just a month and a half I was awarded humanitarian parole. This came with some surprising constraints, including having to fly to the U.S. within ten days and a limit of four months to stay there while applying for asylum. As quickly as events were unfolding, I didn’t even hesitate.

As I was approaching America, I realized that over 5,000 miles of water would now separate me from the most influential forces in my life. The last of these flights took me deep into the center of America, about a third of the way around the planet.

The clock was ticking on my time in America – at some point, factors and decisions outside of my control would deign that I was safe to go back to Afghanistan – so I exhausted every opportunity to obtain knowledge while I was isolated from the forces that would keep me from formal education. I petitioned for an earlier than expected winter enrollment at Arizona State University. In the meantime, I continued my self-education through edX classes (coursework from MIT made available online), as well as with Khan Academy and Coursera.

Phoenix

The answer came back from Arizona State University. They had granted me enrollment for the winter quarter. In December of 2016, I flew to the next state in my journey for intellectual independence and began my first full year of formal education at the largest university in America. Mercifully, my tenure in Phoenix began in the cool winter months. In fact, the climate was very similar to what I knew in Afghanistan.

However, as summer approached, I began to have a much different experience. This was the first time I was living on my own. It took me a while to be accustomed to that. I would generally stay in my room and study, even avoiding classes. The intensifying heat of the Arizona sun ensured that I would stay safely and comfortably encased inside. And I was actually doing okay. At first.

Happy as I was to finally be a part of formal education, it was in direct conflict with the way in which I had trained myself to learn. The rebellious spirit which helped me defy the cultural norms and risk harm to myself and my family, the same fire that I had to continuously stoke for years on my own, also made me rebel against the system that actively wanted me to learn. I constantly felt that I had better approaches to absorb the material and actively ignored the homework assignments. Naturally, my grades suffered and I was forced to make a difficult internal adjustment. I also benefited from advice from Emily, as well as a cousin who was pursuing education in Canada.

As I gritted my teeth and made my best attempts to adopt the relatively rigid structures of formal education, I began to feel more and more isolated. I found myself staying in my room day after day, focused simply on studying. But for what purpose? I was aimless. A machine of insatiable learning, but without any specific direction to guide my curiosity. I did not know it at the time, but I was desperate for something to motivate me.

The ripples from the New York Times piece were still reverberating and Sultana was contacted by author Betsy Devine. Betsy was a writer who had written a couple of books with notable scientists. Betsy was particularly interested in introducing Sultana to her husband, Nobel prize winner in physics, Frank Wilczek.

The first time I met Frank Wilczek was at lunch with with him and his wife. Wilczek enjoys hiking in the mountains overlooking surrounding Phoenix and Betsy suggested that I join Frank on an early morning hike. A hike. With Frank Wilczek. This was someone whose book, A Beautiful Question: Finding Nature’s Deep Design, I had read while in Afghanistan. To say that I was nervous is an understatement, but thankfully we fell into an easy flow of conversation. After going over my background and interests he asked me if I was interested in physics. I told him that I was, but I was principally interested in concepts that could be applied very generally, broadly – so that I could better understand the underpinnings of how society functions.

He told me that I should pursue quantum physics. And more specifically, he got me very excited about the prospects of quantum computers. It felt like I was placed at the start of a whole new journey, but I was walking on clouds. I was filled with a confidence that could only be generated by finding oneself comfortable in casual conversation with a Nobel laureate.

Immediately after the hike I went and collected all of the relevant works Wilczek had suggested, including Dirac’s “The Principles of Quantum Mechanics.”

Reborn

With a new sense of purpose, I immersed myself in the formal coursework, as well as my own, self-directed exploration of quantum physics. My drive was rewarded with all A’s in the fall semester of my sophomore year.

That same winter Nicholas Kristoff had published his annual New York Times opinion review of the previous year titled, “Why 2017 Was the Best Year in Human History.” I was mentioned briefly.

It was the start of the second semester of my sophomore year, and I was starting to feel a desire to explore applied physics. I was enrolled in a graduate-level seminar class in quantum theory that spring. One of the lecturers for the class was a young female professor who was interested in entropy, and more importantly, how we can access seemingly lost information. In other words, she wanted access to the unknown.

To that end, she was interested in gauge/gravity duality models like the one meant to explain the black hole “firewall” paradox, or the Anti-de Sitter space/conformal field theory (AdS/CFT) correspondence that uses a model of the universe where space-time has negative, hyperbolic curvature.

The geometry of 5D space-time in AdS space resembles that of an M.C.Escher drawing, where fish wedge themselves together, end-to-end, tighter and tighter as we move away from the origin. These connections between fish are consistent, radiating in an identical pattern, infinitely approaching the edge.

