Nicole’s guide to writing and editing

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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.

  1. 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.
  2. 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.
        • Example: We substitute from Eq. (1) into Eq. (2).
    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.
  3. 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).
  4. 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 A:
      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 jx^8 in the example under (A). Put the corresponding +, –, \times, 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 O isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an O 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.”
  5. 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.
        • Example of sentence with convoluted, nested structure: Any model of equilibrium and nonequilibrium behaviors of systems observed in tabletop experiments and high-energy colliders must obey the laws of relativistic quantum mechanics.
        • Visualization of the nesting: [Any model of ([(equilibrium and nonequilibrium) behaviors] of {systems observed in [(tabletop experiments) and (high-energy colliders)]})] must obey [the laws of (relativistic quantum mechanics)].
    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.
  6. 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

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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.

  1. “By day” doesn’t mean “by daylight.” Laser labs are almost always in a windowless basement. ↩︎
  2. 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. ↩︎
  3. The coarse-grained momentum flux tensor was anisotropic. ↩︎
  4. Logarithmic derivatives of the parametric transfer matrix generate the conserved charges. ↩︎

Wise guy

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In my closet, in a basket labeled “Random stuff,” sits a bag of quarters. They total only a few dollars, but their worth to me exceeds their monetary value. I received the quarters from Mark Wise.

Mark taught a course about the Standard Model of particle physics at my master’s program at the Perimeter Institute for Theoretical Physics, near Toronto. Perimeter borrowed him from Caltech, to whose faculty he belonged. Mark had grown up in Canada and studied at the University of Toronto; so he didn’t mind visiting Canada even in the depths of winter. 

What would Mark have minded? He projected a mild manner—an innocuousness—that suited his sense of humor, which he often directed at himself. Mark had a bald patch and glasses, and he wore a mustache. Physics jokes and science-fiction references decorated his T-shirts, one of which he wore beneath a black suit jacket to our first class. His voice was nasal; it grated a little. But I relished listening to Mark’s lectures.

Mark’s lecturing exemplified clarity, because he knew particle physics so deeply. When he walked us through its Lagrangians and scattering diagrams, his conclusions seemed inescapable. His lectures’ logic and structure appealed to me as someone who’s been hyper-organized since at least fourth grade.

Yet Mark cared about us students beyond the requirements of pedagogy. His T-shirts invited conversation from those who arrived to class early. Whenever a student answered or asked a question, he tossed them a quarter. Sometimes, he’d pause to examine the quarter, deliberate about whether to toss a Canadian quarter or an American one, or opine about the motto printed on the coin. (Mark confessed to having lower standards than those ingrained in the New Hampshire state motto, “Live free or die.” Where he came from, “We just wanna live!”) 

Some days, Mark found little change in his pocket and announced that he needed to return to the bank for more quarters. The announcements sounded like complaints. He didn’t need to return to the bank, though, as nobody needs to bring doughnuts to the office for sharing.

I discovered the icing on the doughnut two years later, as a PhD student at Caltech. I sat in on part of a quantum course taught by Mark. To every student who completed the course, Mark gave a T-shirt that read, “Licensed quantum mechanic.” I received a T-shirt, although I only sat in on part of the course. I’ve never worn it, because I’ve wanted never to wear it out.

In 2024 and 2025, I co-taught a course on quantum-steampunk creative writing. Students learned about quantum physics, quantum technologies, and thermodynamics. Quanta are discrete units. For example, a photon is a quantum of energy. I illustrated quanta with coins, which are discrete units of money. From then on, I tossed a quarter to every student who answered or asked a question about quantum physics. (I joked that I should have tossed pennies, the minimal units of money, but chose quarters because inflation had been high recently.) I adapted Mark’s tradition to thermodynamics—the study of energy—by tossing Hershey’s kisses—dense packets of energy. 

Before moving out of Caltech, I said goodbye to Mark. He worked among the high-energy theorists, rather than the quantum information or condensed-matter theorists, so I had to hunt down his office. He smiled and made a joke, of course.

Mark passed away this summer. His Caltech colleague John Preskill published a eulogy as a blog post here. (I learned from John’s post that inflation led Mark to upgrade his quarters to dollar coins. So much for feeling generous about upgrading from pennies to quarters.) When asked about the student experience at Caltech, Mark would say, “Caltech is heaven for professors.” Irony would creep into his voice and body language as he’d continue, “Doesn’t that mean it’s heaven for students, too?” I worked my rear off as a student at Caltech and Perimeter, but I’d call both environments fairly heavenly. Mark and his ilk are reasons why.

My friend Mark Wise

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Mark Wise, the John A. McCone Professor of High Energy Physics at Caltech, passed away on July 10 at age 72. At a recent memorial service, John Preskill made these remarks.

I’m John Preskill, Mark’s colleague on the Caltech physics faculty for more than four decades. Our friendship goes back even farther. My wife Roberta and I met Mark and Jackie not long after they arrived at Harvard in 1980. We’ve been friends since then. We attended the bris for both Barry and Jonathan during those Harvard days. Mark and Jackie have two boys and we have two girls who are a few years younger, who were thrilled to connect with Barry and Jonathan when the families would get together for occasions like Passover or Hanukkah or Thanksgiving. When the kids were little, Mark and I would sometimes muse about the potential for forging even closer family ties if those relationships blossomed.

That didn’t happen. But Mark would preside at each Seder with a light hand, sprinkling the occasion with corny jokes as was his style, and Jackie would be determined to make it to the end of that customized family-friendly Haggadah she had meticulously prepared. The children, meanwhile, would be wondering when they’d be able to continue their game of sock baseball.

Many of you know that Mark was deeply dedicated to his family and friends. I’ll make some brief remarks about three facets of Mark I know especially well: Mark the scientist, Mark the teacher and mentor, and Mark the colleague and friend.

Because of his self-deprecating manner, those of you who are not scientists may not appreciate Mark’s stature as a physicist. He was one of the most influential figures in theoretical particle physics of his generation. It was not obvious things would turn out that way. Growing up in Toronto, Mark was an indifferent student, and his poor grades reflected that lack of interest. As a 9th grader, though, it struck him that he better change his ways and figure out how to make something of his life. He liked sports — the possibility of being a professional athlete was briefly considered, but discarded. Somehow he decided that science would be a better fit. I’m not sure why — he had recently failed math. But he worked hard and had inspiring teachers, so by the time he finished high school Mark was an excellent student, and he sailed into the University of Toronto well prepared to major in physics,

At U of T, Mark came under the influence of a young professor, Nathan Isgur, who would later become his close research collaborator. Under Nathan’s guidance, Mark sought admission to the PhD programs of the most prestigious US research universities, intent on a career devoted to deep exploration of the fundamental laws of physics. He was rejected everywhere he applied. He should have been discouraged. But he wasn’t. Mark shrugged and said: “It’s okay. I’ll stay another year in Toronto, I’ll get a master’s degree, I’ll apply again and I’ll get in somewhere.” And that’s what happened. He went to Stanford, where, under the kind tutelage of Fred Gilman, Mark took off like a rocket. Hired to the Caltech faculty in 1982, he was a tenured full professor three years later at the age of 31, and appointed as the John A. McCone Professor of High Energy Physics while still in his 30s.

Mark liked action movies, such as those starring Arnold Schwarzenegger or Clint Eastwood. In serious moments, we would sometimes ponder together why we’re successful at what we do, and Mark would always quote Clint Eastwood as Harry Callahan in Magnum Force: “A man’s got to know his limitations.” We would both laugh, but those were words of wisdom. Mark understood what he did well as a research scientist and what he was less good at. Finding problems he could solve that would have interesting consequences for experiments that had been done or could be done was where he excelled – he did it again and again. Mark never lost his zest for calculating things, often by hand with pen and paper, his head resting on one arm with his glasses pushed up onto his forehead as he scribbled. Getting to an answer that was experimentally relevant never stopped giving him a thrill.

