Nicole’s guide to writing and editing

Posted on August 26, 2026 by Nicole Yunger Halpern

Freshman year of college, I took a writing seminar from German-literature professor Ellis Shookman. Professor Shookman loved Mozart’s music, he told us early in the term. He listened to Mozart on the radio while driving from campus to Boston. Static might mar the transmission, but he could often turn up the volume and continue enjoying the program. Sometimes, the static worsened during the drive. It could worsen and worsen, until Professor Shookman’s frustration outweighed his delight at listening. He’d switch off the radio.

As Professor Shookman loved listening to Mozart’s music, he loved reading about students’ ideas. Yet static can mar a piece of writing: infelicities in grammar, structure, composition, word choice, and more. If enough infelicities obscure the writing, the frustration of reading outweighs the benefits. Professor Shookman will quit reading.

Professor Shookman marked up our essays with a blue pencil that achieved the status of legend among his students. If you’ve written a paper I’ve coauthored, you’ve probably received PDF drafts replete with green highlighting.¹ A sticky note explains the reason for each highlighting: “Singular–plural mismatch.” “Active voice >> passive voice.” “Let’s clue the reader in as to this formula’s meaning before lobbing the math at them.”

Over the past year, I’ve catalogued the suggestions I write most often on paper drafts. The comments embody principles gleaned from Strunk and White’s The Elements of Style; the Physical Review style guide; other writing guides I esteem; literature whose writing I esteem;² collaborations with professional editors; and writing instructors, including Professor Shookman. Each section below begins with more-important principles, shading into more-nuanced ones.

Please use and disseminate these principles. Train your favorite large-language model (LLM) on them, and have the LLM critique your manuscripts. Instruct it to use green highlighting if you wish. Even if the LLM suggests fixes initially, tell it to stop offering suggestions later, so that you can devise the solutions: train not only the LLM, but also yourself. I hope to enjoy your papers as much as Professor Shookman enjoyed his sonatas.

