Stefano Gogioso
@StefanoGogioso
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Mathematician and Computer Scientist.
United Kingdom
Joined August 2009
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Wavefunctions vs Quantum Fields
Wavefunctions are perhaps the most iconic ingredient in the Quantum Mechanical formalism. For a given particle, the wavefunction associates an "amplitude" to each position in space: this is a complex number, encoding both the probability¹ of
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Clifford Rotations
Even the simplest quantum system, a qubit, has an infinite set of states, and an infinite number of ways in which those states can be transformed. Rather conveniently, the possible states of a qubit have the geometry of a sphere, known as the Bloch sphere, and
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Spaces of quantum states
It is well known that the (pure) states of a qubit have the geometrical structure of a sphere, a representation typically known as the Bloch sphere. For quantum systems of higher dimensions, the state space is significantly harder to visualise
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The status of time in quantum mechanics has been controversial since the early days of the field [1-4].
In 1926-1928, Dirac, Heisenberg and Bohr considered time as a quantum observable, conjugate to energy and subject with it to various forms of the uncertainty principle. This
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Black Hole Mining
From a classical point of view, black holes are puzzling but somewhat boring objects: because of the no-hair theorem, an isolated black hole quickly converges to a stable configuration that depends only on its total mass M, total angular momentum J, and total
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The possibility of travelling back in time, and the paradoxes that arise in the attempt to fully flesh out what this would mean, have fascinated humans for centuries. In the mid 20th century, the discovery of relativistic spacetime models with Closed Timelike Curves (CTCs)
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Since I'm travelling today, just a follow-up to the topological characterisation of quantum state spaces that I posted yesterday. On Monday, I'll resume with a post on black hole mining. In the meantime, have a great weekend! 🥳
Specifically, the figure below shows more details
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One of the iconic phenomena in Quantum Theory is the existence of 2nd order interference effects: the pattern from two slits cannot be explained by the patterns from the individual slits. This, however, stops at the 2nd order: patterns from 3 or more slits can be explained by
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From the point of view of information theory, quantum systems are rather peculiar: on the one hand, an n-level quantum system can be in a continuum of definite states; on the other, basis measurements can only extract log2(n) bits of information, at the cost of forever erasing
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The slogan "If you think you understand quantum mechanics, you don't understand quantum mechanics" is condescending, and a plague to quantum science.
We understand quantum mechanics as well as—arguably, much better than—many other physical disciplines.
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No long physics post today (some deadlines to meet), but here is a depiction of the formal hyper-tensor structure of 3x3 matrices, where individual components are further allowed to have different types (e.g. different ranges of allowed values).
Physics posting will resume on
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New paper on the arXiv today! "Categorical Semantics for Feynman Diagrams" by Razin Shaikh and myself, at . A 🧵to describe what we've achieved. (1/25)
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I used to think that a programmer's hell would look like Malbolge, but I am now convinced that it must look like LaTeX.
Malbolge is a programming language so arcane that no-one actually uses it. LaTeX, on the other hand, works just well enough that, at least in mathematical
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Occasional reminder that there are two major topos-theoretic approaches to quantum theory from the late naughties, and that the less well-known one (by Heunen, Landsman and Spitters) is significantly nicer:
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Hot off the press: blog post by
@coecke
on Quantum in Pictures and our educational experiment 🥳
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If a quantum spider bites you, do you become a quantum superhero?⚛️🦸
If you're attending
@IEEEQuantumWeek
and want to acquire diagrammatic reasoning superpowers, come to our "Quantum Pictures in Practice" tutorial tomorrow! (Wed 1pm-4:30pm, Larch room, 3rd floor)
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Our work on Quantum Picturalism, presented earlier this year at QSEEC, is now on the arXiv! 🥳
We describe how diagrammatic methods can be used to democratise learning of quantum theory—including quantum information, computation and communication—by
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Periodic reminder that manifold immersions are a cursed area of mathematics: up to cobordism, every n-dimensional manifold immerses in a sphere whose dimension is 2n...
...minus the number of 1s that appear that appear in the binary expansion of n.
(╯°□°）╯︵ ┻━┻
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It seems that, after a decade of marauding as a quantum physicist, I'm now ready for the next step up the ladder: marauding as a quantum chemist.
