*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.

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.

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.”

**Pasadena**

I spent two exciting days at Caltech with 32 other young women from all over the country on November 1^{st} and 2^{nd} 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.

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.

]]>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. *

^{1}To 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.

I was working with Tony Bartolotta, a PhD student in theoretical physics at Caltech, and Jason Pollack, a postdoc in cosmology at the University of British Columbia. They acted as the souls of consideration. We missed out on dozens of opportunities to bicker—about the paper’s focus, who undertook which tasks, which journal to submit to, and more. Bickering would have spiced up the story behind our paper, because the paper concerns disagreement.

Quantum observables can disagree. Observables are measurable properties, such as position and momentum. Suppose that you’ve measured a quantum particle’s position and obtained an outcome . If you measure the position immediately afterward, you’ll obtain again. Suppose that, instead of measuring the position again, you measure the momentum. All the possible outcomes have equal probabilities of obtaining. You can’t predict the outcome.

The particle’s position can have a well-defined value, or the momentum can have a well-defined value, but the observables can’t have well-defined values simultaneously. Furthermore, if you measure the position, you randomize the outcome of a momentum measurement. Position and momentum disagree.

How should we quantify the disagreement of two quantum observables, and ? The question splits physicists into two camps. Pure quantum information (QI) theorists use uncertainty relations, whereas condensed-matter and high-energy physicists prefer out-of-time-ordered correlators. Let’s meet the camps in turn.

Heisenberg intuited an uncertainty relation that Robertson formalized during the 1920s,

.

Imagine preparing a quantum state and measuring , then repeating this protocol in many trials. Each trial has some probability of yielding the outcome . Different trials will yield different ’s. We quantify the spread in values with the standard deviation . We define analogously. denotes Planck’s constant, a number that characterizes our universe as the electron’s mass does.

denotes the observables’ commutator. The numbers that we use in daily life commute: . Quantum numbers, or operators, represent and . Operators don’t necessarily commute. The commutator represents how little and resemble 7 and 5.

Robertson’s uncertainty relation means, “If you can predict an measurement’s outcome precisely, you can’t predict a measurement’s outcome precisely, and vice versa. The uncertainties must multiply to at least some number. The number depends on how much fails to commute with .” The higher an uncertainty bound (the greater the inequality’s right-hand side), the more the operators disagree.

Heisenberg and Robertson explored operator disagreement during the 1920s. They wouldn’t have seen eye to eye with today’s QI theorists. For instance, QI theorists consider how we can apply quantum phenomena, such as operator disagreement, to information processing. Information processing includes cryptography. Quantum cryptography benefits from operator disagreement: An eavesdropper must observe, or measure, a message. The eavesdropper’s measurement of one observable can “disturb” a disagreeing observable. The message’s sender and intended recipient can detect the disturbance and so detect the eavesdropper.

How efficiently can one perform an information-processing task? The answer usually depends on an entropy , a property of quantum states and of probability distributions. Uncertainty relations cry out for recasting in terms of entropies. So QI theorists have devised entropic uncertainty relations, such as

The entropy quantifies the difficulty of predicting the outcome of an measurement. is defined analogously. is called the *overlap*. It quantifies your ability to predict what happens if you prepare your system with a well-defined value, then measure . For further analysis, check out this paper. Entropic uncertainty relations have blossomed within QI theory over the past few years.

Pure QI theorists, we’ve seen, quantify operator disagreement with entropic uncertainty relations. Physicists at the intersection of condensed matter and high-energy physics prefer out-of-time-ordered correlators (OTOCs). I’ve blogged about OTOCs so many times, *Quantum Frontiers* regulars will be able to guess the next two paragraphs.

Consider a quantum many-body system, such as a chain of qubits. Imagine poking one end of the system, such as by flipping the first qubit upside-down. Let the operator represent the poke. Suppose that the system evolves chaotically for a time afterward, the qubits interacting.* *Information about the poke spreads through many-body entanglement, or *scrambles*.

Imagine measuring an observable of a few qubits far from the qubits. A little information about migrates into the qubits. But measuring reveals almost nothing about , because most of the information about has spread across the system. disagrees with , in a sense. Actually, disagrees with . The represents the time evolution.

The OTOC’s smallness reflects how much disagrees with at any instant . At early times , the operators agree, and the OTOC . At late times, the operators disagree loads, and the OTOC .

Different camps of physicists, we’ve seen, quantify operator disagreement with different measures: Today’s pure QI theorists use entropic uncertainty relations. Condensed-matter and high-energy physicists use OTOCs. Trust physicists to disagree about what “quantum operator disagreement” means.

I want peace on Earth. I conjectured, in 2016 or so, that one could reconcile the two notions of quantum operator disagreement. One must be able to prove an entropic uncertainty relation for scrambling, wouldn’t you think?

You might try substituting for the in Ineq. , and for the . You’d expect the uncertainty bound to tighten—the inequality’s right-hand side to grow—when the system scrambles. Scrambling—the condensed-matter and high-energy-physics notion of disagreement—would coincide with a high uncertainty bound—the pure-QI-theory notion of disagreement. The two notions of operator disagreement would agree. But the bound I’ve described doesn’t reflect scrambling. Nor do similar bounds that I tried constructing. I banged my head against the problem for about a year.

The sky brightened when Jason and Tony developed an interest in the conjecture. Their energy and conversation enabled us to prove an entropic uncertainty relation for scrambling, published this month.^{1} We tested the relation in computer simulations of a qubit chain. Our bound tightens when the system scrambles, as expected: The uncertainty relation reflects the same operator disagreement as the OTOC. We reconciled two notions of quantum operator disagreement.

As *Quantum Frontiers* regulars will anticipate, our uncertainty relation involves weak measurements and quasiprobability distributions: I’ve been studying their roles in scrambling over the past three years, with colleagues for whose collaborations I have the utmost gratitude. I’m grateful to have collaborated with Tony and Jason. Harmony helps when you’re tackling (quantum operator) disagreement—even if squabbling would spice up your paper’s backstory.

^{1}Thanks to *Communications Physics* for publishing the paper. For pedagogical formatting, read the arXiv version.

Over the past few months, I’ve grown to know Yoram better. He had reason to ask about quantum statistical mechanics, because his research stands up to its ears in the field. If forced to synopsize quantum statistical mechanics in five words, I’d say, “study of many-particle quantum systems.” Examples include gases of ultracold atoms. If given another five words, I’d add, “Calculate and use partition functions.” A partition function is a measure of the number of states, or configurations, accessible to the system. Calculate a system’s partition function, and you can calculate the system’s average energy, the average number of particles in the system, how the system responds to magnetic fields, etc.

My colloquium concerned quantum thermodynamics, which I’ve blogged about many times. So I should have been able to distinguish quantum thermodynamics from its neighbors. But the answer I gave Yoram didn’t satisfy me. I mulled over the exchange for a few weeks, then emailed Yoram a 502-word essay. The exercise grew my appreciation for the question and my understanding of my field.

An adaptation of the email appears below. The adaptation should suit readers who’ve majored in physics, but don’t worry if you haven’t. Bits of what distinguishes quantum thermodynamics from quantum statistical mechanics should come across to everyone—as should, I hope, the value of question-and-answer sessions:

One distinction is a return to the operational approach of 19th-century thermodynamics. Thermodynamicists such as Sadi Carnot wanted to know how effectively engines could operate. Their practical questions led to fundamental insights, such as the Carnot bound on an engine’s efficiency. Similarly, quantum thermodynamicists often ask, “How can this state serve as a resource in thermodynamic tasks?” This approach helps us identify what distinguishes quantum theory from classical mechanics.

For example, quantum thermodynamicists found an advantage in charging batteries via nonlocal operations. Another example is the “MBL-mobile” that I designed with collaborators.* *Many-body localization (MBL), we found, can enhance an engine’s reliability and scalability.

Asking, “How can this state serve as a resource?” leads quantum thermodynamicists to design quantum engines, ratchets, batteries, etc. We analyze how these devices can outperform classical analogues, identifying which aspects of quantum theory power the outperformance. This question and these tasks contrast with the questions and tasks of many non-quantum-thermodynamicists who use statistical mechanics. They often calculate response functions and (e.g., ground-state) properties of Hamiltonians.

These goals of characterizing what nonclassicality is and what it can achieve in thermodynamic contexts resemble upshots of quantum computing and cryptography. As a 21st-century quantum information scientist, I understand what makes quantum theory quantum partially by understanding which problems quantum computers can solve efficiently and classical computers can’t. Similarly, I understand what makes quantum theory quantum partially by understanding how much more work you can extract from a singlet (a maximally entangled state of two qubits) than from a product state in which the reduced states have the same forms as in the singlet, .

As quantum thermodynamics shares its operational approach with quantum information theory, quantum thermodynamicists use mathematical tools developed in quantum information theory. An example consists of generalized entropies. Entropies quantify the optimal efficiency with which we can perform information-processing and thermodynamic tasks, such as data compression and work extraction.

