Unifying Classical and Quantum: The Base-60 Infinity Ladder Model

Introduction

The Boredbrains Consortium is developing a new computational model that bridges the gap between classical and quantum computing. This model, incorporating base-60 complex numbers and the concept of an "Infinity Ladder," offers a unique perspective on representing and manipulating quantum states. It leverages the cyclical and iterative nature of the MICT framework for state transitions.

The Challenge

Traditional computing models, while powerful, face limitations when dealing with certain types of complex problems. Quantum computing offers immense potential but presents challenges in terms of stability, error correction, and accessibility. Our model aims to address these challenges by providing a framework that integrates aspects of both classical and quantum computation.

Core Concepts

A. Qubit State

At the heart of our model is the qubit, the fundamental unit of quantum information. Unlike a classical bit, which can be either 0 or 1, a qubit can exist in a superposition of both states simultaneously. This is represented mathematically as:

|ψ⟩ = α|0⟩ + β|1⟩

where α and β are complex numbers representing probability amplitudes. The probabilities of measuring the qubit in state |0⟩ or |1⟩ are given by |α|² and |β|², respectively.

B. Base-60 Complex Numbers

We represent the complex numbers α and β using a base-60 polar form. This draws inspiration from ancient Babylonian mathematics, which used a base-60 system, and from the natural connection between base-60 and angular measurements (degrees, minutes, seconds). A complex number in our base-60 system is represented as:

z60 = (r60, θ60)

where r₆₀ is the magnitude in base-60, and θ₆₀ is the angle in base-60 (degrees, minutes, seconds). This representation may offer advantages in modeling quantum rotations and phases.

C. 3D State Space (S)

We map the base-60 complex numbers representing the qubit state to a 3D state space (S). While the precise mapping is an area of ongoing refinement, we are exploring the use of spherical coordinates, where the base-60 magnitude and angles could define the position of a point within this space.

D. Infinity Ladder (L)

The Infinity Ladder represents increasing levels of precision in our model. It's conceptualized as a sequence of nested intervals within the range [0, 1]. Each level, L_n, provides a finer-grained representation of probabilities. This allows us to model the transition from a probabilistic quantum state to a definite classical state (0 or 1) as we move up the ladder.

E. Probability Function (P)

The probability function P(s, l) defines the probability of being a a point s, in 3D space S, at a level l of the Infinity Ladder, within range [0,1]

MICT Stages within the Model

The MICT (Mapping, Iteration, Checking, Transformation) framework provides a cyclical process for evolving the quantum state and extracting classical information:

Mapping (M)

The initial qubit state (|ψ⟩) is converted to base-60 complex numbers (α₆₀, β₆₀) and then mapped to a point 's' in the 3D state space (S). An initial probability distribution, P(s, 0), is established at the coarsest level of the Infinity Ladder (L₀).

Iteration (I)

The quantum state evolves according to a defined set of quantum dynamics (equations to be further defined). This evolution is represented as a trajectory of 's' within the 3D state space. The probability distribution P(s, l) is updated at each step. The level of the Infinity Ladder (l) is incremented, representing increasing precision.

Checking (C)

The probability distribution P(s, l) is checked for convergence. If the probability of a specific state (|0⟩ or |1⟩) is sufficiently close to 1 (or 0), and a predetermined level of the Infinity Ladder has been reached, the process moves to the Transformation stage.

Transformation (T)

Once convergence criteria are met, a classical bit (0 or 1) is assigned as the final outcome, based on the converged probability distribution. The system transitions from a probabilistic quantum state to a definite classical state.

Further Refinements and Future Work

This model is in its early stages of development. Key areas for further refinement include:

• Precise mathematical mapping from base-60 complex numbers to the 3D state space.
• Definition of the quantum evolution equations within the Iteration stage.
• Establishment of clear convergence criteria for the Checking stage.
• Development of appropriate algorithms for the base-60 usage.

Potential Applications

This model, once fully developed, could have applications in:

• Quantum Computing Research: Providing a new framework for understanding and simulating quantum systems.
• Hybrid Classical-Quantum Algorithms: Developing algorithms that leverage the strengths of both classical and quantum computation.
• Complex Systems Modeling: Modeling complex systems that exhibit both classical and quantum behavior.

Join the Exploration

We invite researchers, developers, and enthusiasts to join us in exploring the potential of this new model. Visit our website at [Link to Website] to learn more and get involved.