Mark also never lost his sense of appreciation for the teachers and mentors who had inspired and helped him. Perhaps that’s why he became such a dedicated teacher and mentor himself. It’s hard to impress Caltech students, but Mark’s lectures where extremely popular, not just for their pedagogical value but also for the humanity and humor he displayed. Students had to pay attention because otherwise one might miss the jokes, which inevitably became known as “Wisecracks.” There is even an account on X with the handle @MarkWiseSays, curated by students who want to preserve Mark’s pithy lessons in physics and in life.

For example, Mark might say: “If you really get depressed, I recommend diagonalizing a 2×2 matrix.” For physics students, this is both funny and sage advice. Or he might say. “This calculation will knock your socks off.” A cliché you might hear from anyone. But who besides Mark would then proceed to remove his shoes, rip off his socks, hurl them at the blackboard, put his shoes back on and resume lecturing?

Most famously, Mark would come to class with an ample supply of coins. He would ask the class questions, sometimes about physics and sometimes random trivia, rewarding a student who gave an answer Mark approved of by tossing a coin. At first the coins were quarters. But Mark, who had a scholarly interest in finance as well as physics, eventually felt that due to inflation he needed to upgrade to dollar coins. These are harder to come by, so it took frequent visits to the bank to make sure he wouldn’t run out. His antics in class made Mark human and approachable, and students responded. Mark felt that many Caltech students don’t fully realize how smart they are. He saw part of his job as building their self-confidence and relieving their stress.

As a colleague and mentor to graduate students and postdoctoral scholars, Mark was highly collaborative. He believed that interactions with others sparked his creativity. He was never at all pompous. I know this started early. When we were in the Harvard Society of Fellows we were obligated to have dinner with the Senior Fellows on Monday nights. It was a rather stuffy occasion. And, though I don’t think they do this anymore, after a sumptuous meal we would literally retire for brandy and cigars. Once, while puffing on his cigar after dinner, Mark had an inspiration. He gathered up a few junior fellows and led them to a theater for a movie he thought everyone should see right away. The movie was Conan the Barbarian. And everyone had a blast. That was a perfect Mark moment.

As the news about Mark has spread, accolades have poured in from physicists all over the world. He was admired not just for his scientific brilliance, but almost as much for his quirky sense of humor and his kindness. Mark was a wonderful friend to many of us. When you were with him, you were sure to laugh and feel good. He touched the lives of countless colleagues, students and friends. We miss him terribly but there are so many memories that we’ll cherish. We are all so very fortunate to have known and loved Professor Mark Wise.

Photo credit: Clara Murgui, Caltech

The physicists of Florence

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A scientist in Florence can’t avoid bumping into colleagues. 

When visiting the Renaissance’s birthplace last summer, I ran into a fellow physicist even on a Saturday morning. I was wandering around the Uffizi Gallery, a museum blessed with some of the greatest hits in western art. A familiar face arrested me on the first floor.

Another colleague cropped up outside the museum. (Some might classify him as an applied physicist or an engineer, but he exhibited a theoretical physicist’s overactive imagination.)

One colleague, I’d been looking forward to meeting for over four years. Jae Dong Noh is a professor of physics at the University of Seoul in South Korea. He’d conducted the first numerical tests (classical-computer simulations) of an idea I’d helped midwife, the non-Abelian eigenstate thermalization hypothesis (NAETH). An earlier blog post described this mouthful, which predicts how certain quantum many-particle systems thermalize, or experience the flow of time. These systems’ dynamics conserve properties, analogous to energy, that are incompatible: one can’t measure the properties simultaneously, as one can’t measure a quantum particle’s position and momentum simultaneously. Because incompatibility helps distinguish quantum from classical physics, such systems’ thermodynamics qualifies as particularly quantum.

Jae Dong modeled such a system and others numerically in a paper. I admired his computational techniques and his grasp of symmetries (for experts: how non-Abelian symmetries affect chaotic quantum systems’ energy-level statistics). My postdoc Aleks Lasek was planning a more thorough numerical test of the NAETH, so I reached out to Jae Dong, and a collaboration crystallized. 

Seoul operates thirteen hours ahead of Maryland, but we managed to Zoom because Jae Dong is a night owl and I’m an early bird.1 Zoom introduced me to a man perpetually dressed in a neat button-down shirt and sweater, silver overriding the black in his hair. The neatness extended to Jae Dong’s explanations: if Aleks and I didn’t understand one of his emails, he’d explain it quietly and calmly, untangling the confusion as though pulling a comb through wool.

The collaboration settled into a rhythm: I’d pose a question or propose a goal, Jae Dong would respond with an analytical calculation,2 I’d find holes in the calculation, Jae Dong would plug the holes, I’d re-check the argument’s logic, and we’d repeat the cycle. Had I been in Jae Dong’s shoes, I’d have swallowed the constant objections as I’ve swallowed grape-flavored cough medicine,3 but he always responded with equanimity—sometimes even good cheer—and a possible solution. Meanwhile, Aleks and then-undergraduate Jade LeSchack checked our analytical arguments numerically.

Florence flaunted a little steampunk during my visit.

So smoothly did the collaboration hum along that we coauthored two papers before ever meeting in person. One demonstrates numerically that two quantum many-body systems (for experts: nonintegrable Heisenberg models) obey the NAETH.4 In the other paper, we derive a symmetry relation from the NAETH. If the 17-syllable NAETH is a mouthful, the symmetry’s name is half a mouthful: a Kubo–Martin–Schwinger (KMS) relation. It’s important because (i) it enables us to calculate how rapidly a thermodynamic system responds to a stimulus, such as a weak magnetic field, and (ii) physicists go gaga over symmetries generally. 

The KMS relation constrains thermal states—essentially, systems that have temperatures. Your typical isolated many-particle quantum system looks thermal if you can observe just a small chunk of it at a time. Accordingly, Jae Dong and collaborators had proved that isolated many-particle quantum systems obey the KMS relation approximately. The larger the system, the more accurate the approximation. 

We extended his argument to systems whose dynamics conserve incompatible properties. Such an extension might sound simple, but its proof filled 24 pages of appendices. (For experts: Clebsch–Gordan coefficients are tricky blighters.) We discovered that, under certain conditions, incompatible conserved quantities can reduce the extent to which a quantum system obeys the KMS relation. Quantum incompatibility can augment deviations from conventional thermodynamics.

Italy’s architecture impressed me.

Jae Dong planned to present about our work at StatPhys, an international statistical-physics conference, which Florence was hosting in 2024. Throughout the two-and-a-half months before the conference, the KMS relation consumed our team. (For experts: Clebsch–Gordan coefficients are very tricky blighters.) I even hid in my hotel room, working and reworking our proofs, during another conference during that time. 

The toil paid off. We submitted our KMS manuscript for public scrutiny the day I flew to Florence—because not only Jae Dong would be representing our team at StatPhys. I was looking forward to meeting him there for the first time.

A corner of the hall where the StatPhys opening ceremony took place.

The StatPhys committee outdid itself. The opening ceremony unfolded in Florence’s Palazzo Vecchio, where members of the Medici dynasty once lived. Giorgio Parisi, who won a Nobel Prize for statistical physics in 2021, lectured at the ceremony.

Giorgio Parisi, with another history maker.

The meat of the conference took place in two other palaces, the Palazzo dei Congressi and the Palazzo degli Affari. In one of them, I met Jae Dong. Although we’d shown that quantum incompatibility can defy thermodynamic predictions, he met my expectations.

We discovered another thermodynamic phenomenon challenged by incompatible conserved quantities, so stay tuned for another paper and blog post. Some colleagues, one can’t avoid; others are worth engaging with again and again.

1 Aleks has confessed to night-owl habits, but physics motivates him to adapt. Some days, he’s emailed me results before even I’ve woken up. Who needs coffee when the thrill of discovery electrifies one minutes after one hops out of bed?

2 An exact calculation written out on paper, as opposed to a numerical, or approximate, calculation performed by a silicon-based classical computer.

