Frontiers of Engineered Coherent Matter and Systems: Proceedings of a Workshop (2025)

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Quantum Information Dynamics: Natural and Synthetic

The workshop’s final session was devoted to the topic of quantum information dynamics in both natural and synthetic systems, albeit with a focus on the synthetic. The session was moderated by Vedika Khemani of Stanford University.

OVERVIEW

Matteo Ippoliti of the University of Texas at Austin opened the session by offering a high-level overview of the dynamics of quantum information. “Obviously, this is going to be highly biased towards the things that I personally find most interesting,” he said, but he said he would try to cover a number of different topics and also set the stage for the following presentations.

As background, he noted that researchers are developing increasingly complex quantum systems that can be collectively referred to as “synthetic quantum matter,” including superconducting qubits, trapped ions, and ultra-cold atoms. What these systems have in common, he said, is that they are developed from the ground up to do information-processing tasks such as quantum computing or simulation. Their information-processing capabilities are made possible by the fact that researchers have developed an unprecedented ability to control and measure microscopic degrees of freedom in many-body quantum systems, allowing a more interactive role for the observer. Furthermore, he continued, while these capabilities have been developed mainly for technological purposes—computing and simulations, mainly—from the physics perspective, these capabilities also open up many research opportunities in many-body physics, both in and out of equilibrium. Noting

that a later talk by Eun-Ah Kim would focus on equilibrium states, Ippoliti said that he would spend most of his time on dynamics.

Ippoliti began by describing some of the non-equilibrium processes that are part of quantum computing. “Computation is a process that takes a simple input to a simple output through some highly complex intermediate states,” so one can think of it as some complex, non-equilibrium process. And in quantum computers, in particular, there are multiple ingredients that come together in novel and interesting ways. These ingredients include drive (unitary operations), open-system effects (noise, errors, decoherence, and dissipation effects), measurement (output read-out, error detection), and feedback (error correction).

The unitary control in a quantum computer is carried out through the elementary logical gates that reflect the quantum algorithm. In terms of many-body processes, Ippoliti said, one can think of the logic gates as “a particularly complex and structured type of drive.” People may be more familiar with simple drives, such as shining light on a solid, he said, but “these are much more complex and microscopically controlled types of drives, where a few qubits can be made to interact with each other in controllable ways, and then a staggered pattern of these few-body interactions can come together to compose a complex computation.”

Obtaining an answer from the computation requires measuring these systems in very sophisticated ways, he said, because the conventional sorts of measurements that people do in many-body physics—for instance, obtaining a single value such as conductivity to describe a mesoscopic system—are not sufficient. “We need to be able to read out all of the qubits one by one to get the answer to our computation,” he observed.

For error correction in quantum computers, Ippoliti said, it is important to make measurements and detect errors not just at the end of the computation but rather throughout it. This makes it possible to take the relevant information, process it separately in a classical computer, and then use the resulting information to feed back into the dynamics in order to correct the effects of the errors that had been detected.

In short, quantum computing opens up a vast new domain for studying many-body physics far from equilibrium, and these new possibilities are generally far beyond what is available in nature. Within that vast space, Ippoliti said, he would focus on a few specific areas, beginning with thermalization.

Thermalization

Thermalization refers to the emergence of thermal equilibrium in isolated many-body systems. In classical physics, Ippoliti said, thermal equilibrium arises because of dynamical chaos. “We have great knowledge of the state of the system, but nonetheless we cannot make predictions for very far into the future,” he explained, “because some small uncertainty today can balloon into very large

uncertainty a few days from now.” That is why it is not possible to make weather forecasts more than a few days ahead, for example.

In quantum physics, however, that does not happen—at least on a naïve level. Because the Schrödinger equation is linear, the amplification of uncertainties that one sees in classical physics does not happen. Instead, the dynamics described by the Schrödinger equation are linear and reversible. “It’s not clear where the information goes,” he said. “It’s always in the system. It’s never destroyed. So how can we end up with an information loss that leads to the emergence of thermodynamic equilibrium?”

The answer arises from quantum entanglement, Ippoliti explained. “While the information is never destroyed, it can be spread out among many degrees of freedom in a collective way, so that if we only focus our attention on some small subsystem of a many-body system, then we have incomplete knowledge about it” (Figure 5-1). Thus, there is some information that cannot be accessed, which opens the way for thermodynamic equilibrium states, even though the information as a whole is not lost but is still present in the system.

This intimate connection between thermodynamic equilibrium and entanglement means, he said, that it is very interesting to understand the ways in which entanglement is generated and propagated in many-body systems. And there are several different paradigms for how this happens, he said. For example, in systems with coherent quasiparticles, the quasiparticles can be responsible for carrying

entanglement and thermalizing. By contrast, in maximally chaotic systems that do not have any coherent excitations and are messy, entangled soups of degrees of freedom, researchers can apply a more geometric picture based on a so-called minimal cut. And in situations where the system fails to achieve equilibrium, Ippoliti said, looking at entanglement can still be very informative; for example, in many-body localization, there is a distinct fingerprint on the entanglement growth that offers information about the mechanism for the failure of equilibration. And this is not just theory, Ippoliti said, as it is now possible to directly measure the process of entanglement growth and propagation associated with equilibration in approximately isolated quantum systems, and these measurements shed light on the microscopic mechanism of thermalization.