Organization
1. Motivate your work; then, present it; and then, explain its physical significance.
2. Begin each paragraph with a topic sentence.
3. Begin each section, apart from the introduction and conclusion, with (i) a statement of the takeaway and (ii) an outline of the section. When outlining a section, hyperlink to each subsection. Similar guidelines concern subsections and subsubsections.
4. Before presenting a piece of math, sketch its meaning and origin. This strategy enables the reader to understand the math as soon as they encounter it. If you throw math at the reader without introducing it, the reader will have to squint at the symbols for a while to figure out what the expression means and where it comes from.
  - Example: To calculate the average work, we substitute the Hamiltonian formula (10) into the definition (12): [equation].
5. Most citations belong at the ends of (i) sentences and (ii) phrases concluded with commas. Put a citation elsewhere only if you have a compelling reason for doing so.
6. Bridge each component of your writing to the next component; the next shouldn’t sound like a non sequitur.
  - Suppose that the next sentence refers to (i) a topic mentioned in the previous sentence and (ii) a new topic. Mention (i) before (ii).
    - Example: Smith et al. applied control theory to the extent possible. The attempt led to intractable equations, unlike our approach.
    - Example of a broken bridge: Smith et al. applied control theory to the extent possible. Our approach does not involve intractable equations, unlike theirs.
7. Whenever you tell a story, tell it from start to finish, step by step. Derivations, proofs, and descriptions of experiments qualify as stories.
  - This guideline extends to descriptions of experimental setups and of mathematical objects. For example, imagine referring to an element of a subgroup of the group generated by some operators. Did you have to read the preceding sentence multiple times to process it? The sentence begins at the end of a story, then rewinds to the story’s beginning. This structure impedes understanding. The subgroup forms the context for the subgroup element, which one can’t grasp until hearing about the subgroup. The subgroup participates in a similar relationship with the group, as does the group with its generators. Therefore, one should introduce the generators, then the group, then the subgroup, and then the subgroup element.
Word choice
1. Use strong, specific words, rather than weak words.
  1. Verbs and nouns are stronger than adjectives and adverbs.
  2. Choose specific verbs (e.g., “prepare,” “evolve,” and “measure”), rather than vague, general verbs (e.g., variants of “to be” and “take,” as in “take a measurement”).
2. Avoid statements such as “we investigate,” “we study,” and “we analyze.” Such statements don’t relate that you’ve accomplished anything. State what you’ve accomplished. Verbs such as “prove,” “test,” “confirm,” “discover,” and “find” achieve this goal.
3. Adverbs such as “importantly” and “remarkably” pollute scientific writing with the authors’ opinions. Demonstrate that a claim is important or that a result is remarkable; then, leave readers to draw their own conclusions. Those conclusions will coincide with yours if you’ve demonstrated your point.
4. Use the active voice, rather than the passive voice. Take responsibility for your work. Editors of high-impact scientific journals have endorsed this advice.
5. Refer to yourself when necessary and only when necessary.
  - Example of unnecessary reference to self: We use the superscript “max” to signify the maximal Fisher information.
    Preferable alternative: The superscript “max” signifies the maximal Fisher information.
  - Example of unnecessary reference to self: Our results establish several opportunities for future research. First, we can implement the experimental proposals.
    Preferable alternative: Our results establish several opportunities for future research. First, one can implement the experimental proposals.
  - You may use the first-person plural when escorting the reader through a derivation.
6. If you’re the only author, don’t use the plural (“we,” “our,” etc.). The usage is inaccurate and misleading. It portrays you as dodging responsibility for your work by dispersing that responsibility across the scientific community.
7. Avoid dangling modifiers.
8. Pair every verb with the appropriate noun.
  - Example of grammatically incorrect text: Equation (1) follows by calculating the sum.
    - One should pair the verb “calculate” with the noun “we,” because “we” undertook the calculating. However, this example’s author omitted the noun out of squeamishness about using the first person in a scientific document. Hence the sentence says that the equation calculates the sum. Equations can’t calculate sums.
  - Examples of correct alternatives
    - We derived Eq. (1) by calculating the sum.
    - Equation (1) follows from the evaluation of the sum.