3 Does anyone like the grape flavor? Why do companies bother producing it?

4 Rohit Patil and Marcos Rigol, too, have checked numerically that a system obeys the NAETH.

Nicole’s guide to handling failure and rejection

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It’s happening. 

Your inbox registers an email from the chair of a faculty-hiring committee. With trembling fingers, you click on the message. “We’ve been grateful for the opportunity to learn about your work…The decision was very difficult…many highly qualified candidates…” Months of labor, soul-searching, strain, and anxiety give way to despair. The committee has filled the position, and not with you.

I recently published advice about how to proceed if you receive an offer of a faculty position. But what if you don’t receive an offer—what if hiring committees reject you? Or scholarship committees, grant committees, admissions committees, or potential advisors? What if a journal referee shreds your magnum opus? Or a program committee declines your submission to a conference?

Failure suffuses science as dinner suffuses a nighttime diaper, for two reasons. First, undertaking science is difficult. By “undertaking science,” I mean formulating and proving theorems, cajoling equipment into working, identifying bugs in code, extracting meaning from noisy data, etc. Second, science doesn’t unfold in a vacuum. Human beings do science within a society fraught with opinions, emotions, and limited resources. These challenges lead to the failures and rejections that this article addresses most. (If you’re a member of my group and you’re facing a failure due to the difficulty of undertaking science, come talk with me.) 

What can you do if failure or rejection ails thee? The rest of this article prescribes, in chronological order, steps that will help you recover. You can even turn your lemons into, if not lemonade, then not-entirely-unappetizing lemon meringue pie.

“O what can ail thee, knight-at-arms, alone and palely loitering?”
“The scientific life is rough.”

Immediately after receiving the news:

  • Read the communication once—or, if you must, twice. Don’t linger over the letter for longer than necessary.
  • Put the communication away, so you won’t see it unless you try to. If the news came via email, remove the message from your inbox.
  • Lower your heart rate. Electrified with anger? Expend that energy; go for a walk, for a run, or to the gym, if possible. If you can’t, breathe deeply for several minutes, extending your exhalations.
  • Review records of your successes. I maintain a folder called “Nice messages.” It contains notices of awards I’ve received, kudos on papers I’ve published, messages such as “Thanks for everything this past semester! Your class was probably my favorite one,” and every compliment I’ve received from the taciturn John Preskill via email. Review evidence—remind yourself—that you’re not a failure even though you’re experiencing failure or rejection.

Avoid thinking about the news for a few days. Imagine cutting yourself on a kitchen knife. The wound may sting and bleed initially. But the blood clots and the stinging diminishes if you cover the wound and don’t aggravate it. 

After experiencing a failure or rejection, invite a colleague to lunch, and ask about their recent reading and travels. Dive into a project that will absorb you. Visit a museum, or watch a movie. Remove the letdown from your thoughts.

Consider seeking input from a mentor, especially if you’ve never experienced a failure or rejection of this type before. Mentors have more experience and so can put obstacles in perspective. For example, suppose you’re a student who’s received an upsetting referee report from a journal. The report might look mild to a faculty member, who’s likely received far more, and more-upsetting, reports. What sounds harsh to you might sound quotidian to an advisor, whose lack of distress might reassure you. 

Also, mentors notice upsides that you’ve overlooked. Perhaps the referee trashed your presentation of an idea but tacitly approved of the idea itself. The trashing might have drowned out the approval during your reading. Yet the approval could justify a resubmission to the journal, upending your belief in the cause’s hopelessness. Relatedly, a mentor can help you identify strategies for moving past the failure. Maybe this journal won’t publish your paper but another journal is soliciting contributions to a relevant special collection.

As a PhD student, I applied for an internship at a company developing a quantum computer. I broke an obligation and flew out of state to interview for the position. No offer materialized. To process the outcome, I spoke with a more-advanced researcher I trusted. He not only offered a mature perspective, but also had access to inside information. The team liked me, he reported, but my interests didn’t overlap with theirs enough. In a sense, I’d grown too independent. As independence marks maturity in PhD students, the rejection came to double as a compliment. Today, I remain on friendly terms with multiple people from that team, and a student of mine just landed an internship at a quantum-computing company.

Someone more experienced than you should have your back. (For the story behind this photo, see this blog post.)

Chart a path past the failure or rejection. Cue the lemon meringue pie. Did a hiring committee reject an application of yours? Email the committee’s chair, thanking them for considering your application. Express your hope of improving your materials so that you can try again the following year. Ask if the chair would provide feedback via phone or Zoom.1 

Pursue the path you’ve charted. To ease the burden, consider undertaking lighter tasks before more-arduous ones. I recently received a laundry list of requests about a manuscript from an editor—and when I say laundry, I mean the equivalent of hauling a multiple-pound bag to a stream, scrubbing everything by hand while the wind attempts to blow cleaned items into the mud, and then hauling everything back. I began with the tasks that required little effort. Dispatching them, I crossed about half the requests off my to-do list. The list looked more manageable as a result; I felt better-equipped to handle the trickiest work.

Celebrate your triumphs. Don’t let failures and rejections consume your attention; carve out time for your successes. If you recognize them, you’ll remember them the next time you face failure; they’ll cushion you when you fall.

I recently failed at a task over a hundred times (yes, I counted). After 28 months of trying, I succeeded. I celebrated by treating myself to lunch at the National Gallery of Art.2 Why not gather ye rosebuds while ye may? This success afforded me the opportunity to fail at a new task.

The National Gallery of Art sold a fairly steampunk book near its Garden Café recently.

Always remember, what matters in the long run is not any one failure or rejection, but your persistence despite failures and rejections. I progressed to the Rhodes Scholarship competition’s final round two years in a row. Each year, a committee interviewed and rejected me. I’d poured time, sweat, and blood into my applications and interview preparations. I felt like I had no more blood to squeeze into the applications I then had to write for graduate programs because of those rejections. Those rejections, though, led me to become a student of John Preskill’s. Over a decade later, I’m not kicking myself.

Result of two failures.

Share about your favorite failures in the comments section below!

1Not via email, which doesn’t offer the same freedom.

2Its Garden Café had enticed me for years, but I can rarely bring myself to pay for food when I can prepare it myself.

How I learned to stop worrying and…no, I’ve always adored entropy

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When I was pursuing a PhD at Caltech, so was my friend Jeremy. He used to throw a dinner party every few months. The email invitations welcomed friends to partake of his cooking and, if we wished, to help him cook. I didn’t help cook; but, when I arrived, the mess of pots and pans drew me to the kitchen like vinegar drawing a pathological fly. I couldn’t sit still while cookware needed cleaning, so I scrubbed and rinsed the pans and spoons and bowls. Jeremy, an applied-physics student, commented on my adeptness at decreasing entropy.

It’s the story of my life, I replied.

In fourth grade, my classmates and I cleaned our desks every Friday afternoon. Once a student finished, my teacher dismissed him or her onto the playground. My neighbor’s desk horrified me like the disaster in a hurricane’s wake, so I neatened his desk after finishing with mine.1 Another friend requested the same favor. A third classmate offered to pay me for cleaning his desk, but I’d have undertaken the chore for its own sake. Ordering the world offered me fulfillment.

From cleaning a fourth-grade desk, I progressed to pursuing a PhD in theoretical physics. The two pursuits might seem to resemble each other no more than Dr. Jekyll and Mr. Hyde; yet, to me, the path between them is but a step. I trained as a theoretical physicist because I love organizing ideas. Caltech paid me to build models, propose definitions and theorems, and structure proofs—to dream up ideas and identify the optimal arrangements for them. I needed that pay, being an adult, as I hadn’t needed my fourth-grade classmate’s desk-cleaning fee. Yet I organized ideas for the same reason that drove me to organize my neighbor’s notebooks.

Many people have called entropy a measure of disorder. To see why, imagine that Jeremy’s crew has used thirty utensils while cooking. The chefs can have scattered the utensils across the kitchen in many ways: they may have dropped forks on the floor, left spoons in the sink, arranged spatulas on the drying rack, or filled a vase with knives like a modern-art bouquet. In few of these configurations do the forks lie in their compartment of the utensil drawer, the spoons lie in their compartment, etc. We call such configurations neat. Most of the other configurations, we call messy. 