In the kinds of experiments where the goal is to detect the equilibration of a subsystem, Ippoliti said, researchers are implicitly throwing away the remainder of the system, which serves as a heat bath holding the degrees of freedom that are somehow hiding the remaining information about the subsystem. However, as Bloch had discussed earlier, researchers have the ability not only to make extremely rich measurements that reveal the state of the local subsystem of interest, but also to take a global snapshot of the state of the entire system—for example, the occupation of all lattice sites on an optical lattice at once. Using all those data and doing new types of analyses based on them leads to some new perspectives on thermalization, Ippoliti said. “In particular,” he continued, “if we look at the state of a subsystem of interest—not by throwing away the measurement on the other sides but rather by conditioning on it, so we look at conditional post-measurement states of a subsystem”—these conditional states can exhibit universal statistics.

This realization spurred a number of theory developments that Ippoliti summarized. The idea, he said, is that when measurements are made on part of a system, because quantum measurement is inherently random, the state of the remainder of the system collapses onto some wave function. That wave function is random and depends on the measurement outcome. Thus, he said, the measurements do not produce a single state but a distribution on a high-dimensional Hilbert space. “The idea is that this distribution in Hilbert space can exhibit universal behavior that goes beyond what is diagnosable just with the standard tools of thermalization,” he said. “The fact that these distributions of wave functions achieve some universal behavior is a stronger version of equilibrium that goes beyond standard paradigms.” In particular, he continued, this distribution not only gives the total amount of entanglement generated between a subsystem and the rest, but it can also provide information about the structure of the entanglement. It can, for example, distinguish between a state that has only a few-body entanglement and a state with a full many-bodies-scrambled entanglement.

This is a very strong form of equilibration, Ippoliti said, and it is a very interesting fundamental topic in its own right, but it also has useful applications in

quantum computing, essentially as a kind of quantum version of a random number generator. In certain quantum computing applications, it can be valuable to use quantum states that are highly random in the sense that they pass some strict tests of randomness—they are unbiased, they have the correct statistical moments, and so on. “This phenomenon, which seems to occur quite generically under chaotic dynamics, offers a shortcut or an efficient way of generating these very high random states and can be exploited for tasks that go from randomized benchmarking to randomized measurements for state learning and quantum simulations and beyond.”

Monitored Dynamics

In monitored dynamics, Ippoliti explained, the idea is for measurement to be a protagonist in the dynamics of a many-body system; that is, one can exert a certain amount of control on the evolution of a many-body quantum system by making measurements at various points. In the case of a circuit model of quantum computing, for instance, one intersperses the computational steps with mid-circuit measurements. “We are free to decide how to do these measurements, how often to do them, or how strong these measurements are,” he said. “And so this is a knob we can turn, and it turns out that upon turning this knob, we can obtain new types of non-equilibrium phases,” that is, phases of matter or information that exist inherently away from stationary equilibrium states and that are defined not by the typical parameters in many body physics, but rather by the structure of quantum correlations and quantum information.

For example, one can use measurements to affect how entangled a local region of the final state is with the remainder of the final state. Specifically, by tuning the density of measurements, one can go from an entangling phase to a disentangling phase, with the density of entanglement entropy going from finite to zero sharply at a critical point.

This theoretical idea has spurred a great deal of research and has various connections to different areas of quantum information science, Ippoliti said, and the intuition behind this phenomenon can be expressed in terms of a network of quantum information. The idea is that in such a network, the interactions between the qubits (the boxes in Figure 5-2), which take place at the unitary gates, can create correlations. “They can propagate information from the past to the future, from the left to the right, and can give rise to this highly connected web of correlations,” he said.

On the other hand, measurements can have a much more complicated and tricky effect, Ippoliti said. They can destroy correlations, and they can move them around and reshuffle them in complicated ways. And the interplay of these processes can give rise to a transition in the connectivity of this network, taking it, for example, from a phase with many, many measurements that ends up having

a weakly connected or disconnected network to a phase with sufficiently sparse measurements where the network is strongly connected.

This transition connectivity of a network shows up in a variety of ways, he said. One, which he had mentioned earlier, is the scaling of entanglement of a subsystem in the final state. Another is the correlation between the initial state and the final state. “In this case, we detect the transition in the memory of initial conditions,” he commented. A third is the correlation between subsystems that are very far apart in the final state—so far apart that they would normally be causally disconnected because of the limit in the circuit on how fast information can travel. However, Ippoliti said, by making measurements, one can get “long-range correlations to emerge in correspondence with this transition.” This is referred to as the many-body teleportation phase transition.

One can unify all of these themes, he said, by thinking about the issue from the coding perspective, a point of view that has much to do with quantum error correction. The idea, he said, is that all of these phenomena can be accessed by asking the question of whether some quantum information is still present in the system. If the information is present in the system, then there should be something that can be done to extract it. In particular, the experimentalist controls both the quantum system and the classical measurement apparatus, so one makes the measurements, writes down the outcomes, performs some analysis, and, based on that analysis, goes back to the system and executes some conditional control, making it possible to extract some information that was present in the system. “And this process succeeds or fails, depending on which side of this phase transition you are,” Ippoliti concluded.