Unbeknownst to me, a friend of that young professor had read the Times opinion article. The article not only mentioned that I had been teaching myself string theory, but also that I was enrolled at Arizona State University and taking graduate level courses. She asked the young professor if she would be interested in meeting me.

The young professor invited me to her office, she told me about how black holes were basically a massive manifestation of entropy, and the best laboratory by which to learn the true nature of information loss, and how it might be reversed. We discussed the possibility of working on a research paper to help her codify the quantum component for her holographic duality models.

I immediately agreed. If there was anything in physics as difficult as understanding human social, religious and political dynamics, it was probably understanding the fundamental nature of space and time. Because the AdS/CFT model of spacetime was negatively curved, we could employ something called holographic quantum error correction to create a framework by which the information of a bulk entity (like a black hole) can be preserved at its boundary, even with some of its physical components (particles) becoming corrupted, or lost.

I spent the year wrestling with, and developing, quantum error correcting codes for a very specific kind of black hole. I learned that information has a way of protecting itself from decay through correlations. For instance, a single logical quantum bit (or “qubit”) of information can be represented, or preserved, by five stand-in, or physical, qubits. At a black hole’s event horizon, where entangled particles are pulled apart, information loss can be prevented as long as less than three-out-of-five of the representative physical qubits are lost to the black hole interior. The original quantum information can be recalled by using a quantum code to reverse this “error”.

By the end of my sophomore year I was nominated to represent Arizona State University at an inaugural event supporting undergraduate women in science. The purpose of the event was to help prepare promising women in physics for graduate school applications, as well as provide information on life as a graduate student. The event, called FUTURE of Physics, was to be held at Caltech.

I mentioned the nomination to Frank Wilczek and he excitedly told me that I must use the opportunity to meet Dr. John Preskill, who was at the forefront of quantum computing and quantum error correction. He reminded me that the best advice he could give anyone was to “find interesting minds and bother them.”

I spent two exciting days at Caltech with 32 other young women from all over the country on November 1st and 2nd of 2018. I was fortunate to meet John Preskill. And of course I introduced myself like any normal human being would, by asking him about the Shor factoring algorithm. I even got to attend a Wednesday group meeting with all of the current faculty and postdocs at IQIM. When I returned to ASU I sent an email to Dr. Preskill inquiring about potentially joining a short research project with his team.

I was extremely relieved when months later I received a response and an invitation to apply for the Summer Undergraduate Research Fellowship (SURF) at Caltech. Because Dr. Preskill’s recent work has been at the forefront of quantum error correction for quantum computing it was relatively straightforward to come up with a research proposal that aligned with the interests of my research adviser at ASU.

One of the major obstacles to efficient and widespread proliferation of quantum computers is the corruption of qubits, expensively held in very delicate low-energy states, by environmental interference and noise. People simply don’t, and should not, have confidence in practical, everyday use of quantum computers without reliable quantum error correction. The proposal was to create a proof that, if you’re starting with five physical qubits (representing a single logical qubit) and lose two of those qubits due to error, you can work backwards to recreate the original five qubits, and recover the lost logical qubit in the context of holographic error correcting codes. My application was accepted, and I made my way to Pasadena at the beginning of this summer.

The temperate climate, mountains and lush neighborhoods were a welcome change, especially with the onslaught of relentless heat that was about to envelope Phoenix.

Even at a campus as small as Caltech I felt like the smallest, most insignificant fish in a tiny, albeit prestigious, pond. But soon I was being connected to many like-minded, heavily motivated mathematicians and physicists, from all walks of life and from every corner of the Earth. Seasoned, young post-docs, like Grant Salton and Victor Albert introduced me to HaPPY tensors. HaPPY tensors are a holographic tensor network model developed by Dr. Preskill and colleagues meant to represent a toy model of AdS/CFT. Under this highly accessible and world-class mentorship, and with essentially unlimited resources, I wrestled with HaPPY tensors all summer and successfully discovered a decoder that could recover five qubits from three.

Example of tensor network causal and entanglement wedge reconstructions. From a blog post by Beni Yoshida on March 27th, 2015 on Quantum Frontiers.

This was the ultimate confidence booster. All the years of doubting myself and my ability, due to educating myself in a vacuum, lacking the critical feedback provided by real mentors, all disappeared.

Tomorrow

Now returning to ASU to finish my undergraduate education, I find myself still thinking about what’s next. I still have plans to expand my proof, extending beyond five qubits, to a continuous variable representation, and writing a general algorithm for an arbitrary N layer tensor-network construction. My mentors at Caltech have graciously extended their support to this ongoing work. And I now dream to become a professor of physics at an elite institution where I can continue to pursue the answers to life’s most confusing problems.

My days left in America are not up to me. I am applying for permanent amnesty so I can continue to pursue my academic dreams, and to take a crack at some of the most difficult problems facing humanity, like accelerating the progress towards quantum computing. I know I can’t pursue those goals back in Afghanistan. At least, not yet. Back there, women like myself are expected to stay at home, prepare food and clean the house for everybody else.

