Single-Qubit Coherence Nudge for ZNE: the Gain Is Not the Signal (the Multi-Qubit One Holds)¶

Variant 2 of issue #243. Script: scripts/noise_mitigation_validation/single_qubit_coherence_zne.py. Negative result.

The library's coherence_predictive_zne_density_matrix lowers the weight of the 3-point Richardson extrapolation when the l1 coherence (Baumgratz, Cramer, Plenio, PRL 113, 140401 (2014)) changes non-linearly across the noise scales, then projects onto a physical state. Here the same core runs on one qubit, on the 2×2 matrix rebuilt from single-qubit tomography (X, Y, Z, K shots each), so the signal is 2|ρ₀₁|.

First test: looks like a clear win¶

Haar-random pure states, 200 seeds, base_p = 0.05, factors 1, 2, 3. Baseline: the same core with nudge_scale = 0 (Richardson plus projection), as in the original validation of the multi-qubit version. Counted on seeds where the nudge activates, as there.

Channel	K	Active	Mean gain (active)	Better / worse	t-test p
phase flip	100	96/200	0.0308	94 / 2	1.0e-23
phase flip	400	100/200	0.0185	100 / 0	3.0e-20
phase flip	1600	95/200	0.0064	94 / 1	3.1e-12
amplitude damping	100	105/200	0.0260	103 / 2	7.4e-18
amplitude damping	400	106/200	0.0144	101 / 5	1.5e-15
amplitude damping	1600	97/200	0.0084	95 / 2	2.0e-20
depolarizing	100	104/200	0.0286	101 / 3	2.8e-20
depolarizing	400	105/200	0.0176	102 / 3	9.8e-21
depolarizing	1600	102/200	0.0087	101 / 1	5.7e-22

Control: the same nudges at random, and one fixed nudge¶

400 seeds, mean gain over all seeds against nudge_scale = 0.

Channel	K	Signal	Same nudges shuffled across seeds	One constant nudge on every seed
phase flip	100	0.0149	0.0117	0.0216
phase flip	400	0.0090	0.0070	0.0141
phase flip	1600	0.0028	0.0023	0.0044
amplitude damping	100	0.0133	0.0126	0.0225
amplitude damping	400	0.0079	0.0070	0.0147
amplitude damping	1600	0.0042	0.0037	0.0074
depolarizing	100	0.0148	0.0118	0.0232
depolarizing	400	0.0084	0.0069	0.0139
depolarizing	1600	0.0038	0.0032	0.0062

Reading it¶

Shuffled nudges keep most of the gain, and a constant nudge on every seed does better than the signal. The improvement comes from extrapolating less aggressively, which amplifies shot noise less, not from the coherence signal choosing when to act.
The signal adds only a little over random placement (0.0005 to 0.003).
A constant nudge trades variance for bias: it moves the estimate toward the noisy data, so at higher noise or with many more shots it can lose. It is not a free improvement and is not promoted.
The multi-qubit version passes the same controls (next section): there the signal does carry the gain.

Re-check of the multi-qubit validation with the same controls¶

Script: scripts/noise_mitigation_validation/coherence_zne_shuffled_control.py. Exactly the setup of #208 (GHZ(4), phase flip, base_p = 0.05, 150 trajectories per scale, seeds 900000 + i, Uhlmann fidelity); it reproduces the original 63/200 active, 63/63 positive, p = 1.07e-8.

Nudge	Mean gain	Better / worse	t-test p
coherence signal (library)	0.00469	63 / 0	1.1e-08
same nudges shuffled across seeds	0.00072	17 / 14	0.16
constant nudge on every seed	0.00243	65 / 43	4.5e-06

The signal beats shuffled placement by 0.0040 (paired p = 5.3e-06) and the constant nudge by 0.0023 (p = 3.9e-06). On the full GHZ(4) state, where the coherence sits in the |0000><1111| element, the l1 signal really chooses when to act; on a single qubit, with one off-diagonal element, it does not.

Details¶

In a GHZ state each single-qubit reduced state is I/2, so the local signal is zero at every scale and never fires; that is why this test uses single-qubit pure states.
Requires ensure_x64() (called by the script) until Dense-Evolution#343 is merged.