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nb: attempting a daily posting cadence as weekly clearly doesn't work. adjust quality priors accordingly

Momentum computing is, as far as I can tell, a reversible computing paradigm which circumvents Landauer's limit by embedding the memory state transitions of some computing device in a physical system which equilibriates slower than the time it takes to do an individual bit-swap, so that the memory state transitions themselves can store information in their "instantaneous momenta" and subsequently perform bit-swaps with near-zero net work.¹

In particular, one can construct toy theoretical energy potentials which implement a $\mathrm{\Delta }W=0$ Fredkin gate.² Consider a particle $(x,p)$ in the 1D potential $V_B(x)=\alpha x^4-\beta x^2.$ The two minimal states are located at $x=\pm \sqrt{\beta /2\alpha },$ and as transitioning from the $x<0$ regime to the $x>0$ regime is exponentially prohibitive in the value of $V(0),$ the system "stores" a bit $\{0,1\}$ depending on the sign of the particle's position.

Now imagine that $V_B$ encodes a thermal bath in which the particle $(x,p)$ is embedded in, and assume that the particle's motion follows harmonic oscillation in the absence of external influence. When the bath is active, the particle's potential is described by $V_B.$ But when the bath is disengaged, the particle behaves according to $V_H(x)=k{x}^2/2,$ which induces a trajectory $x(t)=A\cos (t\sqrt{k/m}+\varphi ),$ which is periodic in time $\tau =2\pi \sqrt{m/k}.$ Therefore, disengaging the bath for time $\tau /2$ swaps the sign of the particle's position, effectively performing a bit-swap.

This theoretical "bit-swap" comes at no work cost because there is no change in potential energy from time $0$ to $\tau /2.$³ However, any physical implementation of this concept must contend with the fact that changing the potentials in a physical system requires energy to be expended, likely in an inefficient manner. How does one get around this? Superconductors!

gradiometric flux logic cells

The theoretical guarantees above require complete & efficient decoupling of the system from the bath. You can get similar results by simply ensuring that the relevant computational timescale is significantly smaller than the energy flux rate between the system and its bath, so that "from the perspective of the computation" there is no coupling.

[RC22] chooses to implement such a system with gradiometric flux logic cells, a kind of circuit utilizing Josephson junctions designed particularly to withstand global magnetic noise fluctuations [I do not really understand GFLCs very well, that will be a topic for another day's post].

With suitable assumptions & parametrizations [such will be the subject of yet another day's post], the GFLCs follow "significantly underdamped Langevin dynamics", which can be described with the following equation: $$d{v}^{\prime }=-\lambda v^{\prime }d{t}^{\prime }-\theta {\partial }_{x^{\prime }}{U}^{\prime }+\nu r(t)\sqrt{2d{t}^{\prime }}$$ where the potential $U$ evolves according to $$U^{\prime }(t^{\prime })=(\varphi -{\varphi }_x(t^{\prime }){)}^2/2+\gamma ({\varphi }_{dc}-{\varphi }_{xdc}(t^{\prime }){)}^2/2+\beta \cos \varphi \cos ({\varphi }_{dc}/2)+\delta \beta \sin \varphi \sin ({\varphi }_{dc}/2)$$ where $x^{\prime }=(\varphi ,{\varphi }_{dc}),v^{\prime }=(\dot{\varphi },\dot{{\varphi }_{dc}})$, $\varphi =({\varphi }_1+{\varphi }_2)/2-\pi ,$ ${\varphi }_{dc}=({\varphi }_2-{\varphi }_1),$ and ${\varphi }_1,{\varphi }_2$ are the phases of the individual Josephson junctions $I_{c1},I_{c2}$ in the above figure [I think]. ${\varphi }_x=2\pi {\psi }_x/{\mathbf{\Phi }}_{\mathbf{0}}\mathbf{-}\pi$ and ${\varphi }_{xdc}=2\pi {\psi }_{xdc}/{\mathbf{\Phi }}_{\mathbf{0}}$ are functions of the external magnetic fluxes ${\psi }_x,{\psi }_{xdc}$ applied to the circuit.

