Category- and Operad-Theoretic Foundations for a Distributed Mashable Semantic Quantum Internet (Part One)

In this essay I will lay out an argument for putting the theoretical foundations for a distributed global biomimicry-inspired hybrid digital/quantum operating system within two closely-related areas of pure mathematics: category theory and operad theory. This is part of a planned series of three posts: the first one giving an overview, the second one on category theory, the third on operad theory.

I will explain the mathematical ideas from scratch, and this is emphatically not a math blog so I won’t get overly technical. These ideas are simple and intuitive (at least the aspects I will discuss) and should not require any mathematical knowledge beyond high school algebra. I occasionally drop in jargon that I don’t fully explain for those already familiar with the ideas, but most jargon will be defined.

My primary inspiration for writing this post has been the distributed operating system Ceptr, the parent project of Holochain that I wrote about in my previous post about quantum resource trading. Thus I will endeavor to explain how many of the ideas behind Ceptr are already category-theoretic and operad-theoretic in nature, and argue that formally placing these ideas in this mathematical context will enable even greater levels of interopability as well as ease eventual integration with the quantum internet. Familiarity with Ceptr will be helpful for understanding the full context of this post but I will explain the main ideas as we go along. The bulk of my argument is not wholly specific to Ceptr, but we use it as a concrete model since its the only project of its kind that I know of. Later edit: I forgot to mention Statebox! This project has a lot in common with Ceptr, but I don’t know enough about it to explain it just yet. There is a great overview here.

This one hour video is probably the best existing introduction to Ceptr, and watching it will give you essential background.

Let’s start with a short, jargon- and picture-filled summary of what is written here, which will then gradually be unpacked as we go along (along with explanations for all of the buzzwords). We will revisit these descriptions in the latter parts of this series.

• Ceptr is an infinitely scalable distributed operating system inspired by biomimicry whose basic computing unit is a highly composable lightweight virtual machine called a “receptor”, which may be plugged into other receptors to form a fractal-like tree of nested receptors. These receptors communicate with one another by exchanging semantic trees, which allows a “protocol for protocols” that ought to virtually eliminate the need for brittle API bridges. The grand vision of Ceptr is to create a “global nervous system” that will enable new forms of “grammatic capacities” (analogous to how writing allowed governments and corporations, and the internet allowed sharing economy apps) which will enable new organizational patterns for humans and ultimately implement the ideals of the Metacurrency Project.

Screencap from the above Ceptr video of the Higher Order Programming Environment, a proof of concept of the receptor-based computing model. Each circle here is a receptor (or a membrane, which filters incoming data to the receptors it contains), though Matryoshka doll nesting of receptors is not depicted here.

• Category theory is a “mathematical model for mathematical models” developed in the 1940’s that gives a common language to most fields of mathematics, and more recently as disparate topics as physics, biology, linguistics, computer science, complex adaptive systems, epistemology, and much more. Communicating in this language enhances the grammatic capacity of the agents which use it, increasing ability for pattern matching of different abstract structure as well as revealing functorial relationships between totally different areas of knowledge.

A particular type of category called an ontology log, recently developed by David Spivak and Robert Kent. Categories consist of “objects” and composable “morphisms/arrows” between them. Nothing more.

• Operad theory is a theory of composable abstract operations that have inputs and outputs. Different operads have different rules for what types of inputs and outputs there are, how such inputs and outputs can be combined, and an attached theory of algebra that describes how data of the particular type described by the operad flows through the system. Operads are a category-theoretic description of universal algebra, which is a theory of all possible consistent mathematical operations on a set. An algebra over an operad is a set equipped with n-ary operations that satisfy the composition laws of the operad.

An operad in action. Empty circles are inputs for operations, circles containing spaghetti and more circles are possible operations to be input, which themselves can have additional other operations put into their empty circles. This figure is also from Spivak.