A system’s entropy is the number of configurations consistent with known large-scale properties of the system, such as the number of forks.2 More configurations are consistent with messiness (and a fixed number of forks and so on) than with neatness (and the same number of forks and so on). Messiness tends to correlate with high entropy. People often say, therefore, that entropy quantifies messiness. Hence Jeremy’s complimenting me on my decreasing of entropy.

Jeremy’s dinner parties came to mind as I read the book The Mattering Instinct, published by Rebecca Newberger Goldstein this January. Rebecca is a philosopher of science and a writer. I had the good fortune to meet her through my undergraduate mentor Marcelo Gleiser, who’s had another cameo or two on Quantum Frontiers. Rebecca’s latest book covers what she calls the mattering instinct: the longing to know that we matter. 

We spend scads of energy and time on securing our “survival and flourishing,” as Rebecca says. We feed ourselves; work to earn money to purchase food; clean, shelter, and clothe ourselves; ingrain ourselves in societies that offer some degree of security; and more. Do we deserve all this effort? We long for assurance that, in the immortal words of L’Oreal, we’re worth it. 

Survival and flourishing, Rebecca writes, requires us to decrease entropy. Every closed, isolated system’s entropy increases or remains constant, according to the second law of thermodynamics. Entropy increases as a system becomes more uniform, loosely speaking. The system’s particles spread out across space, these particles’ temperature comes to equal those particles’ temperature, and so on. In contrast, your body exists because its particles clump together in a certain shape consistently. You withstand heat waves and snow because homeostasis maintains your temperature despite your environment’s temperature. You keep your body’s entropy low to survive. Rebecca therefore casts us as fighting entropy.

As a thermodynamicist, I agree with Rebecca. Yet I also adore entropy. It helps explain why time flows, quantifies uncertainty, and determines the maximal efficiencies with which we can perform tasks such as communication. What versatility and richness! Entropy also embodies tension and subtlety: its mathematical definition looks obscure at first glance, yet entropy helps explain familiar phenomena such as aging. For these reasons, before beginning my PhD, I told a potential advisor that I could imagine devoting the next five years of my life to entropy.

I therefore aspire to rehabilitate entropy’s reputation. Novelist Terry Pratchett endeared mortality to millions of readers through anthropomorphism. His character Death, a mainstay of the Discworld series of novels, elicits empathy and fondness. I won’t anthropomorphize entropy here,3 but I aim to replace conflict with cooperation in the narrative above. To survive and flourish, I hold, we partner with entropy. How? We create oodles of entropy in our environments. This entropy increase offsets the entropy decrease that supports life.

For example, imagine working at a desalination plant. You’d process high-entropy water throughout which salt has spread. You’d concentrate the salt in a tiny region, reducing the water’s entropy. This reduction, producing fresh water, could support your city’s drinking, cooking, and toothbrushing needs.

To reduce the water’s entropy, you’d create loads more entropy. You’d eat breakfast before work, consuming energy stored neatly in your waffle’s chemical bonds. Your body would later break the bonds, releasing the energy. Some energy would power your muscles, so you could program the desalination system, test its output, etc. But much of the chemical energy would transform into heat radiated by your body. The heat would warm up the air molecules around you, magnifying their random jigglings and jostlings. You’d increase the entropy of the air—your environment—to decrease the water’s entropy. The air’s entropy increase would outweigh the water’s entropy decrease.

Organisms survive and flourish by producing entropy in their environments. In fact, organisms have a knack for generating entropy. Entropy and life thereby further each other. A glass-half-full thinker could conclude that we partner with entropy.

So did I partner with entropy as a PhD student, applying it to solve problems in quantum information theory and thermodynamics. So did I partner with entropy in fourth grade and at Jeremy’s apartment, deriving satisfaction from my cleaning. Rebecca would call these activities’ ultimate aim (beyond the aim of, e.g., not sitting beside a pigsty in fourth grade) mattering. She writes that we reduce entropy (within our immediate vicinities) to satisfy the mattering instinct. Rebecca’s proposition describes my behaviors with uncanny precision, I realized upon reading her book.

Which I’ve now finished. So pardon me while I return to washing forks in the quantum kitchen of the universe. 

With thanks to Jeremy for his friendship…and food.

1I also ensured that my neighbor brought home, every afternoon, the sweater he’d brought to school that morning. Before I took charge, he’d ended up with three forgotten sweaters crammed into his cubby.

2At least, one entropy is. Many other entropies exist.

3If you anthropomorphize entropy elsewhere, let me know.

Unleashing the Advantage of Quantum AI

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As experimental capabilities advance rapidly, the quantum computing community faces a critical elephant in the room: What will these quantum machines eventually be useful for? Will they deliver the promised broad societal impact, or will they remain highly specialized devices for exotic tasks known only to the experts?

The elephant in the room

Despite decades of effort, conclusive evidence of large quantum advantage in real-world applications remains confined to a few niche domains, such as simulating quantum materials and cryptanalysis. These problems are either inherently quantum to begin with, or they possess specialized mathematical structure that quantum algorithms can easily exploit. But it seems unlikely that such structures appear broadly in everyday life.

Indeed, most applications of modern computation hinge on the processing of massive, noisy classical data, generated at an unprecedented pace across society. That is the driving force behind the overwhelming success of machine learning and AI. Since the data originates from the macroscopic classical world, there is no obvious reason it should exhibit the delicate, specialized structures that quantum computers require. To playfully adapt Richard Feynman’s famous quote: We live in an effectively classical world, dammit, and maybe classical computers and AI already suffice for most of our problems. (For those unfamiliar, Feynman originally quipped: “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical.”)

The central challenge

To truly unlock the power of a quantum computer, quantum algorithms typically need to access data in quantum superposition, processing many different samples simultaneously in different branches of the quantum multiverse. To use technical jargon, this is called querying a quantum oracle. But in reality, the classical data samples that we want to process are generated from everyday activities in a classical world, and we can only access them one at a time.

Think of the movie reviews you scroll through on a streaming platform. How would you read the plain-text reviews from a million different users all at once in a quantum superposition? This bottleneck—the challenge of efficiently accessing the classical world in quantum superposition—is known as the data loading problem. It has arguably been one of the main obstacles to achieving broadly applicable quantum advantage.

Sketching a quantum oracle

In this new work [1], we provide a solution to this seemingly impossible challenge. We develop a framework, called quantum oracle sketching, that enables us to access the classical world in quantum superposition in an optimal way. Importantly, it automatically handles the noise and correlations in the data, and natively supports flexible data structures like vectors and matrices that enable machine learning applications.

The core mechanism relies on processing data as a continuous stream. For each classical data sample we observe, we apply a carefully designed, small quantum rotation to our system. By sequentially accumulating these quantum rotations, we incrementally build up an accurate approximation of the target quantum oracle, which can then be used in any quantum algorithm for data processing. Because every data sample is processed once and immediately discarded, we completely eliminate the massive memory overhead typically required to store the dataset. The fundamental price to pay for assembling quantum queries from classical data lies in the sample complexity: our algorithm consumes a number of samples that scales quadratically with the number of quantum queries we need to make. We show that this rate is optimal and fundamentally arises from the quadratic relationship between quantum amplitudes and classical probabilities governed by the Born rule.

With the data successfully loaded into the quantum computer, the final challenge is to efficiently read out classical results. To address this, we develop an efficient measurement protocol called interferometric classical shadow. Combined with quantum oracle sketching, it allows us to circumvent the data loading and readout bottleneck to construct exponentially compact classical models from massive classical data with quantum technology.