    - Calculating the sum yields Eq. (1).
9. Avoid empty subjects.
10. Include no unnecessary words.
  1. “So-called” is unnecessary.
  2. “Note that” and “We note that” are unnecessary.
  3. “We have that,” used as a preface to a mathematical statement, is unnecessary. One can better serve the reader by prefacing the mathematical statement with (i) a derivation or (ii) a prose description of the statement.
  4. Never write “is equal to”; “equals” is more concise.
  5. Never write “is able to”; “can” is more concise.
  6. Never write “gives an upper bound to” or “places an upper bound on”; “upper-bounds” is more concise. Analogous statements concern lower bounds.
  7. Never write “a large number of”; “many” is more concise. Never write “a small number of”; “few” is more concise.
  8. Never write “We refer to [symbol] as [name]”; “we call [symbol] [name]” is more concise.
  9. Never write “as long as”; “if” is more concise.
  10. The symbol $>$ means “greater than”; and $\geq$ , “greater than or equal to.” Don’t translate $>$ into “strictly greater than”; the “strictly” is unnecessary. Analogous statements concern $<$ and $\leq$ .
11. Avoid contractions, which are too informal for professional writing.
12. The possessive is not a contraction and belongs in professional writing. It facilitates conciseness.
13. Use the word “for” only when it belongs. Physicists often write “for” when they mean “if,” “per,” “at,” or something else.
  - Example of inappropriate use: The function vanishes for odd arguments.
    Corrected statement: If the argument is odd, the function vanishes.
  - Example of inappropriate use: We performed 10 trials for each parameter value.
    Corrected statement: We performed 10 trials per parameter value.
  - Example of inappropriate use: The function is smaller for small x values.
    Corrected statement: The function is smaller at small x values.
14. Write “we evolve the state,” “we measure,” etc. only if you’re an experimentalist who undertakes those actions. Alternatives include “Consider measuring,” “Suppose the system evolves,” and the command tense (e.g., “One can measure this quantity as follows: prepare the qubit in $\lvert 0\rangle$ . Evolve it under $H$ …”).
15. The condition $x\ll y$ defines a regime, not a limit. The conditions $\lim_{x\to0}$ and $\lim_{y\to\infty}$ define limits and are inequivalent to $x\ll y$ .
16. Write “first,” “second,” “last,” etc., not “firstly,” “secondly,” “lastly,” etc. (I defer in this matter to The Elements of Style.)
17. Humans can assume, suppose, etc. Mathematical expressions, protocols, etc. can’t.
18. One multiplies factors together and sums terms. Don’t call factors terms and vice versa.
19. If you mean “X equals Y,” say so. Don’t write “X agrees with Y,” “X matches Y,” or “we identify X with Y.” The latter three phrases are vaguer, and two of them contain more words, than “X equals Y.”
20. Regarding the words “general” and “generally”:
  1. A general object subsumes every example of that object. If any example behaves unlike a supposedly general object, don’t call the object general.
  2. Many claims contain the term “general,” “generally,” or “in general” but don’t need the term.
    - Example of a sentence that contains “generally”: The terms generally commute.
    - Equivalent, more concise sentence: The terms commute.
  3. Physicists tend to use the words “general” and “generic” differently. By “general,” physicists usually mean “subsuming every example.” By “generic,” we usually mean “typical,” or “common.”
21. “Then” makes sense (i) in discussions of chronology and (ii) in if–then statements. Don’t use “then” outside these contexts.
  - Example of inappropriate use: “Define $X:=\ldots$ Then Y.”
  - Examples of appropriate alternatives
    - Define $X:=\ldots$ This definition implies Y.
    - If $X:=\ldots \, ,$ then Y.
    - Define $X:=\ldots \, ,$ such that Y.
22. Don’t justify any equation with “we used that [such-and-such is true],” which violates the rules of grammar. Grammatically correct alternatives include “We applied [a property],” “The equation follows from [a property],” and “…since [such-and-such is true].”
23. Regarding tense:
  1. When describing what you’ve accomplished, use only one tense.
  2. Experiments happened in the past, so describe them in the past tense.
  3. When describing a proof’s steps, use the present tense.
    - Example: We Taylor-approximate the function about $x=0$ . Substituting into Eq. (1) yields [equation].
24. Nouns, verbs, and adjectives should agree about whether a quantity is singular or plural.
  - Example of singular–plural mismatch: The equations are a rule for evolving the cellular automaton.
  - Example alternative: The equations form a rule for evolving the cellular automaton.
25. “Admit of” means “allow for,” or “permit.” The phrase needs the “of.”
  - Example: The formula admits of the following interpretation.
Punctuation