Most statistical-mechanics researchers use just the Shannon and von Neumann entropies, and , and perhaps the occasional relative entropy. These entropies quantify optimal efficiencies in large-system limits, e.g., as the number of messages compressed approaches infinity and in the thermodynamic limit.

Other entropic quantities have been defined and explored over the past two decades, in quantum and classical information theory. These entropies quantify the optimal efficiencies with which tasks can be performed (i) if the number of systems processed or the number of trials is arbitrary, (ii) if the systems processed share correlations, (iii) in the presence of “quantum side information” (if the system being used as a resource is entangled with another system, to which an agent has access), or (iv) if you can tolerate some probability that you fail to accomplish your task. Instead of limiting ourselves to and , we use also “-smoothed entropies,” Rényi divergences, hypothesis-testing entropies, conditional entropies, etc.

Another hallmark of quantum thermodynamics is results’ generality and simplicity. Thermodynamics characterizes a system with a few macroscopic observables, such as temperature, volume, and particle number. The simplicity of some quantum thermodynamics served a chemist collaborator and me, as explained in the introduction of https://arxiv.org/abs/1811.06551.

Yoram’s question reminded me of one reason why, as an undergrad, I adored studying physics in a liberal-arts college. I ate dinner and took walks with students majoring in economics, German studies, and Middle Eastern languages. They described their challenges, which I analyzed with the physics mindset that I was acquiring. We then compared our approaches. Encountering other disciplines’ perspectives helped me recognize what tools I was developing as a budding physicist. How can we know our corner of the world without stepping outside it and viewing it as part of a landscape?

^{1}The title epitomizes clarity and simplicity. And I have trouble resisting anything advertised as “the information-theoretic approach to such-and-such.”

Video games have been a part of my life for about as long as I can remember. From *Paperboy* and *The Last Ninja* on the Commodore 64 when I was barely old enough to operate a keyboard, to *Mario Kart 8* and *Zelda* on the Nintendo Switch, as a postdoc at Caltech, working on quantum computing and condensed matter physics. Up until recently, I have kept my two lives separate: my love of video games and my career in quantum physics.

The realization that I could combine quantum physics with games came during an entertaining discussion with my current supervisor, Gil Refael. Gil and I were brainstorming approaches to develop a quantum version of Tetris. Instead of stopping and laughing it off, or even keeping the idea on the horizon, Gil suggested that we talk to Spyridon (Spiros) Michalakis for some guidance.

This is not the story of Quantum Tetris (yet), but rather the story of how we made a quantum version of a much older, and possibly more universally known game. This is a new game that Spiros and myself have been testing at elementary schools.

And so I am super excited to be able to finally present to you: Quantum TiqTaqToe! As of *right now*, the app is available both for Android devices and iPhone/iPad:

Gil and I knew that Spiros had been involved in prior quantum games (most notably qCraft and Quantum Chess), so he seemed like the perfect contact point. He was conveniently located on the same campus, and even in the same department. But more importantly, he was curious about the idea and eager to talk.

After introducing the idea of Quantum Tetris, Spiros came up with an alternative approach. Seeing as this was going to be my first attempt at creating a video game, not to mention building a game from the ground up with quantum physics, he proposed to put me in touch with Chris Cantwell and help him improve the AI for Quantum Chess.

I thought long and hard about this proposition. Like five seconds. It was an amazing opportunity. I would get to look under the hood of a working and incredibly sophisticated video game, unlike any game ever made: the only game in the world I knew of that was truly based on quantum physics. And I would be solving a critical problem that I would have to deal with eventually, by adapting a conventional, classical rules-based game AI for quantum.

My first focus was to jump on Quantum Chess full-force, with the aim of helping Chris implement a new AI player for the game. After evaluating some possible chess-playing AI engines, including state-of-the-art players based off of Google’s AlphaZero, we landed on Stockfish as our best candidate for integration. The AI is currently hot-swappable though, so users can try to develop their own!

While some of the work for implementing the AI could be done directly using Chris’s C++ implementation of Quantum Chess, other aspects of the work required me to learn the program he had used to develop the user interface. That program is called Unity. Unity is a free game development program that I would highly recommend trying out and playing around with.

This experience was essential to the birth of Quantum TiqTaqToe. In my quest to understand Unity and Quantum Games, I set out to implement a “simple” game to get a handle on how all the different game components worked together. Having a game based on quantum mechanics is one thing; making sure it is fun to play requires an entirely different skill set.

Classic Tic-Tac-Toe is a game in which two players, called X and O, take turns in placing their symbols on a 3×3 grid. The first player to get 3 of their symbols in a line (diagonally, vertically or horizontally) wins. The game goes as far back as ancient Egypt, and evidence of the game has been found on roof tiles dating to 1300 BC [1].

Many variations of the game have existed across many cultures. The first print reference to a game called “tick-tack-toe” was in 1884. In the US the game was renamed “tic-tac-toe” sometime in the 20th century. Here’s a random fun fact: in Dutch, the game is most often referred to as “Butter-Cheese-and-Eggs” [2]. In 1952, computer scientist Alexander S. Douglas at the University of Cambridge turned it into one of the first computer games, featuring an AI player that could play perfect games against a human opponent.

Combinatorics has determined that whoever plays first will win 91 out of 138 possible board combinations. The second player will win in 44 boards. However, if both players play *optimally*, looking ahead through all the possible future outcomes, neither player should ever win and the game always ends in a draw, in one of only 3 board combinations.

*In Quantum TiqTaqToe, with the current ruleset, we don’t yet know if a winning strategy exists.*

I explicitly refer to the *current ruleset* because we currently limit the amount of quantumness in the game. We want to make sure the game is fun to play and ‘graspable’ for now. In addition, it turns out there already is a game called Quantum TicTacToe, developed by Allan Goff [3]. That version of TicTacToe has similar concepts but has a different set of rules.

A typical game of Quantum TiqTaqToe will look very much like regular Tic-Tac-Toe until one of the players decides to make a quantum move:

At this point, the game board enters into a superposition. The X is in each position with 50/50 chance; in one universe the X is on the left and in the other it is on the right. Neither player knows how things will play out. And the game only gets more interesting from here. The opponent can choose to place their O in a superposition between an empty square and a square occupied by a quantum X.

Et voilà, player O has entangled his fate with his opponent’s. Once the two squares become entangled, the only outcomes are X-O or O-X, each with probability ½. Interestingly, since the game is fully quantum, the phase between the two entangled outcomes can in principle be leveraged to create interesting plays through destructive and constructive interference. The app features a simple tutorial (to be updated) that teaches you these moves and a few others. There are boards that classically result in a draw but are quantumly “winnable”.

The squares in TiqTaqToe are all fully quantum. I represent them as qutrits (like qubits, but instead of having states 0 and 1 my qutrits have states 0, 1 and 2), and moves made by the players are unitary operations acting on them. So the game consists of these essential elements:

- The squares of the 3×3 grid are turned into qutrits (Empty, X, O). Each move is a unitary gate operation on those qutrits. I’ll leave the details of the math out, but for the case of qubits check out Chris’ detailed writeup on Quantum Chess [4].
- Quantum TiqTaqToe allows you to select
*two*squares in the grid, providing you with the option of creating a superposition or an entangled state. For the sake of simplicity (i.e. keeping the game fun to play and ‘graspable’ for now), no more than 3 squares can be involved in a given entangled state.

I chose to explicitly track sets of qutrits that share a Hilbert space. The entire quantum state of the game combines these sets with classical strings of the form “XEEOXEOXE”, indicating that the first square is an X, the second is Empty, etc.

So, when does the game end if these quantum states are in play? In Quantum TiqTaqToe, the board collapses to a single classical state as soon as it is full (i.e. every square is non-empty). The resulting state is randomly chosen from all the possible outcomes, with a probability that is equal to the (square of the) wave-function amplitude (basic quantum mechanics). If there is a winner after the collapse, the game ends. Otherwise, the game continues until either there *is* a winner or until there are no more moves to be made (ending in a draw). On top of this, players get the option to forfeit their move for the opportunity to cause a partial collapse of the state, by using the collapse-mode. Future versions may include other ways of collapse, including one that does not involve rolling dice! [5]

Due to quantum physics and the collapse of the state, the game is inherently statistical. So instead of asking: “Can I beat my opponent in a game of Quantum TiqTaqToe?” one should ask “If I play 100 games against my opponent, can I consistently win more than 50 of them?”

You can test your skill against the in-game quantum AI to see if you’ve indeed mastered Quantum TiqTaqToe yet. At the hardest setting, winning even 30% of the time after, say, 20 games may be extraordinary. The implementation of this AI, by the way, would have been a blog-post by itself. For the curious, I can say it is based on the ExpectiMiniMax algorithm. As of the moment of this writing, the hardest AI setting is not available in the app yet. Keep your eyes out for an update soon though!