Exponential quantum advantage in machine learning

Using this new approach, we are finally able to find exponential quantum advantage in processing classical data and machine learning. We rigorously prove that a small quantum computer can perform large-scale classification and dimensionality reduction on massive classical data by processing samples on the fly. In contrast, any classical machine achieving the same prediction performance requires exponentially larger size. When the classical machine does not have the required exponentially large memory size, it needs super-polynomially more samples and time relative to our protocol running on a quantum device. Remarkably, this illustrates that quantum technology enables us to construct compact and accurate classical models out of classical data, which is impossible with classical machines alone unless given exponentially larger memory.

The true scale of this exponential memory advantage is staggering. A quantum processor with 300 logical qubits can outperform a classical machine built from every atom in the observable universe. Of course, to actually see such a comical contrast, we would also need universe-scale datasets and processing time.

To contextualize these results in realistic scenarios, consider a large-scale scientific experiment, like a large particle collider. Each experimental run generates a colossal volume of data. With a quantum computer, we can keep squeezing all the data into this tiny quantum chip to perform downstream machine learning tasks such as classification and dimensionality reduction. But if we only have classical machines, we would need to build massive, energy-consuming data centers to store the raw data to match the performance. Without this massive memory overhead, classical machines simply couldn’t extract the same clear signals from a single run, forcing us to repeat the massive, expensive experiment many more times to compensate. To put this into perspective, the Large Hadron Collider (LHC) at CERN generates petabytes (millions of gigabytes) of data per hour, but the data storage bottlenecks force researchers to discard all but a tiny fraction—retaining perhaps only one in a hundred thousand events.

We validated these quantum advantages on real-world datasets, including movie review sentiment analysis and single-cell RNA sequencing. In these public datasets, we demonstrate four to six orders of magnitude (ten thousand to a million times) reduction in memory size with fewer than 60 logical qubits. Given the rapid advancements in high-rate quantum error correction codes and experimental techniques, quantum computers capable of demonstrating such applications are foreseeable in the near future. Crucially, the quantum advantage we propose likely carries a clearer positive impact for society and likely arrives sooner than the applications in cryptanalysis, where the current best estimate requires a thousand logical qubits.

Towards Quantum AI

Our results provide strong evidence that the utility of quantum computers extends far beyond specialized tasks, opening a path for quantum computers to be broadly useful in our everyday life. Rather than fearing that classical AI will “eat quantum computing’s lunch,” we now have rigorous evidence pointing towards a much more exciting prospect: quantum-enhanced AI overpowering classical AI.

Of course, there is still a long way to go towards the dream of quantum intelligence. Our current results establish the provable supremacy of quantum machines in foundational machine learning tasks, such as high-dimensional linear classification and dimensionality reduction. They do not yet imply immediate utility for modern generative AI such as large language models.

That said, our results give me a strong feeling that we are living in an age strikingly reminiscent of the traditional machine learning era—an age dominated by support vector machines and random forests; an age when we relied on rigorous statistical analysis because we lacked the computational resources for large-scale heuristic exploration; an age that ultimately heralded the birth of deep learning and the AI revolution. Today, quantum AI seems to sit at a similar historical position. I cannot wait to see what quantum AI will become once we are capable of unconstrained heuristic exploration on large-scale fault-tolerant quantum computers.

To accelerate this dawn of quantum AI, we invite physicists, computer scientists, developers, and machine learning practitioners to join our efforts and help us push the boundaries of what quantum AI can achieve. To bridge the gap between abstract quantum theory and hands-on machine learning practice, we are open-sourcing our core framework. Our numerical implementation of quantum oracle sketching is built in JAX, natively supporting GPU/TPU acceleration and automatic differentiation to integrate nicely with modern machine learning pipelines. Check out the code, run the simulations, and help us shape the future of quantum AI at github.com/haimengzhao/quantum-oracle-sketching!

References

[1]. Haimeng Zhao, Alexander Zlokapa, Hartmut Neven, Ryan Babbush, John Preskill, Jarrod R. McClean, and Hsin-Yuan Huang. Exponential quantum advantage in processing massive classical data, arXiv:2604.07639, 2026.

The FeMo-cofactor and classical and quantum computing

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Recently, my coworkers and I put out a preprint “Classical solution of the FeMo-cofactor model to chemical accuracy and its implications’’ (Zhai et al. 2026). It is a bit unusual to write commentary on one’s own scientific article. However, in this case, given the many inquiries I have had about the work in the context of quantum computing, many of which have contained similar questions (and often similar misunderstandings), I thought it would be useful to provide some perspective that we could not provide in the original preprint, in an informal manner.

What is FeMo-co?

I will start with some background on the FeMo-cofactor (FeMo-co). This cofactor is the reaction center of nitrogenase, an enzyme found in certain soil-dwelling bacteria. Nitrogenase’s claim to fame is that it converts atmospheric dinitrogen, which is held together by a strong N-N triple bond, into a reduced form (ammonia) which can then be taken up by plants and thereby be passed onto the rest of the living biomass. In terms of incorporating nitrogen into biomass, nitrogenase is believed to be responsible for about 2/3 of biological nitrogen, with the remainder coming from fertilizers. Because it plays this critical role, it is sometimes referred to as the enzyme that feeds the planet.

The chemistry of how dinitrogen is reduced at the FeMo-cofactor is still largely unknown. The basic stoichiometry of the reaction is often written as

\mathrm{N}_2 + 8\mathrm{H}^{+} + 8e^- + \mathrm{16MgATP} \to 2 \mathrm{NH}_3 + \mathrm{H}_2 + 16\mathrm{MgADP} + 16\mathrm{P}_i

but this just a sketch of the process. In particular, the above equation contains, nominally, a large number of molecular reactants, and clearly they do not all just come together in a bang! The role of the cofactor, and the enzyme more generally, is to coordinate the protons, electrons, biological energy source (ATP), and the dinitrogen molecule, into a sequence of well-defined steps, known as the reaction mechanism. Since the work of Lowe and Thorneley (Thorneley and Lowe 1984), the most common proposal for the nitrogenase reaction mechanism contains 8 intermediate steps (corresponding roughly to 8 sequential proton and electron additions). However, due to the difficulty in isolating the intermediate states of FeMo-co, as well as challenges in using experimental probes to deduce what these states are, the Lowe-Thorneley cycle still remains an unproven hypothesis. Biochemists, spectroscopists, as well as a few theoretical quantum chemists, are today actively engaged in observing, computing, deducing (and arguing about) the nitrogenase mechanism (Jiang and Ryde 2023; Lancaster et al. 2011; Einsle and Rees 2020; Badding et al. 2023; Thorhallsson et al. 2019).

So how did nitrogenase become so widely discussed in the setting of quantum computing? In 2016, an article “Elucidating reaction mechanisms on quantum computers’’, that has since become one of the most cited papers in the nitrogenase field, arguably started this all (Reiher et al. 2017). The article included a number of proposals, including (1) that the ‘promise of exponential speedups for the electronic structure problem’ could be applied to elucidate the nitrogenase reaction mechanism that had so far proved intractable for classical computation, and (2) that solving this problem would be an example of how quantum simulation could be ‘scientifically and economically impactful’. (Similar proposals can also be found repeated in less technical language and settings, see e.g. ‘Why do we want a quantum computer’). An important technical contribution of the article was to provide a detailed quantum resource estimate for a simulation of chemistry. The problem statement was to compute the ground-state energy of a specific ‘54 orbital’ (108 qubit) model of FeMo-co, to an accuracy of 1 kcal/mol, referred to as chemical accuracy. It is important to note the word ‘model’ in the problem statement. Electrons move in continuous space, and thus quantum chemical Hamiltonians are formulated in the continuum, while quantum computation requires discretization of this space. This discretization, in terms of a so-called active space set of orbitals that the electrons can variously occupy, constitutes the model. We will return to the definition of the model below. By compiling a Trotter-Suzuki implementation of the quantum phase estimation algorithm within a fault-tolerant resource model for their specific FeMo-co model Hamiltonian, Ref. (Reiher et al. 2017) provided a T-gate resource estimate. Combined with some assumptions about the quantum architecture, this provided perhaps the first concrete time-cost to solve an interesting chemistry problem on a quantum computer. This work has since served as an inspiration for many subsequent quantitative resource estimation efforts in the quantum computing for chemistry field.