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@davidad
Tensors and their contraction operation should be the foundation of linear algebra, at least for CS applications (AI, but also graphics, modelling, etc).
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The 24 Clifford rotations of a single qubit, shown as rotations of a cube. The little cube pictures in the Cayley graph show what the cube looks like from the front after the rotation. (From some of my old Quantum Computing teaching materials.)
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Quantum Circuits are like imperative programs: they give you a recipe to follow, but you need something else to perform the computation.
ZX Diagrams are like functional programs instead: they provide both the recipe and the substrate upon which the computation is performed.
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Hurray! My new book with
@coecke
, "Quantum in Pictures", is finally available on Amazon! 🥳🍻⚛🎨📚
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Today's entry for "Data Structures from Hell": the hierarchy of Dependent Hyper-tensor types.
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As the final Quantum Computing assignment for this year, I asked my students to reproduce the results of .
I feared it might be too abstract, but it turned out to be the best cohort I've ever had: a median mark of 75 and a max of 100. I'm so proud! 🥳
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I'll shortly be talking at
#QPL2023
about joint work with
@NicolaPinzani
on the fascinating interplay between causality, non-locality and contextuality.
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Randomness is relative. Erasure is an illusion. Entropy only exists because of coarse-graining.
Welcome, to the Church of the Larger Hilbert Space.
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Excited to announce that "The Topology of Causality" is now on the arXiv 🥳. In it,
@NicolaPinzani
and I extend the sheaf-theoretic framework for non-locality/contextuality by Abramsky and Brandenburger to arbitrary causal orders, static and dynamic.
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First copies of my first book! 🤯 "Quantum in Pictures", by
@coecke
and myself, is coming out soon 🥳🥳. Get ready to learn quantum the fun way! 🖌️🧮🕷️⚛️
It exists. First copy, gonna take a few more weeks before you people can buy it, but this is gonna be a game changer...
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If I had to pick a favourite picture from my PhD thesis, it would be this one. It shows, using Eilenberg-Moore algebras and the qudit ZX calculus, how a quantum system turns into a quantum clock when we impose an energy constraint on the larger clock which govers its dynamics.
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I am pleased to announce that the 19th International Conference on Quantum Physics and Logic (QPL 2022) will take place in Oxford, UK from the 27th of June to the 1st of July 2022.
For all information, see our conference website:
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Starting from a theory of global states and reversible transformations (read: the universal wavefunction and the unitaries acting upon it), it is possible to explain the emergence of local systems and processes, both reversible and irreversible, using the Purification Principle
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@smartereveryday
If I got what you're doing, it should be (C3+C4) for the cos and i(C3-C4) for the sin.
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All the necessary diagrams now out of the way, ready for VQE and QAOA! ⚛️🚀
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When I revisit some of my work after a while, occasionally I go through a cycle of:
1. Confusion: "How does this work, again?"
2. Disbelief: "Seriously, this can't possibly work."
3. Panic: "Damn, this is a big mistake, and it's been out there for months."
4. (Many hours go by,
I have come across a "It is straightforward to check..." which I have spent hours attempting to straightforwardly check
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GM! An animation of Bernstein-Vazirani using the ZX calculus, from my Wed tutorial at
@IEEEQuantumWeek
. Thank you
@IEEEQuantum
for organising this event, it was a blast 🤩⚛️
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Lowersets and Static Causality
To first approximation, static¹ no-signalling constraints can be modelled by imposing a causal order—mathematically, a preorder²—on a set of events.
A closer look, however, reveals that no-signalling constraints are actually captured by the
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It's great to finally see the VQLS (Variational Quantum Linear Solver) algorithm published in
@quantumjournal
!
For those who have never come across it, the VQLS is a variational quantum algorithm which attempts to solve large linear problems Ax=b where the matrix A is decomposed
Now in Quantum: Variational Quantum Linear Solver by Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles
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@davidad
The pseudoinverse is just so elegant: numerically stable, easily derived from the SVD, returns a preimage with least square error. It coincides with the inverse when the matrix is actually invertible, so barely any reason to teach straight inverses, tbh.
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A quantum computer scientist is a machine which turns pizza into explanations of measurement-based quantum computing.
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The pace and results at
@comma_ai
is. just. phenomenal. They are truly the standard bearer for getting cool shit done! 🤩🏎️
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The artists are going first, but programmers are coming soon. Most OS licences, even very permissive ones, require at the very least that the license itself be preserved. We're talking about GitHub-wide violations for both OpenAI and Microsoft, and it's gonna be a huge problem.
