# Classical Shadow Tomography & Quantum Learning
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**Anchor paper:** K. Agarwal, P.G. Rozon, N. Bao, "Optimal twirling depth for classical shadows in the presence of noise," *Phys. Rev. Lett.* 133, 130803 (2024). [arXiv:2311.10137]
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## Background
Classical shadow tomography estimates many properties of an unknown quantum state from few measurements by applying random "twirling" unitaries before a fixed measurement basis, then classically post-processing the snapshots — sample complexity independent of system size for finite-weight observables. Prior noiseless-case work showed shallow-depth random circuits achieve optimal sample complexity, but real devices introduce gate/measurement noise during twirling itself, degrading snapshot fidelity. Open question: does an optimal (finite) twirling depth exist under realistic noise, and how does it depend on noise strength?
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## New results
Under single-qubit noise (provably reducible to an effective depolarizing channel with one damping parameter f, regardless of underlying noise type), the authors show there is generically a finite optimal twirling depth rather than an arbitrarily deep circuit. For noise strength f and qudit dimension q there's a threshold f_th below which purely local twirling (depth 0) is optimal, and above it an upper bound on optimal depth that stays remarkably small even for weak noise (e.g. depth 3 for qubits at f=0.99) — giving experimentalists a practical, noise-dependent prescription for how deep to twirl.
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![Optimal twirling depth vs. noise strength](https://arxiv.org/html/2311.10137v2/x2.png)
*Fig. 2a — upper bound on optimal twirling circuit depth vs. noise damping parameter f, for qudit dimensions q=2-6, showing the noise thresholds below which local twirling is optimal.*
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## Related work in this direction
- P.G. Rozon, K. Agarwal, "Learning shadows to predict quantum ground state correlations," arXiv:2508.00052 (2025); K. Agarwal, "Learning quantum ground states in the space of measurement outcomes" (solo, 2026) — extend this noise-aware shadow framework into a variational method representing quantum ground states as "bags of measurement snapshots," learning correlations without reconstructing an explicit density matrix.
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**Quantum Hopfield Networks** and **Toy Model for Black Hole Decoherence** are both nested here: the former shares the broader "quantum-enhanced information processing/learning" framing, and the latter shares both a collaborator (N. Bao) and the underlying question of how much quantum information survives noise and decoherence.
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# Parents
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* Topological Matter & Quantum Information⏎