This kind of conditional quantum control is very important in quantum technologies, he said, but it is not yet at a stage where it can always be implemented. “In the absence of that feedback control,” he said, “one can replace this final step by purely classical post-processing, and this gives rise to so-called hybrid or quantum classical order parameters.” An order parameter, he noted, is a quantity that distinguishes the two sides of a phase transition. “But here,” he said, “this order parameter is obtained by cross correlating data from the quantum experiment with classical simulation. That’s why we call it hybrid.”

Ippoliti finished his section on monitored dynamics by mentioning a couple of experiments enabled by the coding perspective. In one of those experiments (Google Quantum AI and Collaborators 2023), the team was able to scale up to 70 superconducting qubits, which is the largest demonstration to date, he said. “The point of view we take,” he explained, “is this teleportation point of view, where there is a qubit over here, a bunch of measurements in the middle, and then some correlation with qubits on the far end that naively would not be possible if we didn’t have the measurements, because these subsystems haven’t yet had a chance to exchange signals before the measurements.”

Learning

Ippoliti’s final topic was what he called the learning perspective, which is taking the view of an experimentalist who cannot directly access or influence the quantum-many body state, but instead is limited to the classical data that have been extracted from measurements on the quantum system. Given that situation, how much can the experimentalist learn from the measurement record about some unknown quantum state in this system at the beginning of the experiment? A key idea, he said, is that any information should either be safely stored in the system or it should have leaked out into the measurement data, but not both. “So we expect that the transition in the coding perspective should be dual to transition in this learning perspective, where the two sides get exchanged,” he said, “so in the side of the phase where there are a lot of measurements, we should be able to learn efficiently about this quantum state, whereas on the side with fewer measurements, we should fail to do so.” He noted that this perspective had recently been implemented in an experiment performed on Quantinuum trapped-ion hardware (Agrawal et al. 2024).

The idea of learning is becoming increasingly important in the age of programmable quantum matter, Ippoliti said, as scientists are able to make increasingly complex quantum states, and it is important to benchmark them and to efficiently characterize properties of these complex states. “To do that,” he said, “there has been a very successful paradigm that has emerged recently that turns the standard approach of experimental physics on its head.” In the typical experimental paradigm, he explained, a physicist goes into a laboratory and makes a measurement to answer

a question, “but in this field of randomized measurements, the rule is to measure first and ask questions later.” The idea, he said, is to go into the lab, make a large number of measurements in a set of randomized bases on a quantum state, and then store those data and process them to predict properties of the state at a later time. “So there is a distinction between a data-acquisition phase and a prediction phase that allows one to measure complex properties like entanglement and so on.”

In closing, Ippoliti said that the randomized measurements approach can also greatly benefit from an understanding of the physics of information in many-body systems. “In particular,” he continued, “if we understand the way that information and correlations are created and transported in many body systems, we can then design the best and most efficient ways of taking that information out and learning from it.”

LEARNING FROM NOISE

Next, Sarang Gopalakrishnan of Princeton University spoke about noise in quantum systems and what can be learned from noise about interesting many-body effects.

As background, he addressed the question of whether one really needs a quantum computer to simulate quantum dynamics and how much of it could be done classically. If one wants to exactly predict all possible observables, he said, then yes, one would need a quantum computer to do the simulations. But suppose one only cares about physical observables and only wants to get them approximately to some desired accuracy—how difficult is that to do classically?

You can ask this question about classical dynamics as well, Gopalakrishnan noted. How well can one predict, for instance, the trajectories of a number of particles moving around in a box and colliding with each? Chaos can make predicting the exact trajectories of all the particles difficult, if not impossible. “Even if you have just single particle bouncing around your regular shaped box,” he continued, “you’re going to find it exponentially difficult to predict it out to later times.” On the other hand, if one just wants to understand the physics of a system like this, it is not necessary to predict the positions of all the particles; hydrodynamics provides the necessary details on the macroscopic, coarse-grain properties of such a system. The limitation of this approach, Gopalakrishnan said, is that it is not very informative about what the system’s microscopic processes are, and someone wishing to understand what is causing the system to behave the way it does may not find the answers there.

These limitations raise the question of exactly what sorts of tasks one might care about in a quantum simulation context, he said, and he chose two axes with which to classify these tasks: the structure/number of moving parts and the task complexity. As an example of a task that is high on the number of moving parts but

relatively low on complexity, he mentioned superconductivity in a perovskite. It is a problem with an enormous amount of structure—various atoms in a crystalline structure, phonons, bands of electrons, and so on—but the questions being asked are relatively simple, he said—wanting to know about transport or some expectation value.

By contrast, he continued, the experiments that are currently being done in a quantum computing context are low on moving parts—the computations are dealing with almost structureless systems—but they are high on task complexity, as the questions being asked are incredibly detailed questions, such as, What is the full distribution of all the bit string probabilities? So these questions, in some sense, do not really learn any structure, Gopalakrishnan said. There really is no structure to learn. “They tell you, yes, you’re simulating a system correctly, and that’s pretty much it,” he said. They offer no help to researchers interested in learning about chemistry or physics from experiments and quantum devices.

