Single-Qubit ZNE with Bloch-Ball Projection

Variant 1 of issue #243. Script: scripts/noise_mitigation_validation/single_qubit_bloch_ball_zne.py.

A single-qubit state is fixed by its Bloch vector r = (<X>, <Y>, <Z>), which needs only three measurement settings. Zero-noise extrapolation of each component can return |r| > 1, a state that does not exist. For one qubit the eigenvalues of the density matrix are (1 ± |r|)/2, so the eigenvalue projection onto the simplex of Smolin, Gambetta and Smith (arXiv:1106.5458) is exactly "shrink r to length 1". The script checks this against the library's project_to_physical: maximum difference 7.8e-16 over 200 random matrices.

Setup

• Haar-random pure single-qubit states, 200 seeds per row.
• Channels: depolarizing, amplitude damping, phase flip, base_p = 0.05, noise factors 1, 2, 3.
• K shots per basis per noise factor (binomial sampling of each Pauli expectation).
• Estimators at the same shots: plain 3-point Richardson; Richardson then clipping each component to [-1, 1] (the post hoc clipping discussed by Miranskyy, Sorrenti, Thind, Gravel, arXiv:2604.24475, Sect. III.D and VI.A); Richardson then Bloch-ball projection.
• Metric: trace distance to the ideal state, |r - n|/2, used only for grading.

Results

Channel	K	Projection acts	Mean gain vs Richardson	Max gain	Mean gain vs clipping	Ball better / worse than clipping	Wilcoxon p
depolarizing	100	137/200	0.0537	0.417	0.0353	132 / 5	4.1e-23
depolarizing	400	117/200	0.0223	0.231	0.0193	117 / 0	3.1e-21
depolarizing	1600	113/200	0.0061	0.095	0.0057	113 / 0	1.4e-20
amplitude damping	100	125/200	0.0483	0.345	0.0326	118 / 7	3.7e-21
amplitude damping	400	129/200	0.0237	0.137	0.0199	126 / 3	1.4e-22
amplitude damping	1600	114/200	0.0068	0.069	0.0066	114 / 0	9.6e-21
phase flip	100	143/200	0.0631	0.348	0.0427	137 / 6	1.6e-24
phase flip	400	117/200	0.0143	0.151	0.0117	115 / 2	6.5e-21
phase flip	1600	112/200	0.0085	0.086	0.0079	108 / 4	5.1e-20

Reading the table

• The projection acts on more than half of the seeds at every shot count: extrapolating a nearly pure state often overshoots the Bloch sphere.
• The gain over plain Richardson is never negative. This is guaranteed, not measured luck: the Bloch ball is convex and contains the ideal state, and a metric projection onto a convex set never moves a point further from any point of that set. The table measures how often and how much it helps.
• Against component clipping the ball projection wins in most active seeds and loses in a few (at most 7/200). Clipping only enforces each |<P>| ≤ 1, so the clipped vector can still lie outside the ball.
• The gain shrinks as shots grow (about 0.05 at K = 100, under 0.01 at K = 1600): the projection mostly removes overshoot caused by shot noise amplified by extrapolation.

Details

• project_to_physical gives the same matrices only with JAX 64-bit enabled; without it the library computed in 32-bit and differed by 4.5e-07. The script calls ensure_x64() first. The library's mitigation functions do not call it themselves, which is worth fixing in Dense-Evolution.
• Not yet tested: the multi-qubit product-state case (one projection per qubit) and real-device noise.