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# Black Hole Mechanics in Quantum Image Compression
## Hawking Radiation and Information Theory The `HawkingProjector` class uses black hole physics as inspiration for quantum data compression. The fundamental concept relates to how information behaves near a black hole's event horizon, particularly drawing from Hawking radiation theory.
$$ \mathcal{H}(\rho) = -\text{Tr}(\rho \log \rho) $$
Where $\mathcal{H}(\rho)$ represents the von Neumann entropy of a quantum state $\rho$, analogous to the entropy of a black hole.
## Black Hole Information Paradox The compression algorithm draws inspiration from the black hole information paradox, where information that falls into a black hole appears to be lost but is theoretically preserved in a scrambled form. In the code, this is implemented as:
$$ \text{entropy} = -\sum_{i} p_i \log_2(p_i) $$
This is calculated in the `analyze_entropy` method, which computes the Shannon entropy as an approximation of von Neumann entropy.
## Quantum Circuit Encoding The encoding process uses quantum circuits that preserve structural information, similar to how information might be encoded on a black hole's event horizon:
$$ |\psi\rangle = \sum_{i=0}^{2^n-1} \alpha_i |i\rangle $$
Where $|\psi\rangle$ is the quantum state representing encoded image data, and $\alpha_i$ are complex amplitudes.
## Holographic Principle The `create_quantum_circuit` method implements a form of the holographic principle, where the information content of a region is encoded on its boundary:
$$ S \propto \frac{A}{4G\hbar} $$
This equation from black hole thermodynamics relates entropy ($S$) to the area ($A$) of the event horizon. In the code, this manifests as spatial locality preservation in the quantum encoding:
``` # Custom entanglement pattern that preserves locality sqrt_n = int(np.sqrt(self.num_qubits)) ```
## Information Scrambling with Controlled Rotation The quantum circuits use controlled rotation gates to simulate the scrambling of information that occurs near a black hole:
$$ R_y(\theta) = e^{-i\theta\sigma_y/2} = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix} $$
Where $\theta$ is the `ROTATION_ANGLE` parameter that controls information scrambling. This is implemented as:
```python # Controlled scrambling with configurable rotation angle for i in range(self.num_qubits):
qc.ry(self.config['ROTATION_ANGLE'], i)
```
## Weighted Reconstruction Model The reconstruction algorithm uses a weighted combination of quantum states, inspired by how information might be retrieved from Hawking radiation:
$$ \text{reconstructed\_bit}_i = \begin{cases} 1 & \text{if } \sum_{j} p_j \cdot \text{bit}_j^i \geq \text{threshold} \\ 0 & \text{otherwise} \end{cases} $$
Where $p_j$ are the probabilities of different quantum states, and $\text{bit}_j^i$ is the $i$-th bit of the $j$-th state.
## Entropy Ratio Analysis The code calculates an entropy ratio that measures how close the system is to maximum entropy:
$$ \text{entropy\_ratio} = \frac{\text{entropy}}{\text{max\_entropy}} = \frac{\text{entropy}}{n} $$
Where $n$ is the number of qubits. This relates to the idea that black holes represent maximum entropy configurations.
## Context-Aware Reconstruction The spatial relationships in the image are preserved using a context-aware reconstruction method that treats adjacent blocks as entangled systems:
$$ \text{reconstructed\_array}[y:y+h, x:x+w] += \text{block\_array} \cdot \text{weight} $$
This is implemented in the `reconstruct_with_context` method.
## Performance Metrics Inspired by Black Hole Physics The reinforcement learning environment uses a reward function that incorporates metrics inspired by black hole information theory:
$$ \text{reward} = w_1 \cdot \text{pixel\_match\_rate} + w_2 \cdot \text{edge\_preservation} + w_3 \cdot \text{detail\_preservation} + \ldots $$
These metrics measure how well the information is preserved through the compression and reconstruction process, similar to how information might theoretically be preserved despite passing through a black hole.