“What you want to learn is something in between,” Gopalakrishnan said. “Ultimately, we want to design better and more interesting materials, and we want to know what about a material makes it do useful and interesting stuff, and so we need observables that help us locate mechanisms for the useful and interesting phenomena that we care about.” Knowing about transport on its own is a very crude probe, but predicting the full bit string distribution is completely irrelevant for these purposes. So, he said, it is objects of intermediate complexity which would be the sorts of things he would be talking about.

In particular, he continued, he would focus on the dynamics of fluctuations in interacting many-body systems. They carry information about dynamical mechanisms, make it possible to explore the boundaries of validity of classical descriptions, provide insights into new universality classes of fluctuations, and make it possible to do experiments into things that are relevant to understanding many body physics and reveal things that researchers did not know or expect. To illustrate, he told three “stories” about experiments where his team “acquired new information about the world by doing the experiment and thinking about it.”

Pump–Probe Spectroscopy

Gopalakrishnan first addressed the question, Are there long-lived excitations in disordered interacting systems? To study this, the team carried out a solid-state experiment on a lightly doped semiconductor using a probe that was sensitive to higher-order fluctuations. As background, he noted that linear response at finite frequencies does not tell a researcher very much about the distinction between a metal and an insulator. “Both a metal and insulator absorb energy from the drive,” he said. “They just absorb it in totally different ways.” In a metal, the absorption happens everywhere at a weak level, whereas in a disordered insulator the absorp-

tion is localized at a few strong resonance sites that absorb a lot because they are in resonance with the drive (Figure 5-3).

However, although the two look very similar in their linear response, they look very different if one goes beyond the linear response, Gopalakrishnan said. In the case of the insulator, “you fry your resonances after a while, and it doesn’t absorb anymore.” So the experiment, which was done by collaborators of Gopalakrishnan, was one of the simplest nonlinear response sorts of experiments, a pump–probe experiment. “You whack the system very hard,” he explained, “then you wait a little bit of time, and you measure the transport coefficient for a two-time correlation function after a wait time of τ. Intuitively what is happening, he said, the experiment as a function of τ shows how quickly the system recovered from being whacked by the pump in the first place. An analysis of the two time variables employing Fourier transforms results in a two-dimensional plot that separates out density effects from the lifetime of the excitations that are being driven.

The experiment was done in a relatively simple system, phosphorus-doped silicon, which is a disordered system with coulomb interactions. Increasing the amount of phosphorus leads eventually to the system transitioning from a metal to an insulator. Surprisingly, Gopalakrishnan said, as one gets close to the metal–insulator transition, the absorption lines get increasingly sharp. “It’s like the particle–hole excitations you’re exciting with a pump get increasingly long-lived near the metal–insulation transition,” he said. He characterized this as a phenomenon that he never would have guessed at without doing the experiments

and as something that has still not been completely explained by theory. “That’s one example of where you learn something from going beyond linear response.”

Full Counting Statistics in Quantum Spin Chains

The second experiment that Gopalakrishnan described addressed the question, What causes divergent spin conductivity in the Heisenberg spin chain model? The experiment involved looking at full counting statistics in quantum spin chains, a concept that Bloch had mentioned in his presentation.

As background, Gopalakrishnan noted that the Heisenberg model has been around for a long time, and it is one of the simplest interacting quantum models, consisting of a closed chain of quantum spins that can point in any direction and interact with their neighbors. Hans Bethe wrote down an exact formal solution for finite systems in 1931, but there are various difficulties in analyzing these spin chains; for example, spin conductivity at no=nzero temperatures has remained an open question.

Gopalakrishnan then described a series of theoretical and experimental efforts to understand the divergent spin conductivity in quantum spin chains, beginning with theoretical work by Marko Žnidarič (2011) in which he used simulations to calculate that the conductivity σ was proportional to ω^−⅓, where ω is the angular frequency. The exponent −⅓ was something of a mystery, Gopalakrishnan said, but in 2018, he and Rama Vasseur developed a theory to explain that exponent. Shortly after that, another group numerically computer the full dynamic structure factor and showed that it fits very well to a Kardar–Parisi–Zhang (KPZ) scaling form. The appearance of the KPZ factor was mysterious because it is something from classical stochastic physics, but then experimental work from another group confirmed the proportionality to ω^−⅓, and it seemed as if the problem was solved.

At that point, Gopalakrishnan said, he had a conversation with Immanuel Bloch, who was studying the Heisenberg model in his laboratory using his quantum gas microscope, and Bloch asked what could be done in the system that was not just computing the correlation function that one would compute in some other platform. The two came up with the idea of studying full counting statistics. As Gopalakrishnan explained, the experimental protocol for full counting statistics begins with the initialization of two half-systems separated by a barrier, each with a known number of particles, and then the barrier is lowered, and the dynamics are run to a particular time. At that point, the experimentalist measures all the particle positions, and that measurement provides a histogram of outcomes. The mean of the outcomes is the average transport, he said, and the higher moments are information beyond the average transport—they are information about fluctuations. “And the reason I wanted to do this,” he said, “was that in the KPZ universality class, there is a sharp prediction for what this entire distribution should be.”

What followed was what Gopalakrishnan referred to as the “era of fluctuations,” when a number of groups published theoretical analyses and experimental studies looking further into this, and the ultimate conclusion was that the transport was not KPZ. When Gopalakrishnan went back to look at where his theoretical work, which was based on two coupled continuity equations, went wrong. He found that his error arose from the way noise was put into the hydrodynamic equations.

