1. INTRODUCTION
We are going to discuss data representations where we attempt to shed light on how to bridge the gap between systems that are fully Quantum to systems that are Pseudo-Quantum (or Quantum like) to systems that are classical in which data is represented by classical bits but that offer some of the characteristics of Quantum Computing, such as quantization (quantization of data, versus quantization of information versus quantization of knowledge), superposition (read several bits in unit time, or, access multiple facets of information or achieve knowledge access in a way that simultaneously cross-cuts several aspects of knowledge) and entanglement (in which entities are correlated in indirect ways, such as the analog to the way a hologram re-creates the whole image from a fragment of its parts which might be seen as the entanglement of the whole throughout its parts).
I cannot possibly cover all this ground in a single blog. So I am going to do it in several blogs. My aim is to cover the conceptual ground and then bring it to actual implementation ground.
First, the Wikipedia definition:
A quantum computer is a computation system that makes direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data..We'll start with the concept of superposition.
2. Superposition of States
Wikipedia tells us about on definition of Quantum Superposition:
Quantum Superposition
Quantum superposition is a fundamental principle of quantum mechanics that holds a physical system—such as an electron—exists partly in all its particular theoretically possible states (or, configuration of its properties) simultaneously; but when measured or observed, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics).
This definition presupposes a physical design and a foundation in the axioms of quantum theory. But, let us pause to ask what it is about superposition that is important, as properties, and then return to its concept in order to pave the way for a discussion of design and implementation.
Accord to Quantum Superposition from Princeton:
Quantum superposition refers to the quantum mechanical property of a particle to occupy all of its possible quantum states simultaneously. Due to this property, to completely describe a particle one must include a description of every possible state and the probability of the particle being in that state.
Superposition, from a classical perspective is available through a number of representations that allow us to bijectively represent a collection of states in a single observation. Here are some examples:
(i) Fourier Theory: states are represented by individual sines and the aggregate observation is a waveform ( Fourier Series ).
(ii) Prime Number Theory: states are represented by prime numbers and the aggregate observation is a number that is the sequence of primes multiplied together. In 1679, Leibniz invented his method for representing aggregate concepts in terms of simpler concepts in terms of prime factors. The interested reader can find more here: Leibniz's Characteristica Universalis and also here: Gottfried Leibniz
(iii) Hereditarily Finite Sets: bijective encoding of hereditarily finite sets onto natural numbers where these sets encode the states and the observable is the natural numbers ( Moto-o Takahashi. A Foundation of Finite Mathematics. Publ. Res. Inst., Math. Sci., 12(3):577–708, 1976 ; and see also Paul Tarau. 2010. Declarative modeling of finite mathematics. In Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming (PPDP '10). ACM, New York, NY, USA, 131-142. DOI=10.1145/1836089.1836107 http://doi.acm.org/10.1145/1836089.1836107)
(iv) Combinatorial Geometric Objects: finite structures that encode the combinations or permuations (aka configurations) of objects as a state: permutohedra, combinohedra and relations between such structures and volumes or angles or surface-areas. For example see "The Amplituhedron".
(v) Multidimensional Venn Diagrams: can also provide a superposition as noted here: especially Multideminsional Venn Diagrams or Venn Diagram of Particle Interactions and also here: Venn
The issue with Venn Diagrams is that we tend to forget that they exits and are useful logical objects.
(vi) Quantum Neural Networks in which a single layer network is trained concurrently with several other single layer network to produce a single state multilayer network. In this case, the superposition occurs through the layering of networks. See for example: Quantum Neural Net and Quantum Inspired Nets.
(vii) Density Matrix: A density matrix is the name given to the operator which is represented as a matrix. This matrix represents a the mixed state as a sum of expectation values of each state weighed by a probability factor. However, while the density matrix represents the distribution of states in terms of probabilities it is not itself the actual superposition: the superposition is given by the computation of the state in terms of the results of the interfering wave function's amplitudes. Therefore, there is a relationship between the superposition (which is seen after the act of measurement) and the probabilities of the states but the point is important to distinguish.
(viii) Phasors: a less well known re-representation for superposition of states is to use a phasor which is essentially the motion of a point on the circumference of a circle driven by the behavior of the wavefunction (as it rises and falls). Phasors are vectors and can be composed or decomposed to produce superpositions of states represented by the driving waves. This dynamical aspect, which is that properties of the medium can be factored into the way the driving signal (wave) is operating is an important point sometimes not well defined in descriptions, for example, and has to be inferred, as here on the Wikipedia page: Phasor
(ix) Wavelets: the superposition (of wavelet pyramids) is used classically in image representations. Wavelets
(x) Twistor Structures: a linear superposition of dynamical processes can be described by different twistor spaces in which the (dynamical) state represents behaviors. For example, in scattering amplitudes (as seen in collisions between particles that then break up into other particles) a positive or negative orientation is assigned to each particle that is either outgoing or incoming. However twistors can, in general represent aribtrary superposition structures of dynamical processes. They are not extensively discussed in contemporary literature as a representation for quantum computing or quantum-like computing processes. To learn more about this fascinating structure, have a look here: Twistor Theory
(xi) R-Functions: an interesting option for representing superpositioning is the use of Rvachev-Functions, an interesting variant of implicit function theory, and one that I myself like to use for its fast computational properties and relation to Boolean logic: Rvachev Function and a for a good primer, here: Primer
(xii) Topology: superposition represented using ideas in knot-theory and topology. This is another exotic model for superpositioning of states and one that also has physical promise. For the curious see here: Computing with Quantum Knots and here for a more advanced version: Topological Quantum Computing or for an Arvix paper: Quantum Knots
There are likely several more ways to encode a superposition of states, and, I will try to add to the methods.
However, what I would like to draw attention to is that it is not any one method for superposition representation but that the method connects with other concepts, like entanglement, in order to become useful in the quantum inspired models of computing.
More next time, in Science-II.
No comments:
Post a Comment