## Line Chart: vNMSE Performance Comparison Across Dimensions
### Overview
The chart compares the vNMSE (von Neumann Entropy Measure of State) performance of two quantum algorithms, **TurboQuant QJL (1-bit)** and **EDEN-unbiased (1-bit)**, across increasing dimensions (8 to 4096). The y-axis represents vNMSE values, while the x-axis represents computational dimensions. The chart includes a shaded uncertainty region for TurboQuant QJL.
### Components/Axes
- **X-axis (Dimension)**: Logarithmically spaced values: 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096.
- **Y-axis (vNMSE)**: Linear scale from 0.4 to 1.6, with increments of 0.2.
- **Legend**:
- **Blue line**: TurboQuant QJL (1-bit)
- **Green line**: EDEN-unbiased (1-bit)
- **Shaded Region**: Uncertainty bounds for TurboQuant QJL (light blue).
### Detailed Analysis
#### TurboQuant QJL (1-bit)
- **Trend**: Starts at ~1.42 (dimension 8) and increases gradually to ~1.58 (dimension 4096), with a shaded uncertainty region that narrows as dimension increases.
- **Key Data Points**:
- Dimension 8: 1.42 ± 0.05 (uncertainty)
- Dimension 16: 1.48 ± 0.04
- Dimension 32: 1.53 ± 0.03
- Dimension 64: 1.55 ± 0.02
- Dimension 128: 1.56 ± 0.02
- Dimension 256: 1.57 ± 0.01
- Dimension 512: 1.58 ± 0.01
- Dimensions 1024–4096: 1.58 ± 0.01 (stable)
#### EDEN-unbiased (1-bit)
- **Trend**: Remains nearly flat across all dimensions, starting at ~0.48 (dimension 8) and stabilizing at ~0.56 (dimension 128 onward).
- **Key Data Points**:
- Dimension 8: 0.48
- Dimension 16: 0.52
- Dimension 32: 0.54
- Dimension 64: 0.55
- Dimension 128: 0.56
- Dimensions 256–4096: 0.56 (no change)
### Key Observations
1. **TurboQuant QJL** exhibits a consistent upward trend in vNMSE as dimension increases, with diminishing uncertainty at higher dimensions.
2. **EDEN-unbiased** maintains a stable, low vNMSE across all dimensions, outperforming TurboQuant QJL by ~1.02 units at maximum dimension.
3. The shaded uncertainty region for TurboQuant QJL decreases in width as dimension increases, suggesting improved confidence in estimates at higher dimensions.
### Interpretation
- **Performance Implications**: EDEN-unbiased demonstrates superior stability and lower error propagation across dimensions, making it more reliable for high-dimensional computations. TurboQuant QJL’s performance degrades slightly with increasing dimension but stabilizes at ~1.58 vNMSE, with reduced uncertainty at larger scales.
- **Uncertainty Analysis**: The narrowing shaded region for TurboQuant QJL indicates that measurement confidence improves as dimensions grow, though its baseline error remains higher than EDEN-unbiased.
- **Algorithmic Insight**: The divergence between the two algorithms suggests fundamental differences in error handling or state representation, with EDEN-unbiased potentially offering a more robust framework for quantum state estimation.
