Quantum Inspired Computing (QUIC) : The Bloch Sphere and its Variations

 Quantum Inspired Computing (QUIC) - 1

Last time we made a list of the types of models for data structures for quantum entities.  The first item on the list was entitled "Bloch-Hyperspheres in the sense of extensions of the Bloch-Sphere".

Let's unpack what that means in the context I am thinking about.

In terms of a short list it means how to represent quantum entities in software using models drawn from:

1. The traditional Bloch Sphere (aka Poincaré sphere aka Qubit ).
2. The Riemann Sphere
3. The Majorana Representation ( and as points on the Riemann Sphere )
4. The Quasi-Probabilistic Frame Representation
5. Other Possibilities for representation (including ad-hoc computer languages)

1. Bloch-Sphere Qubit

 The two-level quantum bit is called a Qubit and the data representation that is useful computations is known as the Bloch or Poincaré sphere depending on whether or not you are talking to mathematicians or physicists.  There is already a lot of information online about the Bloch Sphere that the reader can find many articles and introductions to this basic representation.

The problem with the Bloch sphere is that it is solely a representation for a two-level system.

2. The Riemann Sphere

The Riemann Sphere generalizes the Bloch Sphere and can handle d-dimensional quantum entities ( Qudits ).   The advantage is that the representation is a well-known mapping so all the tools of algebra and geometry as well as directional statistics can be used.

3. Majorana Representation 

The Majorana Representation provides a fully generalized representation for quantum entities. The representation dates back to 1932 and has a permutation and symmetry underpinning for which there are nice mathematical properties useful in software such as ease of implementation and visualization.  In fact, a good and easy to read review can be found here. The Majorana representation forms a complete visually appealing representation of the wave function and therefore carries complete quantum information about the state of the system.

4. Quasi-Probabilistic Frame Representation 

The paper at Arvix by C. Ferrie and his thesis provides the best introduction to the idea of extending probabilistic frames to the quantum case.   The representations hinge on the connection between classsical phase space and quantum states through quasi-probability distributions.  A tutorial can be found here.  As stated in the tutorial, "It furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations."

5. Other Possibilities for representation  

There are several other more esoteric and less well understood models for representation that the daring may seek.  For example, the paper Quantum Theta Functions and Gabor Frames for Modulation Spaces expresses one of these more esoteric approaches.  For an overview of using Haskell to represent quantum computing, see this paper or to logic programming in pure prolog as a way to compute or this interesting paper for quantum inspired Interclausal Variables.

Which Choice?

At this point it is still too early to state a choice because choices will invariably tied to contexts of data processing or the types of problems being addressed.

Until next time!

Data and Algorithm Part-1 (Continued): Quantum Inspired System Semantics (QUISS) - Overview

Quantum Inspired Computing (QUIC)

There is a lot of information available on the Internet for the definitions of Qubits, Qutrits and Qudits so there is no need to repeat those here.

What is not discussed is that these objects are all geometric in nature and have more than one dimensionality to them.

There are many options for quantum-inspired computing (QUIC) and for a QUIC list of options, we can choose to represent the properties of data using one or some combination or all of our own short selection of ten such options:

1. Bloch-Hyperspheres in the sense of extensions of the Bloch-Sphere;
2. Amplituhedral structures and Grassmanians in the sense of Trnka, Postnikov, Bourjaily et.al in which volumes produce probabilities
3. Polyhedral and Combinatohedral structures (e.g. Permutohedron) in which directional probabilities are represented by permutation polyhedra where each vertex represents a permutation (there are N! vertices for an N-element permutation);
4. Topological structures such as Topoi, simplices and generalized maps in which involutions and functions with higher order symmetries (complex involutions) define the skeletal structures.
5. Non-Binary Base Numbers (Complex, Figural, Tree, Functional and Mixed Radix) in which properties of big numbers that can represent the Goedel numberings of various structures are combined with probabilities (for example, the real parts and the imaginary parts treated as on single whole but entwining different conceptual bases);
6. Quantum random walk and quasi-quantum like stochastic walks on classical structures like graphs or lattices represent properties of the data of interest.
7. Field Structured Representations and quasi-quantum/analog representations such as particle swarms which are represented in the complex plane as well as the real plane.
8. Quantum-like entanglement defined as any correlation in complementary bases of representation of information.  For example, measures of discord or mutual ignorance may co-correlate with measures of informativeness and these may produce some quantum-like effects (though not true quantum entanglement ... we are, after all, working on QUIC).
9. Virtual machine designs and architectures that represent quantum like properties such as variables that entangle (as high-order "sharing")  or produce uncertainty between clause definitions or reflective, simulations of quantum particles as computational analogs of quantum processes (such as treating text in terms of Bose-Einstein condensates)
10. Genetic, parallel, distributed systems as quantum analogs or real quantum systems.

These are just some ideas for some top level concepts for representing quantum inspired data structures that encapsulate the desirable properties of being able to quantize, superpose and correlate at higher dimensions (i.e. entangle) information in a possibly useful way.

For other sources of data structures and examples the following sources are also very useful and inspiring:

Quantum Computing since Democritus  and its companion website online course.
Quantum Machine Learning: What Quantum Computing Means to Data Mining
Principles of Quantum Artificial Intelligence

There are, of course many sources around but we shall attempt to look into a few algorithms and reprsentations in the coming months to see what the possibilities can be.