Predicting Many Properties of a Quantum System from Very Few Measurements

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Claim: a simple protocol called classical shadows stores inverted post-measurement snapshots that are unbiased estimators of ρ, then uses median-of-means to predict many linear targets with only O((max shadow‑norm(O))/ε^2 · log m) single‑copy measurements. Using global Clifford sampling the shadow‑norm reduces to ∥O∥_2^2 and snapshots are scaled stabilizer states; using local Clifford (random Pauli) the cost is independent of dimension but scales ≲4^k with observable locality k. Numerics show constant sample cost up to >100 qubits and ~10^4× faster classical post‑processing than neural‑network tomography; Renyi-R predictions extend via U‑statistics with an R factor.

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" chapters (12)

1. 0:00Intro
2. 2:38Table of Contents
3. 7:14Quantum state tomography
4. 10:08Shadow tomography
5. 16:05Main question
6. 23:23Predicting a single linear property
7. 28:59Predicting many linear properties
8. 34:36General result
9. 37:23Example l: global Clifford circuits
10. 48:33Example 1: local Clifford circuits
11. 56:26Numerics I+II: runtime comparison
12. 57:45Classical shadows: asymptotic optimality