Little do they know how terrible I am at housework – and how much I love math.

# Yes, seasoned scientists do extraordinary science.

Imagine that you earned tenure and your field’s acclaim decades ago. Perhaps you received a Nobel Prize. Perhaps you’re directing an institute for science that you helped invent. Do you still do science? Does mentoring youngsters, advising the government, raising funds, disentangling logistics, presenting keynote addresses at conferences, chairing committees, and hosting visitors dominate the time you dedicate to science? Or do you dabble, attend seminars, and read, following progress without spearheading it?

People have asked whether my colleagues do science when weighed down with laurels. The end of August illustrates my answer.

At the end of August, I participated in the eighth Conference on Quantum Information and Quantum Control (CQIQC) at Toronto’s Fields Institute. CQIQC bestows laurels called “the John Stewart Bell Prize” on quantum-information scientists. John Stewart Bell revolutionized our understanding of entanglement, strong correlations that quantum particles can share and that power quantum computing. Aephraim Steinberg, vice-chair of the selection committee, bestowed this year’s award. The award, he emphasized, recognizes achievements accrued during the past six years. This year’s co-winners have been leading quantum information theory for decades. But the past six years earned the winners their prize.

Peter Zoller co-helms IQOQI in Innsbruck. (You can probably guess what the acronym stands for. Hint: The name contains “Quantum” and “Institute.”) Ignacio Cirac is a director of the Max Planck Institute of Quantum Optics near Munich. Both winners presented recent work about quantum many-body physics at the conference. You can watch videos of their talks here.

Peter discussed how a lab in Austria and a lab across the world can check whether they’ve prepared the same quantum state. One lab might have trapped ions, while the other has ultracold atoms. The experimentalists might not know which states they’ve prepared, and the experimentalists might have prepared the states at different times. Create multiple copies of the states, Peter recommended, measure the copies randomly, and play mathematical tricks to calculate correlations.

Ignacio expounded upon how to simulate particle physics on a quantum computer formed from ultracold atoms trapped by lasers. For expert readers: Simulate matter fields with fermionic atoms and gauge fields with bosonic atoms. Give the optical lattice the field theory’s symmetries. Translate the field theory’s Lagrangian into Hamiltonian language using Kogut and Susskind’s prescription.

Even before August, I’d collected an arsenal of seasoned scientists who continue to revolutionize their fields. Frank Wilczek shared a physics Nobel Prize for theory undertaken during the 1970s. He and colleagues helped explain matter’s stability: They clarified how close-together quarks (subatomic particles) fail to attract each other, though quarks draw together when far apart. Why stop after cofounding one subfield of physics? Frank spawned another in 2012. He proposed the concept of a time crystal, which is like table salt, except extended across time instead of across space. Experimentalists realized a variation on Frank’s prediction in 2018, and time crystals have exploded across the scientific literature.1

Rudy Marcus is 96 years old. He received a chemistry Nobel Prize, for elucidating how electrons hop between molecules during reactions, in 1992. I took a nonequilibrium-statistical-mechanics course from Rudy four years ago. Ever since, whenever I’ve seen him, he’s asked for the news in quantum information theory. Rudy’s research group operates at Caltech, and you won’t find “Emeritus” in the title on his webpage.

My PhD supervisor, John Preskill, received tenure at Caltech for particle-physics research performed before 1990. You might expect the rest of his career to form an afterthought. But he helped establish quantum computing, starting in the mid-1990s. During the past few years, he co-midwifed the subfield of holographic quantum information theory, which concerns black holes, chaos, and the unification of quantum theory with general relativity. Watching a subfield emerge during my PhD left a mark like a tree on a bicyclist (or would have, if such a mark could uplift instead of injure). John hasn’t helped create subfields only by garnering resources and encouraging youngsters. Several papers by John and collaborators—about topological quantum matter, black holes, quantum error correction, and more—have transformed swaths of physics during the past 15 years. Nor does John stamp his name on many papers: Most publications by members of his group don’t list him as a coauthor.

Do my colleagues do science after laurels pile up on them? The answer sounds to me, in many cases, more like a roar than like a “yes.” Much science done by senior scientists inspires no less than the science that established them. Beyond their results, their enthusiasm inspires. Never mind receiving a Bell Prize. Here’s to working toward deserving a Bell Prize every six years.

With thanks to the Fields Institute, the University of Toronto, Daniel F. V. James, Aephraim Steinberg, and the rest of the conference committee for their invitation and hospitality.

You can find videos of all the conference’s talks here. My talk is shown here

1To scientists, I recommend this Physics Today perspective on time crystals. Few articles have awed and inspired me during the past year as much as this review did.