Olivia
Online
Transmon Qubit
@freefallframe
Physics enthusiast. Currently reading Scott Aaronson’s Quantum Information lecture notes and Anthony Zee’s Group Theory in a Nutshell for Physicists.
Joined May 2020
2.4K Following
296 Followers
489 Posts
Much of the puzzling nature of quantum mechanics can be understood through one simple but profound fact: particles of the same kind are truly identical
1/9
A remarkable paper appeared on arXiv tonight by Thomas Bloom, Will Sawin, Carl Schildkraut and Dmitrii Zhelezov. In this paper, they prove that there exists c>0 and arbitrarily large finite sets A of real numbers such that max(|A+A|,|AA|)≤|A|^{2-c}. This disproves the well-known sum-product conjecture over the real numbers. The sum-product conjecture considers the two most basic operations: addition and multiplication. A+A is the set of all pairwise sums of two elements in A while AA is the set of all pairwise products of two elements in A. (1/5)
In 1980, two years before Feynman's famous Caltech lecture on Quantum Computing, a 43-year-old Soviet mathematician named Yuri Manin published a slim 128-page popular-science book called Вычислимое и невычислимое — Computable and Noncomputable — through the Moscow publishing house Sov. Radio. Manin was not a computer scientist. He was already one of the great algebraic geometers of his generation: a Lenin Prize laureate (1967), professor of algebra at Moscow State University, principal researcher at the Steklov Mathematical Institute, the mathematician behind the Gauss–Manin connection and the Mordell conjecture for function fields. He had been forbidden from foreign travel since 1968. The book was written in Russian, never officially translated for nearly thirty years, and its argument about quantum computation took up barely three pages of the introduction...
https://t.co/pOZ0460JWB
What's striking about Manin's framing — and what got almost entirely lost when the Western quantum computing canon formed around Benioff, Feynman, and Deutsch — is the direction of the argument.
Feynman's 1982 case for quantum computers was pragmatic and engineering-flavored: classical machines can't efficiently simulate quantum systems, therefore we should build quantum machines that can.
Manin came at it from the opposite end. He looked at molecular biology — at protein synthesis on messenger RNA, at the absurd information density and energetic efficiency with which living cells perform what looks structurally like Turing-machine computation — and concluded that nature had already solved the problem. Classical physics, he argued, simply cannot account for what biology does. The mathematical theory of quantum automata must already be implicit in the substrate of life. Engineering quantum computers wasn't the goal; it was the obvious downstream consequence of taking biology's existence-proof seriously.
That places Manin in a different intellectual lineage than the one quantum computing eventually inherited. He was downstream of Schrödinger's What Is Life? (1944) and the broader Soviet tradition of treating life as a physical system whose laws had not yet been written — Vernadsky, Lyapunov, the cybernetics revival under Berg and Glushkov.
The West built quantum computing as an engineering discipline of qubits-as-fabricated-systems, and pushed biology off into a separate and often-dismissed sub-field called "quantum biology."
Forty-five years later, with the work emerging on microtubules, tryptophan networks, ordered water, and coherent processes in neural lattices, the field is, in a real sense, finally catching up to its own actual origin.
The translation below is from pages 13–15 of the introduction.
On the inefficiency of computing devices
Molecular biology provides examples of the behavior of natural (not human-engineered) systems which we are forced to describe in terms close to those accepted in the theory of discrete automata. The figure below depicts the scheme of protein synthesis on messenger RNA: it closely resembles the depiction of a Turing machine copying information from one tape to another.
Classical continuous systems governed by differential equations can imitate discrete automata only when their phase space has an exceptionally complex structure — an abundance of stability regions separated by low energy barriers. Loading a program carves out a sophisticated system of passages through these barriers, predetermining the motion of the phase trajectory through this labyrinth. As a physical system, the computing device must be highly unstable, since an error of a single character in the program generally leads to an entirely different trajectory. Yet the computational process itself must be exceptionally stable — that is, spontaneous errors (transitions of the trajectory across a barrier that should remain closed, as a result of fluctuations) must have very low probability. It is well known that these requirements — combined with slowness of operation and the exponential growth of dissipated energy as complexity increases — erected the barrier that halted the development of mechanical computers.