1. Consider any list that contains at least three items. If no item contains a comma, separate the items with commas. If any item contains a comma, separate the items with semicolons.
2. In American English, periods and commas belong inside quotation marks. (Example: She told me, “Have a good day.”) In British English, periods and commas belong outside quotation marks. (Example: She told me, “Have a good day”.)
3. To write quotation marks in LaTeX, don’t use your keyboard’s quotation-mark key; use the appropriate keys.
4. Regarding hyphens:
  1. The hyphen (-) feeds into punctuation of three types: the hyphen (-), the en dash (–), and the em dash (—).
  2. The hyphen appears in some compound words, as in “non-negative.”
  3. In American English, em dashes can separate ideas within a sentence. Don’t separate any em dash from neighboring text with a space.
    - Example of appropriate use: The sample—the only product of this experiment—barely survived.
    - Example of inappropriate use: The sample — the only product of this experiment — barely survived.
    - Example of appropriate use: He told me only one sample had survived—hardly what I wanted to hear.
  4. This article specifies how to use the en dash. One use is “to separate the names of two or more people used as a compound modifier.”
    - Example: Feynman–Kitaev clock
  5. Hyphenate compound adjectives.
  6. If an adverb ends in “-ly,” it probably shouldn’t precede a hyphen.
    - Example of inappropriate hyphenation: strongly-coupled systems
  7. Follow a prefix with a hyphen if and only if the Physical Review style guide indicates that you should.
5. A complete clause must follow any semicolon (unless the semicolon separates items in a list).
Math
1. Introduce only necessary notation, which readers will have enough trouble remembering. If a mathematical symbol appears only once, eliminate it. If a symbol appears only twice, try to eliminate it.
2. Every sentence must obey the rules of English grammar, punctuation, and syntax, regardless of whether the sentence contains mathematical symbols. All math-containing sentences must end with punctuation marks. If a sentence contains a list of mathematical expressions, precede the final expressions with an “and.” If the list contains at least three mathematical expressions, separate them with commas.
3. Introduce almost every mathematical symbol before you use it. If you introduce a symbol after using it, the reader will encounter the first use, stop, feel confused for a while, tentatively continue, find the definition, return to the earlier use to understand it, and then progress again. This back-and-forth breaks up the reading process. You may define a mathematical symbol after using it only if (i) the symbol is very common, known to nearly all physicists, and unmistakeable and (ii) defining the symbol earlier would disrupt the text’s flow.
4. If you define a new function, denote it by only one letter. (I defer in this matter to the Physical Review style guide.)
  - Example: $f(x,y,z)$
  - Examples of disallowed notation: $fxn(x,y,z)$ , ${\rm fxn}(x,y,z)$
5. Suppose that a superscript or subscript stands for a word or phrase without representing any variable or constant. The superscript/subscript must not be italicized. (I defer in this matter to Physical Review style guide.)
  - Example: Let $x_{\mathrm{meas}}$ denote the measurement outcome.
6. If a variable or constant appears in a superscript, parenthesize it. The parentheses communicate that the superscript isn’t an exponent.
  - Example: Let $\sigma_z^{(j)}$ denote the Pauli- $z$ operator of qubit $j.$
  - If a superscript is not italicized (stands for a word or phrase), don’t parenthesize it.
7. Regarding the definition of a symbol :
  1. If you write $A$ alone on one side of a defining equation, use \coloneqq or \eqqcolon: $A$ \coloneqq [expression], or [expression] \eqqcolon $A$ . The symbols \coloneqq and \eqqcolon relate more information than does $\equiv$ , encoding directionality.
  2. Use $\equiv$ if $A$ does not appear alone on its side of the equation: [function of $A$ ] $\equiv$ [result of replacing $A$ with its definition in the equation’s left-hand side].
8. Refer to the Cartesian axes using the formatting “[italicized letter]-axis.” Don’t include any hat, boldface, or \vec symbol.
  - Example: $x$ -axis
9. Avoid denoting any index by $i$ , which means $\sqrt{-1}$ to physicists. Use $j$ instead, unless you’re writing for engineers (who denote $\sqrt{-1}$ by $j$ ).
10. Don’t use the lowercase letter l (“ell”) as an index; readers might mistake it for a one. Use $\ell$ (\ell) instead.
11. Give every set-off equation a number. Readers (and coauthors) may want to refer to the equation easily when discussing the paper. Save them (and us) from having to say, e.g., “that equation halfway down page three.”
12. When writing a set-off mathematical expression, use the align environment, not the equation environment. Using the align environment, one can easily extend an expression across multiple lines.