While Sam Altman is fearmongering over godlike AI that could doom humanity or create utopia, the adults are talking in the other room.
Artists are asking for 3 simple things.
Permission. Credit. Compensation.
#CreateDontScrape
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@d_feldman
@0xabad1dea
This sounds like the incipit for a cryptography paper that should totally be written. With all other generation parameters at hand, as well as the net running on one's machine, I'm confident it'd be possible to recover the password from the picture.
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@mattecapu
It is also a fact that a lot of caffè al bar in Italy tastes sour and horrible. North, centre, south, all the same: depends on the machine, the beans, and on the good taste of the barpeople.
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Excited to announce that "The Geometry of Causality" is now on the arXiv 🥳. In it,
@NicolaPinzani
and I characterise the geometric structure of causal polytopes for arbitrary causal orders and arbitrary measurement covers. Lots of examples included!
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Look at them spiders, looking slick while popping squares on a small b/w screen 🤩🕷️
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On March 18th, I'll be talking about diagrammatic languages for quantum computing at the Zen4Quantum meetup, based on my recent book with
@coecke
"Quantum in Pictures" ⚛️🎨📚.
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Love the open mapping theorem but don't appreciate your spheres doubling out of control?
Look no further than the Axiom of Dependent Choice!
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Q: Quantum computers: what are they good for?
A: For now, absolutely nothing.
A sober new position from the journal that once boldly titled "Quantum Supremacy" 🤣
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On Sep 20, I will be delivering a tutorial on diagrammatic methods for quantum computing at IEEE Quantum Week 2023, together with Lia Yeh, Thomas Cervoni and
@harnywang
. If you're attending the conference, come say hi!
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To conclude my day, one thousand stabilizer ZX spiders, seen from inside with a fisheye lens.
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When I moved to the 🇬🇧 in 2009, it was an example of cultural openness, economic opportunity, and academic excellence. Over 14 years of British life, I witnessed Tory governments relentlessly work to destroy it all, oppositions too weak-willed to stop them.
They. Must. All. Go.
Spanish woman deported from UK after returning from Christmas holiday
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Category theory: the mathematical field where boasting about "having epic strength" may well be a legit technical statement.
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The LLM/GPL argument is fairly simple: if a LLM can be made to produce verbatim copies of GPL-licensed code, it contains (some encoding of) the code, and hence a strict reading of the the GPL license dictates that the model itself must be released under GPL terms.
The GPL
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@bgavran3
Personally, I find it easiest to remember that ⊕ vs ⊗ in FVect arises from + vs × in Set, in the sense that f⊕g is freely defined on the disjoint union basis of K^{n+m}, while f⊗g is freely defined on the Cartesian product basis of K^{n×m}.
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@MichaelAlleged
@davidad
I think the reality is that there's just no time to teach everything, so it's important to choose the approach that is most relevant to CS today. Tensors are useful for all sorts of topical CS stuff: AI, cryptography, graphics, simulations, quantum, to name a few.
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TIL that
@hackmdio
supports using \𝚗𝚎𝚠𝚌𝚘𝚖𝚖𝚊𝚗𝚍 to define LaTeX macros. Total game changer for me.
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Four randomly sampled balanced affine partitions of the tesseract 𝔽₂⁴
(testing some work with
@fabgenovese
)
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@gunsnrosesgirl3
It's worth noting that touching sea anemones this way is safe out of water (for most species, at least), but not necessarily underwater: the tentacles can quickly extend upon stimulation and reach thinner areas of skin, where the venom is able to attack human flesh.
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A very interesting thread about the recently announced aperiodic tiling, formed by a "hat" tile (cf. ). The tile itself is composed of 8 kite tiles, and if you focus on the individual kites you see the deltoidal trihexagonal tiling appearing underneath.
Decorating hats. Spent the weekend decorating the kite shapes that make up the "hat" - the aperiodic monotile announced last week. Some interesting results that might open the way to new board games and art.
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A GIF showing how a CZ gate can be carved from a cluster state, using Z measurements to remove sites (red 1-leg spiders) and X measurements to keep sites (green 1-leg spiders). The possible π-spider errors, to be corrected in later stages of computation, are omitted for clarity.