The details of this are very interesting, still confusing to me, and this is by no means an exhaustive parametrization of the underlying models. However, below (Fig. 2) showcases that varying $\varphi$ and ${\varphi }_{dc}$, the sum and difference of the Josephson phase parameters, recovers the potential geometry associated with costless bitswaps.

further considerations

• I really want to understand the interface between the theoretical dynamics and the physical implementation better. Why is this theory so substrate independent? Why does it matter that our memory state transitions are modeled by CTHMCs instead of CTMCs? Why are we using superconductors?
• How do we actually get efficient circuit modeling of the kind described here? I couldn't readily find a Github repository associated with the paper, so I want to write my own library and replicate their results. They find that the efficiency of their circuits are largely dependent on "circuit hyperparameters", and it would be interesting to investigate their structure.
• Benchmarks for algorithms that can be implemented with momentum computing and with typical CMOS/transistor logic, and developing simulations that can accurately predict efficiency differences. Still not sure how to think about this! Will absolutely be the topic of a later post.
• Fermi estimates of all the physical quantities at play here. What is "one Landauer" at STP? How much does it cost to make a gradiometric flux logic cell? Etc. Etc.

All credit goes to the coauthors of the two papers cited in this post.

TCP implementations should follow a general principle of robustness: be conservative in what you do, be liberal in what you accept from others.

The original TCP specification is explicitly designed to be agnostic to IP implementation. It's only supposed to handle byte transfer from application to application, not the specifics of packet delivery between hosts. In practice, this ideal is unattainable¹, but it points at a deep truth about healthy integration of parts in systems.

Modularity is a natural consequence of system complexification. Internal bottlenecks on information transfer necessitate internal specialization². Dually, effective modularity requires partwise efficiency: judicious outputs and robustness to inputs, through either filtration or error-correction.

Cells have well-defined outputs and boundaries which protect them from harmful inputs. Organs as well. Intuitively, the "conservative output, liberal input" principle could be reformulated as a "specific function, high survivability" dogma which we find exhibited in biological systems.

Admittedly, TCP has more in common with the blood-brain barrier than it does with the liver. But the Internet is special in that the design of transportation organs is centralized while the design of substantive organs is not, and as a result boundary-manufacture is a centralized process. Insofar as the success of biological systems is to be attributed to organ-organ boundaries, those same properties may be reflected in TCP.

As the Internet scaled to tens of thousands of simultaneous hosts operating on the same network, the RFC 761 implementation could no longer support reasonable host to host communication without congestion collapse, requiring updates to the default TCP. Yet modern TCPs are backwards compatible with TCPs from 1980, in part because the abstraction was so well-designed.

"The first thing you need to know about Cyprus is that everything is Hotel. School is Hotel. Restaurant is Hotel. Home is Hotel." ¹

Leptos Estates owns everything on the island, as well as NUP, so naturally NUP is in a hotel. Because Leptos Estates is in the business of making and running hotels. Given that, it was not that strange to find the former CEO of Yandex giving a talk on AI and mathematics in a ramshackle, moldy hotel in Paphos, but it certainly was not expected.

The talk itself had relatively milquetoast content (at least by scaling standards), but the more interesting anthropological aspect of it has to do with its attendees and post-remarks. Everyone who attended (except me) was Slavic. There were no native Cypriots in the lecture hall.

Part of this is downstream of the ongoing Russification of Cyprus. Yet Paphos is ostensibly one of the two last Greek bastions of the country (the other being Nicosia, the capital), and given the great selection pressures on the listeners it is unlikely that this is the primary contributing factor.