[Citing Poplavsky's 1975 paper on thermodynamic models of information processes:] A genuinely instructive calculation can be found there: the quantum-mechanical description of the methane molecule by the lattice method requires computation at 10⁴² points. If we assume only 10 elementary operations are performed at each point, and suppose all computations are carried out at ultra-low temperature, then even so the calculation of the methane molecule would require expending energy roughly equal to that produced on Earth over a century.
On quantum automata
It is possible that for a better understanding of such phenomena a mathematical theory of quantum automata is lacking. The mathematical model of such objects must exhibit highly unusual properties compared with deterministic processes. The reason is that the capacity of the quantum state space is dramatically greater: where in the classical case there are N discrete states, in quantum theory — which permits their superposition — the state space lies in Cᴺ. When classical systems are combined, their state-counts N₁ and N₂ simply multiply; in the quantum case one obtains C^(N₁·N₂).
These rough estimates show that systems exhibiting quantum behavior are potentially far more complex than their classical counterparts. For example, since the system has no unique decomposition into parts, the state of a quantum automaton may be regarded in many different ways as states of entirely different virtual classical automata.
In carrying out such a program, the first difficulty will be finding the right balance between mathematical and physical principles. The quantum automaton must be abstract: its mathematical model should use only the most general quantum principles, without prejudging physical implementations. Then the model of evolution is a unitary rotation in finite-dimensional Hilbert space, and the virtual decomposition into subsystems corresponds to the tensor-product decomposition of that space. Somewhere in this picture the place of interactions — traditionally described by Hermitian operators and probabilities — must still be found.
Notes on this translation:
The C in "Cᴺ" is the field of complex numbers; Cᴺ is N-dimensional complex Hilbert space. C^(N₁·N₂) reflects the tensor product H₁ ⊗ H₂ — the structure that gives quantum systems their entanglement-driven computational advantage.
The Poplavsky reference is to R.P. Poplavsky, "Thermodynamical models of information processing," Uspekhi Fizicheskikh Nauk 115:3 (1975), 465–501.
![JosephJacks_'s tweet photo. In 1980, two years before Feynman's famous Caltech lecture on Quantum Computing, a 43-year-old Soviet mathematician named Yuri Manin published a slim 128-page popular-science book called Вычислимое и невычислимое — Computable and Noncomputable — through the Moscow publishing house Sov. Radio. Manin was not a computer scientist. He was already one of the great algebraic geometers of his generation: a Lenin Prize laureate (1967), professor of algebra at Moscow State University, principal researcher at the Steklov Mathematical Institute, the mathematician behind the Gauss–Manin connection and the Mordell conjecture for function fields. He had been forbidden from foreign travel since 1968. The book was written in Russian, never officially translated for nearly thirty years, and its argument about quantum computation took up barely three pages of the introduction...
https://t.co/pOZ0460JWB
What's striking about Manin's framing — and what got almost entirely lost when the Western quantum computing canon formed around Benioff, Feynman, and Deutsch — is the direction of the argument.
Feynman's 1982 case for quantum computers was pragmatic and engineering-flavored: classical machines can't efficiently simulate quantum systems, therefore we should build quantum machines that can.
Manin came at it from the opposite end. He looked at molecular biology — at protein synthesis on messenger RNA, at the absurd information density and energetic efficiency with which living cells perform what looks structurally like Turing-machine computation — and concluded that nature had already solved the problem. Classical physics, he argued, simply cannot account for what biology does. The mathematical theory of quantum automata must already be implicit in the substrate of life. Engineering quantum computers wasn't the goal; it was the obvious downstream consequence of taking biology's existence-proof seriously.
That places Manin in a different intellectual lineage than the one quantum computing eventually inherited. He was downstream of Schrödinger's What Is Life? (1944) and the broader Soviet tradition of treating life as a physical system whose laws had not yet been written — Vernadsky, Lyapunov, the cybernetics revival under Berg and Glushkov.
The West built quantum computing as an engineering discipline of qubits-as-fabricated-systems, and pushed biology off into a separate and often-dismissed sub-field called "quantum biology."
Forty-five years later, with the work emerging on microtubules, tryptophan networks, ordered water, and coherent processes in neural lattices, the field is, in a real sense, finally catching up to its own actual origin.
The translation below is from pages 13–15 of the introduction.
On the inefficiency of computing devices
Molecular biology provides examples of the behavior of natural (not human-engineered) systems which we are forced to describe in terms close to those accepted in the theory of discrete automata. The figure below depicts the scheme of protein synthesis on messenger RNA: it closely resembles the depiction of a Turing machine copying information from one tape to another.