13. Regarding a set-off mathematical expression that extends across multiple lines:
  1. Format the expression as follows by default.
    1. Put an & symbol immediately leftward of the first = sign or analogous symbol (e.g., $\leq$ ).
    2. If any subsequent line begins with another = sign (or analogous symbol), put an & immediately leftward of the symbol. (I’ll stop writing “or analogous symbol.”)
    3. Suppose that a subsequent line begins with a $+$ , –, $\times$ , or $/$ . Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.
    - Example:
  2. Modify the default formatting if necessary (a) to reduce the number of lines used in a PRL submission or (b) if the initial = appears awkwardly far to the right.
    - Example of (b):
  3. Suppose a new line begins with a term or factor, such as the in the example under (A). Put the corresponding , –, , or at the beginning of the new line, not at the end of the previous line.
    - Examples of inappropriate placement:
14. The symbol $\approx$ means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.
15. Use big-O-type notation or ~ symbols, not both; they’re partially redundant.
16. \ldots, rather than \cdots, should stand in for elements that fit a pattern.
  - Example: $x_1,x_2,\ldots,x_n$
17. When using \ldots as in the previous rule, present at least two initial examples of the pattern. One can’t define the pattern.
  - Contains insufficient examples: $x_1,\ldots,x_n \, .$ For example, if $n$ is odd, then $x_1, x_2, \ldots, x_n$ and $x_1, x_3, \ldots, x_n$ fit the template.
18. Parentheses (), square brackets [], and curly braces {} are delimiters. If you nest them, do so in the order dictated by the Physical Review style guide.
19. If delimiters enclose a symbol, it shouldn’t protrude above or below them (unless the delimiters would have to be grotesquely enormous). Use the \left and \right commands if the delimiters appear on the same line.
20. An operator isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an and a matrix. An arrow can.
  - Example: $O\to\begin{bmatrix}1&0\\0&2\end{bmatrix}$
21. Every real number is complex. Don’t say “complex” if you mean “nonreal.”
22. Consider introducing a mathematical symbol in a prose sentence without using a comma or colon. Put the symbol immediately after the word that names the object represented by the symbol.
  - Example of inappropriate placement: the set of real numbers $\{ a, b \}$
  - Examples of appropriate placements
    - the set $\{a, b\}$ of real numbers
    - the set of real numbers $a$ and $b$
    - Recall the set of real numbers, $\{a, b\}$ , in Lemma 1.
More mechanics of writing
1. Use concise sentences, as advocated for in The Elements of Style. The reader can hold only so many ideas in their head at once.
2. Structure sentences simply, as advocated for in The Elements of Style. The reader should be able to grasp each sentence easily.
  - Avoid nesting ideas within a sentence, to avoid convoluting the sentence’s structure.
3. The ideal paper title has the structure of a newspaper headline: it presents a claim, containing a subject and a predicate.
4. Regarding abbreviations:
  1. Don’t abbreviate the first word in any sentence.
  2. Abbreviate “Figure,” “Section,” “Professor,” and “Appendix” if such a word appears partway through a sentence.
  3. Don’t abbreviate “Sections.”
5. Regarding acronyms:
  1. Write every acronym in capital letters, as per the Physical Review style guide.
  2. Introduce each acronym the first time you use it.
  3. Thereafter, use only the acronym, not the spelled-out phrase, throughout the rest of the document’s main text. You may spell out the phrase in section, figure, and table titles if doing so improves the document’s clarity.
6. Every paragraph should contain at least three sentences.
7. Wherever you insert a blank line into your LateX code, a new paragraph begins in the corresponding PDF. Insert a blank line only if you wish to begin a new paragraph. This advice applies immediately before and after set-off equations.
8. Never begin a subsection immediately after a section title. Between the two titles, overview the section. Analogous rules govern subsections and subsubsections.
9. Put the word “only” in the appropriate place.
  - For example, suppose you’ve sampled data at a point $x=0$ in parameter space and sampled data at no other points. “We sampled data only at $x=0$ ” is correct; “We only sampled data at $x=0$ ” is probably not. The latter claim means that (i) you might have sampled data at $x=0$ and (ii) you did nothing to the $x=0$ data apart from sample it: you didn’t analyze the $x=0$ data, discuss the $x=0$ data, etc.
When in doubt, consult the Physical Review style guide or The Elements of Style.
- If those references don’t contain the information you seek, search for it in online writing guides. Not all such guides have equal merit, however. Lean toward guides written by human editors or published by college writing centers.

