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Can AI Generate Theories? A Simplified Fibonacci-Based Quantum Optimization Model Part 1

Can AI generate theories was the question...

Robert Mourey Jr

Robert Mourey Jr

— 4 min read
Can AI Generate Theories? A Simplified Fibonacci-Based Quantum Optimization Model Part 1
Generated by Adobe Firefly AI using a prompt created by Robert Mourey Jr.

Can AI Generate Theories was the question I asked myself the other day. And so I thought why not see what Google Gemini can do with a few original ideas and some tinkering. Below we have the results of such a chat. In the Fibonacci-Based Quantum Optimization Model series, we'll explore how wecould theoretically develop a simplified, Fibonacci-based quantum optimization model. Could Fibonacci, once again, prove to be a key to unlocking secrets of our universe?

Problem Selection: Quadratic Unconstrained Binary Optimization (QUBO)

For our simplified model, let's focus on the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is a well-studied problem in optimization and has applications in various fields, including machine learning and finance. A QUBO problem can be expressed as:

minimize Q(x) = sum(sum(Qij * xi * xj))
subject to xi ∈ {0, 1}

where Q is a symmetric matrix of coefficients, x is a vector of binary variables, and the objective is to find the binary vector x that minimizes the quadratic form Q(x).

Implementing a Fibonacci-Based Quantum Optimization Model in Rust

Understanding the Scope

Before diving into the code, let's clarify the scope:

  • Quantum Simulator: We'll create a basic quantum simulator to simulate quantum circuits.
  • Fibonacci Integration: We'll explore various ways to incorporate Fibonacci numbers into circuit parameters and structure.
  • Optimization: We'll implement a basic optimization loop to find optimal circuit parameters for a given QUBO problem.
  • Security Considerations: We'll implement basic obfuscation techniques to protect potential vulnerabilities.

Rust Data Structures and Modules

To structure our code, we can use the following modules and data structures:

mod qubit;
mod gate;
mod circuit;
mod optimizer;

#[derive(Debug, Clone, Copy)]
struct Qubit {
    // ...
}

#[derive(Debug, Clone, Copy)]
enum Gate {
    X,
    Y,
    Z,
    H,
    CNOT,
    // ...
}

struct Circuit {
    qubits: Vec<Qubit>,
    gates: Vec<Gate>,
    // ...
}

Fibonacci Integration

We can create a module to handle Fibonacci-related calculations:

mod fibonacci {
    pub fn fib(n: usize) -> u64 {
        // ...
    }

    pub fn golden_ratio() -> f64 {
        // ...
    }

    // Other Fibonacci-related functions
}

To incorporate Fibonacci numbers into circuit parameters, we can:

  • Fibonacci-Based Angle Generation:
    fn fibonacci_angles(num_angles: usize) -> Vec<f64> {
    let mut angles = Vec::new();
    for i in 0..num_angles {
        let fib_num = fibonacci::fib(i) as f64;
        let angle = fib_num * std::f64::consts::PI / 180.0; // Convert to radians
        angles.push(angle);
    }
    angles
}
  • Fibonacci-Based Circuit Structure:
    fn fibonacci_circuit(num_qubits: usize) -> Circuit {
    // Use Fibonacci numbers to determine number of layers, gates per layer, etc.
}
  • Fibonacci-Based Parameterization:
    fn fibonacci_parameters(num_params: usize) -> Vec<f64> {
    // Use Fibonacci numbers or ratios to generate parameters
}

Optimization and Security

The optimizer module will handle the optimization process. We can use a simple gradient-based optimizer or explore more advanced techniques.

For security, we can:

  • Randomize Circuit Structure: Introduce random variations in the circuit structure.
  • Noise Injection: Add random noise to circuit parameters.
  • Parameter Encryption: Encrypt sensitive parameters before storing or transmitting them.

Exploring Fibonacci Integration Techniques

Fibonacci-Based Parameterization

We've already touched on using Fibonacci numbers to generate angles for single-qubit rotations. Let's delve deeper into this:

  • Golden Ratio as a Scaling Factor: We can use the golden ratio (φ) to scale the Fibonacci numbers before converting them to angles. This can introduce additional variability in the parameter space.
    fn fibonacci_angles_with_golden_ratio(num_angles: usize) -> Vec<f64> {
    let golden_ratio = fibonacci::golden_ratio();
    // ...
}
  • Fibonacci Sequence as a Phase Shift: Instead of using Fibonacci numbers directly for angles, we can use them as indices into a precomputed phase shift sequence. This can introduce more complex patterns.
    fn fibonacci_phase_shifts(num_shifts: usize) -> Vec<f64> {
    // ...
}

Fibonacci-Based Circuit Structure

  • Fibonacci-Inspired Layer Structure: We can use Fibonacci numbers to determine the number of gates in each layer of the circuit.
  • Fibonacci-Based Qubit Coupling: We can use Fibonacci numbers to determine which qubits are coupled in entangling gates.

Fibonacci-Based Quantum Gate Definitions

  • Fibonacci-Parameterized Gates: We can create custom quantum gates whose parameters are derived from Fibonacci numbers.
    struct FibonacciGate {
    // ...
}

Combining Techniques

We can combine these techniques to create more complex and potentially more effective quantum circuits. For example:

  • Use Fibonacci numbers to determine the number of layers and qubits in the circuit.
  • Use golden ratio scaled Fibonacci numbers for rotation angles.
  • Introduce Fibonacci-based phase shifts for additional complexity.

Security Considerations

While we'll focus on the core implementation first, it's important to keep security in mind. Some potential obfuscation techniques:

  • Dynamic Fibonacci Sequences: Generate Fibonacci sequences with random starting values.
  • Parameter Encryption: Encrypt circuit parameters before storing or transmitting them.
  • Circuit Structure Randomization: Introduce random variations in the circuit structure.

Implementing Basic Quantum Gates and Circuits

Qubit Representation

We'll represent qubits as complex numbers in the form of a vector:

use num_complex::Complex;

#[derive(Debug, Clone, Copy)]
struct Qubit {
    state: Complex<f64>,
}

Quantum Gate Implementation

We can define basic quantum gates as functions that operate on qubits or pairs of qubits:

fn x(qubit: &mut Qubit) {
    // ...
}

fn h(qubit: &mut Qubit) {
    // ...
}

fn cnot(control: &mut Qubit, target: &mut Qubit) {
    // ...
}

Quantum Circuit Implementation

A quantum circuit can be represented as a sequence of gates applied to a set of qubits:

struct Circuit {
    qubits: Vec<Qubit>,
    gates: Vec<Box<dyn Fn(&mut Vec<Qubit>)>>,
}

Fibonacci Integration and Security

We can incorporate Fibonacci-based parameters and security measures as we build out the circuit:

fn create_fibonacci_circuit(num_qubits: usize) -> Circuit {
    let mut circuit = Circuit::new(num_qubits);

    // Generate dynamic Fibonacci sequence
    let fib_seq = generate_dynamic_fibonacci_sequence();

    // Use Fibonacci numbers and golden ratio for parameterization
    // ...

    // Apply gates with encrypted parameters
    // ...

    circuit
}

In the Next Article

We'll need to:

  • Implement the logic for the X, H, and CNOT gates.
  • Define the Circuit::new constructor.
  • Implement the generate_dynamic_fibonacci_sequence function.
  • Explore methods for parameter encryption.

Read more