“We still haven’t solved the KPZ case,” Gopalakrishnan said, but the work there led him to studies of a new universality class of diffusion with anomalous non-Gaussian fluctuations. He started with the question: Suppose you have diffusion, but it is a diffusion where the noise that is causing the normally random movement of individual particles actually has ballistic correlations. Imagine, he said, that each particle is undergoing Brownian motion because they are colliding with sound waves which are coming in from the right or left with equal probability. Thus, any given particle will seem to be bouncing around randomly, but all of the particles are feeling the same sound waves, so the fluctuations felt by different particles at different space-time points are strongly correlated with each other. This produces strongly non-Gaussian distributions even in the case where the average transport is diffusive, he said. In contrast with normal hydrodynamics where noise is uncorrelated in space-time, in this case, the noise has ballistic correlations, leading to a new type of diffusion.

Hydrodynamics of Fluctuations

The third experiment, which Gopalakrishnan discussed only briefly, dealt with the question, When does a system approach local thermal equilibrium? Normally, one would expect that if a system is disturbed by a very short-wavelength density wave, it should come back into equilibrium pretty fast because it has no long-wavelength density gradient. But when one of Gopalakrishnan’s colleagues, David Weiss, looked at this in the laboratory in a Bose gas, he found something surprising—very long timescales for the relaxation of certain parts of the momentum distribution, specifically the lower momentum modes.

Why are there such long timescales? The basic idea, Gopalakrishnan said, is that it is wrong to say that the state relaxes fast. “This state relaxes fast at the level of averages,” he explained, “but the initial state and the final state to which it relaxes have completely different profiles of fluctuations at long wavelengths.” The longer-timescale phenomenon occurs because the long-wavelength fluctuations have to build up by ballistic or diffusive transport.

After Weiss’s work, Immanuel Bloch and Monika Aidelsburger of the Ludwig-Maximilian University in Munich carried out a much more detailed study of how fluctuations equilibrate, and they found that the hydrodynamics of fluctuations has some quite deep connections with hydrodynamics of averages, although

Gopalakrishnan did not have time to go into any details. “But once again,” he concluded, “we learned something new about relaxation after a quench that I think you would never have thought of asking if you’re thinking about averages.”

Summarizing, he said that experiments probing fluctuations have led to surprising insights into many-body physics. The insights from his three stories included the following: the existence of marginally stable excitations in disordered systems, an inconsistency between average spin transport (KPZ) and higher-order fluctuations (not KPZ) in the Heisenberg model, and the discovery of generic, long timescales after spatially uniform/short-wavelength quenches.

He closed with a look to the future. “One thing that would be really nice is to really make the derivation of hydrodynamics somewhat more systematic,” he said, “so we didn’t have to make so many ad hoc physical assumptions when we’re doing these derivations.” One way to do that would be to learn hydrodynamic equations from experiments. Researchers in his group have recently measured current fluctuations as well as charge fluctuations in optical lattices, he said, and by measuring both of these, one can ask such things as whether a diffusion holds as an operator identity. And learning that, he continued, would allow one to determine, for instance, the noise in a many-body system.

A second approach to making the derivation of hydrodynamics more systematic, he said, would be to come up with better numerical methods. That raises the question, What are the inherent limits to the classical simulation of quantum systems? “To what extent,” he asked, “can you really simulate the questions you care about in quantum dynamics using classical algorithms, where, for example, you put in a bit of noise? You say that the questions you care about shouldn’t be sensitive to that little bit of noise, but it makes simulation vastly easier.”

Another interesting frontier, he said, is complex systems with more interesting network connectivity than one-dimensional systems. “We don’t really know a whole lot classically about how to deal with them,” Gopalakrishnan said, “but can we come up with classical algorithms that are better than the existing brute force tensor network ones to deal with them?” Finally, what interesting phenomena occur in these systems? The only way to find out will be through experiments.

NON-ABELIAN ANYON-BASED QUANTUM COMPUTING

Eun-Ah Kim of Cornell University spoke about a particular approach to quantum computing that uses non-Abelian anyons on synthetic platforms. She began by defining non-Abelian anyons. An anyon is a particular type of quasi-particle—in essence a group of particles that behaves as if it were a single particle—that so far has only been observed in two-dimensional systems. Non-abelian anyons are those for which the order in which the quasi-particles are exchanged matters.

To illustrate, Kim introduced the concept of braided world lines (Figure 5-4). Imagine there are two identical particles, one of which circles around the other (top left-hand side of Figure 5-4). Imagine further that the first particle circles the second in an adiabatic way that is gentle enough that it does not disturb the surroundings of those two particles. Then if one maps the motion in time of the two particles (bottom left-hand side of Figure 5-4), then one sees that the world-lines of the two particles have braided. Thus, one talks about braiding world lines.

If one measures the particle that has made the circle around the other, since the particle is exactly where it has started, one would expect that the state of the particle is identical to what it was before it started around in the circle. But that is not what happens with anyons. First, assume that there are additional spectator particles, α₁ and α₄. If α₂ is a non-Abelian anyon, once it has completed going around in the circle, the many-body quantum state of the system is not equivalent to the state before α₂ went around in the circle; indeed, the ending many-body quantum state is not even proportional to the initial state. Instead, the circling of α₂ around α₃ transforms the many-body quantum state by multiplication by a matrix. Since matrix multiplication is generally not commutative—that is, the order of the multiplication matters—the particles are non-Abelian.