“*Perhaps kids who grow up playing quantum games will acquire a visceral understanding of quantum phenomena that our generation lacks*.” – John Preskill, in his recent article [6].

From the get-go, Quantum TiqTaqToe (and Quantum Chess) have had outreach as a core motivation. Perhaps future quantum engineers and quantum programmers will look back on their youth and remember playing Quantum TiqTaqToe as I remember my Commodore 64 games. I am convinced that these small steps into the realm of Quantum Games are only just the beginning of an entirely new genre of fun and useful games.

In the meantime, we are hard at work implementing an Online mode so you can play with your fellow human friends remotely too. This online mode, plus the option of fighting a strong quantum AI, will be unlockable in-game through a small fee (unless you are an educator who wishes to introduce quantum physics in class through this game; those use cases are fee-free courtesy of IQIM and NSF). Each purchase will go towards supporting the future development of exciting new Quantum TiqTaqToe features, as well as other exciting Quantum Games (Tetris, anyone?)

Just in case you missed it: the app is available both for Android devices and iPhone/iPad *right now*:

I really hope you enjoy the game, and perhaps use it to get your friends and family excited about quantum physics. Oh, and start practicing! You never know if the online mode will bring along with it a real Quantum TiqTaqToe Tournament down the road

[1] https://en.wikipedia.org/wiki/Tic-tac-toe

[2] The origin of this name in Dutch isn’t really certain as far as I know. Alledgedly, it is a left-over from the period in which butter, cheese and eggs were sold at the door (so was milk, but that was done separately since it was sold daily). The salesman had a list with columns for each of these three products, and would jot down a cross or a zero whenever a customer at an address bought or declined a product. Three crosses in a row would earn them praise from the boss.

[3] https://en.wikipedia.org/wiki/Quantum_tic-tac-toe

[4] https://arxiv.org/abs/1906.05836

[5] https://en.wiktionary.org/wiki/God_does_not_play_dice_with_the_universe

]]>Merchandise spilled outside shops onto the streets, restaurateurs parked diners under trees, and ice-cream cones begged to be eaten on park benches. People thronged the streets, markets filled public squares, and the scents of flowers wafted from vendors’ stalls. I couldn’t blame the city. Its sunshine could have drawn Merlin out of his crystal cave. Insofar as a city lives, Barcelona epitomized a quotation by thermodynamicist Ilya Prigogine: “The main character of any living system is openness.”

Prigogine (1917–2003), who won the Nobel Prize for chemistry, had brought me to Barcelona. I was honored to receive, at the Joint European Thermodynamics Conference (JETC) there, the Ilya Prigogine Prize for a thermodynamics PhD thesis. The JETC convenes and awards the prize biennially; the last conference had taken place in Budapest. Barcelona suited the legacy of a thermodynamicist who illuminated open systems.

Ilya Prigogine began his life in Russia, grew up partially in Germany, settled in Brussels, and worked at American universities. His nobelprize.org biography reveals a mind open to many influences and disciplines: Before entering university, his “interest was more focused on history and archaeology, not to mention music, especially piano.” Yet Prigogine pursued chemistry.

He helped extend thermodynamics outside equilibrium. Thermodynamics is the study of energy, order, and time’s arrow in terms of large-scale properties, such as temperature, pressure, and volume. Many physicists think that thermodynamics describes only equilibrium. Equilibrium is a state of matter in which (1) large-scale properties remain mostly constant and (2) stuff (matter, energy, electric charge, etc.) doesn’t flow in any particular direction much. Apple pies reach equilibrium upon cooling on a countertop. When I’ve described my research as involving nonequilibrium thermodynamics, some colleagues have asked whether I’ve used an oxymoron. But “nonequilibrium thermodynamics” appears in Prigogine’s Nobel Lecture.

Another Nobel laureate, Lars Onsager, helped extend thermodynamics a little outside equilibrium. He imagined poking a system gently, as by putting a pie on a lukewarm stovetop or a magnet in a weak magnetic field. (Experts: Onsager studied the linear-response regime.) You can read about his work in my blog post “Long live Yale’s cemetery.” Systems poked slightly out of equilibrium tend to return to equilibrium: Equilibrium is stable. Systems flung far from equilibrium, as Prigogine showed, can behave differently.

A system can stay far from equilibrium by interacting with other systems. Imagine placing an apple pie atop a blistering stove. Heat will flow from the stove through the pie into the air. The pie will stay out of equilibrium due to interactions with what we call a “hot reservoir” (the stove) and a “cold reservoir” (the air). Systems (like pies) that interact with other systems (like stoves and air), we call “open.”

You and I are open: We inhale air, ingest food and drink, expel waste, and radiate heat. Matter and energy flow through us; we remain far from equilibrium. A bumper sticker in my high-school chemistry classroom encapsulated our status: “Old chemists don’t die. They come to equilibrium.” We remain far from equilibrium—alive—because our environment provides food and absorbs heat. If I’m an apple pie, the yogurt that I ate at breakfast serves as my stovetop, and the living room in which I breakfasted serves as the air above the stove. We live because of our interactions with our environments, because we’re open. Hence Prigogine’s claim, “The main character of any living system is openness.”

JETC 2019 fostered openness. The conference sessions spanned length scales and mass scales, from quantum thermodynamics to biophysics to gravitation. One could arrive as an expert in cell membranes and learn about astrophysics.

I remain grateful for the prize-selection committee’s openness. The topics of earlier winning theses include desalination, colloidal suspensions, and falling liquid films. If you tipped those topics into a tube, swirled them around, and capped the tube with a kaleidoscope glass, you might glimpse my thesis’s topic, quantum steampunk. Also, of the nine foregoing Prigogine Prize winners, only one had earned his PhD in the US. I’m grateful for the JETC’s consideration of something completely different.

When Prigogine said, “openness,” he referred to exchanges of energy and mass. Humans can exhibit openness also to ideas. The JETC honored Prigogine’s legacy in more ways than one. Here’s hoping I live up to their example.

]]>How do the laws of thermodynamics apply in the quantum regime? Thanks to novel ideas introduced in the context of quantum information, scientists have been able to develop new ways to characterize the thermodynamic behavior of quantum states. If you’re a *Quantum Frontiers* regular, you have certainly read about these advances in Nicole’s captivating posts on the subject.

Asking the same question for *quantum channels*, however, turned out to be more challenging than expected. A quantum channel is a way of representing how an input state can change into an output state according to the laws of quantum mechanics. Let’s picture it as a box with an input state and an output state, like so:

A computing gate, the building block of quantum computers, is described by a quantum channel. Or, if Alice sends a photon to Bob over an optical fiber, then the whole process is represented by a quantum channel. Thus, by studying quantum channels directly we can derive statements that are valid regardless of the physical platform used to store and process the quantum information—ion traps, superconducting qubits, photonic qubits, NV centers, etc.

We asked the following question: If I’m given a quantum channel, can I transform it into another, different channel by using something like a miniature heat engine? If so, how much work do I need to spend in order to accomplish this task? The answer is tricky because of a few aspects in which quantum channels are more complicated than quantum states.

In this post, I’ll try to give some intuition behind our results, which were developed with the help of Mario Berta and Fernando Brandão, and which were recently published in Physical Review Letters.

First things first, let’s worry about how to study the thermodynamic behavior of miniature systems.

One of the important ideas that quantum information brought to thermodynamics is the idea of a resource theory. In a resource theory, we declare that there are certain kinds of states that are available for free, and that there are a set of operations that can be carried out for free. In a resource theory of thermodynamics, when we say “for free,” we mean “without expending any thermodynamic work.”

Here, the free states are those in thermal equilibrium at a fixed given temperature, and the free operations are those quantum operations that preserve energy and that introduce no noise into the system (we call those *unitary operations*). Faced with a task such as transforming one quantum state into another, we may ask whether or not it is possible to do so using the freely available operations. If that is not possible, we may then ask how much thermodynamic work we need to invest, in the form of additional energy at the input, in order to make the transformation possible.

Interestingly, the amount of work needed to go from one state *ρ* to another state *σ* might be unrelated to the work required to go back from *σ* to *ρ*. Indeed, the freely allowed operations can’t always be reversed; the reverse process usually requires a different sequence of operations, incurring an overhead. There is a mathematical framework to understand these transformations and this reversibility gap, in which generalized entropy measures play a central role. To avoid going down that road, let’s instead consider the macroscopic case in which we have a large number *n* of independent particles that are all in the same state *ρ*, a state which we denote by . Then something magical happens: This macroscopic state can be reversibly converted to and from another macroscopic state , where all particles are in some other state *σ*. That is, the work invested in the transformation from to can be entirely recovered by performing the reverse transformation:

If this rings a bell, that is because this is precisely the kind of thermodynamics that you will find in your favorite textbook. There is an optimal, reversible way of transforming any two thermodynamic states into each other, and the optimal work cost of the transformation is the difference of a corresponding quantity known as the *thermodynamic potential*. Here, the thermodynamic potential is a quantity known as the *free energy* . Therefore, the optimal work cost per copy *w* of transforming into is given by the difference in free energy .