Exponential speedup and societal impact

Before proceeding further in this story, it is worth examining the two key propositions made in Ref. (Reiher et al. 2017). I start with the question of exponential speedup. Quantum algorithms for the ground-state energy, such as quantum phase estimation, essentially perform a projective measurement of the energy (encoded in a phase). Thus, it is essential to prepare a good initial state, i.e. with large overlap with the desired eigenstate, to measure the correct energy. This, however, is a strong constraint, if we are seeking asymptotically large quantum advantage. For example, if such an initial state is first determined classically, as is often suggested, then exponential quantum advantage in a given problem requires that finding good classical guesses is easy, while improving them classically to fully solve the problem becomes exponentially hard as the problem size increases. Unfortunately, convincing evidence that chemically relevant electronic structure problems, including the problem of cofactor electronic structure exemplified by FeMo-co, fall into this category has not yet been found, as discussed in detail in Refs. (Lee et al. 2023; Chan 2024).

The second proposition, that elucidating the reaction mechanism of nitrogenase will lead to a transformative societal impact, is similarly nuanced. The claim originates in the observation that the competing industrial process for fertilizer production via nitrogen reduction, namely, the Haber-Bosch process, takes place at high temperatures and pressures and consumes a significant percentage of the world’s energy. Bacteria, on the other hand, can do this process at room temperature.

While it is true that the nitrogenase enzyme functions at ambient temperature and pressure, it is simply false that it consumes much less energy. This is because the large amount of energy required for nitrogen fixation mainly originates from thermodynamics, i.e. one needs energy to break the strong nitrogen triple bond. In fact, taking into account the physiological conditions and the ATP cost, bacteria arguably expend more energy to reduce ammonia (Chan 2024) than a modern efficient industrial implementation of the Haber-Bosch process. Thus the real hope behind trying to understand the nitrogenase mechanism in the context of societal impact is that we may one day engineer a variant of it with more desirable properties, e.g. with higher turnover, or with a lower carbon footprint, or which is more selective for nitrogen reduction. Whether this is actually possible remains to be seen, and certainly requires much more than knowing the ground-state of FeMo-co, or even the full reaction mechanism.

Which FeMo-cofactor model?

I now return to the question of FeMo-cofactor models. Ref. (Reiher et al. 2017) introduced a particular cofactor model, which I will refer to it as RWST, following the names of the authors. As we soon found out, simulating the ground-state of the RWST model was actually very easy classically, and in fact (as reported in  (Li et al. 2019)) could be done using standard quantum chemistry methods with a few hours of calculation on a laptop. This was because although the RWST model was a 108 qubit model, and (in the worst case) a 108 qubit ground-state cannot be stored classically, the RSWT model Hamiltonian was constructed in such a way to not capture any of the difficult features of the FeMo-cofactor ground-state. This highlights the importance of not assuming worst case complexity about physical problems!

What makes the electronic structure of the FeMo-cofactor (relatively) complicated is the presence of many ‘unpaired’ electrons. In simple molecules, we can describe the ground-state as one where all the electrons sit in pairs in orbitals. Since an orbital can only carry a pair of electrons at a time, the ground-state is simply described by filling the lowest energy orbitals with pairs. However, in molecules with transition metals, there are typically ‘unpaired’ electrons (so-called open-shells), and then we need to consider whether and how they pair up, which orbitals are singly versus doubly occupied, and so on. The RSWT model ground-state had no unpaired electrons! It was therefore unrepresentatively easy to solve for the ground state classically.

Because of the problems with the RWST model, my group formulated a more suitable 76 orbital/152 qubit model of FeMo-co in Ref. (Li et al. 2019), which I will refer to as the LLDUC model, again by the names of the authors. Although the LLDUC model is still a significant truncation of the true electronic structure of FeMo-co, we verified that it contains the correct open-shell character of the cofactor, and thus has a ‘representative’ complexity in its ground-state. Since we published the LLDUC model, it has become the most common benchmark model of FeMo-co used in quantum resource estimates for new quantum chemistry ground-state algorithms (Wan et al. 2022; Berry et al. 2019; Luo and Cirac 2025; Low et al. 2025).

Heuristics in the classical solution of the LLDUC FeMo-cofactor model

This brings me now to the recent work in Ref. (Zhai et al. 2026), where, through a sequence of classical calculations, we could produce a classical estimate of the ground-state energy of the LLDUC model to chemical accuracy. How was this achieved?

Classical electronic structure methods (aside from exact diagonalization) are heuristic algorithms. Much like quantum algorithms, they implicitly or explicitly start from an initial state. In chemical applications, this can be viewed as a product state or set of product states: for tensor network algorithms, such as the density matrix renormalization group (when not considering topological order) this is the set of D=1 states (specified by the underlying basis) which connect to the space of slightly entangled states with D>1 that the algorithm naturally explores. In coupled cluster methods, this is the initial reference state to which excitations are applied. Although many classical heuristics are exact with exponential effort, e.g. by increasing the bond dimension D in a tensor network or excitation level in coupled cluster theory, in practical computational chemistry, classical heuristics are used with the assumption that so long as the initial state is chosen appropriately, they will converge rapidly to the true ground-state without exponential effort. I analyze this heuristic working assumption in Ref. (Chan 2024) where I name it the classical heuristic cost conjecture. However, finding the good classical initial state is an NP hard problem, and this is often the crux of where the challenge in simulation actually lies.

In FeMo-co, unlike in simpler molecules, it is not at all obvious what product state to start from. To address this, in Ref. (Zhai et al. 2026), we devised an enumeration and filtering protocol. The relevant manifold arises from the orbital and spin degrees of freedom of the Fe ions: which Fe orbitals are occupied, by how many electrons, and with which spins. One technical point is that the resulting product states do not generally conserve the global S^2 spin symmetry. However, as recognized by Anderson decades ago, for magnetic order in large systems, the eigenstates can be chosen to break S^2 symmetry due to the tower of symmetry preserving eigenstates at an energy scale of \mathrm{const} \times V^{-1} (where V is the system volume). For a finite energy resolution \Delta \gg \mathrm{const} \times V^{-1} we can equally use a broken symmetry description of the states, an example of the fragility of entanglement effects in physical systems.

Because applying the highest level of classical approximation to all enumerated product states was far too expensive, we used a filtering funnel, where product states were ranked at different levels of theory, passing promising candidates to higher levels of classical computation. In the end, the final most accurate calculations were performed on only 3 candidates, which we deduced to all be essentially degenerate to within chemical accuracy.

There are other important technical details in Ref. (Zhai et al. 2026) which I have not mentioned: the use of unrestricted orbitals, the systematic extrapolations to obtain the final energies and estimated errors, and the benchmarking required to be confident about the protocol. However, recognizing that the FeMo-co ground-state problem could be reduced essentially to a ranking problem was the essence of what made the estimate possible.

Implications of the classical solution for chemistry

From a chemical and biochemical perspective, computing the ground-state energy of a model to some specified accuracy – even chemical accuracy – is a highly artificial target. Most chemical calculations that have an impact on our understanding never achieve or even target chemical accuracy in the total energy. In addition, chemistry does not depend on the the total energy, but the relative energy of different chemical configurations, which typically differ only by O(1) changes in the bonding and chemistry.

The main take-home from our work then is that there is nothing especially mysterious about FeMo-co’s electronic structure. The story of the FeMo-co ground-state is not one of multiconfigurational electronic structure (i.e. where the states are not at all close to product states), but one of multiple configurations (i.e. many competing product states). Indeed, this is basically how nitrogenase chemists have long reasoned about the electronic structure of iron-sulfur clusters and FeMo-co (Lovell et al. 2001; Yamaguchi et al. 1990). Our work thus now provides extensive and rigorous numerical support for this picture.

