So much to do, so little time. Tending to one task is inevitably at the cost of another, so how does one decide how to spend their time? In the first few years of my PhD, I balanced problem sets, literature reviews, and group meetings, but at the detriment to my hobbies. I have played drums my entire life, but I largely fell out of practice in graduate school. Recently, I made time to play with a group of musicians, even landing a couple gigs in downtown Austin, Texas, “live music capital of the world.” I have found attending to my non-physics interests makes my research hours more productive and less taxing. Finding the right balance of on- versus off-time has been key to my success as my PhD enters its final year.

Of course, life within physics is also full of tradeoffs. My day job is as an experimentalist. I use tightly focused laser beams, known as optical tweezers, to levitate micrometer-sized glass spheres. I monitor a single microsphere’s motion as it undergoes collisions with air molecules, and I study the system as an environmental sensor of temperature, fluid flow, and acoustic waves; however, by night I am a computational physicist. I code simulations of interacting qubits subject to kinetic constraints, so-called quantum cellular automata (QCA). My QCA work started a few years ago for my Master’s degree, but my interest in the subject persists. I recently co-authored one paper summarizing the work so far and another detailing an experimental implementation.

QCA, the subject of this post, are themselves tradeoff-aware systems. To see what I mean, first consider their classical counterparts cellular automata. In their simplest construction, the system is a one-dimensional string of bits. Each bit takes a value of 0 or 1 (white or black). The bitstring changes in discrete time steps based on a simultaneously-applied local update rule: Each bit, along with its two nearest-neighbors, determine the next state of the central bit. Put another way, a bit either flips, i.e., changes 0 to 1 or 1 to 0, or remains unchanged over a timestep depending on the state of that bit’s local neighborhood. Thus, by choosing a particular rule, one encodes a trade off between activity (bit flips) and inactivity (bit remains unchanged). Despite their simple construction, cellular automata dynamics are diverse; they can produce fractals and encryption-quality random numbers. One rule even has the ability to run arbitrary computer algorithms, a property known as universal computation.

In QCA, bits are promoted to qubits. Instead of being just 0 or 1 like a bit, a qubit can be a continuous mixture of both 0 and 1, a property called superposition. In QCA, a qubit’s two neighbors being 0 or 1 determine whether or not it changes. For example, when in an active neighborhood configuration, a qubit can be coded to change from 0 to “0 plus 1” or from 1 to “0 minus 1”. This is already a head-scratcher, but things get even weirder. If a qubit’s neighbors are in a superposition, then the center qubit can become entangled with those neighbors. Entanglement correlates qubits in a way that is not possible with classical bits.

Do QCA support the emergent complexity observed in their classical cousins? What are the effects of a continuous state space, superposition, and entanglement? My colleagues and I attacked these questions by re-examining many-body physics tools through the lens of complexity science. Singing the lead, we have a workhorse of quantum and solid-state physics: two-point correlations. Singing harmony we have the bread-and-butter of network analysis: complex-network measures. The duet between the two tells the story of structured correlations in QCA dynamics.

In a bit more detail, at each QCA timestep we calculate the mutual information between all qubits i and all other qubits j. Doing so reveals how much there is to learn about one qubit by measuring another, including effects of quantum entanglement. Visualizing each qubit as a node, the mutual information can be depicted as weighted links between nodes: the more correlated two qubits are, the more strongly they are linked. The collection of nodes and links makes a network. Some QCA form unstructured, randomly-linked networks while others are highly structured.

Complex-network measures are designed to highlight certain structural patterns within a network. Historically, these measures have been used to study diverse networked-systems like friend groups on Facebook, biomolecule pathways in metabolism, and functional-connectivity in the brain. Remarkably, the most structured QCA networks we observed quantitatively resemble those of the complex systems just mentioned despite their simple construction and quantum unitary dynamics.

What’s more, the particular QCA that generate the most complex networks are those that balance the activity-inactivity trade-off. From this observation, we formulate what we call the Goldilocks principle: QCA that generate the most complexity are those that change a qubit if and only if the qubit’s neighbors contain an equal number of 1’s and 0’s. The Goldilocks rules are neither too inactive nor too active, balancing the tradeoff to be “just right.”  We demonstrated the Goldilocks principle for QCA with nearest-neighbor constraints as well as QCA with nearest-and-next-nearest-neighbor constraints.

To my delight, the scientific conclusions of my QCA research resonate with broader lessons-learned from my time as a PhD student: Life is full of trade-offs, and finding the right balance is key to achieving that “just right” feeling.

# Life, cellular automata, and mentoring

One night last July, IQIM postdoc Ning Bao emailed me a photo. He’d found a soda can that read, “Share a Coke with Patrick.”

Ning and I were co-mentoring two Summer Undergraduate Research Fellows, or SURFers. One mentee received Ning’s photo: Caltech physics major Patrick Rall.

“Haha,” Patrick emailed back. “I’ll share a Coke.”

Patrick, Ning, and I shared the intellectual equivalent of a six-pack last summer. We shared papers, meals, frustrations, hopes, late-night emails (from Patrick and Ning), 7-AM emails (from me), and webcomic strips. Now a senior, Patrick is co-authoring a paper about his SURF project.

The project grew from the question “What would happen if we quantized Conway’s Game of Life?” (For readers unfamiliar with the game, I’ll explain below.) Lessons we learned about the Game of Life overlapped with lessons I learned about life, as a first-time mentor. The soda fountain of topics contained the following flavors.