A summary of the synthesis of these ideas: semantic trees are objects in a category of semantic trees (which has many subcategories of particular sorts of semantic trees), and can also be seen as categories themselves where the objects are the data and the morphisms are the semantic relations. Computations on semantic trees (of the sort a receptor would do) can be formalized as functors between such categories. Receptors are formalized as operads, with the inputs and outputs being various types of allowed semantic trees that are processed or sent out by the receptor. Algebras over such “receptor operads” are sets of semantic trees and the operations of the operad are implemented as transformations on the semantic trees.

ARPANET in 1969.

To give context, I will describe what I mean by quantum internet in operational terms. Instead of “distributed mashable semantic quantum internet”, I will abbreviate “quantum internet”. With that phrase I include the digital internet as a subpart, not a separate entity. There are three recognized epochs of the quantum internet (apart from the distributed mashable semantic aspect) according to experts (though I’m not sure that anyone has named them as such yet):

• Epoch 1: Quantum computers can be accessed and computed on remotely over the cloud. We are already here! IBM Q Experience, Rigetti Forest, and the University of Bristol all allow anyone in the world to run code on actual experimental quantum computers over the cloud. Pick an algorithm and go do that right now! In particular, Q Experience is very user friendly (with a GUI) and I’ve used it for a four hour introduction to quantum computing course.
• Epoch 2: Qubits can be exchanged between quantum computers via the quantum teleportation protocol and small numbers of qubits may be entangled across spatially separated quantum computers. We are already at this point in the lab, but I would say we are not yet at the equivalent point of the first 1969 ARPANET message. Quantum scientists in the Netherlands aim to have the equivalent of that event in 2020. I find it reasonable to think that we will see Epoch 2 reach maturity in our lifetimes (in that every networked quantum computer of whatever architecture becomes dominant can exchange qubits with every other one), and that we will have made serious progress in this direction by 2030. This is the highest epoch needed to achieve all but the most fantastic possibilities in this series of posts. This is the also the epoch needed for my previous article on quantum resource trading over Holochain to become possible.
• Epoch 3: Quantum computers can have many entangled qubits across a distance. The physical separation between them has little relevance, and many quantum processors can behave as though it were a single quantum processor. This is some serious science fiction stuff. I would hazard a guess that this could not happen on a large scale before 2050, but I also think it is possible that this may never happen. Nevertheless, the ideas in this post become most relevant at this point. They are still useful for the previous epochs, but the full generality of the formalism does not become relevant until this epoch.

The quantum equivalent of the 1969 ARPANET, coming to a spacetime neighborhood near you in 2020.

Besides quantum resource trading over Holochain, I have also written about the quantum internet in Quantum + Decentralized. I plan to write a post in the near future specifically about the quantum internet as I realize this idea is even more foreign than quantum computers are. On the other hand, perhaps we are more intimately familiar with the quantum internet than we think, if the thesis statement of this blog is true (see below).

Why do this?

I posit that many of the ideas contained in Ceptr are already implicitly category- and operad-theoretic in nature, but none of the documentation is currently written in this language. It is always very fascinating when separate parties arrive at the same ideas independently — and there are numerous examples in the history of mathematics and physics where mathematicians and physicists invented the same ideas independently for different reasons. In this instance, I think that the creators of Ceptr were thinking very carefully about complex adaptive systems and computational models coming from nature and arrived at category- and operad-theoretic thinking because they really are the right way to ponder these topics. At the same time, there has been a cottage industry of applied category theorists working over the past several years on agent-centric distributed computing models as well as complex adaptive systems in these terms, a wonderful 7-part overview of which has been written by John Baez on his blog which I will consider as required reading to grok the full content of these posts. This convergence is inevitable.

Putting the ideas in the right mathematical formalism will serve to clarify and refine the right form for these ideas, which should ultimately result in wider understanding, better code, and profoundly enhanced grammatic capacity. As I will explain in the next two posts in this series, these foundations will also make eventual integration with the quantum internet more seamless, as quantum information processing is also naturally described in terms of categories and operads. These theoretical foundations will make the transition from Ceptr to “Quantum Ceptr” a natural transformation (which is a category theory joke that will be explained later, but maybe it will actually be true in a sense!).