¹ Collaborators have wondered why I use green; a student guessed it’s my favorite color. It isn’t; but I bleed green, having graduated from the Big Green, also known as Dartmouth College. Sometimes, I highlight certain pieces of text for one reason (e.g., to point out logical inconsistencies) and other text for another reason (e.g., to point out grammatical inconsistencies). Green distinguishes the first highlightings, while orange distinguishes the second: when not bleeding Dartmouth green, I bleed Caltech orange.

² Don’t learn how to write from physics papers.

Interacting collaborators reveal noninteracting fermions

Posted on August 9, 2026 by Logan Hillberry

By day, I work as an experimentalist on laser-cooling molecules¹, but I’ve never fully surrendered my theoretical-physics license. I started as an undergraduate in Lincoln Carr’s group at the Colorado School of Mines in Golden, CO. I learned from his expertise in simulations and complex systems. Since then I’ve moonlighted as a theorist while also pursuing an unrelated PhD and, now, an unrelated postdoc position. With Nicole Yunger Halpern and other collaborators, we devised a quantum circuit whose dynamics looked complex when run on a quantum computer. It took six years and five collaborators across four countries to discover that, for the right settings, these complex dynamics could be understood when viewed from the right angle.

Some time ago, I told you about quantum cellular automata (QCA). These quantum machines are built from one-dimensional strings of qubits. A qubit changes its state depending on the state of its two nearest neighbors. Different rules are encoded into three-qubit gates that change a central qubit based on the state of its left and right neighbors. Some rules induce change for many combinations of neighbor states. Others, less. We apply this neighborhood-constrained update in two waves, first to every other qubit, then to the ones skipped in the first wave. This is a common quantum circuit structure called a brickwork pattern. We call one rule the Goldilocks QCA: A qubit is updated if one of its neighbors is a 0 while the other is a 1 (activity); otherwise the qubit does not change its state (inactivity).

The first figure from our recent paper illustrating the Goldilocks QCA brickwork circuit. Orange boxes represent unitary gates. Half-white-half-black circles represent the Goldilocks neighborhood constraint. Some choices for the unitary gate result in free fermion dynamics. Most choices are consistent with chaos.

Repeating brickwork layers of the Goldilocks rule, we found, balances activity and inactivity to be “just right,” as Goldilocks might say. Striking this balance produced surprisingly rich patterns of quantum correlation. The same type of network structure is found in complex classical systems like metabolic pathways, social networks, and brain activity. What’s more, the observed patterns of connectivity persist through thousands of circuit layers while other QCA tend towards uniformity.

Goldilocks in a state of activity. Published by The Grolier Society, 1912

Our new paper, Integrability of Goldilocks quantum cellular automata, answers a question that’s been lurking underneath that first result for the last several years. Why does this balance produce such rich and persistent structure? Some Goldilocks QCA, we prove, map onto free fermions, one of the simplest examples of exactly solvable quantum dynamics. How does uncovering this simplification explain the persistent complex patterns? The answer follows from the concept of conservation laws. Piecing together this understanding required assembling an international team of experts who generously shared their knowledge and time. I’ll tell a bit of this scientific story through the lens of our collaboration’s history.

A key inspiration for this work started with a May 2020 video call with Norman Margolus, an MIT-affiliated researcher and pioneer of using cellular automata to model real systems. In the 1980s he worked on a custom computer chip called CAM-6, and later CAM-8, that was dedicated to simulating massive arrays of cellular automata with the limited computational resources of the era². He proudly showed us beautiful pictures of cellular automata simulating phenomena like optical refraction and chemical reactions.

Cellular automata book by Norman Margolus. His coauthor’s name may also be familiar to those with quantum-circuit experience. Published by MIT Press, 1987.

He told us a story about trying to mimic fluid flow with the simple local rules of classical cellular automata. These models, called lattice gas automata, were first defined on a square lattice. While they did show fluid-like behavior, these models did not quite correctly conserve momentum³. Moving to a hexagonal lattice fixed up these problems and the community was able to devise cellular automata that quantitatively modeled continuum fluid flow.

The author’s primitive lattice-gas cellular automaton showing an initial high-density region displaying wave-like propagation, reflection, and diffusion into a low-density background.

Part of that story stuck with me: conservation laws are fundamental ingredients of a physical model. Our Goldilocks quantum cellular automata, we observe, exhibit persistent complex structures. Could some conservation law be behind these observations? If found, could these conservation laws be harnessed for more efficient simulations? Going even further, could there be enough conservation laws to exactly solve the dynamics (at least in principle)? This property would buy the system membership in a special class called integrable systems.

An integrable system conserves enough quantities, often called charges in the quantum setting, that you can compute its future state from its conservation laws and its initial conditions. Two-body gravitational orbits are a classic example. The initial positions and velocities set the orbital energy and angular momentum in the center-of-mass reference frame. Those two conserved quantities let you write down an exact equation for the orbit’s shape.

