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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.11557 |
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| _version_ | 1866910211132882944 |
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| author | Kam, Chon-Fai Cadet, Xavier Bessafi, Miloud Cadet, Frederic |
| author_facet | Kam, Chon-Fai Cadet, Xavier Bessafi, Miloud Cadet, Frederic |
| contents | While world models learn compact representations of complex environments, they lack a physics-grounded metric to assess the structural fidelity of their latent spaces. We identify the wavelet scaling exponent $α$ as a critical diagnostic, proposing optimal representations satisfy variance equipartition ($α\approx 1/2$) -- mirroring Kolmogorov's inertial range. We establish $α= 1/2$ as a sharp transition boundary for the classical simulability of amplitude-encoded quantum kernels. Using tensor-network theory, we prove latents with $α> 1/2$ reside in an area-law phase admitting efficient classical emulation, while $α< 1/2$ triggers a volume-law phase where the Matrix Product State bond dimension $χ$ grows exponentially with qubit count $n$. Analyzing pre-trained VideoMAE latents reveals a dichotomy: spatial tokens approach the equipartition limit ($α\approx 0.423$), but permutation-invariant feature channels exhibit unstructured disorder ($α\approx -0.123$). This forces real-world latents deep into the volume-law phase, providing a data-driven necessary condition for simulation hardness. Finally, we apply Weingarten calculus to derive the exact variance of the scrambled transition probability under a 2-design ensemble. We prove this variance scales strictly as $\Var[X] = Θ(d^{-2})$. We confirm this numerically with a log-log slope of $-1.881$ ($R^2 = 0.999$), identifying a formidable shot-noise wall demanding a measurement budget of $M = Ω(d^2)$ that constrains quantum machine learning scalability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11557 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wavelet Variance Equipartition as a Threshold for World-Model Quality and Quantum Kernel TN-Simulability Kam, Chon-Fai Cadet, Xavier Bessafi, Miloud Cadet, Frederic Quantum Physics While world models learn compact representations of complex environments, they lack a physics-grounded metric to assess the structural fidelity of their latent spaces. We identify the wavelet scaling exponent $α$ as a critical diagnostic, proposing optimal representations satisfy variance equipartition ($α\approx 1/2$) -- mirroring Kolmogorov's inertial range. We establish $α= 1/2$ as a sharp transition boundary for the classical simulability of amplitude-encoded quantum kernels. Using tensor-network theory, we prove latents with $α> 1/2$ reside in an area-law phase admitting efficient classical emulation, while $α< 1/2$ triggers a volume-law phase where the Matrix Product State bond dimension $χ$ grows exponentially with qubit count $n$. Analyzing pre-trained VideoMAE latents reveals a dichotomy: spatial tokens approach the equipartition limit ($α\approx 0.423$), but permutation-invariant feature channels exhibit unstructured disorder ($α\approx -0.123$). This forces real-world latents deep into the volume-law phase, providing a data-driven necessary condition for simulation hardness. Finally, we apply Weingarten calculus to derive the exact variance of the scrambled transition probability under a 2-design ensemble. We prove this variance scales strictly as $\Var[X] = Θ(d^{-2})$. We confirm this numerically with a log-log slope of $-1.881$ ($R^2 = 0.999$), identifying a formidable shot-noise wall demanding a measurement budget of $M = Ω(d^2)$ that constrains quantum machine learning scalability. |
| title | Wavelet Variance Equipartition as a Threshold for World-Model Quality and Quantum Kernel TN-Simulability |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.11557 |