“The point,” Kim said, “is that there it’s possible in two spatial dimensions for there to be objects or particles whose statistics are such that when you take one particle and go move it around the other, you have transformed the state in the Hilbert space as a transformation through a multiplication matrix.”

This has implications for quantum computing, Kim explained, because of the role that anyons can play in memory for a quantum computer. In particular, it is

necessary to be able to perform operations on a computer memory in order to manipulate that memory. “Because with non-Abelian anyons, you are truly operating on the Hilbert space of these many-body states,” she explained, “the creation of anyons itself would be a creation of memory, and the braiding of those particles would act as applying gate operations.”

The reason that this approach to memory could be useful, she said, is that the non-Abelian anyons are realized against a background of many qubits, “so the vacuum that they live in is not trivial.” And, she continued, “in that vacuum you are encoding this information in a way that is redundant because you’re using many degrees of freedom to encode, for instance, one logical bit,” and serve to provide quantum error correction and make the memory robust.

Topological quantum computing is based on this idea of using topological properties—for example, properties related to the braiding of these anyons—to create logical qubits that are robust against local perturbations. Researchers were trying to produce manifest realizations of non-Abelian anyons even it was realized that they could be used for quantum computing, Kim said, but after the idea of topological quantum computing was developed, the pursuit of non-Abelian anyons became much more energized. However, realizing non-Abelian anyons experimentally proved to be quite difficult. In 2005, for instance, researchers proposed a particular type of quantum state that would allow the production of non-Abelian anyons, but it has been only recently that researchers’ ability to produce the 5/2 quantum Hall states proposed in that paper has improved to the level that it may be possible to perform interferometry, Kim said.

Other ways to realize the braiding of non-Abelian anyons have also been proposed, she added, such as one that would employ p_x + ip_y superconductors and would grab the anyons and move them around with an atomic force microscope. However, she said, “We’re not quite there in any kind of solid-state platform yet.”

Designing many-body states with the desired properties is a complicated process, but work on stabilizer states has advanced that effort, Kim said. An example of a stabilizer state is a toric code, a topological quantum error correcting code which lives on a two-dimensional spin lattice. Then Kim offered two examples of designing states in which geometric intuition was used to come up with many-body states that could realize anyonic states on quantum hardware.

Ising Non-Abelian Anyons

The first example Kim offered was of Ising non-Abelian anyons. The roots of this work lie in the toric code, originally proposed by Alexei Kitaev (2003). “He noted that this model would be a good code in the sense that you have the stabilizers that all commute,” she said, “and you can define a state, and you can encode one bit of information if you have a surface code on a finite size system, but this model by

itself has two Abelian anyons defined by violation of the stabilizers.” The model had been around for nearly two decades before the Google AI group was able to realize the toric code as Kitaev had proposed it (Satzinger et al., 2021).

Kim became involved when Pedram Roushan, one of the leaders of the group that had performed the experiment, came to Cornell to give a talk. It was that talk, she said, that made her realize that the field had fully entered the era of designing quantum states and building the hardware to realize them. Then, working with her postdoc Yuri Lensky, she decided to try to realize non-Abelian anyons in a system. The idea, she said, was to make minimal changes to what had already been done but nevertheless introduce Ising non-Abelian anyons.

Their work started with a 2010 paper by Héctor Bombin in which he observed that “if you think of the toric code as a checker board, if you introduce a lattice dislocation, this lattice dislocation acts as Ising anyon,” she said. That was exciting because it meant there was a way to deform the toric code, which had now been realized, and get non-Abelian anyons. But it was also daunting, she continued, because “anybody who’s thought about solid-state systems and lattice dislocations knows lattice dislocations are not something that you can pick up and move around. They happen. Usually you don’t want them, but you don’t know how to get rid of them.” So her team had to figure out how to move the dislocations around in order to braid the anyons.

Lensky came up with an idea of how to do that. He suggested stepping away from the rigid picture of a lattice, Kim said, because “if you want to be dynamic with irregularities or defects, you want to think away from lattice.” So they developed a new code which they call plaquette surface code (PSC), and which Kim explained in contrast with the normal surface code (Figure 5-5). The surface code has stabilizers defined for every plaquette (thick edges in the left-hand figure). Violating a stabilizer produces anyons of two types, e and m, depending on which plaquette is violated. The two types of anyons are both Abelian.

“If we are interested in thinking about defects, which are defined by connectivity,” Kim said, “we know that a perfect surface code that we’ve been used to thinking about [i.e., the left-hand side of the figure] has a situation where every qubit [the small squares in the figure] is coupled to four neighboring qubits.” However, she continued, if the goal is to treat defects not as abnormalities but rather as a natural part of the system, one needs to think in terms of a graph rather than a lattice. Specifically, each qubits in the graph should be connected with two, three, or four other qubits and not always exactly four other qubits. Then Kim defined a plaquette surface as a graph on an orientable manifold with degree two, degree three, and degree four connectivity.