In fact, NUP (Neapolis University Pafos) contains a contingent of self-exiled faculty members from the Steklov Mathematical Institute, who have constructed a remarkable mathematics computer science and AI program in the middle of touristic hell. Students live in hotels in the off-season, and in on-campus dorms during the summer.

I don't really understand why the lecture was conducted English? No one spoke in anything but Russian afterwards: the Jetbrains developer contingent as well as the students were either Russian, Ukrainian, or Russian Israelis. The course itself is also taught in English, to keep up pretenses I suppose.

Paphos smells like India. I suspect this has to do with the marked lack of certain gasoline additives that decreases its olfactory pungency, or perhaps the shared high humidity. Its cost of living is comparable to that of Western European nations.

Aphrodite's birthplace is a temple ruin. It is common to utilize the leftover limestone rubble for one's own purposes.

• Astor Piazzolla, Tango: Zero Hour
• Javier Perianes, Debussy: Préludes Book I; Estampes
• Popol Vuh, Hosianna Mantra
• Captain Beefheart, Trout Mask Replica
• François-Frédéric Guy, Liszt: Harmonies poétiques et religieuses & Sonata in B minor
• Matti Raekallio, Prokofiev: Complete Piano Sonatas / Visions Fugitives
• Peter Rösel, Bach: Busoni Transcriptions
• Faust, Faust
• KASHIWA Daisuke, program music I
• Venetian Snares, Rossz Csillag Alatt Született
• Leonard Bernstein, The Complete Mahler Symphonies
• Antti Siirala, Beethoven: Piano Sonatas Nos. 30-32
• Ivo Pogorelich, Ravel: Gaspard de la Nuit / Prokofiev: Piano Sonata No. 6
• Kazuhito Yamashita, The Complete Suites For Solo Cello BWV 1007-1012 [Guitar Version]
• Uncle Chris, Uncle Chris
• Philip Glass, Koyaanisqatsi
• Arthur Rubenstein, Chopin: Mazurkas

next week, the blog shall return to its regularly scheduled technical programming

• Copper brooch with tardigrade & tesseract engravings. Gifted to me by an eco-technologist acting as an avant-garde musician at an Edge Esmeralda closing event. 500 extant copies: if you own one, I would love to meet you.
• Studies in Hegelian Cosmology by J. M. E. McTaggart. 1903 edition, greenbacked & hardcover & originally archived at Trinity College, Cambridge. "On loan" from the Rhodes Library; saddened by the likely indefinite postponement of return.
• U.S. Passport. Visas to Turkiye, India. Entry and exit stamps from U.K., Czechia, France, Germany, Switzerland, Japan, Taiwan, India, Turkiye, Poland, Finland, Ireland, Georgia, Ukraine.
• Turkish Antarctic expedition pin, with decorated wooden bird. 17th birthday present from a dear friend of mine.
• Framework laptop, 13". Ran NixOS. Was approaching a faithful representation of my own brain; I lost an extension of myself.
• Black Leuchtterm notebook. Maths & rambles & private keys.
• Silver necklace with misshapen star. Acquired in a jewelry store in the Mission, originally intended for my partner. Ultimately not for me.
• Clothes: checkered flannel shirt, also acquired in the Mission. Partner's beanie. Foresight hoodie, MATS jacket. Beloved boots.
• Bike. Estimated ~$15k new, a kind mechanic gave it to me for free. Was the way I got to school for a while; suspension perfect for mountain biking. Stolen in Healdsburg.
• Letters. From S to me, written and drawn in Prague on self-Narrativism; from me to S, written in Taiwan as poetry; from R to me, written in print English in the Queen's College; from G to me, first on a pink sticky-note and next on a four-inch thumbpad; so, so many between L and I. ESPR/WARP/Atlas/ASPR/Rise notes.
• My phone, twice: firstly in Warsaw airport, and secondly in a San Francisco Airbnb. The second was a replacement for the first, however, and I found the first when I went through Warsaw again. Ended up back where I started.
• My glasses.