The AI & Quantum Computing Chronicle

Modular forms are mathematical functions defined by extremely restrictive symmetries. This rigidity makes it possible to reconstruct their global behavior from a fundamental region and to transform complex problems into far more tractable structures. This is why they have played a decisive role in number theory, in the proof of Fermat’s Last Theorem, and in counting problems through generating functions. Their applications extend to quantum physics and string theory, where they help study states and interactions; graph theory, through the construction of expander graphs; and combinatorial problems, where they allow sequences to be encoded through generating functions and exact formulas to be obtained for their coefficients. From a perspective applied to portfolio management, which is not developed in the article, the most fertile idea would be this: a sufficiently strong structure can reduce global complexity, identify invariants, and make many local decisions governable without analyzing each one in isolation. Link https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/

Thermodynamic Computing What if thermal noise, rather than being a problem, could actually help us compute? This new form of computing harnesses natural energy fluctuations to explore possible solutions, moving through an “energy landscape” until it reaches more stable states. It follows a logic similar to the process that allows proteins to fold correctly inside a cell. Companies such as Normal Computing and Extropic are already testing circuits and chips based on this principle for AI, optimization, and simulation tasks. Unlike quantum computing, which relies on superposition and entanglement but requires highly delicate hardware and often cryogenic temperatures, thermodynamic computing could operate through simpler systems that take advantage of physical noise itself. Both approaches aim to overcome the limits of traditional computing, but thermodynamic computing may reach practical applications sooner because it faces fewer technical barriers. Its promise is as ambitious as it is elegant: to consume less energy, generate less heat, and turn disorder into a computational resource. https://www.quantamagazine.org/thermodynamic-computers-go-with-the-energy-flow-20260715/?utm_source=chatgpt.com

Large language models learn by building associations between words, encoded as relationships in a high-dimensional embedding space. Like Hebb's principle in neuroscience, correlated activity strengthens connections, bringing related concepts closer together. But association alone cannot tell a system when reality has moved beyond its model. Faced with something genuinely novel, it does not recognise the boundary of its knowledge. It interpolates. The question, then, is what makes a world model open rather than closed. Biological intelligence appears to rely on more than association alone. It continually assigns value: this matters, this is dangerous, this is surprising, this deserves to be learnt. Those valuation signals may do more than guide behaviour. They may be what allows a world model to recognise when it no longer fits reality, and to change accordingly. If that's true, then emotion is not the opposite of reason, nor merely an evolutionary add-on. It is part of the architecture that makes intelligence capable of extending itself beyond what it already knows. This essay develops that idea through Hebbian learning, reward prediction error, hallucination, black swan events, and the distinction between solving problems within an existing mathematical framework and creating an entirely new one. Intelligence may not just be developing a world model, but also recognising when the map is no longer adequate. If so, simply scaling a closed world model may never be enough. https://vzocca.substack.com/p/feeling-right-why-emotions-may-be

Knowing When to Stop Every decision is risky business. Selecting the best time to stop and act is crucial. When Microsoft prepares to introduce Word 2020, it must decide when to quit debugging and launch the product. When a hurricane veers toward Florida, the governor must call when it’s time to stop watching and start evacuating.   “While much of mathematics has roots that reach back millennia to Euclid and even earlier thinkers, the history of probability is far shorter. And its lineage is, well, a lot less refined. Girolamo Cardano’s famed 1564 manuscript De Ludo Aleae, one of the earliest writings on probability and not published until a century after he wrote it, primarily analyzed dice games. Although Galileo and other 17th-century scientists contributed to this enterprise, many credit the mathematical foundations of probability to an exchange of letters in 1654 between two famous French mathematicians, Blaise Pascal and Pierre de Fermat. They too were concerned with odds and dice throws—for example, whether it is wise to bet even money that a pair of sixes will occur in 24 rolls of two fair dice. Some insisted it was, but the true probability of a double six in 24 rolls is about 49.1 percent.” “The history of optimal-stopping problems, a subfield of probability theory, also begins with gambling. One of the earliest discoveries is credited to the eminent English mathematician Arthur Cayley of the University of Cambridge. In 1875, he found an optimal stopping strategy for purchasing lottery tickets. The wider practical applications became apparent gradually. During World War II, Abraham Wald and other mathematicians developed the field of statistical sequential analysis to aid military and industrial decision makers faced with strategic gambles involving massive amounts of men and material. Shortly after the war, Richard Bellman, an applied mathematician, invented dynamic programming to obtain optimal strategies for many other stopping problems. In the 1970s, the theory of optimal stopping emerged as a major tool in finance when Fischer Black and Myron Scholes discovered a pioneering formula for valuing stock options. That transformed the world’s financial markets and won Scholes and colleague Robert Merton the 1997 Nobel Prize in Economics. (Black had died by then.)” “The Black-Scholes formula is still the key to modern option pricing, and the optimal-stopping tools underlying it remain a vigorous area of research in academia and industry. But even elementary tools in the theory of optimal stopping offer powerful, practical and sometimes surprising solutions.”   Link https://www.americanscientist.org/article/knowing-when-to-stop