Classical continuous systems governed by differential equations can imitate discrete automata only when their phase space has an exceptionally complex structure — an abundance of stability regions separated by low energy barriers. Loading a program carves out a sophisticated system of passages through these barriers, predetermining the motion of the phase trajectory through this labyrinth. As a physical system, the computing device must be highly unstable, since an error of a single character in the program generally leads to an entirely different trajectory. Yet the computational process itself must be exceptionally stable — that is, spontaneous errors (transitions of the trajectory across a barrier that should remain closed, as a result of fluctuations) must have very low probability. It is well known that these requirements — combined with slowness of operation and the exponential growth of dissipated energy as complexity increases — erected the barrier that halted the development of mechanical computers.
[Citing Poplavsky's 1975 paper on thermodynamic models of information processes:] A genuinely instructive calculation can be found there: the quantum-mechanical description of the methane molecule by the lattice method requires computation at 10⁴² points. If we assume only 10 elementary operations are performed at each point, and suppose all computations are carried out at ultra-low temperature, then even so the calculation of the methane molecule would require expending energy roughly equal to that produced on Earth over a century.
On quantum automata
It is possible that for a better understanding of such phenomena a mathematical theory of quantum automata is lacking. The mathematical model of such objects must exhibit highly unusual properties compared with deterministic processes. The reason is that the capacity of the quantum state space is dramatically greater: where in the classical case there are N discrete states, in quantum theory — which permits their superposition — the state space lies in Cᴺ. When classical systems are combined, their state-counts N₁ and N₂ simply multiply; in the quantum case one obtains C^(N₁·N₂).
These rough estimates show that systems exhibiting quantum behavior are potentially far more complex than their classical counterparts. For example, since the system has no unique decomposition into parts, the state of a quantum automaton may be regarded in many different ways as states of entirely different virtual classical automata.
In carrying out such a program, the first difficulty will be finding the right balance between mathematical and physical principles. The quantum automaton must be abstract: its mathematical model should use only the most general quantum principles, without prejudging physical implementations. Then the model of evolution is a unitary rotation in finite-dimensional Hilbert space, and the virtual decomposition into subsystems corresponds to the tensor-product decomposition of that space. Somewhere in this picture the place of interactions — traditionally described by Hermitian operators and probabilities — must still be found.
Notes on this translation:
The C in "Cᴺ" is the field of complex numbers; Cᴺ is N-dimensional complex Hilbert space. C^(N₁·N₂) reflects the tensor product H₁ ⊗ H₂ — the structure that gives quantum systems their entanglement-driven computational advantage.
The Poplavsky reference is to R.P. Poplavsky, "Thermodynamical models of information processing," Uspekhi Fizicheskikh Nauk 115:3 (1975), 465–501.](https://pbs.twimg.com/media/HHCIl1ib0AAPeMD.jpg)
Next in who after the Ramanujan Series? In Nov 2025, the universe became a little less symmetrical. A 90 yr old woman passed away in Chicago, largely unnoticed by the country of her birth. She was the hidden architect of Quantum Symmetry, the woman who took the raw, chaotic numbers of Ramanujan’s city & forced them into the perfect geometric elegance of Group Theory. We lost the final living link to the golden age of Indian logic, & most of us did not even know she was holding the map. Bhama Srinivasan (1935-2025)
Born in 1935 in Madras, Bhama grew up in the same intellectual ecosystem that produced the Madras School of mathematicians. She completed her B.A. & MSc. from the University of Madras, where the legacy of Ramanujan was still a fresh, guiding force. Like many Indian titans of that era, she moved to the UK for higher studies. She earned her PhD in 1960 from the University of Manchester, working under J.A. Green, a world leader in group theory.
She taught at the University of British Columbia & Ramanujan Institute of Mathematics in Madras before eventually settling in the US, where she became a central figure in the American Mathematical Society (AMS).
BBhama Srinivasan’s core work was in the representation theory of finite groups, especially finite groups of Lie type (the finite analogues of Lie groups over finite fields). This area lets us take abstract symmetries like those in crystals/atoms/quantum systems & represent them concretely using matrices & characters.
She made important contributions to the modular representation theory of these groups & collaborated extensively with Paul Fong. Her research connected to the broader Deligne-Lusztig theory pioneered by George Lusztig, whose groundbreaking work on representations of finite groups of Lie type revolutionized the field & linked it to algebraic geometry & quantum groups.