Can we repeat the same story for quantum channels? Suppose that we’re given a channel , which we picture as above as a box that transforms an input state into an output state. Using the freely available thermodynamic operations, can we “transform” into another channel ? That is, can we wrap this box with some kind of procedure that uses free thermodynamic operations to pre-process the input and post-process the output, such that the overall new process corresponds (approximately) to the quantum channel ? We might picture the situation like this:

Let us first simplify the question by supposing we don’t have a channel to start off with. How can we implement the channel from scratch, using only free thermodynamic operations and some invested work? That simple question led to pages and pages of calculations, lots of coffee, a few sleepless nights, and then more coffee. After finally overcoming several technical obstacles, we found that in the macroscopic limit of many copies of the channel, the corresponding amount of work per copy is given by the maximum difference of free energy *F* between the input and output of the channel. We decided to call this quantity the *thermodynamic capacity* of the channel:

Intuitively, an implementation of must be prepared to expend an amount of work corresponding to the worst possible transformation of an input state to its corresponding output state. It’s kind of obvious in retrospect. However, what is nontrivial is that one can find a single implementation that works for all input states.

It turned out that this quantity had already been studied before. An earlier paper by Navascués and García-Pintos had shown that it was exactly this quantity that characterized the amount of work per copy that could be extracted by “consuming” many copies of a process provided as black boxes.

To our surprise, we realized that Navascués and García-Pintos’s result implied that the transformation of into is reversible. There is a simple procedure to convert into at a cost per copy that equals . The procedure consists in first extracting work per copy of the first set of channels, and then preparing from scratch at a work cost of per copy:

Clearly, the reverse transformation yields back all the work invested in the forward transformation, making the transformation reversible. That’s because we could have started with ’s and finished with ’s instead of the opposite, and the associated work cost per copy would be . Thus the transformation is, indeed, reversible:

In turn, this implies that in the many-copy regime, quantum channels have a macroscopic thermodynamic behavior. That is, there is a thermodynamic potential—the thermodynamic capacity—that quantifies the minimal work required to transform one macroscopic set of channels into another.

Resource theories that are reversible are pretty rare. Reversibility is a coveted property because a reversible resource theory is one in which we can easily understand exactly which transformations are possible. Other than the thermodynamic resource theory of states mentioned above, most instances of a resource theory—especially resource theories of channels—typically produce the kind of overheads in the conversion cost that spoil reversibility. So it’s rather exciting when you do find a new reversible resource theory of channels.

Quantum information theorists, especially those working on the theory of quantum communication, care a lot about characterizing the capacity of a channel. This is the maximal amount of information that can be transmitted through a channel. Even though in our case we’re talking about a different kind of capacity—one where we transmit thermodynamic energy and entropy, rather than quantum bits of messages—there are some close parallels between the two settings from which both fields of quantum communication and quantum thermodynamics can profit. Our result draws deep inspiration from the so-called *quantum reverse Shannon theorem*, an important result in quantum communication that tells us how two parties can communicate using one kind of a channel if they have access to another kind of a channel. On the other hand, the thermodynamic capacity at zero energy is a quantity that was already studied in quantum communication, but it was not clear what that quantity represented concretely. This quantity gained even more importance as it was identified as the entropy of a channel. Now, we see that this quantity has a thermodynamic interpretation. Also, the thermodynamic capacity has a simple definition, it is relatively easy to compute and it is additive—all desirable properties that other measures of capacity of a quantum channel do not necessarily share.

We still have a few rough edges that I hope we can resolve sooner or later. In fact, there is an important caveat that I have avoided mentioning so far—our argument only holds for special kinds of channels, those that do the same thing regardless of when they are applied in time. (Those channels are called *time-covariant*.) A lot of channels that we’re used to studying have this property, but we think it should be possible to prove a version of our result for any general quantum channel. In fact, we do have another argument that works for all quantum channels, but it uses a slightly different thermodynamic framework which might not be physically well-grounded.

That’s all very nice, I can hear you think, but is this useful for any quantum computing applications? The truth is, we’re still pretty far from founding a new quantum start-up. The levels of heat dissipation in quantum logic elements are still orders of magnitude away from the fundamental limits that we study in the thermodynamic resource theory.

Rather, our result teaches us about the interplay of quantum channels and thermodynamic concepts. We not only have gained useful insight on the structure of quantum channels, but also developed new tools for how to analyze them. These will be useful to study more involved resource theories of channels. And still, in the future when quantum technologies will perhaps approach the thermodynamically reversible limit, it might be good to know how to implement a given quantum channel in such a way that good accuracy is guaranteed for any possible quantum input state, and without any inherent overhead due to the fact that we don’t know what the input state is.

Thermodynamics, a theory developed to study gases and steam engines, has turned out to be relevant from the most obvious to the most unexpected of situations—chemical reactions, electromagnetism, solid state physics, black holes, you name it. Trust the laws of thermodynamics to surprise you again by applying to a setting you’d never imagined them to, like quantum channels.

]]>Quantum physics influences cognition insofar as (i) quantum physics prevents atoms from imploding and (ii) implosion inhabits atoms from contributing to cognition. But most physicists believe that useful entanglement can’t survive in brains. Entanglement consists of correlations shareable by quantum systems and stronger than any achievable by classical systems. Useful entanglement dies quickly in hot, wet, random environments.

Brains form such environments. Imagine injecting entangled molecules *A* and *B* into someone’s brain. Water, ions, and other particles would bombard the molecules. The higher the temperature, the heavier the bombardment. The bombardiers would entangle with the molecules via electric and magnetic fields. Each molecule can share only so much entanglement. The more *A* entangled with the environment, the less *A* could remain entangled with* B*. *A* would come to share a tiny amount of entanglement with each of many particles. Such tiny amounts couldn’t accomplish much. So quantum physics seems unlikely to affect cognition significantly.

Yet my PhD advisor, John Preskill, encouraged me to consider whether the possibility interested me.

*Try some completely different research*, he said. *Take a risk. If it doesn’t pan out, fine. People don’t expect much of grad students, anyway. Have you seen Matthew Fisher’s paper about quantum cognition?** *

Matthew Fisher is a theoretical physicist at the University of California, Santa Barbara. He has plaudits out the wazoo, many for his work on superconductors. A few years ago, Matthew developed an interest in biochemistry. He knew that most physicists doubt whether quantum physics could affect cognition much. But suppose that it could, he thought. How could it? Matthew reverse-engineered a mechanism, in a paper published by *Annals of Physics* in 2015.

A PhD student shouldn’t touch such research with a ten-foot radio antenna, says conventional wisdom. But I trust John Preskill in a way in which I trust no one else on Earth.

*I’ll look at the paper*, I said.

Matthew proposed that quantum physics could influence cognition as follows. Experimentalists have performed quantum computation using one hot, wet, random system: that of nuclear magnetic resonance (NMR). NMR is the process that underlies magnetic resonance imaging (MRI), a technique used to image people’s brains. A common NMR system consists of high-temperature liquid molecules. The molecules consists of atoms whose nuclei have quantum properties called *spin*. The nuclear spins encode quantum information (QI).

Nuclear spins, Matthew reasoned, might store QI in our brains. He catalogued the threats that could damage the QI. Hydrogen ions, he concluded, would threaten the QI most. They could entangle with (decohere) the spins via dipole-dipole interactions.

How can a spin avoid the threats? First, by having a quantum number . Such a quantum number zeroes out the nuclei’s electric quadrupole moments. Electric-quadrupole interactions can’t decohere such spins. Which biologically prevalent atoms have nuclear spins? Phosphorus and hydrogen. Hydrogen suffers from other vulnerabilities, so phosphorus nuclear spins store QI in Matthew’s story. The spins serve as qubits, or quantum bits.

How can a phosphorus spin avoid entangling with other spins via magnetic dipole-dipole interactions? Such interactions depend on the spins’ orientations relative to their positions. Suppose that the phosphorus occupies a small molecule that tumbles in biofluids. The nucleus’s position changes randomly. The interaction can average out over tumbles.

The molecule contains atoms other than phosphorus. Those atoms have nuclei whose spins can interact with the phosphorus spins, unless every threatening spin has a quantum number . Which biologically prevalent atoms have nuclear spins? Oxygen and calcium. The phosphorus should therefore occupy a molecule with oxygen and calcium.