In the last few days we have read about how an AI disproved a conjecture that had stood since 1946 (https://www.scientificamerican.com/article/ai-just-solved-an-80-year-old-erdos-problem-and-mathematicians-are-amazed/). What they don't tell you is that It could not tell you whether its own proof was right. That gap is the whole story, and most of the coverage missed it. Recently an OpenAI model produced a counterexample to the Erdős unit distance conjecture, open for almost 80 years. Nine leading mathematicians verified it. The conjecture is refuted. A real result, and widely misread. Three things that rarely survive the headline: → It could not check its own work. Humans did that. The model had no way to tell this result apart from the confident, plausible-but-wrong proofs these systems produce all the time, and nobody has said how many attempts it took. → It invented no new mathematics. Every technique already existed; the originality was the combination. It overturned Erdős's answer while depending entirely on Erdős's question. → It is a disproof, not a solution. We now know the answer is not what Erdős expected. We still do not know what it is. The mathematicians who verified it call it a milestone. We may need to be more cautious. A genuine result? Yes. Intelligence? No. A very fast machine assembling known pieces and, this once, landing on something true: it may be closer to luck than to insight. https://vzocca.substack.com/p/a-discovery-by-accident

The Fourier Transform allows us to take a complex signal and decompose it into elementary frequencies. It turns something apparently “chaotic” into an interpretable structure of components. The Quanta article explains this through very clear examples: the ear separating sounds, heat spreading along a rod, JPEG compression, noise filtering, images as 2D functions, and the connection with quantum mechanics through position and momentum. But the deeper lesson goes beyond signals: Fourier teaches us that sometimes the problem is not in the data itself, but in the domain from which we observe it. In traditional Data Science, this is essential: it transforms temporal, spatial, or sequential data into representations where certain patterns become more separable, compressible, and interpretable. It applies to feature engineering, noise reduction, anomaly detection, images, audio, sensors, and more. In quantum computing and Machine Learning, Fourier appears naturally when we speak about amplitudes, waves, changes of basis, and alternative representations. In quantum mechanics, it connects position and momentum, directly relating to the uncertainty principle. And in portfolio management, the analogy is powerful: a portfolio can also look like a chaotic signal made of initiatives, risks, dependencies, value, debt, operational noise, and external shocks. Perhaps looking only at its “visible state” is not enough. Perhaps we need to decompose it into its dominant frequencies: value cycles, risk patterns, recurring frictions, strategic signals, and noise. Link https://www.quantamagazine.org/what-is-the-fourier-transform-20250903/

What is a decision? It seems trivial until you try to define it. Decision theory explains how we choose, but rarely what a decision is. Choosing is just the outcome. Ronald A. Howard defined it as an allocation of resources. But what truly drives a decision is uncertainty. If the future were known, we would not decide. Warren B. Powell goes further: a decision is a form of information. It defines how we respond to a world we do not fully control. The real challenge is not choosing, but identifying what decisions actually exist. In complex systems, many remain invisible. In AI and quantum systems, this becomes critical. It is no longer about choosing actions, but building structures that decide under uncertainty. Link https://castle.princeton.edu/makingdecisions/

This paper provides a complete, ground-up explanation of how Large Language Models (LLMs) work. A central section walks through the entire forward pass with concrete numbers: embedding lookup, Query/Key/Value matrices, the dot-product score matrix, softmax weights, weighted Value sums, layer-by-layer refinement, and the final word prediction. It explains how backpropagation adjusts every matrix in response to prediction errors. It also covers fine-tuning and RLHF in detail. The paper concludes by applying this understanding to explain precisely why LLMs fail at constraint satisfaction. https://vzocca.substack.com/p/how-large-language-models-work