A familiar integrable system from classical mechanics: the two-body gravitational orbit. Angular momentum *L=r x p* is conserved. So are the total energy and the Runge-Lenz vector A.

A chaotic system, by contrast, may conserve energy and even a few other quantities, but not enough for us to solve for the state arbitrarily far in the future. To find out what a chaotic system does, you have to evolve the equations of motion approximately—one small time step at a time. Chaotic systems are the norm in nature; integrable ones are rare. To illustrate their qualitative differences, compare the regularity of the above orbit to the trend towards uniformity in the above lattice-gas simulation. In the quantum regime, physicists still don’t fully agree on the precise definition of integrability, though conservation of many independent quantities is a strong indicator.

In August 2020, Nicole emailed Lorenzo Piroli about his preprint on QCA, now published as Phys. Rev. Lett. 125, 190402. Lorenzo was a postdoc at the Max Planck Institute for Quantum Optics in Garching, Germany when we first met. He is now an associate professor at the University of Bologna and expert in many-body quantum dynamics. The correspondence that unfolded set the blueprint for the research effort that followed. One of us would ask a question, and Lorenzo would respond incredibly fast with accurate and useful detail. He started working with us to understand why the Goldilocks QCA dynamics appeared so unique. Lorenzo would suggest computations, I would implement them, and we would discuss what the results meant.

Then came an echo of the collaboration’s inception. In May 2021, Nicole pointed out a relevant preprint from Tomaž Prosen, now published in Chaos 31, 093101. Tomaž is a Slovenian physicist at the University of Ljubljana and a leading researcher in the fields of quantum chaos and integrability. I sent an email about the connections between our work and his. He responded with enthusiasm. He shared some code that would, through exhaustive search, find quantities conserved by our QCA.

The code’s brute-force approach meant the algorithm could only find conservation laws defined over, at most, a 5-qubit subsystem. A tantalizing signal emerged: the number of conserved quantities supported by 5 qubits exceeded the number supported by 3 qubits. Having more and more conserved quantities as you look at larger neighborhoods is a signature of integrability. Soon after, Tomaž proved one of our Goldilocks QCA is integrable using a well-established toolkit from statistical mechanics called Yang-Baxter integrability. He built a parametric transfer matrix, essentially a machine that spits out a new conserved quantity every time you turn its mathematical crank⁴.

Rodney Baxter’s classic textbook. Published by Academic Press, 1982

But there was a wrinkle. The transfer matrix generates charges that mutually commute, meaning you can measure them simultaneously. For example, you can know a quantum particle’s kinetic energy and momentum simultaneously because those operators commute. Yet, the search algorithm kept finding charges that did not commute with each other, like a particle’s position and momentum. The only explanation was that our QCA has more charges than the transfer matrix method guarantees, and more than are minimally required for integrability. This extra-conservation-law property, called superintegrability, also shows up in two-body gravitational orbits. In addition to energy and angular momentum, orbits conserve the Runge-Lenz vector. Nicole is an expert on noncommuting charges, so this is where one of her main research efforts entered the QCA collaboration.

Next came a key insight from Lorenzo: the automaton we had been considering was one member of a larger family of integrable Goldilocks QCA. He showed this using a Jordan-Wigner transformation, a mathematical dictionary that translates between the language of qubits and the language of fermions. Complexity in the qubit language transformed into simplicity in the fermion language. Under this translation, our QCA mapped to noninteracting, or free, fermions: particles that never bump into or influence each other. That lack of interaction is what makes free-fermion dynamics easy to calculate. A system of free fermions is a well-known example of superintegrability.