Matthew designed this molecule to block decoherence. Then, he found the molecule in the scientific literature. The structure, , is called a *Posner cluster* or a *Posner molecule. *I’ll call it a *Posner*, for short. Posners appear to exist in simulated biofluids, fluids created to mimic the fluids in us. Posners are believed to exist in us and might participate in bone formation. According to Matthew’s estimates, Posners might protect phosphorus nuclear spins for up to 1-10 days.

How can Posners influence cognition? Matthew proposed the following story.

Adenosine triphosphate (ATP) is a molecule that fuels biochemical reactions. “Triphosphate” means “containing three phosphate ions.” Phosphate () consists of one phosphorus atom and three oxygen atoms. Two of an ATP molecule’s phosphates can break off while remaining joined to each other.

The phosphate pair can drift until encountering an enzyme called *pyrophosphatase*. The enzyme can break the pair into independent phosphates. Matthew, with Leo Radzihovsky, conjectured that, as the pair breaks, the phosphorus nuclear spins are projected onto a singlet. This state, represented by , is maximally entangled.

Imagine many entangled phosphates in a biofluid. Six phosphates can join nine calcium ions to form a Posner molecule. The Posner can share up to six singlets with other Posners. Clouds of entangled Posners can form.

One clump of Posners can enter one neuron while another clump enters another neuron. The protein VGLUT, or BNPI, sits in cell membranes and has the potential to ferry Posners in. The neurons will share entanglement. Imagine two Posners, *P* and* Q*, approaching each other in a neuron *N*. Quantum-chemistry calculations suggest that the Posners can bind together. Suppose that *P* shares entanglement with a Posner *P’* in a neuron *N’*, while *Q* shares entanglement with a Posner *Q’* in *N’*. The entanglement, with the binding of *P* to *Q*, can raise the probability that *P’* binds to *Q’*.

Bound-together Posners will move slowly, having to push much water out of the way. Hydrogen and magnesium ions can latch onto the slow molecules easily. The Posners’ negatively charged phosphates will attract the and as the phosphates attract the Posner’s . The hydrogen and magnesium can dislodge the calcium, breaking apart the Posners. Calcium will flood neurons *N* and *N’*. Calcium floods a neuron’s axion terminal (the end of the neuron) when an electrical signal reaches the axion. The flood induces the neuron to release neurotransmitters. Neurotransmitters are chemicals that travel to the next neuron, inducing it to fire. So entanglement between phosphorus nuclear spins in Posner molecules might stimulate coordinated neuron firing.

Does Matthew’s story play out in the body? We can’t know till running experiments and analyzing the results. Experiments have begun: Last year, the Heising-Simons Foundation granted Matthew and collaborators $1.2 million to test the proposal.

Suppose that Matthew conjectures correctly, John challenged me, or correctly enough. Posner molecules store QI. Quantum systems can process information in ways in which classical systems, like laptops, can’t. How adroitly can Posners process QI?

I threw away my iron-tipped medieval lance in year five of my PhD. I left Caltech for a five-month fellowship, bent on returning with a paper with which to answer John. I did, and *Annals of Physics* published the paper this month.

I had the fortune to interest Elizabeth Crosson in the project. Elizabeth, now an assistant professor at the University of New Mexico, was working as a postdoc in John’s group. Both of us are theorists who specialize in QI theory. But our backgrounds, skills, and specialties differ. We complemented each other while sharing a doggedness that kept us emailing, GChatting, and Google-hangout-ing at all hours.

Elizabeth and I translated Matthew’s biochemistry into the mathematical language of QI theory. We dissected Matthew’s narrative into a sequence of biochemical steps. We ascertained how each step would transform the QI encoded in the phosphorus nuclei. Each transformation, we represented with a piece of math and with a circuit-diagram element. (Circuit-diagram elements are pictures strung together to form circuits that run algorithms.) The set of transformations, we called *Posner operations.*

Imagine that you can perform Posner operations, by preparing molecules, trying to bind them together, etc. What QI-processing tasks can you perform? Elizabeth and I found applications to quantum communication, quantum error detection, and quantum computation. Our results rest on the assumption—possibly inaccurate—that Matthew conjectures correctly. Furthermore, we characterized what Posners could achieve if controlled. Randomness, rather than control, would direct Posners in biofluids. But what can happen in principle offers a starting point.

First, QI can be teleported from one Posner to another, while suffering noise.^{1} This noisy teleportation doubles as superdense coding: A trit is a random variable that assumes one of three possible values. A bit is a random variable that assumes one of two possible values. You can teleport a trit from one Posner to another effectively, while transmitting a bit directly, with help from entanglement.

Second, Matthew argued that Posners’ structures protect QI. Scientists have developed quantum error-correcting and -detecting codes to protect QI. Can Posners implement such codes, in our model? Yes: Elizabeth and I (with help from erstwhile Caltech postdoc Fernando Pastawski) developed a quantum error-detection code accessible to Posners. One Posner encodes a logical qutrit, the quantum version of a trit. The code detects any error that slams any of the Posner’s six qubits.

Third, how complicated an entangled state can Posner operations prepare? A powerful one, we found: Suppose that you can measure this state locally, such that earlier measurements’ outcomes affect which measurements you perform later. You can perform any quantum computation. That is, Posner operations can prepare a state that fuels universal measurement-based quantum computation.

Finally, Elizabeth and I quantified effects of entanglement on the rate at which Posners bind together. Imagine preparing two Posners, *P* and *P’*, that share entanglement only with other particles. If the Posners approach each other with the right orientation, they have a 33.6% chance of binding, in our model. Now, suppose that every qubit in *P* is maximally entangled with a qubit in *P’*. The binding probability can rise to 100%.

I feared that other scientists would pooh-pooh our work as crazy. To my surprise, enthusiasm flooded in. Colleagues cheered the risk on a challenge in an emerging field that perks up our ears. Besides, Elizabeth’s and my work is far from crazy. We don’t assert that quantum physics affects cognition. We imagine that Matthew conjectures correctly, acknowledging that he might not, and explore his proposal’s implications. Being neither biochemists nor experimentalists, we restrict our claims to QI theory.

Maybe Posners can’t protect coherence for long enough. Would inaccuracy of Matthew’s beach our whale of research? No. Posners prompted us to propose ideas and questions within QI theory. For instance, our quantum circuits illustrate interactions (unitary gates, to experts) interspersed with measurements implemented by the binding of Posners. The circuits partially motivated a subfield that emerged last summer and is picking up speed: Consider interspersing random unitary gates with measurements. The unitaries tend to entangle qubits, whereas the measurements disentangle. Which influence wins? Does the system undergo a phase transition from “mostly entangled” to “mostly unentangled” at some measurement frequency? Researchers from Santa Barbara to Colorado; MIT; Oxford; Lancaster, UK; Berkeley; Stanford; and Princeton have taken up the challenge.

A physics PhD student, conventional wisdom says, shouldn’t touch quantum cognition with a Swiss guard’s halberd. I’m glad I reached out: I learned much, contributed to science, and had an adventure. Besides, if anyone disapproves of daring, I can blame John Preskill.

*Annals of Physics published “Quantum information in the Posner model of quantum cognition” here.** You can find the arXiv version here and can watch a talk about our paper here.*

^{1}Experts: The noise arises because, if two Posners bind, they effectively undergo a measurement. This measurement transforms a subspace of the two-Posner Hilbert space as a coarse-grained Bell measurement. A Bell measurement yields one of four possible outcomes, or two bits. Discarding one of the bits amounts to coarse-graining the outcome. Quantum teleportation involves a Bell measurement. Coarse-graining the measurement introduces noise into the teleportation.

In 2013, I was attending a workshop on noise, information and complexity at the Ettore Majorana Center in beautiful Erice, Sicily, a medieval town sitting on top of a steep hill overlooking the western part of the island. The town, a network of tiny, winding streets lined mostly with medieval buildings, was foggy most days. The Center I was visiting, apart from its awe-inspiring location, is said to have played an important role in fostering relationships between scientists of the West and the East during the Cold War. As a proof of its openness to hosting even the most unexpected of visitors, the Center proudly displays a picture of Pope John Paul II seated behind a version of Dirac’s equation missing an all-important , the unit of imaginary numbers.

One afternoon, the hosts of the workshop drove us down to Palermo for sightseeing. We toured a number of churches, whose layered styles and decorations reflected the different cultures that flourished on the island over the centuries. The last stop on our tour was the Martorana Church, an Italo-Albanian church of the 12th century, where to this day Mass is held in ancient Greek (yes, it is a complicated history). And while everybody had their noses up in the air, admiring the golden mosaics on the ceilings and the late baroque decorations, I was mesmerized by what lied underneath my feet. I am not talking about some forgotten crypt or creepy burial vault: I was looking at triangles – colorful, 12th century triangles.

What I was looking at, was a 12th century version of a fractal figure which is known today as the Sierpinski triangle, a geometric pattern named after Wacław Sierpiński, the Polish mathematician who studied it eight centuries later, in 1915.