Her work was so deep into the backend of logic that even her colleagues in other branches of science find it daunting. To her family, she is a pioneer; to the world, she is the woman who mapped the symmetry of the finite universe.
Bhama Srinivasan represents the Full Circle. She started in the city where Ramanujan was discovered, she mastered the abstract structures that Ramanujan hinted at, she became a global leader, yet lived with the same intellectual humility as the ghosts before her. She lived through 90 yrs of the most radical changes in human history, yet her focus on the Symmetry of Groups never wavered. #WhoAfterRamanujan
Karpathy told Dwarkesh that a 1 billion parameter model, trained on clean data, could hit the intelligence of today's 1.8 trillion parameter frontier.
That is a 1,800x compression claim. The math behind it is more defensible than it sounds.
When researchers at frontier labs look at random samples from their training corpus, they see stock ticker symbols, broken HTML, forum spam, autogenerated gibberish. Not Wikipedia. Not the Wall Street Journal. The actual pretraining dataset is mostly noise, and the model is burning parameters to vaguely remember all of it.
One estimate pegs Llama 3's information compression at 0.07 bits per token. Well-structured English carries around 1.5 bits per token of real information. The trillion-parameter model is holding a roughly 5% resolution image of the internet it trained on.
So when a lab ships a 1.8 trillion parameter model, the overwhelming majority of those weights are handling rough memorization. They are compression overhead for a noisy training set, taking up capacity that could be doing reasoning instead.
Karpathy's proposal is to separate the two. Build a cognitive core: a small model that contains only the algorithms for reasoning and problem-solving, stripped of encyclopedic memorization. Pair it with external memory the model queries when it needs a fact. A 1 billion parameter reasoner plus retrieval beats a 1.8 trillion parameter model trying to do both.
The data already supports this direction. GPT-4o runs at roughly 200 billion parameters and outperforms the original 1.8 trillion GPT-4. Inference costs for GPT-3.5 level performance fell 280x between 2022 and 2024, driven almost entirely by smaller, cleaner, better-architected models. The trend line is pointing where Karpathy says it should.
The real implication for anyone tracking the AI trade: data quality is the actual constraint. The companies winning the next phase will be the ones who figured out what to train on, and what to throw away.
We are thrilled to announce that Professor Emeritus and Nobel Laureate, David Gross received the 2026 Special Breakthrough Prize in Fundamental Physics.
Details here: https://t.co/k6Xrso5DDU
Evergreen.
This sentence.
In 1948, a 32-year-old at Bell Labs published a paper nobody fully understood.
Engineers found it too mathematical. Mathematicians found it too engineering-focused. One prominent mathematician reviewed it negatively.
That paper - "A Mathematical Theory of Communication", became the founding document of the digital age.
The man was Claude Shannon. Father of Information Theory.
At 21, he wrote the most important master's thesis of the 20th century.
Working at MIT on an early mechanical computer, Shannon noticed its relay switches had exactly two states - open or closed. He had just taken a philosophy course introducing Boolean algebra, which also operated on two values: true and false.
Nobody had ever connected these two things.
His 1937 thesis proved that Boolean algebra and electrical circuits are mathematically identical, and that any logical operation could be built from simple switches.
Howard Gardner called it "possibly the most important, and also the most famous, master's thesis of the century."
Every digital computer ever built traces back to this insight.
At 29, he proved that perfect encryption exists.
During WWII, Shannon worked on classified cryptography at Bell Labs. His work contributed to SIGSALY, the secure voice system used for confidential communications between Roosevelt and Churchill.
In a classified 1945 memorandum, he mathematically proved the one-time pad provides perfect secrecy, unbreakable not just computationally, but provably, permanently, against an adversary with infinite power.
When declassified in 1949, it transformed cryptography from an art into a science. It laid the foundations for DES, AES, and every modern encryption standard.
At 32, he defined what information is.
His 1948 paper introduced one equation:
H = −Σ p(x) log p(x)
Shannon entropy. The average uncertainty in a probability distribution. The minimum bits required to encode a message.
Three things followed:
> He defined the bit - the fundamental unit of all information. His colleague John Tukey coined the name.
> He proved the channel capacity theorem, every communication channel has a maximum rate of reliable transmission. You can approach it. You can never exceed it.