Along the way, Lorenzo recruited his friend and collaborator Eric Vernier, a CNRS researcher based in Paris, France. He is an expert on vertex models. The classical version of the six-vertex model was developed in the 1930s to explain a troubling mystery: Water ice appears to have more entropy than permitted by the third law of thermodynamics at near-zero temperature. In the six-vertex model, a water molecule’s oxygen atom is envisioned at every vertex in a square lattice. Each molecule contributes two hydrogen ions, to use Baxter’s terminology, that fall along the lattice edges. Intermolecular hydrogen bonds between adjacent molecules slightly alter the intramolecular O-H bonds. To maintain electrical neutrality, each oxygen (lattice vertex) has two nearby and two far-away hydrogen ions (four edges), leading to six possible ice vertices. The vertices are commonly visualized in three ways: 1) as the dots representing hydrogen ions located on edges near or far from each vertex, 2) as electric dipole arrows pointing into (“ion is close”) or out of (“ion is far”) each vertex, or 3) as thick (downward- and leftward-pointing dipoles) and thin (upward- and rightward-pointing dipoles) edges. Despite the model’s simplicity (2D square lattice) compared to real ice (3D tetrahedral lattice), it agrees with experimentally measured entropy values to better than 2%.

This figure appears in chapter 8 of R.J. Baxter’s book. It shows three visualizations of the same ice crystal.

More recently, vertex models have been adapted from two-dimensional classical crystals to one-dimensional quantum systems that evolve in time. Eric showed us how the ice vertices relate to QCA circuit rules. In doing so, Eric uncovered an even larger set of integrable Goldilocks QCA than that found by Lorenzo. Eventually, Lorenzo’s Jordan-Wigner transformation method and Eric’s six-vertex method agreed on the complete family of integrable Goldilocks QCA.

Representation of the six ice vertices from our recent paper (rotated 45 degrees from the lattice shown above). The a, b, and c variables represent the classical statistical weight or the quantum transition amplitude for each vertex type.

We finally had our Avengers-style collaboration: individual heroes brought together to wield their unique strengths. With Lincoln’s supervision, I developed the QCA models and performed the computations. Lorenzo found the Jordan-Wigner transformation. Tomaž found the first signals of integrability and delivered a set of conservation laws. Nicole brought her expertise in quantum thermodynamics, clarifying how the noncommuting charges constrain dynamics. Eric made the six-vertex connection. We drafted and redrafted the paper until it balanced the scientific story, the analytical derivations, and the numerical evidence.

Our team collaborated over six years.
Art by Barry Windsor-Smith. Published by Titan Comics, 2024

Because the discovered family of Goldilocks QCA maps to free fermions, we can efficiently simulate them classically. I simulated 256 qubits on my laptop this way. These large simulations were satisfying: I had worked with this model for years with an order of magnitude fewer qubits and even saw the dynamics implemented on Google’s Sycamore-era hardware with 23 qubits. Most Goldilocks QCA are consistent with chaos rather than integrability, and therefore hard to simulate classically. Therefore, our work gives experimentalists a tunable model: dial in integrable dynamics for something checkable at large qubit number. Set up chaotic dynamics for a potential demonstration of quantum advantage.

While preparing this post, I opened my old email account to check the timeline set out above. I looked through nearly six years of email chains, some with hundreds of messages, full of logistics for coordinating each author’s ever-changing time zone, and dozens of calculations and results that never made it into the paper. This collaboration helped me grow as a researcher in a big way.

I found old emails where Nicole was coaching me on messaging potential collaborators. I can hardly believe she dedicated so much effort to mentoring me. We have never met in person, despite our shared work starting when I was an undergraduate and she was a graduate student more than a decade ago. If you know Nicole, you can probably believe it easily. I had similar moments with each collaborator. They all gave their time and expertise generously over the many years this paper took to come together.

As I continue my efforts in experimental physics, I will pay forward the effort and generosity shared with me by this collaboration. I may even keep my theoretical-physics license for a while longer.

“By day” doesn’t mean “by daylight.” Laser labs are almost always in a windowless basement. ↩︎
CAM-6 featured 32 kB of cell-state memory (CAM-8 had 8 MB ), far less than the memory currently used by this author’s numerous open browser tabs. ↩︎
The coarse-grained momentum flux tensor was anisotropic. ↩︎
Logarithmic derivatives of the parametric transfer matrix generate the conserved charges. ↩︎

Quantum Frontiers

A blog by the Institute for Quantum Information and Matter @ Caltech

Monthly Archives: August 2026

Nicole’s guide to writing and editing

Interacting collaborators reveal noninteracting fermions

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