Quantons (aka Kyndi's Model for Quantum Inspired Computing)
First, I would like to start off with a quote:
"Tensor networks provide a mathematical tool capable of doing just that. In this view, space-time arises out of a series of interlinked nodes in a complex network, with individual morsels of quantum information fitted together like Legos. Entanglement is the glue that holds the network together. If we want to understand space-time, we must first think geometrically about entanglement, since that is how information is encoded between the immense number of interacting nodes in the system." - Tensor Networks
Why I mention this is that the MERA (multiscale entanglement renormalization ansatz) model has potentially interesting surrogates in terms of classical structures. One of the computer science components at the heart of the concepts are variable orderings and permutations with respect to network structures (specifically tree tensor network diagrams). However, none of this matters if it cannot be done fast. So the crux is to identify the mappings between structural regularity and analytic precision between representations like tensor networks and surrogates that are faster but essentially equivalent.
One way to approach this is through geometry.
The idea of tensor networks, and in general, matroids, could be seen from the viewpoint of multiplihedra, associahedra and related polytopes (zonotopes) have a very interesting structure.
More on tensor networks and matroids in another posting - there are several intertwined concepts that still need to be surfaced. Mathematical ideas without understanding, in my view, is junk factoids - so I will do my best to craft a path from where we are (with introducing ideas, specifically an interesting object, the Quanton) and where we need to get - so please bear with me.
One of the interesting structures is the Birkhoff polytope because it corresponds to permutations (i.e. it is what is called an extended representation of the Permutohedron).
Given a Birkhoff polytope, for example, for 4 integers, it will encode the 24 outcomes of the permutation of the 4 integers as an extended representation of the 4-Permutohedron. Of course, this structure, has 24 vertices, but, there are also 24! ways of choosing the ordering of vertices (in other words, there are a very large number of different Permutohedra!).
The interesting element is when the polytopes are embedded into manifolds, which could be (hyper)spherical or (hyper)torii - imagine each vertex is associated with a probability density, and tangent-density polytope is the tangent space at the vertex of the polytope associated to a probability density (the tangent space of the sphere providing a nice linear subspace if you like).
What this really is about, is precompiling a number of general relationships about a data structure, such as a certain kind of Orbitope so that the operations are effectively "parallel" across the various general relationships that are codified within the structure: to do this, it usually means you have to also come up with a way to associate a nicely computable manifold with the orbitope - that is where the hard work comes in. However, once you can precompile an orbitope and its embedding, then some unique and powerful properties, including the ability to effectively approximate some specialized quantum operations, most notable of which are Topological Quantum Computing and the operations of braiding to build gates.
I will get into how this can be achieved as a sort of "poor-man's" surrogate for a real topological quantum computing fabric (pun intended) in which the gate processes are woven (punning again!) together to provide a computing model along the lines of permutational quantum computing.
An excercise for the reader is the following question: what orbitopes admit an analytic representation in terms of "good enough" on a manifold as embeddings for simplified calculations that are reversible? More on this in the near future. I have a conference talk to give and promised I would save this for later.
Quantum, Classical and Quantum Inspired Computing Revisited
Today, physical theory tend to dominate the progress in the field of quantum computing as theoretical quantum science is only really in its infancy. The basic thrust, unproven, is that real physically implemented quantum computers that run quantum algorithms will solve the computational complexity problems and open realms of possibilities beyond classical computing. Unfortunately it isn't so.
The realms of possibilities still remain to be discovered in the classical realm, though perhaps, taking inspiration from the quantum world.
With respect to so-called pure or true quantum algorithms there are no proofs that I have found anywhere that show that any quantum algorithm will always outclass a well designed algorithm for the task.
If you want quantum-like computing then, perhaps, it will be achievable by new algorithms and methods as well as through more processor cores on a chip: we still don't understand the dynamics of dense core communications, behaviours, quasi-chaos, distribution and access of resources in such systems. Again, we are in our infancy.
Building a real hardware quantum computer is rife with experimental difficulties such as fault tolerance, scalability (beyond a few qubits) and a demonstration that entanglement can surmount complexity issues (since you can have classical systems that perform with quantization, superpositioning and parallelism - all that really makes purified quantum computing unique is entanglement). Of course, some approaches, like Topological Quantum computing, promise to alleviate many issues.
But even for entanglement, we can ask if there are surrogate ideas in the form of correlation functions (not in the sense of quantum, but mimicking the fact that two observables can be strongly related) and whether or not these can be maximized in number to act-as-if there were deeper parallelism.
We suffer significant limitations even in classical complexity theory and no proofs to guide the design of algorithms for classically hard problems though there are lots of proofs for upper and lower bounds --- where we still have to innovate is in proofs about design. Proof of classical hardness is notoriously difficult and there have been many false starts but proofs about design are still non-existent (anyone out there know if this is truly the case? Any pointers welcome!).
Quantons
So in the realm of quantum-inspired, we are really forging a bridge between interconnected, strongly co-dependent data structures on which certain kinds of computations are extremely efficient and that mimic what our ideas might be about the benefits of quantum computing: in other words, the hard work is about clever algorithm design by drawing upon analogies from the hard core science of quantum and field theoretic physics.
So what are quantons?
Well, my blog time is over for today but I promise to introduce and explore the quantum inspired concept of Quantons.
Until next time.
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