Because of this simplicity, the full richness of classical quantum chemistry methods can now be brought to bear on FeMo-co electronic structure beyond the LLDUC model. Assuming the model already captures the qualitative complexity of the cluster’s electronic structure, we expect such investigations to provide quantitative corrections to the picture we have obtained. We took some initial steps to confirm this in our manuscript, considering larger orbital spaces, the effect of protein fluctuations, and the interpretation of certain spectroscopies. In the future, connecting these simulations to more spectroscopic measurements will be an exciting possibility. In addition, now that the electronic structure is on a conceptually sound footing, we have a foundation to support the central question of resolving the reaction mechanism. This opens up a whole new set of scientific challenges associated with observing reactions on extremely slow timescales.

Implications of the classical solution for quantum computing in chemistry

Because of the success of classical heuristic methods for this problem, one may naturally wonder what these results mean for the application of quantum computers in chemistry. Here I address some commonly asked questions.

Is the classical simulation of the LLDUC model a ‘last hurrah’ for classical methods?

I have seen the analogy drawn between the FeMo-co result and the classical tensor network simulations for random circuit sampling experiments. In that case, while the famous Google Sycamore experiment (Arute et al. 2019) could be replicated by classical tensor network simulations (Gray and Kourtis 2021), subsequent improvements in quantum processors, soon led to random circuit sampling experiments outpacing the capabilities of classical simulations.

However, the situation here is quite different. There is strong evidence that generating samples from a random quantum circuit (without noise) is actually exponentially hard to do using a classical algorithm, and indeed, the classical simulations used for the task were (mostly) brute force simulations with exponential cost in circuit size. In contrast, the theoretical support for exponential quantum advantage in the FeMo-co problem is much weaker, and as an empirical fact, most of the methods used in the FeMo-co simulation (namely the coupled cluster methods for a given excitation level) are polynomial cost algorithms. Since a similar simulation strategy has also been successfully applied across the series of 2, 4, and 8 metal iron-sulfur complexes (Sharma et al. 2014; Li et al. 2019; Zhai et al. 2023, 2026), we have no reason to expect a radically different situation if we consider larger analogous complexes in this series.

And in any case, chemistry does not provide an endless scaling of problem size; FeMo-co is the largest enzyme cofactor in terms of the number of transition metals. Materials simulations provide a setting to scale the problem size, but one still faces the question as to whether the relevant states observed are truly that complicated classically. For example, classical simulations of the ground-state orders of the 2D Hubbard model currently show no exponential increase in difficulty when going to larger system sizes (Chen et al. 2025; Liu et al. 2025).

Has the availability of a classical strategy for FeMo-co changed your enthusiasm for quantum computers in chemistry?

Again, my answer is no. There is an entire community of nitrogenase scientists: experimental spectroscopists, synthetic chemists, and of course computational chemists, who are working to map out the reaction mechanism, none of whose research is predicated on using a quantum computer. Personally, I have never thought that to understand nitrogenase we would first have to build a quantum computer, otherwise I would not work on the problem!

At the same time, any computational tool brings new capabilities that will be useful. Quantum algorithms come with theoretical guarantees; for example, so long as the initial state is well prepared, we know the error in the energy that we measure from a quantum algorithm, which is more reliable than the classical estimates of error we obtain from extrapolations. Similarly, initial state preparation for a quantum computer, even for classically tractable problems, is probably easier than solving the entire problem classically, since only a ‘rough’ guess is needed. And finally, a polynomial or even constant factor speedup is exciting, so long as the speedup is large enough!

Thus, I am in fact excited to see quantum computers applied to this problem, I am just not waiting for them to be built first.

How should one think about past work on quantum algorithms that has used FeMo-co as a target?

FeMo-co was amongst the earliest examples of a chemical problem for which a case for quantum advantage was made. For this reason, it is overrepresented in the literature of quantum computing for chemistry. Should fully fault-tolerant quantum computers be available, they will naturally be applied to a wider set of systems (Chan 2024; Babbush et al. 2025).

Also, one must recognize that the availability of a single concrete optimization target has led to undeniable advances in quantum algorithms for quantum chemistry. In most cases, prior work to improve quantum resources estimates for FeMo-co involve techniques that apply to other systems as well. Thus, there’s no need to throw away those papers!

What are some lessons and conclusions to draw?

The first is obviously that, just because something has not been solved, or appears hard to solve classically, does not mean it is the best problem to choose for a quantum computer. The classical solution strategy for FeMo-co essentially involved a complicated classical state preparation problem, which is a shared challenge with ground-state estimation algorithms in quantum computers, and thus not perhaps an optimal choice of problem.

My second main conclusion is that since classical solutions in complex problems are possible because they use some understanding of the problem, for quantum algorithms to have maximum impact, they should use the same knowledge. In fact most chemistry is not about truly mysterious quantum systems, but more about ordinary quantum matter where we know roughly what is going on, but where detailed simulations are still required. If quantum computing algorithms can target this ‘mundane’ regime, they will have maximum impact on chemistry as it is practiced today. In recent work, we have taken some steps in this direction by proposing quantum algorithms for electronic structure that work within the same heuristic framework as most current quantum chemistry methods (Chen and Chan 2025).

Finally, I wish to emphasize that, from the perspective of understanding nitrogenase, and maximising societal impact, the choice of computational algorithm and hardware to solve the problem is irrelevant. The fact that FeMo-co electronic structure is not so mysterious is an enormously positive thing, as it means that making progress on the larger problem of the mechanism using computation no longer seems so impossible. I have seen some of the brightest minds in the world helping to advance quantum algorithms for this problem. If any of this brainpower can be devoted to the chemical question itself, I believe we can be very optimistic about the future solution of the nitrogenase problem.

References

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Babbush, Ryan, Robbie King, Sergio Boixo, et al. 2025. “The Grand Challenge of Quantum Applications.” arXiv Preprint arXiv:2511.09124.
Badding, Edward D, Suppachai Srisantitham, Dmitriy A Lukoyanov, Brian M Hoffman, and Daniel LM Suess. 2023. “Connecting the Geometric and Electronic Structures of the Nitrogenase Iron–Molybdenum Cofactor Through Site-Selective 57Fe Labelling.” Nature Chemistry 15 (5): 658–65.
Berry, Dominic W, Craig Gidney, Mario Motta, Jarrod R McClean, and Ryan Babbush. 2019. “Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization.” Quantum 3: 208.
Chan, Garnet Kin-Lic. 2024. “Spiers Memorial Lecture: Quantum Chemistry, Classical Heuristics, and Quantum Advantage.” Faraday Discussions 254: 11–52.
Chen, Ao, Zhou-Quan Wan, Anirvan Sengupta, Antoine Georges, and Christopher Roth. 2025. “Neural Network-Augmented Pfaffian Wave-Functions for Scalable Simulations of Interacting Fermions.” arXiv Preprint arXiv:2507.10705.
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Nicole’s guide to navigating faculty-position offers

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It’s happening. 

Your inbox registers an email from the chair of a faculty-hiring committee. With trembling fingers, you click on the message. “We were very impressed…we’re delighted to offer…” Months of labor, soul-searching, strain, and anxiety give way to jubilation. You hug your partner/roommate/mom/dog; throw an impromptu dance party; and forward the email, prefaced with five exclamation points, to your mentor.1 

As your heart rate returns to a level less likely to alarm a cardiologist, a new source of uncertainty puckers your brow. You’ve received an offer of a faculty position. What happens now? How should you proceed?

This article will address those questions. It follows my guide to faculty interviews, which follows my guide to writing research statements. Like the former guide, this one pertains most to theoretical physicists seeking assistant professorships at R1-level North American universities. Yet all the advice pertains to candidates outside this pool.

The institution will bring you (and, if relevant, your partner) over for a visit. Yes, you visited to interview; but you’re now visiting for another purpose. Assess whether you and your family could flourish if you accepted the offer. Which neighborhoods might you like to live in? Could you tolerate the commute to campus? Vide infra for more questions to keep in mind.