Ceptr is an operating system currently under development (with the active development branch currently being in Holochain), so it may seem like jumping the gun to already be trying to work out what Quantum Ceptr could be (as if “quantum Holochain” wasn’t crazy enough!). I actually do not think this is the case at all. A simulacrum of the quantum internet is already accessible to us! Simulaqron is an already complete Python package designed to simulate the quantum internet. Computers on a network run simulations of quantum computers (which are basically just fancy linear algebra engines), as well as simulate quantum communication channels, all the way up to the level of Epoch 3 described above! Not only is the quantum internet inevitable, we can already develop for it, and Simulaqron is designed to make the code written for it to eventually be easily adaptable to run on actual quantum computers.

Ceptr is already mind-bogglingly forward-looking and deep in its core design principles and is a project that aims at nothing less than complete restructuring of the digital internet via the biomimicing agent-centric computing model, and human society as a whole via the creation of new grammatic capacities and patterns of organization. The inevitable quantum internet is hurdling towards us, also ready to transform human society in perhaps even more profound ways than the original internet. Simulaqron allows us to play with and develop for a fledgling quantum internet as if it already existed. So why not incorporate quantum internet readiness into Ceptr while it is still in infancy, rather than bolt it on in 10 years, which would inevitably result in years and years of foundational restructuring to accommodate the still utterly alien paradigm of quantum computing?

One very interesting topic we will have to gloss over is the connection between the diagrammatic thinking of category theory, linguistics, and quantum physics. Given this connection and the fact that much of the thinking behind Ceptr has linguistics in mind (in the form of grammatic capacities), it becomes even less surprising to discover the connections with category theory and quantum computing are already implicit.

The thesis statement for this blog that is always gradually being refined is that biological organisms are large scale distributed hybrid classical/quantum computers that are consciously experienced via mechanisms of qualia computing — in other words, biological organisms themselves are a form of distributed mashable semantic quantum internet, with a poorly understood connection with qualia computing. This “biological quantum/qualia information processing hypothesis” is nowhere near being proven, but with developments like Fisher’s quantum brain, quantum topology and protein folding, the Bio-Inspired Quantum Technologies group at Oxford, and the growing field of quantum biology, I see this thesis statement as having a lot of evidence to support it. If we want to truly reflect in the outside world that which is within and create a biologically inspired global nervous system, quantum internet capability needs full consideration from conception, and with Simulaqron we have the capability to do so as if the quantum internet already existed.

Most people think they do not know anything about categories or operads because most people have never heard of these things. Yet they are natural and simplistic at their core — categories simply consist of “objects” and composable transformations between objects. The flow charts that everyone knows and uses are frequently already categories (a sort of ontology log), with boxes being the objects and arrows being composable transformations. Cooking food is operadic in nature — the inputs of a given operation, such as “make lemon filling”, must be lemon, butter, and sugar, and the output is lemon filling. The algebra over this operad tells you the conversion rate from lemon, butter, and sugar to lemon filling.

An element of a cooking operad. From Seven Sketches in Compositionality.

I aim to bring these esoteric-seeming topics out of the shadows and show that you secretly already think in these terms. To avoid introducing too many new ideas at once, I will probably not be as detailed on the connection with quantum computing and leave that for a later series of posts. Stay tuned to dive deep into categories and operads!

Hopefully, if quantum processing is actually operational in the brain, it should be possible to co-opt the biological “machinery” to duplicate the processing in a test tube — by analogy with genetic engineering. The recipe for the simplest “synthetic quantum brain” would be as follows: Fill a test tube with simulated body fluid (minus the phosphate ions), add pyrophosphate, add the enzyme pyrophosphatase to create free phosphates that can bind with calcium ions to form pseudospin entangled Posner molecules, add calcium indicators (molecules that fluoresce when calcium ions bind), pour half of the fuid into a second test-tube, drop a bit of hydrochloric acid into each test tube to encourage the Posner molecules to disassociate, and detect the emitted fluorescence from the calcium-indicators. Matthew Fisher — Are we quantum computers, or merely clever robots?