In 1935, the Austrian physicist Erwin Schrödinger showed the absurdity of common interpretations of quantum mechanics with his famous cat-based thought experiment. The cat is put into a box with vial of poison, which will be released if a radioactive atom decays. If the box remains isolated from its environment, the atom exists in a superposition of both decayed and not-decayed, and until observed, the cat is an undefined state of both dead and alive. In the real world, objects eventually become too complex or interact too much to maintain a superposition, an idea known as decoherence. But there are also extensions to quantum mechanics, known as collapse theories, that suggest that beyond a certain point, a system will inevitably reduce to a classical state, even in isolation. These theories were picked by 4% of researchers as their favourite interpretation of quantum mechanics in a 2025 Nature survey. https://www.scientificamerican.com/article/quantum-physicists-just-supersized-schroedingers-cat/

The article explains how mathematicians have achieved an important breakthrough in a classic problem of Fourier analysis, one of the mathematical pillars for understanding waves and signals. More than half a century ago, Sarvadaman Chowla posed a puzzle about sums of waves (cosines): given a set of numbers, how low can the sum of their cosines go? While the maximum is clear (it is simply the size of the set), the minimum has proven extremely difficult to predict. That minimum is not a technical detail. It describes the worst possible case of destructive interference. It marks how far a wave system, classical or quantum, can self-cancel even in the absence of noise or external errors. In quantum mechanics, this result does not define a physical bound, but it can be read as a conceptual limit on superposition: how far amplitudes can cancel before any measurement takes place. It points to a boundary of internal cancellation, not caused by the environment, but by the very combination of possible states. In telecommunications, conceptually, it sets the extreme scenarios of signal cancellation. It does not explain when everything works well, but what is the worst that can happen even in a perfectly designed system. Link https://www.quantamagazine.org/networks-hold-the-key-to-a-decades-old-problem-about-waves-20260128/

A century ago, Erwin Schrödinger formulated an equation that describes how a quantum system evolves when it is not being observed. This equation, now known as the Schrödinger equation, remains one of the fundamental tools for understanding the quantum world. 🔹 The equation allows physicists to calculate the so called wave function, which does not describe a single outcome but rather all the possible states of a system. However, when a measurement is made, the wave function appears to “collapse” into a single result. Why this collapse happens is still one of the great mysteries of physics. 🔹 One hundred years later, physicists continue to explore new ideas, such as including the observer and the measuring instruments within the quantum framework itself. This leads to rethinking basic concepts like time and even the meaning of observation. 👉 In short, the Schrödinger equation is still essential, but its deepest implications continue to challenge our understanding of what it really means to describe quantum reality. It reminds us that no matter how dense the mathematics or how hard the physics, the technology is still in motion and its foundations remain unsettled. A sobering thought when bold industrial claims are made. Link https://www.scientificamerican.com/article/the-schroedinger-equation-just-turned-100-and-quantum-physicists-are-still/

This paper offers a broad perspective on how machine learning is being integrated into the business world, not from a deep technical standpoint but through its impact on management, innovation, and decision making. Through a large scale analysis of scientific publications, it shows that the focus is no longer only on algorithms, but on how organizations adopt these technologies within their real operational structures. Rather than serving as a practical guide, the work functions as a strategic warning: ML adoption should not be driven by trends or competitive pressure, but by gradual evaluations that consider social, labour, and regulatory implications. The underlying message feels timeless, integrating technology requires balance and judgement, avoiding both total rejection and impulsive adoption without a clear structural vision. https://www.nature.com/articles/s41599-025-04361-6#Fig10

For decades, reinforcement learning has been studied in artificial settings: simple tasks, discrete states, and clearly defined rewards. This paper starts from a direct critique of that approach: the real world does not work this way, and yet humans make surprisingly effective decisions in complex, noisy, and changing environments. Learning in the real world is not about optimization, but about exploiting structure. Humans do not learn by trying everything or by exhaustively maximizing rewards, as many classical RL models assume. In natural environments, that would be computationally impossible. Human decision-making is inseparable from context and embodiment. Naturalistic learning cannot be separated from the context in which it occurs. Unlike laboratory settings, the real world is continuous, partially observable, embodied, and driven by implicit rather than explicit goals. Link https://www.sciencedirect.com/science/article/pii/S1364661323002127