Politely notify the other hiring committees that interviewed you and that are still considering your application. You’ll do the other committees a kindness: their chances of hiring you have narrowed. If they wish to lure you, they’ll need to act quickly. The notice may bump you up in their priority lists. Did the first institution request that you decide about its offer by some deadline? If so, notify the other institutions.

Gather all the information you need. The department may offer to put you in touch with faculty members, deans, and more. Request more connections if necessary. Approach each conversation with a list of questions, and take notes. How will the tenure process unfold? How do early-career faculty members characterize their experiences with it? To what extent does the department shield early-career faculty from administrative duties (serving on committees)? How do the institutions’ policies address parental leave and elder care? If you have a child as an assistant professor, will your tenure clock pause for a year (will you be able to build your credentials for an extra year before applying for tenure)? In which neighborhoods should you search for a house?  

List your priorities. Rank them. Measure each offer against each criterion. Here are example priorities that you might wish to include:

  • Salary
  • Startup package
  • Type of environment: Do you want to live in a city, in the suburbs, or in the country? Do you drive, or would you learn to drive?
  • Length of commute
  • Geographical location: Do you prefer to live near family? 
  • Proximity, and means of transportation, to an airport: You might commute to and from that airport many times to participate in conferences, present seminars, etc. How much time and exhaustion would the experience cost?
  • Local school system: If you have or might have children, where would they learn?
  • Partner’s needs: Do you have a partner who would need to find a job near yours?
  • Proximity of faculty with whom you could collaborate
  • Courtesy positions in other departments: Suppose you’re a physicist who studies quantum computation. You might want to recruit students from the computer-science or math department occasionally. Could you? Would you need a courtesy position in the other department? A courtesy appointment offers you limited privileges at the cost of limited responsibilities: you probably won’t be able to vote in the other department’s faculty meetings. On the other hand, you probably won’t need to spend time on those faculty meetings.
  • Academic quality of undergraduate/graduate population
  • Presence of an institute/center dedicated to your specialization
  • Lab space: location, size, quality, renovations available, how soon and quickly the university would undertake those renovations
  • Help with finding housing: Some universities have apartments that new faculty can rent for a year or two. Other universities offer real-estate-agent services or help faculty obtain mortgages (*cough* San Francisco Bay area *cough*). 
  • Administrative assistance for you and your research group
  • Protection from onerous service to the department until you reach tenure
  • Teaching relief granted en route to tenure: At some universities, a new faculty member can avoid teaching their usual course load during one or two semesters. Such relief frees you to buff up your research program while pursuing tenure.
  • Deferral: Deferring an offer, you postpone the time at which you take up the new mantle. When I accepted a permanent position, I was completing year two of a three-year postdoctoral fellowship. I wanted to complete the final year before assuming my new role: I was still finishing projects with the community to which I belonged, and I wanted to continue deepening my ties with that community. Also, I enjoyed undertaking research without the distraction of a primary investigator’s administrative responsibilities. Other people defer their start dates for other reasons. For example, a partner might need time to fulfill a contract where they live and work. In my experience, people tend to defer PI positions for approximately twelve months, give or take six months. Some institutions don’t offer deferrals, though.

Identify everything you’ll need in a startup package. A startup package helps cover your research program’s costs until you’ve won your first grants. Multiple organizations within a university might contribute to a startup package—for example, a department and an institute that cuts across departments. The hiring committee might propose a startup package to you, or the committee might ask you what you need. Either way, you can (and should) negotiate the package. 

View the negotiation in terms of the question “What do I need to succeed?” List every item, and estimate its cost. Don’t skimp on rigor: estimate prices to the single-dollar level of precision. Such precision helps demonstrate the thoroughness of the research behind your list—helps demonstrate that you need every dollar you seek. 

Request more funding than you believe you’ll need, because you will need more than you believe. Build the breathing room into your estimates. For example, assume that your academic visitors will fly from across the country or across the world. Assume that you’ll fly such distances to present talks. Estimating how much you’ll pay a student or postdoc throughout the next few years? Don’t forget that the university might raise salaries and benefits under a cost-of-living adjustment (COLA) every year. COLAs fluctuate across years, so assume you’ll face steep ones. 

Here are examples of items that a startup package can include:

  • Summer salary: Your institution won’t pay your salary during the summer; you’ll need to fund yourself through grants. A startup package can cover the initial summers.
  • Lab equipment
  • Computers and tablets for you and your group members: Don’t forget protective cases, AppleCare or a non-Apple equivalent, implements for writing on tablets, external mice, and external monitors. Check whether your department or institute has spare mice or monitors that you can requisition.
  • Other computational resources: Does the department have a computational cluster that you intend to use? Do you need to access a national lab’s supercomputer or a quantum computer available on the cloud? How much will you pay per unit time and memory?
  • Postdoc costs: These costs include a salary, benefits, and the cost of moving to your institution. The salary will increase from year to year if the institution implements a COLA. The benefits include healthcare, dental care, and the like. Administrators might call benefits “fringe,” as I discovered after considerable confusion. 
  • Graduate-student costs: These costs include a research assistantship, benefits, and possibly tuition. The salary might increase as a student progresses through the stages of their PhD, particularly once they achieve candidacy. Their need for tuition might change, too. Check whether domestic students cost more than international students, and budget for international students.
  • Undergraduate researchers: Do you plan to employ an undergraduate during the summer? Throughout the academic year?2 
  • Travel for yourself: Budget several trips per year for yourself. You’ll need to spread the word about your research and to grow your network en route to tenure.
  • Travel for your postdocs and students: A mentor shared that she covers one conference per year per group member. You might want to budget also for a seminar or two per group member per year.
  • Visitors: Visitors can boost your research program. Budget for week-long visits if your institution can accommodate them.

Negotiate. Even if your dream school has offered you your dream job. Even if you receive only one offer. You might still garner resources that can help your research program and family to thrive. Don’t feel shy, sheepish, or ashamed to negotiate. If you remain polite and considerate, you won’t offend anyone. Besides, the hiring committee, department chair, and dean expect you to negotiate. The department chair might even hope that you do so; vide infra. 

When I was a PhD student, Caltech offered a workshop about negotiation to women grad students. The workshop helped participants build skills, knowledge, and self-assurance that would benefit us when we negotiated contracts. I recommend attending a workshop, taking a course, reading a book, or watching videos about negotiation. Contact your institution’s professional-development office about opportunities and suggestions. If you’re reading this blog post before applying for jobs—any jobs—start now.

What can you negotiate for? Many of the items on your list of priorities. Certain institutions might lack the freedom to negotiate certain items, though. For example, a union might determine salaries. Don’t let such a discovery discourage you; explore the options thoroughly.

View the department chair as an ally. The department chair negotiates on your behalf with administrators higher up in the university hierarchy, such as deans. The chair aims to garner as many resources as possible for you—and, by extension, for their department. Explain to the department chair (or to the committee chair who might explain to the department chair) what you need and why you need it, to help strengthen their argument.

As soon as you know you’ll decline an offer, decline it politely. Your notification will free the committee to attract another candidate. Imagine you’re Candidate #2 on the priority list. Wouldn’t you want the current offer recipient to decline their offer as soon as their conscience allows? Now, imagine you’re the hiring-committee chair. You’re worried that Candidate #1 will decline—and, by the time they decline, other institutions will have snapped up the other top candidates. As Candidate #1, demonstrate toward the committee chair and toward Candidate #2 the consideration that you’d value if in their shoes.

Savor the moment. You’ve just survived the faculty-application process, one of the most stressful periods of your life. The faculty life is no walk in the park, either. Nor will you necessarily sleep soundly between the receipt of your first offer and your signing of a contract. The prospect of more offers could leave you in limbo. If you receive multiple offers, choosing between them—choosing the course of your and your family’s life—may stress you as much as applying did. So remember to feel grateful for the source of your anxiety. Give yourself credit for your accomplishment. 

Congratulations!

1Please do! They’ll want to celebrate with you.

2I recommend targeting undergrads who’ll work with you for more than a summer. Training an undergrad takes nearly a summer; you and the student will benefit from having time to take advantage of that training.