Tying Up Loose Ends: Topology Swallows Quantum Computing

The central visual image representing the art & science of Alchemy is the Ourobourus: a dragon or snake swallowing its own tail. This symbol deftly reflects the totality of the alchemical process, the “great circulation” of Nature, by which what was “above” as the very subtle spirit descends and becomes the “below,” the earthy and fixed; then by reversing the process, what is below ~ fixed, heavy & earthy ~ ascends and gradually becomes volatile and subtle as the above. Hence, the circle is closed: the snake/dragon has indeed swallowed its tail. Homer, the Greek philosopher, referred to this process as the “golden chain.” And what is represented, along with the individual internal process of the yogi or yogini, is also the chain of alchemical adepts (those able to link Heaven and Earth), beginning ~ in the classical Alchemy lineage ~ with the Egyptian Hermes Trismegistus, who wrote: “Make what is above become like what is below, then what is below returns to what is above, thus creating the miracles of one thing.”

This same process ~ the descent of consciousness into matter, and the subsequent ascent of matter back into its more rarefied state (and, in nondual traditions, the linking of these two processes in a continuous loop) ~ is described in various other mystical traditions (most notably in Taoist Inner Alchemy texts, but also in, for instance, Patanjali’s Yoga Sutras).

In the language of Topology ~ the branch of mathematics that looks at what happens when certain geometric figures are stretched, twisted & folded (or, more precisely, when the space around these figures is distorted in these ways) ~ the Ourobourus is what’s called a “knot” (a closed curve in three-dimensional space). A salient characteristic of knots is that they can indeed be stretched, twisted & folded into all sorts of superficially “different” shapes, while retaining, mathematically, their “sameness.” The central question explored within Topology is how to know when one shape is simply a “distorted” version of another? The (very short!) answer is that this depends upon identifying what are called “invariants” ~ characteristics which remain, across variations in a given shape/knot.

Topology has recently been employed in the service of the newly-emerging science of quantum computing, as a resource for addressing the problem of decoherence. In quantum computers (which are at this point mostly still theoretical), information is stored on sub-atomic particles. The major problem with these so-called qubits is their susceptibility to decoherence: disturbance of their quantum space/structure by the Newtonian environment. Each time such a disturbance happens, and the quantum state of the particles is affected, the information stored on them is lost or distorted.

And this is where Topology, potentially, is able to come to the rescue. For information stored on topological structures is impervious to local error, since small pieces of “lost” information can easily be reconstructed, given what is known of the overall pattern/shape of which it was a part. (Imagine a donut, and a small piece on the surface of the donut being lost … Since we can still see what the overall shape of the donut is, we can easily replace that missing link.) So what is needed then, in terms of quantum computing, are sub-atomic particles which can be mapped topologically.

And it just so happens … that when an electron liquid on a two-dimensional crystalline surface is subjected to extremely cold temperatures and a strong magnetic force, sub-atomic particles called anyons are created. These anyons move about the surface ~ they “dance” ~ in patterns which physicists call “braids,” and which can be mapped topologically. As a result, these anyon braids (defining what’s called a Quantum Hall fluid) are very good candidates for storing information in a quantum computer, in a way which side-steps the problem of decoherence. (Another interesting thing about anyons is that they carry a fractional charge. Whereas protons carry a single positive charge, and nuetrons a single negative charge, anyons carry charges corresponding to any real number. Different fractional states, i.e. charges, define different “flavors” of Quantum Hall fluid, and determine how complex a computation is possible, using that fluid as a carrier of information in a quantum computer.)

So assuming this all works out, and we’re able to use these anyon braids to store information … What would be the advantage of such a topological quantum computer? For one thing: speed. A classical/Newtonian computer operates via bits (housed in circuits or silicon chips) that are either “on” or “off.” In line with the so-called “weirdness” of the quantum world, qubits (say, in the form of anyon braids housed in Quantum Hall fluid) exist in combinations of “on” and “off,” simultaneously (a phenomenon known as the superposition of states). This property allows a quantum computer to perform multiple computations, and get multiple answers, all at the same time. So a calculation that might take a classical computer millions or billions of years to complete, would be completed via a quantum computer in a matter of minutes!

A second advantage of such a computer is, perhaps, more aesthetic (or spiritual/philosophical) than practical. There’s a sense in which ~ by employing sub-atomic particles as our “hardware” (and software, simultaneously) ~ we’re using nature as our computer! For the movements of the anyons has an intelligence “of its own” … similar to when we throw a ball up in the air, the physics of its motion is “calculated” by the medium (air, gravity, etc.) through which it travels. It doesn’t require us to transmit the appropriate mathematical formulas to it, in order to continue along its trajectory …

So how is the creation of a topological quantum computer (the projection by our collective human consciousness of a radically different model of computing/information processing) related to Alchemy & the Ourobourus (where we began this essay)? Or, in relation to my larger body of work, how is it related to the Alchemies of Taoist, Buddhist, or Hindu Yogas?

There area number of Buddhist as well as Taoist teachers who use the metaphor of a computer to describe the working of the human bodymind. Which, at this point, then begs the question: How might we (as Buddhist, Taoist or Hindu Yogis) transform ourselves (the way that we process information, i.e. relate with the intelligence/energy of the manifest world) from Newtonian/classical to topological quantum computers? Such a change will require, as we’ve seen above, a change not only in our “software” but also, and perhaps more fundamentally, in our “hardware” or in the relationship between hardware & software. How might I evolve from being a Newtonian machine that relates to my world in terms of a series of “yes’s” and “no’s” (0’s and 1’s) to a being defined by fractional charges, by superpositions, by the capacity to perform multiple simultaneous calculations (to hold simultaneously multiple points of view)?

And perhaps a part of the answer lies ~ as it does within the realm of quantum computing ~ in finding an information storage & processing medium (a bodymind!) which operates via physical/energetic patterns more than via individual bits of information … so that a stretching, bending or twisting (a “distortion”) of the medium will not sacrifice its integrity. So that small losses of information, small fluctuations in the immediate environment can easily be compensated for by the wisdom & stability of the larger pattern. (What might this mean in terms of a physical practice such as yoga or qigong?)

And could this be part of what is being transmitted to us via the alchemical symbol of the Ourobourus? That the creation (or recognition) of patterns, and more specifically, closed loops, of so-called “opposites” finding their unity, recognizing themselves to be two ends of the same stick … is an important key to the whole alchemical process, to spiritual and well as technological awakening? Taoist Inner Alchemy (as well as Kashmir Shaivite and other mystical) texts remind us, again and again, that the entire universe exists within and as our human bodies. The stars & all the galaxies are alive within the very structure of our cells. (According to which the Theory of Relativity should be able ~ yes? ~ to “swallow the tail” of Quantum Mechanics, but alas …)

And perhaps what’s being pointed to also is the fact that we are indeed ~ in our manifestations within the phenomenal world ~ part of Nature, partaking of the wisdom, love & limitations of the plant, mineral & animal kingdoms … And it is in this association (rather than the escape from it, into, say, the world of machines, of “artificial” intelligence) that lies our greatest power …