Stabilizers in the PSC are defined in the same way as for the surface code, she continued. That is, stabilizers are defined for every plaquette, and one can talk about the violation of the plaquette stabilizers. “But what we are really interested

in,” she said, “is the irregularity, or freedom—that freedom for dynamics, freedom for deformation—that we’re introducing by allowing us ourselves to think about a more diverse set of vertices, as opposed to just degree-four vertices.”

Because the code is defined by the graph, Kim said, calculating the amount of information that can be encoded in a given PSC is straightforward enough that she was able to explain to her son who is in high school. One takes into account the numbers of degree-two, degree-three, and degree-four vertices, which is related to the degrees of freedom in the graph, which is in turn related to the amount of information—i.e., the number of qubits—that can be encoded in the graph. “If I take the total number of degrees of freedom, subtract the number of constraints, that tells me how much room I have left to encode information, and that’s precisely related to the number of degree-three vertices divided by two,” she said. “That is, two degree-three vertices together give me one bit of information, the room to encode one bit of information.”

Then, showing an illustration of a defect—a dislocation—in a square lattice, she pointed out that the dislocation created a degree-three vertex. Thus, she said, the use of the PSC is “a way to think about the dislocation without worrying about the fact that it is affecting the whole system, but we can think locally about the degree-three vertex.”

Next, she illustrated how a degree-three vertex can be moved around a graph by showing a video with edges that would swing from one vertex to another, changing the degree of each vertex. For instance, if an edge is moved from a degree-four vertex to a degree-three vertex, the result is that the degree-four vertex has become

degree-three, and the degree-three vertex has become degree-four. In this way, a degree-three vertex, which is where information is encoded, can be moved around the graph, ultimately interchanging the position of two anyons.

“Once I have an understanding of how I want to do this geometrically with a notion of graphs,” Kim said, “the next thing I have to figure out is, how do I do it through hardware?” So she and her team developed unitary operators that reflected in quantum terms the edge-swinging rules that they had developed geometrically and then realized those in a superconducting processor. Specifically, she said, what her team did was start with a surface code and introduce four degree-three vertices—the Ising anyons—and then they moved them around following the rules that she had described. “That got us to a position where we started from vacuum, where there were no violations, and we had created four Ising anyons. We braided the middle two twice around each, making one go around the other, and then we fused the Ising anyons back with the two spectator Ising anyons. And the result is that we got fermions.” They were also able to show that there had been a change of the state of the logical qubit.

In one further demonstration, the group created three logical qubits and braided the three of them to entangle the three logical qubit states and arrive at a Greenberger-Horne-Zeilinger state, confirming through measurement that they had arrived at that state.

Fibonacci Anyons

At the end of her presentation, Kim spoke briefly about Fibonacci anyons as a second example of designing many-body states that could realize anyonic states on quantum hardware. She began by contrasting Fibonacci anyons with other particles by looking at the Hilbert state of many identical particles. In the case of fermions, the dimension of a Hilbert space of n fermions is 2ⁿ, and the space is totally factorizable. In the case of Ising anyons, or majoranas, one can factor down to pairs of majoranas, and the dimension of the Hilbert space of n majoranas is 2^n/2. A Fibonacci anyon has the property that when two are fused, it creates yet another anyon, in contrast with Ising anyons, which produce a fermion or just the identity when they are fused. Thus, a collection of n Fibonacci anyons forms a many-body state that cannot be factorized. It is also the case that when one braids Fibonacci anyons, it is possible to implement universal gates and not just logical X, which is what can be done with Ising anyons.

Offering just a few details on how her group has been working with Fibonacci anyons, Kim said that once again her team was able to make use of geometric insights and think in terms of graphs “instead of the daunting task of working on a lattice and a 12-qubit interaction, which is what people long have thought is necessary.” They were able to start with a baby stringnet state with just 3 qubits and then were

able to grow the stringnet to sample chromatic polynomials and braid the Fibonacci anyons. And what she is really excited about, she said, is that it offers the challenge of something in the middle in terms of the degree of entanglement or complexity, as Gopalakrishnan had talked about in his presentation. When one looks at a sampling of the Fibonacci stringnet state, it is not as spread out as a Porter-Thomas distribution on a random circuit, but it is still quite a bit spread out, she said. “And because it is complex enough,” she continued, “you can sample this and evaluate a chromatic polynomial, which is known to be #P-hard.” So this offers a different problem for a quantum computer that is known to be very difficult to solve classically. “We are not there yet,” she said, “but this is an interesting challenge.”

DISCUSSION

Workshop chair Charlie Marcus opened the discussion period by asking Kim how she liked being an experimental physicist and how it felt different from being a theorist. “I’ve always wanted to be an experimentalist,” Kim answered, and she spoke about her desire to see things from her theory work in a real physical system. “It’s really, really exciting to see something in data that overlaps with my simple theoretical understanding,” she said.

Then Khemani asked a broad question of the panelists: “If you had to pick one idea from quantum information theory which to you has been the most interesting idea that has influenced many-body physics in some way, what would you choose?” She added that respondents could limit the answer to the last 5 years.

