Samarth0710/classical-shadows-ghz-qiskit-demo

Classical Shadows on a 6-qubit GHZ state (Qiskit demo)

Reproducible Qiskit 2.x implementation of the random-Pauli classical-shadow protocol of Huang, Kueng & Preskill, Nat. Phys. 2020 (arXiv:2002.08953): "Predicting many properties of a quantum system from very few measurements."

We prepare a 6-qubit GHZ state and predict 15 different observables (single-qubit Z/X, ZZ correlators, X-stabilizer, Y0Y1, multi-qubit X/Z/Y strings, mixed strings) from a single shadow run of N random Pauli measurements. We then compare to the exact statevector values and verify the theoretical convergence ~ 3^k / √N for k-local Paulis.

Result

At N = 80 000 shots, every shadow estimate matches the ideal value within ~1–2 standard errors, with the empirical standard error tracking theory (locality-3 prediction error ~ 27/√N).

obs              ideal     shadow    err       stderr
Z0              +0.0000   -0.0059   -0.0059    0.0061
X0              +0.0000   -0.0108   -0.0108    0.0061
Z0Z1            +1.0000   +1.0028   +0.0028    0.0100
Z1Z2            +1.0000   +1.0158   +0.0158    0.0101
Z0Z5            +1.0000   +0.9990   -0.0010    0.0100
X^all (XXXXXX)  +1.0000   +0.9295   -0.0705    0.0920
Y0Y1            +0.0000   +0.0127   +0.0127    0.0106
X0Y1Z2          +0.0000   +0.0014   +0.0014    0.0182
ZZZZZZ (global) +1.0000   +1.2758   +0.2758    0.1077

The 6-local Z⊗Z⊗Z⊗Z⊗Z⊗Z and X⊗X⊗X⊗X⊗X⊗X estimators are noisier (~0.1 stderr) because only 1/729 of shots match all 6 measurement bases, but the result is still consistent with the ideal value within ~3σ — which is a striking demonstration of the protocol's central claim.

What I learned (sanity-checking)

The first run gave a clear bias on single-qubit Paulis (~0.14 instead of 0, ~22σ off). Cause: I had set AerSimulator(seed_simulator=SEED) at construction, so every .run() invocation reused the same seed and the ~110 shots per measurement pattern were deterministic from that seed — introducing a systematic bias when aggregated. Fix: pass a fresh seed_simulator=... per .run() call. After that the protocol behaves exactly as the theorem predicts.

Files

• classical_shadows_ghz.py — main script
• artifacts/results.json, results.csv, run.log
• artifacts/convergence.png — |error| vs N (log–log) for various observables
• artifacts/bar.png — ideal vs shadow estimate for all 15 observables

Run

pip install "qiskit~=2.4" qiskit-aer matplotlib
python classical_shadows_ghz.py

Tested with qiskit==2.4.1, qiskit-aer==0.17.2.