Qubit Superposition & Rotation Simulator
Adjust the rotation angle θ using the slider below to observe how the state vector
|ψ⟩ = α|0⟩ + β|1⟩ shifts and probabilities
|α|² vs |β|² evolve.
State Amplitudes
|ψ⟩ = 0.71|0⟩ + 0.71|1⟩ Probability Distributions
1. Overview
Quantum Genetic Algorithms (QGA) apply quantum computational principles — specifically qubit superposition and unitary rotation gates — to evolutionary optimization on classical architectures.
This lab explores how a population of quantum chromosomes searches multi-dimensional solution spaces with higher diversity and faster convergence than standard bitstring genetic algorithms.
2. The Problem: Combinatorial Explosion
In the 0-1 Knapsack problem, selecting from $n$ candidate items presents a search space of $2^n$ combinations. For $n = 50$, the state space contains over $10^{15}$ configurations. Classical genetic algorithms frequently get trapped in deceptive local optima because binary crossover destroys beneficial allele combinations (the Hamming cliff problem).
3. Quantum Mechanics Theory
Qubit State Vector
A qubit state $|\psi\rangle$ exists in a continuous linear superposition of the basis vectors $|0\rangle$ and $|1\rangle$:
$$|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$$
where $|\alpha|^2$ and $|\beta|^2$ represent the probability of observing $|0\rangle$ and $|1\rangle$, constrained by:
$$|\alpha|^2 + |\beta|^2 = 1$$
In polar coordinates:
$$\alpha = \cos(\theta), \quad \beta = \sin(\theta)$$
Quantum Rotation Gate
To evolve the quantum chromosome toward optimal configurations, we apply a rotation gate matrix $U(\Delta \theta)$:
$$\begin{bmatrix} \alpha’ \ \beta’ \end{bmatrix} = \begin{bmatrix} \cos(\Delta \theta) & -\sin(\Delta \theta) \ \sin(\Delta \theta) & \cos(\Delta \theta) \end{bmatrix} \begin{bmatrix} \alpha \ \beta \end{bmatrix} = \begin{bmatrix} \cos(\theta + \Delta \theta) \ \sin(\theta + \Delta \theta) \end{bmatrix}$$
4. Architecture & Simulation Loop
┌────────────────────────────────────────────────────────┐
│ Initialize Quantum Chromosome Pop │
│ (All qubits α = β = 1/√2) │
└───────────────────────────┬────────────────────────────┘
│
▼
┌────────────────────────────────────────────────────────┐
│ Probabilistic Collapse & Measurement │
│ (Generate classical bitstrings X via |β|²) │
└───────────────────────────┬────────────────────────────┘
│
▼
┌────────────────────────────────────────────────────────┐
│ Evaluate Knapsack Fitness f(X) │
│ (Track global best individual B) │
└───────────────────────────┬────────────────────────────┘
│
▼
┌────────────────────────────────────────────────────────┐
│ Update Amplitudes via Rotation Gates │
│ (Rotate toward best state B by angle Δθ) │
└───────────────────────────┬────────────────────────────┘
│ Repeat until convergence
▼
5. Python Implementation Snippet
import numpy as np
class QuantumChromosome:
def __init__(self, num_qubits: int):
# Initialize each qubit in equal superposition: |α|² = |β|² = 0.5
self.num_qubits = num_qubits
self.alpha = np.full(num_qubits, 1.0 / np.sqrt(2))
self.beta = np.full(num_qubits, 1.0 / np.sqrt(2))
def measure(self) -> np.ndarray:
"""Collapse qubits into classical binary bits based on |β|² probability."""
probs = self.beta ** 2
rand = np.random.random(self.num_qubits)
return (rand < probs).astype(int)
def rotate(self, delta_theta: np.ndarray):
"""Apply unitary rotation gate U(Δθ) to all qubits."""
cos_dt = np.cos(delta_theta)
sin_dt = np.sin(delta_theta)
new_alpha = cos_dt * self.alpha - sin_dt * self.beta
new_beta = sin_dt * self.alpha + cos_dt * self.beta
self.alpha, self.beta = new_alpha, new_beta
6. What I Learned
- Quantum representation allows a tiny population ($N = 10$) to maintain the genetic diversity of a classical population ten times its size.
- Dynamic rotation angle scheduling ($\Delta \theta$ decaying from $0.05\pi$ to $0.005\pi$) prevents oscillations around global optima.
- Provides a bridge toward native gate programming for NISQ (Noisy Intermediate-Scale Quantum) devices.