The article explains how, in mathematics, major problems are not tackled all at once, but by building a tower of implications: first simpler results, then progressively more complex ones. This “tower” means that a major conjecture, such as the Riemann Hypothesis, sits at the top, while lower levels consist of more accessible statements that would follow from it. Proving these lower-level results does not establish the main conjecture, but it strengthens the logical structure and often provides new techniques and perspectives. The Riemann Hypothesis, which is related to the distribution of prime numbers through the zeta function, is one of the great enigmas crowning many of these towers of ideas. The text also notes that some researchers have even constructed “higher floors,” introducing objects linked to hypothetical quantum systems that, if they existed, would imply the truth of the hypothesis. In summary: in mathematics, the hardest problems are not solved in a single stroke, but by patiently building a scaffold of related results that together help to understand the problem better and sometimes bring us closer to a proof. Link https://www.linkedin.com/pulse/solve-math-problem-build-tower-quanta-magazine-1xmge/?trackingId=I24XGkGDTSaPX5uLS%2FpyQw%3D%3D

From this channel, we wish you a 2026 full of challenges that sharpen our thinking, projects that demand the best of us, and experiments that reward depth, clarity, and discipline.

How diamonds are powering a new quantum revolution Exactly a century after German scientist Werner Heisenberg built a mathematical framework for explaining quantum physics, the world is launching into what scientists call a “second quantum revolution”. This is where the diamond — the hardest naturally occurring substance on Earth — comes in. It is resistant to vibration because of its rigid crystal lattice of carbon atoms, linked by strong chemical bonds. Most of the carbon atoms have properties intrinsic to their nuclei that make for a magnetically “quiet” environment for quantum effects to take place. https://tinyurl.com/2hdv652e

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Quantum Portfolio Management (QPM): Foundations in Motion This version consolidates the first formal foundations of Quantum Portfolio Management (QPM) and introduces an initial structural validation. The framework is instantiated as a modular symplectic decision architecture and evaluated through a seven-stage mini benchmark, designed to test coherence, stability, robustness to stochastic noise, multi-stakeholder composition, and scalability under controlled perturbations. QPM is not a product, an algorithm, or a ready-to-use solution. It is a formal architecture in development, shared openly through versioned preprints, whose purpose is to explore how coherence, rather than pointwise optimization, governs the stability or collapse of decisions in complex and multiscale environments (Based on PMI principles). 🔗 Official DOI (Zenodo): https://zenodo.org/records/17750878

The article describes a new mathematical result whose proof draws on ideas originating in string theory, even though the problem it resolves is purely mathematical. The work addresses a long-standing impasse in algebraic geometry: how to understand and classify certain highly complex geometric spaces for which classical methods no longer suffice. The key to the breakthrough does not lie in using physics directly, but in applying the concept of duality, originally developed in string theory, to reformulate the problem in an alternative representation where the underlying structure becomes accessible. In this “mirror” representation, calculations that are intractable in the original framework become manageable, and the equivalence between the two descriptions is then established rigorously. The result is considered brilliant because it opens an entirely new path forward, and baffling because its logic does not emerge from the usual mathematical tradition. The article illustrates how physical intuitions can inspire new formal architectures without introducing physics into the final result. https://www.quantamagazine.org/string-theory-inspires-a-brilliant-baffling-new-math-proof-20251212/

For decades we have assumed that quantum mechanics needs imaginary numbers. The famous i, which never appears in any real physical measurement, sits at the heart of Schrödinger’s equation and the structure of the theory. But is it truly essential or simply a useful mathematical choice? In 2021 a special kind of Bell-type experiment seemed to settle the matter. A version of quantum theory built only with real numbers could not reproduce certain observed results. The conclusion looked final: without i, quantum mechanics fails. Yet in 2025 several research teams showed that this is not the whole story. By modifying some basic assumptions that we usually take for granted, such as the standard rule for combining quantum systems, they managed to rebuild the entire theory without imaginary numbers and still obtain the same predictions as the complex version. The interesting part is that even though i disappears from the equations, its effects remain. We still see rotations, phases and interference. The real-number formulation behaves as if an imaginary component were hidden inside another mathematical structure. This raises a deep question. Does nature truly require complex numbers or do we prefer them because they make the theory clearer, more elegant and more intuitive? Even the authors of these new models admit that the standard complex formulation still feels the most natural. And that might be the real lesson: mathematics does not only describe physics. Sometimes it reveals the geometry that makes the physical world possible. https://www.quantamagazine.org/physicists-take-the-imaginary-numbers-out-of-quantum-mechanics-20251107/