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Diagrammatic Languages ≈ Compositionality + Locality of Information
Diagrammatic Compositionality ≈ Type Safety + Causality
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100% agree. Diagrammatic reasoning is the future of a whole lot of truly useful mathematics.
My intuition tells me that categorical string diagrams, as described in Quantum in Pictures, is the future of human thinking.
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@bgavran3
@_julesh_
@tangled_zans
For me, Twitter's main selling point has been the same for the past few years: info from a variety of different sources all flowing to me through the feed. It's harder to get that from Mastodon, at least at the moment.
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Some sweet cluster state carving for today's
#WorldQuantumDay
celebrations at
@IUBloomington
. With
@rctmpx
,
@JavaFXpert
and
@Bingmanson
.
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When our work on the Combinatorics of Causality [1,2] came out last year, it was somewhat disappointing to me and
@NicolaPinzani
that we could not fully characterise the hierarchy of dynamical causal orders beyond 3 events.
The algorithm we used at the time had to be terminated
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Academic papers should be more like open source projects: data should be available, results should be verifiable/reproducible, everything should be adequately documented. No need for peer review when the users themselves can easily decide what works and what doesn't.
In machine learning, where there’s 100 papers per day on arXiv and it’s kind of socially important for non-experts to have some idea what’s going on, this obviously could never work. Maybe letting NeurIPS reviewers throw darts at a wall blindfolded is better than the alternative.
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Spreading the quantum thought contagion at last night's talk for
@pintofscience
London, sponsored by
@QuantinuumQC
. We had a blast 🍻!
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If you're reading "The Topology of Causality" (or "The Geometry of Causality" ), please note that there's a typo in the definition of an open cover. Corrected definition, with thanks to
@interpretantion
for spotting it 👍.
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A sneak-peak from the upcoming book by
@coecke
and myself, teaching quantum computing to a younger audience 🤩🤓⚛️
For book/paper writing purposes I tend to hook up with superstar illustrators. For dodo book it was
@AleksKissinger
, my QNLP sidekick
@konstantinosmei
made some amazing ZX-animations (). For coming book (December release) it’s
@StefanoGogioso
.
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A 4-term Fermionic operator in the ZX Calculus, from ongoing work with
@aymanamrn
on the optimisation of ansatzë for strongly correlated electronic systems.
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@paulg
You're right in your original point, but why did you rate the community note as not helpful? This may have been clear in your mind from the start, but I would wager that a large portion of your US audience isn't aware of the current asymmetry in visa rules and costs.
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Chomsky: “colourless green dreams sleep furiously" is a semantically nonsensical sentence.
#midjourneyAi
: hold my beer.
#midjourney
#AIart
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The QPL 2023 abstracts for my quantum causality work with
@NicolaPinzani
have been submitted...
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"Every vector space has a basis" is equivalent to the Axiom of Choice, but Hahn-Banach is weaker. So it might be possible to swap AC for an axiom that results in spaces of continuous functionals which are not empty and yet don't have a basis. I wonder if someone tried...
There was a debate between Hilbert (an algebraist) and Brouwer (an analyst) which Hilbert won, so now every vector space has a basis and every ring has a maximum ideal, whereas not all functions are continuous and not all sets are measurable. It could have been different!
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I'm now convinced that sigma types and pi types are just special cases of a broader quantification construction, where we consider the set of dependent partial functions having domain in a given open cover (an antichain). Discrete cover gives sigma, indiscrete cover gives pi.
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Making some jupyter notebooks to accompany the release of our upcoming library for causal analysis (the one used by all my work with
@NicolaPinzani
).
The plots below demonstrate the the operation of "replacement" in static causal orders, where each event of a causal order (top)
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In summary, we introduce the basic machinery to talk about Feynman diagrams in categorical quantum mechanics (CQM). We are able to compose Feynman diagrams sequentially and in parallel, obtaining the summands of the composition via diagrammatic commutation rules alone. (2/25)
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Still waiting for scientific evidence that the real numbers are more useful than just using the rationals.
Still waiting for scientific evidence that manifolds are more useful than just using R^n
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The proof of convergence to the geometric mean in the zero limit is especially delicious, with a log sum turning into a prod log. Yum yum.
Generalized means interpolate between the min, the geometric mean, the arithmetic mean and the max.
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