Classical Shadow Tomography & Quantum Learning

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]

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?

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.

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.

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.

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.