You might think this famous tiling pattern was a fluke back then, a random pattern appearing only on the floor of this particular church. It turns out that this type of decoration existed all over the floors of Italy and Europe and was due to a family of Roman artists known as the Cosmati. If you find this fascinating (and you definitely should), I recommend reading “Sierpinski triangles in stone, on medieval floors in Rome”, by Conversano and Tedeschini Lalli, *J. Appl. Math* 4 (2011). Or you can simply maze through the pictures of these pavements on Wikipedia.

Since I was a little kid, I was fascinated by tilings. I would spend hours looking at them (don’t all kids do?), trying to figure out which set of tiles was sufficient to reproduce the whole thing (which, to my great surprise, did not always coincide with the way the tiles were cut). I didn’t know at the time that what I was looking for was the *period* of the tiling, the minimum set of tiles needed to cover the whole space in a periodic fashion. To illustrate this concept, let’s have a look at these beautiful Ottoman tiles from the city of İznik, Turkey.

Here, we quickly realize that there are two different kinds of tiles: the top right and bottom left tiles are the same, whereas the ones on the diagonal are mirror reflections of the off-diagonal ones. The artist who made these had to actually paint two different kinds of tiles, preparing two separate stacks, one for each kind. If the tiles were made of thin, translucent glass, only one stack would have been necessary (why?)

While it is the drawings that make these tiles beautiful, if we wish to study how they can be composed, we might as well forget about the particular details of the drawings for a moment, and just focus on how each tile can be attached to its neighbor while preserving the continuity of the picture (this is something we do a lot in science, trying to focus on important features by filtering out unnecessary details). Since each square tile has four neighbors, we can think of these two different kinds of tiles in the following way:

From this new point of view, one kind of tile is just a square with four quadrants labeled 1,2,3,4 in a clockwise fashion, and the other kind of tile (the reflection of the first kind) has four quadrants labeled, -1, -2, -3, and -4, also in a clockwise fashion (as if looking at the first kind of tiles from the other side). The tiling rule is such that neighboring tiles sum to zero across their common edge. Now it is easy to see that, if we were given only one type of tiles, we could not do much with them, since the sum would always be positive (for the positive tiles), or negative (for the negative ones) across any edge, but never zero. But if we have access to both types, then we can cover an arbitrarily large surface.

But, how do we know that we can actually keep going and fill up any rectangular region, no matter how big it is? The trick is, there is a pattern which repeats: every second tile (both horizontally and vertically), the colors repeat, so we can keep making the same choice over and over again. There is a 2×2 square which is our *period*, and once we obtain it we can simply copy-and-paste this period as many times as we need. Notice that a period is the smallest tiling whose sum is zero along each of the two dimensions.

The Sierpinski tiling, on the other hand, does not have a period.

Try to focus on the pattern of the small drank green triangles. In the top row, they appear fairly often, but already in the second row they are spaced further apart, and then in the middle of the picture there is a big segment (the light green triangle) where they don’t appear. In other words, since we have larger and larger triangles appearing, there **cannot** be a period, since we would eventually find a triangle larger than the period itself! Tilings of this kind are called **aperiodic**.

While the Sierpinski triangle does not have a period that could cover the whole plane as the triangle gets bigger and bigger, if we use a Sierpinski triangle of a fixed size, we can actually generate a simple periodic tiling of the plane, as follows: Attach upside-down versions of the original triangle to its left and right, repeating the process in both directions ad infinitum. Then, take this infinite row of triangles, flip it upside-down and glue it to the original row below, stacking copies of these two rows on top of each other to fill an infinite plane. The aperiodicity of the Sierpinski triangle was a choice of how the smaller triangles tiled the inside of the Sieprinski triangle as it got larger and larger. The same set of triangles would tile the plane periodically if we used the procedure outlined above. In other words, aperiodicity was by choice, not of a necessity. But could there be a particular set of tiles for which no periodic tiling could ever exist?

In 1961, Hao Wang conjectured that, at least for the case of square tiles (which are now called Wang tiles), this is not the case: If a set of square tiles can cover an arbitrarily large rectangle, then there is a way to do so in a periodic fashion. Wang was not interested in floor tilings (at least, we don’t know of any floors decorated by him). Instead, he cared about the *decidability* of the tiling problem: given a set of tiles, is there an algorithm which can tell whether these tiles can be used to tile an infinitely large floor? If Wang’s conjecture about square tiles was true, we could set up a computer program that explored all the possible ways of covering a 1×1 square, then a 2×2 square, then a 3×3 square, and so on. The program would simply try every possible combination: while there are a lot of combinations, for any *n-by-n* square there is a finite number of tilings, so the computer could just check every single one of them. Specifically, at some point in the computation, one of two things would happen and the program would stop:

- The computer would find a square which could not be covered with the given tiles, or
- The computer would find a square which contained a period.

When either of the above happened, the program would stop. In the first case, finding a square which cannot be covered by our tiles implies that any larger square is also impossible to tile. In the second case, since we have found a period, just like in the case of the tiles from İznik, we can tile any rectangular region by repeating the period as needed. The computer might take a long time to decide whether 1. or 2. is the case for our set of tiles, but we know that we will always get an answer, with certainty, at some point. You may be thinking by now that there is a third possibility that I skipped over: *The tiles could cover the whole space, but not in a periodic way.* And you would be correct in thinking that.

If Wang’s conjecture were to be false, and there is a set of tiles which only generates aperiodic tilings, then our computer program would keep exploring larger and larger squares, without ever being able to give us a definitive answer whether we could tile the plane with this set of tiles. It would keep calculating, using more and more resources, until either it ran out of memory, or the heat generated by the computation boiled the oceans and the Earth and the tiles themselves.

So is Wang’s conjecture true? In 1964, a student of Wang, Robert Berger, showed in his PhD thesis that this conjecture is false: he constructed a set of 20,426 tiles which cover the plane, but can only do so aperiodically! Even worse than that, he actually managed to show that the tiling problem was *undecidable*: no computer ever built could predict with certainty whether a given set of tiles covered the plane or not!

Before I explain how Berger’s proof works, let me digress a bit and focus on his aperiodic tiling. Clearly, 20,426 are too many to be shown in a blog post, but since his result first appeared, other examples of smaller sets of aperiodic tiles have been found. Berger himself lowered the number to 104, Donald Knuth (of Computer Science fame) to 92, Hans Läuchli to 40, and finally, Raphael Robinson in 1971 produced a set of 6 tiles with the same property! Robinson tiles look like this (they are not depicted as exactly square tiles here, but they can be made into squares easily).

The pattern they create looks like this.

So, here we do not have triangles but squares, but apart from this it looks very similar to the Sierpinski triangle. Focus on the orange squares: there are some smaller ones, and they are sitting at the corners of slightly larger squares, which are in turn at the corners of even larger squares, and so on. While at a first look it might seem like a periodic pattern, it is not, since larger and larger squares keep appearing. We will come back to this orange squares in a while, keep them in mind.

In 1974, Roger Penrose found a set of just 2 aperiodic tiles, but which are not squares.

Penrose also had this cute idea that one could make a puzzle game out of these shapes, and he even got a patent for that! (“The tiles of the invention may be used to form an instructive game or as a visually attractive floor or wall-covering or the like”). At some point such a puzzle game was actually produced, but it is unfortunately out of production now. If you ever stop by the Newton Institute in Cambridge, UK, they own a copy (and they let you play with it!)

One of the characteristics of Penrose’s tiling is that with it one can obtain patterns with a 5-fold rotational symmetry, which means that you can rotate the tiling by 72°, which is 1/5th of 360°. This is interesting because a beautiful, and elementary, argument from Linear Algebra shows that in periodic tilings you can only get 2-, 3-, 4- or 6-fold symmetries (which corresponds to all n-fold symmetries for which is an integer), so having a 5-fold symmetry is a very unique thing! And just like the case of Sierpinski, there are traces of Penrose’s tiling in art, for example in the Darb-e Imam shrine in Isfahán, Iran.

Going back to Wang’s problem of whether the tiling problem is decidable: how did Berger prove his undecidability result? There are a lot of technical details he had to take care of, but the essence of his proof was to map each step of adding tiles to an ever-growing tiling, to the steps taken by a computer when running an algorithm (also known as a computer program). Each step of running the algorithm would correspond to instructions on which tile to add next and where. Specifically, Berger was interested in simulating the behavior of a very simple, yet very general computer – a Turing machine.

A Turing machine is basically a model for a machine that can run a particular computer algorithm, reduced to the bare minimum. It comprises of four main ingredients:

- A tape of arbitrary length on which the machine can write (and overwrite) symbols,
- A “head” which
- can read/write one symbol at a time (like a scanner/printer combo)
- can move the tape left/right one position at a time
- can store a finite amount of information (in internal memory)

- A program (table of instructions), which tells the “head” what to do next given the symbol it reads on the tape and the current internal memory state.
- An initial internal state (which tells the “head” how to start moving), as well as a final (halting) internal state (which tells the “head” when to stop).

