Monthly Archives: April 2026

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.