Entanglement: Quantum Inspired Computing (QuIC) Approach to Precompilation of Entanglers

Entanglement and Entanglers

Quantum entanglement is an observable phenomenon that his dependent on strong correlations being measurable between superficially disparate things.  In a nutshell, entangled particles are precompiled (i.e. prepared) in states in which entanglement is observable and measurable.

I think of entanglement and quantum state preparation in particular as a precompilation step.  The preparation process is, in my mind, just precompilation.

"The best way to predict the future is to invent it" - Alan Kay

Note of Caution

I personally hate redefining or using new terms except and unless where they contribute a conceptual mental model that could not otherwise be easily implemented.

My reading recommendation is to review Baez's Crackpot Index or Siegels Quack Index.

With those words of wisdom and caution, I would like to add a few conceptual colors to terms such as Entangler and Precompilation.

Entangler

An entangler is a process that accepts un-entangled data representations and outputs entangled representations.  The representations entering and leaving the entangler may have the same or different representation models as long as the qualities of entanglement are observable and measurable after the entangler has operated on the inputs.

Precompiler

So what is precompilation?

Precompilation is a way to think of the concept of preprocessing and partial evaluation as a step of setting up the state of a computation so that compilation and execution are efficient.

I have not seen a lot of dialog on quantum computing and the concept of precompilation though, implicitly, any state preparation could be considered precompilation.

A compiler, in the quantum sense, then is the configuration of the computing material that has been prepared and then configured into a form that runs the computational model by either generating or accepting data which produces the results of a simulation or computation.

Generic Quantum Inspired Precompilation

Okay so here is the juice:  taking some inspiration from Pitowski Correlation Polytopes and the relationships between bicomplex numbers and the vertices of various interesting polytopes such as the Birkhoff Polytope, an extended formulation of the Permutohedron, then it struck me that using Pitowski and a few other gems (which I will get into next post) that you formulate a series of approximations that appear to provide some of the inspirational features of quantum computing.

For example, by association a complementary probability density to the vertices of a correlation polytope using an extended formulation, then an approach using an Estimation of Distribution Algorithm (EDA) would seem to fit the purpose of getting from joint probabilities of concurrent or seriated events as components of a solution space:  the applications would apply to Travelling Salesman, Matrix Permanents, Consecutive Ones, Seriation, K-Sums, and many, many other problem sets.

In short, by precompiling a probability distribution with a correlation polytope in terms of the extended formulation of a combinatorial polyhedron the opportunity seems to exist that we could do some quite clever coding to produce a non-commutative geometrical probability density that could be useful for problem solving through approximation.

In other words, the propositions at the vertices of such a polytope are characterized by the probability density as a function of the geometry rather then the other way around.   In the limit, these distributions are spherical directional non-commutative probability densities on the sphere (or torus or some other similar smooth manifold).

I would also bet that the tangent spaces would be better at linearity than the actual surface kissing point of the vertices but I will defer those thoughts to another day.

For now, let's just call these new objects (Probability) Density Polytopes since they are a combination of correlation polytope with probability distribution.  If anyone out there knows what these things can be called, please let me know!

Density polytopes could have differing kinds of distributions and functions so they could come in a variety of specifications: von Mises Fisher, Mallows, Watson, Gaussian distributions, etc...

The Entangler

Therefore, the entangler in this case becomes the EDA itself and the objects become the populations of density polytopes.  The features of quantum superposition and interference can occur as the by products of choosing  probability distribution function mixtures and ensuring a means to filter and detect those potentially useful quantum features.  In any case, it would seem that the total system is classical and superficially only is inspired by quantum thinking.

Until next time: the Density Polytope.