Update rules: Till last spring, I’d been burrowing into two models for out-of-equilibrium physics. PhD students burrow as no prairie dogs can. But, given five years in Caltech’s grassland, I wanted to explore. I wanted an update.

Ning and I had trespassed upon quantum game theory months earlier. Consider a nonquantum game, such as the Prisoner’s Dilemma or an election. Suppose that players have physical systems, such as photons (particles of light), that occupy superposed or entangled states. These quantum resources can change the landscape of the game’s possible outcomes. These changes clarify how we can harness quantum mechanics to process, transmit, and secure information.

How might quantum resources change Conway’s Game of Life, or GoL? British mathematician John Conway invented the game in 1970. Imagine a square board divided into smaller squares, or cells. On each cell sits a white or a black tile. Black represents a living organism; white represents a lack thereof.

Conway modeled population dynamics with an update rule. If prairie dogs overpopulate a field, some die from overcrowding. If a black cell borders more than three black neighbors, a white tile replaces the black. If separated from its pack, a prairie dog dies from isolation. If a black tile borders too few black neighbors, we exchange the black for a white. Mathematics columnist Martin Gardner detailed the rest of Conway’s update rule in this 1970 article.

Updating the board repeatedly evolves the population. Black and white shapes might flicker and undulate. Space-ship-like shapes can glide across the board. A simple update rule can generate complex outcomes—including, I found, frustrations, hopes, responsibility for another human’s contentment, and more meetings than I’d realized could fit in one summer.

Modeled by Conway’s Game of Life. And by PhD students.

Initial conditions: The evolution depends on the initial state, on how you distribute white and black tiles when preparing the board. Imagine choosing the initial state randomly from all the possibilities. White likely mingles with about as much black. The random initial condition might not generate eye-catchers such as gliders. The board might fade to, and remain, one color.*

Enthusiasm can fade as research drags onward. Project Quantum GoL has continued gliding due to its initial condition: The spring afternoon on which Ning, Patrick, and I observed the firmness of each other’s handshakes; Patrick walked Ning and me through a CV that could have intimidated a postdoc; and everyone tried to soothe everyone else’s nerves but occasionally avoided eye contact.

I don’t mean that awkwardness sustained the project. The awkwardness faded, as exclamation points and smiley faces crept into our emails. I mean that Ning and I had the fortune to entice Patrick. We signed up a bundle of enthusiasm, creativity, programming skills, and determination. That determination perpetuated the project through the summer and beyond. Initial conditions can determine a system’s evolution.

Long-distance correlations:  “Sure, I’d love to have dinner with you both! Thank you for the invitation!”

Lincoln Carr, a Colorado School of Mines professor, visited in June. Lincoln’s group, I’d heard, was exploring quantum GoLs.** He studies entanglement (quantum correlations) in many-particle systems. When I reached out, Lincoln welcomed our SURF group to collaborate.

I relished coordinating his visit with the mentees. How many SURFers could say that a professor had visited for his or her sake? When I invited Patrick to dinner with Lincoln, Patrick lit up like a sunrise over grasslands.

Our SURF group began skyping with Mines every Wednesday. We brainstorm, analyze, trade code, and kvetch with Mines student Logan Hillberry and colleagues. They offer insights about condensed matter; Patrick, about data processing and efficiency; I, about entanglement theory; and Ning, about entropy and time evolution.

We’ve learned together about long-range entanglement, about correlations between far-apart quantum systems. Thank goodness for skype and email that correlate far-apart research groups. Everyone would have learned less alone.

Long-distance correlations between quantum states and between research groups

Time evolution: Logan and Patrick simulated quantum systems inspired by Conway’s GoL. Each researcher coded a simulation, or mathematical model, of a quantum system. They agreed on a nonquantum update rule; Logan quantized it in one way (constructed one quantum analog of the rule); and Patrick quantized the rule another way. They chose initial conditions, let their systems evolve, and waited.

In July, I noticed that Patrick brought a hand-sized green spiral notepad to meetings. He would synopsize his progress, and brainstorm questions, on the notepad before arriving. He jotted suggestions as we talked.

The notepad began guiding meetings in July. Patrick now steers discussions, ticking items off his agenda. The agenda I’ve typed remains minimized on my laptop till he finishes. My agenda contains few points absent from his, and his contains points not in mine.

Patrick and Logan are comparing their results. Behaviors of their simulations, they’ve found, depend on how they quantized their update rule. One might expect the update rule to determine a system’s evolution. One might expect the SURF program’s template to determine how research and mentoring skills evolve. But how we implement update rules matters.

Caltech’s 2015 quantum-information-theory Summer Undergraduate Research Fellows and mentors

Life: I’ve learned, during the past six months, about Conway’s Game of Life, simulations, and many-body entanglement. I’ve learned how to suggest references and experts when I can’t answer a question. I’ve learned that editing SURF reports by hand costs me less time than editing electronically. I’ve learned where Patrick and his family vacation, that he’s studying Chinese, and how undergrads regard on-campus dining. Conway’s Game of Life has expanded this prairie dog’s view of the grassland more than expected.

I’ll drink a Coke to that.

Glossary: Conway’s GoL is a cellular automatonA cellular automaton consists of a board whose tiles change according to some update rule. Different cellular automata correspond to different board shapes, to boards of different dimensions, to different types of tiles, and to different update rules.

*Reversible cellular automata have greater probabilities (than the GoL has) of updating random initial states through dull-looking evolutions.

**Others have pondered quantum variations on Conway’s GoL.