Kim said that the biggest thing to her is that her work has made her re-appreciate the fact that quantum dynamics and the Schrödinger equation are about dynamics. “When I learned quantum mechanics, it was about solving differential equations and boundary conditions,” she said, “and you forget the fact that it’s actually about dynamics.” But the capabilities she had in her research allowed her to think about dynamics in a broken-down way and ask new questions, and now she has a very tangible way to talk about what is adiabatic. “It’s not waving hands,” she said. “Adiabatic is if I can keep moving my anyon and keep the system in that ground state manifold.”

Ippoliti agreed with Kim’s point and added that random circuit methods have taught him a lot. Noise is something else that has led him to think about physics differently, he said. Noise is always present in quantum experiments, he explained, and doing these sorts of experiments forces him to think about it. In other situations he might not think much about noise “because it feels like an annoying thing you have to deal with,” he said, “but actually it’s often instructive and opens up some new kind of directions for phases and so on.”

Gopalakrishnan said that error correction has had an enormous impact on how physicists think about many-body states. Ten years ago, he continued, it was

not something that people studying many-body physics paid much attention to, but now it is central, and when people think about, for instance, topological order, they think about it in terms of coding properties.

Laumann pointed to the effect that entanglement has had on how physicists think about thermalization. People now characterize a pure state being thermal by cutting it and looking locally. “People don’t even give that as part of their introduction to a talk anymore,” he said. “Everybody knows it. It’s just sort of become a standard way to think about it. So in that sense, I think that has actually been the most impactful in reorganizing how our generation thinks about thermalization.”

Pedram Roushan of Google asked why there is not a single problem in quantum simulation that everyone agrees is the most important one to solve. By contrast, he said, people interested in high-temperature superconductors see the discovery of the pairing mechanism between electrons as the holy grail.

Kim answered that quantum simulation is a method, while high-temperature superconductivity is a phenomenon, so the latter has an obvious mystery surrounding it. But quantum simulation, being a method, is defining a new kind of platform and system. So she is learning what sorts of questions she can ask. “It’s not like you can simulate anything [you want],” she said. Some things are much harder than others, and right now it is important to know what is possible.

Next, James Thompson triggered a long discussion by asking, “Since I’m surrounded by many-body people, does the many-body physics community still believe in spontaneous symmetry breaking? I mean, come on, it’s quantum measurement, right? I mean, the concept of spontaneous symmetry breaking, is that still a thing?”

“It’s essentially the only thing we know for sure,” Kim said. “If you take that from us, like, we are nothing.” Thompson replied, “Well, I just think something has to break it. And it’s the quantum measurement process, right? Like, you look at your system, and it collapses.” In short, he was questioning the idea of “spontaneous” symmetry breaking because there should be no symmetry breaking without some measurement causing the breaking.

When Gopalakrishnan said he was not sure what Thompson meant by “spontaneous symmetry breaking,” Thompson replied that this was his point. “I’m just poking the field a little bit” and trying to get some of the physicists there to ask themselves on their plane rides home if they were sure that spontaneous symmetry breaking was not just quantum measurement.

“It really is not,” Kim replied, “because spontaneous symmetry breaking is a notion of equilibrium in the thermodynamic limit. In the thermodynamic limit, there is a singularity that you can calculate and see that there is a divergence. You don’t have to measure anything, but when you do measurements and affect the system, you’re not in equilibrium. It’s not an isolated system.”

Ana Maria Rey of JILA brought up the role that the thermodynamic limit plays. “When you are taking the thermodynamic limit, you are breaking somehow the symmetry,” she said. Then Gopalakrishnan suggested that the mathematical explanation of spontaneous symmetry suggests that both Thompson and Rey can be correct. “The phenomenon of spontaneous symmetry breaking,” he said, “is that the limit of taking the symmetry-breaking field to zero and taking the thermodynamic limit don’t commute, right? That’s the mathematical phenomenon that we call spontaneous symmetry breaking. I think it’s compatible with everything James is saying and everything Ana Maria is saying.”

To close the session, Khemani asked the panelists which idea from quantum information they thought had been the most hyped in its application to many-body physics. Roushan said he thought it is a healthy field in which there are some bad ideas, but those bad ideas do not generally have a long lifetime, so there is no reason to worry about them. Ippoliti pointed to some of the variation algorithms or quantum machine learning methods that were very popular a few years earlier but have since lost favor. Gopalakrishnan said that the influence of variational algorithms and adiabatic optimization was ultimately good because it got people in the field thinking about such things as the adiabatic algorithm and adiabatic ramp processes, which are interesting many-body problems, he said, “even if they’re not going to optimize anything.”

Marcus prefaced his comment by mentioning a book by David Kaiser called How the Hippies Saved Physics. “It was a great book,” he said, “because it had a philosophical underpinning, which was that when the hippies asked the question in the 1970s whether or not mental telepathy and Uri Geller’s spoon bending could be explained by quantum entanglement, it led to a crisp answer of no, and that scientific progress should not be measured in those questions which are answered with a yes.” So, he concluded, the community should celebrate all of this adiabatic quantum processing “because we now understand so much more about that problem, about small gaps, and the relationship between small gaps and complexity in the whole spectrum.”

REFERENCES

Satzinger, K.J., Y.J. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, et al. 2021. “Realizing Topologically Ordered States on a Quantum Processor.” Science 374(6572):1237–1241.

Žnidarič, M. 2011. “Spin Transport in a One-Dimensional Anisotropic Heisenberg Model.” Physical Review Letters 106:220601.