While being a really simple object, Turing machines are capable of running any computer algorithm, no matter how complex, so they can come in handy when you need something simple and extremely versatile at the same time!

For example, we could have a Turing machine which can only read/write the symbols 0 and 1, has 6 internal states labeled with letters A, B, C, D, E, F, and has the following program:

A | B | C | D | E | F | |

0 | 1RB | 1RC | 1LD | 1RE | 1LA | H |

1 | 1LE | 1RF | 0RB | 0LC | 0RD | 1RC |

Here is how to read this table: Assume the initial state of the machine is A and the tape is filled with the symbol 0. The head of the machine will check the entry in the table corresponding to (0,A) and find the instruction “1RB”, which instructs it to write the symbol 1 (flipping the 0 that was already there to a 1), move the tape to the right, and change the internal state of the head to B. The head will now look up the new instructions for (0,B) (since, after moving the tape to the right, the new symbol under the head will be a 0 again), find “1RC” on the table of instructions, change the 0 into a 1, move the tape to the right once again, and change the internal state to C. It will repeat this process, reading one symbol at a time, checking its table of instructions to decide what to do next, until it reads a 0 while being in state F. If that happens, the special instruction “H” tells the machine to stop its execution: it has reached the “halting” state.

You can try to simulate the execution of this machine on a piece of paper, at least for the first few steps (you might need quite a lot of paper if you want to keep going). Or you could use a computer to simulate it. But you may find that after ten, or a thousand, or a million steps the machine has not halted yet. What if we kept going for another million steps? What about a billion? Can we be sure that the machine will halt eventually?

In his landmark work of 1936, Alan Turing showed that analyzing the behavior of this type of machines is outside the reach of any algorithmic computation: there cannot exist any algorithm which, given the description of a Turing machine’s program, can decide whether the machine will eventually halt, or if the machine will keep running forever! This is known as the halting problem.

Berger’s idea was to simulate a Turing machine using a set of tiles. For each possible symbol, the machine could read or write on the tape, he associated a corresponding color for each edge on the tile borders, as well as one color for each of the possible internal states of the machine. As you can probably guess, for two tiles to be neighbors, their common borders had to have the same color. Then he defined a set of tiles which “implemented” the transitions of the machine’s program, in such a way that each horizontal line was one “time step” of the tape during the execution of the machine. The resulting tiles looked like this, and the rule for the arrow is: two tiles can be next to each other only if the head of each arrow matches with the tail of another arrow.

Imagine we start our tiling with a row describing the initial state of the machine, which means having a “blank tape” (for example, a tape filled with the symbol 0), and one tile where the head of the machine is. It would look like this.

Then there is only one way we can extend this tiling further: for each of the tiles we have put down, there is only one tile that can go on top of that (try to check it yourself!). This is because the Turing Machine only has one possible transition, starting from the symbol 0 and state A. So after we add an extra layer, the pattern looks like this.

And then we repeat. Each time we put down a new tile, there is only one choice possible: we have to respect the transition rules of the Turing Machine, and our tiling will describe the state of the tape at the various steps of the execution.

If the machine halts at some point because it has completed its task, then there will be no way to add new tiles. In order to be sure that we could tile an arbitrarily large area, we would need to know in advance that the Turing machine defined by these tiles (converted into a set of fixed Turing instructions via Berger’s, or Robinson’s mapping) never halted. But, as I mentioned earlier, Turing showed that no algorithm can ever tell us such a thing. Which means you might regret having chosen these tiles for your new bathroom floor (you definitely should have chosen the ones with the flowers instead).

So, why is the aperiodic tiling so important for Berger’s and Robinson’s proofs? We assumed that we started the tiling with a special line, representing the tape in the “blank” state, and this has forced every other choice in the tiling. But using only the alphabet tiles with a single symbol, we get a periodic tiling which can always fill any region! In order to really force our tiling to have a description of the execution of a Turing Machine, we need to guarantee that the tiling is started with that special initialization line. In Robinson’s construction, this is possible using the orange squares as guides (go back and look at the picture of Robinson’s aperiodic tiling if you don’t visualize them), forcing the initialization to happen along the lower edge of each orange square which appears in the pattern. But remember, the Turing machine needs to have access to arbitrarily long segments of tape (we cannot predict how much it will need in case it halts), so we need to have arbitrarily large squares in our tiling. And this means, we really need an aperiodic tiling in order to have all possible tape lengths at our disposition! Any periodic tiling would have restricted the maximum amount of tape the machine could have used before repeating itself.

You might be wondering: what does all of this have to do with physics (you are, after all, reading the *Quantum Frontiers* blog and not *The IKEA Catalogue 2019*). The answer is: tiling problems can be converted into Hamiltonian groundstate energy problems. Think of a square lattice, where to each edge we can assign one of the possible edge configurations of your set of tiles. We can force edges of a square to come from one of the valid tiles by defining a plaquette interaction which gives an energy penalty to non-valid configurations. In this way, we can tile a region of the plane with our set of tiles if and only if this Hamiltonian has a *frustration-free* groundstate: a groundstate which simultaneously satisfies all the local plaquette constraints, or in other words, one that has zero energy. Deciding whether of not this special kind of groundstate exists is undecidable!

You do not need quantum mechanics for this, as this is a completely classical problem, but you soon realize that the number of possible configurations of the edges in the lattice is arbitrarily large! If you want to write down the matrix which represents this Hamiltonian interaction, you have to resort to larger and larger matrices.

Here is where quantum mechanics comes to the rescue! In a celebrated result, Toby Cubitt, David Pérez-García and Michael Wolf proved that you can have a similar result, this time for the spectral gap of a local Hamiltonian (the problem of deciding whether the spectrum of the Hamiltonian has a constant gap above the groundstate energy), using only a fixed number of local degrees of freedom. Their result is definitely not easy to explain: the first version of the paper was 146 pages long – luckily they managed to simplify it down to 127 pages… But I can try to give a very minimal explanation of how they managed to do this. The key part of their construction is to encode the rules of the Turing machine not directly in the tiling, but in a complex phase (complex number of unit length) which multiplies a certain fixed set of local Hamiltonian terms. They then use the quantum phase estimation algorithm to read off this phase, feeding this input into a Universal Turing Machine (a programmable Turing machine which can simulate any algorithm). In this way, the number of degrees of freedom needed is fixed, and by varying the complex phase mentioned above, they are able to simulate all possible classical Turing machines!

Now that we have entered the realms of local Hamiltonian problems, one might wonder if what is going on here is specific to 2 dimensional systems. Clearly, the same phenomena can happen in 3 or more dimensions, since we can simply take multiple slices of 2D systems and stack them on top of each other. But what about 1 dimensional systems? Can we make this construction work on a line?

Interestingly, Wang’s conjecture in 1D is true: every tiling of a line necessarily has a period. Since we are tiling a line, we can think of each tile as essentially a connection between its left-edge color and its right-edge color. Any set of tiles (and associated edge colors) then defines an oriented graph whose vertices are the colors and whose edges are given by the tiles. The rule is again that tiles can be neighbors if their corresponding edges are the same color. The longest (oriented) path we can find in the graph is then the length of the longest segment which can be tiled. It turns out that this length will be infinite if and only if there is a cycle in the graph. In other words, if there is a period.

So we can’t construct aperiodic tilings in 1D, and the tiling problem is decidable. One might be tempted to guess that the same should happen with the spectral gap of local Hamiltonians: We can look at the terms defining the Hamiltonian and decide if a uniform spectral gap exists, as the size of our quantum system increases. After all, in many cases, 1D systems behave “nicely”: we have the DMRG algorithm, polynomial time algorithms for computing groundstates of gapped Hamiltonians, area laws and matrix product state approximations, no thermal phase transitions or topological order, and so on.

But against all odds, in a paper with Johannes Bausch, Toby Cubitt, and David Perez-Garcia, we showed that the spectral gap problem is still undecidable in 1D. How did we get around the lack of aperiodic tilings in 1D?

The key idea was to construct a Hamiltonian whose groundstate would be periodic in the (state of the) spins of an arbitrarily long spin chain, but with a period depending on the halting time of an algorithm (modeled as a Turing machine) encoded (in binary) in the complex phase multiplying each Hamiltonian term. Roughly speaking, this is how we set this up: We partitioned the set of spins into segments. On each segment, we introduced a special Hamiltonian, known as the Feynman-Kitaev history state Hamiltonian, which made sure that the groundstate on that segment was a transcription of the tape during the execution of the classical Turing machine defined by the complex phase (as discussed above).

