QuIC (Quantum Inspired Computing): Amplituhedral structures and Grassmanians

QuIC (Quantum Inspired Computing):  Amplituhedral structures and Grassmanians

Last time I looked at some structures based on the Bloch Sphere and its variants.  While the Amplituhedron is not yet a popular or well-known structure for performing quantum inspired computation, it is an amazing machinery for simplifying thousands of pages of equations that would be needed for scattering amplitude calculations (which you can read about in terms of the Amplituhedron here) into just a few (with the commensurate speedup in computation).

The reference work is:

Scattering Amplitudes and the Positive Grassmannian

And the best background available is this (download the "other formats" compressed file to get the Mathematica notebook): 

Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica

Of course, there is comedy online in the form of Scott Aaronson's Unitarihedron post (more here:  http://www.scottaaronson.com/blog/?p=1537 )

With all the kidding aside, there is perhaps a use for such a structure and it is possibly in representing a probabilistic infon that is itself a multivariate probability density over permutations  (the objects of the permutation can represent things like rules, identifications, tracks, stories, scenarios, and other combinatorially complex data).

One possibility is to combine or be inspired by geometric approaches (which the Amplituhedron is) with other models such as stabilizer states, and possible extensions of novel graph based formalisms, within some physically inspired model (such as the Orbital Angular Momentum). 

The main problem is to identify the appropriate surrogate structures that would enable a quantum inspired computing software implementation to be realized.  A rather nice view is presented in Gil Kalai's blog (and all of his blogs are a must-read).

Okay, so here is the prospective connection as a data structure:  we can represent any permutation and combinations in terms of geometric structues that have a local neighborhood that represents a probability density, as long as this object (for example, a Permutohedron or Combinohedron kind of object) lives in a real subspace that acts as its embedding (this is what the the Grassmannian in effect provides). 

For example, for the permutations of the set {1,2,3}, there are six in total and the geometry is that of a polygon, a simple hexagon.  The permutations live at the vertices but we can can define this object (the hexagon) in terms of its lines and subspaces (in 2D or 3D) through which its vertices are defined implicitly.

For example, we can represent any point within the hexagon in terms of an equation that amounts to a linear combination of its vertices or any of its sides in terms of its partioning k-planes (hyperplanes, which in this case are lines) that act as the generators of the hexagon.   This is expressed as a positivity condition.

Finally, we note that any combination or permutation can be associated to this type of geometry and that if the faces represented probability densities, then, in the case of the hexagon, it is the lines connecting the vertices and their subspaces (i.e. the triangles within) that determine the probability distributions of the permutation space.

In this way, we can use the Amplituhedron-like model to connect arbitrary, possibly hierarchical and nested structures (aka an "amplitu-nestohedron" like structure) that represent conditional probability densities of factorially large spaces and for which, thanks to the Grassmannian, we can always build some embedding.  And then, we can ask wild questions as food for thought on whether there is a Majorana Representation or an inspiration for one can be connected to such a geometric representation given that in the Majorana representation, a spin-J system state is modelled as a set of 2J points on the Bloch sphere.

Ideally, for a quantum inspired computation model we might like to see how we can integrate the concepts from Bloch-Spheres and its variants into models inspired by the Amplituhedron.  Of course, realizing that this is a stretch and not an easy task, it is left as an exercise for the reader ;)

Until next time!