> He unified telegraph, telephone, and radio into a single mathematical framework for the first time.
Robert Lucky of Bell Labs called it the greatest work "in the annals of technological thought."
Where his equation lives in AI today:
Cross-entropy loss - the function training every classifier and language model, is derived directly from H. Decision tree splits use information gain, which is H applied to data. Perplexity, the standard LLM evaluation metric, is an exponentiation of cross-entropy.
Every time a neural network trains, Shannon's formula runs inside it.
He also built the first AI learning device.
In 1950, Shannon built Theseus, a mechanical mouse that navigated a maze through trial and error, learned the correct path, and repeated it perfectly. Mazin Gilbert of Bell Labs said: "Theseus inspired the whole field of AI."
That same year he published the first paper on programming a computer to play chess. He co-organized the 1956 Dartmouth Workshop, the founding event of AI as a field.
The man:
He rode a unicycle through Bell Labs hallways while juggling. He built a flame-throwing trumpet, a rocket-powered Frisbee, and Styrofoam shoes to walk on the lake behind his house.
He called his home Entropy House.
When asked what motivated him: "I was motivated by curiosity. Never by the desire for financial gain. I just wondered how things were put together."
In 1985, he appeared unexpectedly at a conference in Brighton. The crowd mobbed him for autographs. Persuaded to speak at the banquet, he talked briefly, then pulled three balls from his pockets and juggled instead.
One engineer said: "It was as if Newton had showed up at a physics conference."
He died in 2001 after a decade with Alzheimer's, the cruel irony of information slowly leaving the mind of the man who defined what information was.
Claude, the AI model, is named after Claude Shannon, the mathematician who laid the foundation for the digital world we rely on today.
Weinberg: “I found, to my amazement, teaching myself some physics and mathematics in high school, that you could do powerful things with this intellectual apparatus. I remember calculating the shape of the cables that hold up a suspension bridge. It's a catenary, and deriving the equation for a catenary, I thought, "Wow, I'm not building a bridge, but I can see why it takes the shape it does." That sense of the power of mathematical reasoning was intoxicating.”
Mathematics is very meditative—it heals a man by involving him (in his entirety) in a purposeful objective. If pursued relentlessly, it empowers you to dissociate yourself from life’s difficulties & direct your energy to constructivism; in doing so, you understand yourself better
Edward Witten on the mass gap in strong interactions. (An excerpt from his 2009 paper "The Problem of Gauge Theory", which is available on the net.) #physics #QuantumTheory
Charles Bennett and Gilles Brassard have been named the winners of the A.M. Turing Award, one of the highest honors in computing, for their work establishing the foundations of quantum information theory. The award comes with a $1 million prize. https://t.co/PFTEye35cu
You can understand more about how learning works from Ravi Vakils page for PhD students than 90% of writing on this topic
I remember this moment very vividly. Apologies in advance for the arrogant and self-indulgent personal anecdote...
I originally went to university with the intention of becoming a pure mathematician, and for the first couple of years this dream seemed to be pretty safe. (1/10)
Professor Terence Tao analyzes how Erdős Problem 1026 was solved "through an interesting combination of existing literature, online collaboration, and AI tools."
The conjecture was first proven in Lean using @HarmonicMath's Aristotle. Further patterns discovered via @GoogleDeepMind's AlphaEvolve.
"It was only through the combined efforts of all the contributors and their tools that all these key inputs were able to be assembled within 48 hours."
➡️ https://t.co/R2voaOM0ix
Congratulations to the finalists of the XPRIZE Quantum Applications competition. These 7 teams demonstrate the potential to pioneer quantum algorithms that can outperform classical computers and solve real-world problems. https://t.co/uuuFpmF33q
1/6 Cold 2D electron liquids in transverse magnetic fields so strong and fine-tuned that there are exactly an integer (generally: rational) number of flux quanta per electron exhibit a "topologically ordered" ground state where vortices are anyons. This is famous. But now...
My favorite books to self study pure math!!
When I was in undergrad, I spent a lot of time self-studying pure math. But I wasted a lot of time because I didn't have a roadmap.
So here's a list of my favorite books, videos, and problem sets that you can use to self-study many of the key topics in undergrad-level math: a 🧵 1/n
“It is in this continuous effort to articulate the inarticulable, to define what is yet unclear, that the particular dynamics of mathematical work (and perhaps as well all creative intellectual work) is perhaps found.”
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