If at some point the machine has not halted and is running out of tape, so that the segment is not large enough to contain the complete transcription of its execution, then the machine can “push” the delimiter a bit further away, “stealing” some tape space from its neighbor (more technically: the resulting configuration with a larger tape segment is more energetically favorable than the previous one). But once the machine halts, the tape segment shrinks exactly to the minimal size required for the machine to reach its halting state. So, in case the machine halts, the line is divided up into periodic segments, whose length is exactly the optimal length for the machine to halt. If on the other hand the machine does not halt, then the best configuration is the one where there is a unique tape segment, and only one machine running on it.

To recap, the groundstate of this Hamiltonian looks very different depending on whether the Turing machine (encoded in the phase parameter) eventually halts or not. If it does, the groundstate will look periodic, with the period being determined by the halting time. It is therefore a product state, if we think of each segment as a single, huge, particle. If instead the machine never halts, then the groundstate will have a single, very long segment, with a big Kitaev-Feynman history state, which is a highly entangled state.

Even more interestingly, we can set up the different energy scales in the system to behave as follows: for system sizes where the machine has not halted (because it still does not have enough tape to do so, or because it will never do), the single tape segment groudstate has vanishing (but positive) energy, while after it halts, each segment has a small, negative energy. These negative energies in the halting case keep accumulating, so that the thermodynamic groundstate has strictly negative energy density. We can use this difference in energy density between the two cases to construct a “switch”: we introduce two other Hamiltonians to the system (introducing extra local degrees of freedom), one gapped and one gapless. We couple them to everything else we had already set up (the tape segment and the Kitaev-Feynman history state Hamiltonians), in such a way that only one of them controls the low-energy properties of our system. We can set up the switch based on the difference in the energy density in such a way that, before halting, the system is gapped, and it becomes gapless only after the Turing machine has halted (and we cannot predict if this will ever happen!) Hence, the spectral gap is undecidable!

As is the case for 2D system, we need a very large local Hilbert space dimension to make this construction work (so large we did not even care to compute an exact number – but we know it is finite!) On the other extreme end, we know if the local dimension is 2 (we have qubits on a line), and the Hamiltonian has a special property called frustration freeness, then the spectral gap problem is easy to solve. Contrast this with the aperiodic tiling constructions: first Berger found a highly complicated case (with 20,426 tiles), then his construction was refined and simplified over and over, until Robinson got it down to 6 and Penrose showed a similar one with only 2 tiles.

Can we do the same for the undecidability of the spectral gap? At which point does the line become complex enough that the spectral gap problem is undecidable? Can we find some sort of “threshold” which separates the easy and the impossible cases? We need new ideas and new constructions in order to answer all these questions, so let’s get to work!

]]>The authors kindly invited me to write a foreword for the book, which I was happy to contribute. That foreword is reproduced here, with the permission of the publisher.

**Foreword**

In 1989 I attended a workshop at the University of Minnesota. The organizers had hoped the workshop would spawn new ideas about the origin of high-temperature superconductivity, which had recently been discovered. But I was especially impressed by a talk about the fractional quantum Hall effect by a young physicist named Xiao-Gang Wen.

From Wen I heard for the first time about a concept called* topological order*. He explained that for some quantum phases of two-dimensional matter the ground state becomes degenerate when the system resides on a surface of nontrivial topology such as a torus, and that the degree of degeneracy provides a useful signature for distinguishing different phases. I was fascinated.

Up until then, studies of phases of matter and the transitions between them usually built on principles annunciated decades earlier by Lev Landau. Landau had emphasized the crucial role of symmetry, and of local order parameters that distinguish different symmetry realizations. Though much of what Wen said went over my head, I did manage to glean that he was proposing a way to distinguish quantum phases founded on much different principles that Landau’s. As a particle physicist I deeply appreciated the power of Landau theory, but I was also keenly aware that the interface of topology and physics had already yielded many novel and fruitful insights.

Mulling over these ideas on the plane ride home, I scribbled a few lines of verse:

Now we are allowed

To disavow Landau.

Wow …

Without knowing where it might lead, one could sense the opening of a new chapter.

At around that same time, another new research direction was beginning to gather steam, the study of *quantum information*. Richard Feynman and Yuri Manin had suggested that a computer processing quantum information might perform tasks beyond the reach of ordinary digital computers. David Deutsch formalized the idea, which attracted the attention of computer scientists, and eventually led to Peter Shor’s discovery that a quantum computer can factor large numbers in polynomial time. Meanwhile, Alexander Holevo, Charles Bennett and others seized the opportunity to unify Claude Shannon’s information theory with quantum physics, erecting new schemes for quantifying quantum entanglement and characterizing processes in which quantum information is acquired, transmitted, and processed.

The discovery of Shor’s algorithm caused a burst of excitement and activity, but quantum information science remained outside the mainstream of physics, and few scientists at that time glimpsed the rich connections between quantum information and the study of quantum matter. One notable exception was Alexei Kitaev, who had two remarkable insights in the 1990s. He pointed out that finding the ground state energy of a quantum system defined by a “local” Hamiltonian, when suitably formalized, is as hard as any problem whose solution can be verified with a quantum computer. This idea launched the study of *Hamiltonian complexity*. Kitaev also discerned the relationship between Wen’s concept of topological order and the *quantum error-correcting codes* that can protect delicate quantum superpositions from the ravages of environmental decoherence. Kitaev’s notion of a *topological quantum computer*, a mere theorist’s fantasy when proposed in 1997, is by now pursued in experimental laboratories around the world (though the technology still has far to go before truly scalable quantum computers will be capable of addressing hard problems).

Thereafter progress accelerated, led by a burgeoning community of scientists working at the interface of quantum information and quantum matter. Guifre Vidal realized that many-particle quantum systems that are only slightly entangled can be succinctly described using *tensor networks*. This new method extended the reach of mean-field theory and provided an illuminating new perspective on the successes of the *Density Matrix Renormalization Group* (DMRG). By proving that the ground state of a local Hamiltonian with an energy gap has limited entanglement (the *area law*), Matthew Hastings showed that tensor network tools are widely applicable. These tools eventually led to a complete understanding of gapped quantum phases in one spatial dimension.

The experimental discovery of *topological insulator*s focused attention on the interplay of symmetry and topology. The more general notion of a *symmetry-protected topological (SPT) phase* arose, in which a quantum system has an energy gap in the bulk but supports gapless excitations confined to its boundary which are protected by specified symmetries. (For topological insulators the symmetries are particle-number conservation and time-reversal invariance.) Again, tensor network methods proved to be well suited for establishing a complete classification of one-dimensional SPT phases, and guided progress toward understanding higher dimensions, though many open questions remain.

We now have a much deeper understanding of topological order than when I first heard about it from Wen nearly 30 years ago. A central new insight is that topologically ordered systems have* long-range entanglement*, and that the entanglement has universal properties, like *topological entanglement entropy*, which are insensitive to the microscopic details of the Hamiltonian. Indeed, topological order is an intrinsic property of a quantum state and can be identified without reference to any particular Hamiltonian at all. To understand the meaning of long-range entanglement, imagine a quantum computer which applies a sequence of geometrically local operations to an input quantum state, producing an output product state which is completely disentangled. If the time required to complete this disentangling computation is independent of the size of the system, then we say the input state is *short-ranged entangled*; otherwise it is *long-range entangled*. More generally (loosely speaking), two states are in different quantum phases if no constant-time quantum computation can convert one state to the other. This fundamental connection between quantum computation and quantum order has many ramifications which are explored in this book.

When is the right time for a book that summarizes the status of an ongoing research area? It’s a subtle question. The subject should be sufficiently mature that enduring concepts and results can be identified and clearly explained. If the pace of progress is sufficiently rapid, and the topics emphasized are not well chosen, then an ill-timed book might become obsolete quickly. On the other hand, the subject ought not to be *too* mature; only if there are many exciting open questions to attack will the book be likely to attract a sizable audience eager to master the material.

I feel confident that *Quantum Information Meets Quantum Matter* is appearing at an opportune time, and that the authors have made wise choices about what to include. They are world-class experts, and are themselves responsible for many of the scientific advances explained here. The student or senior scientist who studies this book closely will be well grounded in the tools and ideas at the forefront of current research at the confluence of quantum information science and quantum condensed matter physics.

Indeed, I expect that in the years ahead a steadily expanding community of scientists, including computer scientists, chemists, and high-energy physicists, will want to be well acquainted with the ideas at the heart of *Quantum Information Meets Quantum Matter*. In particular, growing evidence suggests that the quantum physics of spacetime itself is an emergent manifestation of long-range quantum entanglement in an underlying more fundamental quantum theory. More broadly, as quantum technology grows ever more sophisticated, I believe that the theoretical and experimental study of highly complex many-particle systems will be an increasingly central theme of 21st century physical science. It that’s true, *Quantum Information Meets Quantum Matte*r is bound to hold an honored place on the bookshelves of many scientists for years to come.

John Preskill

Pasadena, California

September 2018

]]>