Extending i-th Order Tensor Characterization to Non-Diagonal Clifford Hierarchy Operators

A computational investigation of whether the tensor framework of Bauer et al. (arXiv:2601.15396) extends beyond diagonal operators in the Clifford hierarchy.

351 Operators Tested

1.0 Diagonal Verification

-0.822 Diag Weight Correlation

136 Conjugation Tests

90.56s Runtime

Key Findings

Finding 1: All 85 diagonal operators achieve perfect tensor fits (rate 1.0), confirming Proposition 6.1.

Finding 2: Mean monomial residual for non-diagonal operators is 1.051, indicating the strict (q, epsilon) framework does not extend directly.

Finding 3: Diagonal weight anticorrelates with monomial residual (r = -0.822 for d=4).

Finding 4: Tensor form preserved at rate 1.0 under Clifford conjugation across all 136 tests.

Operator Classification

Classification Breakdown

Classification Summary

Category	Count	Fraction
Diagonal + Tensor	21	5.98%
Monomial + Tensor	30	8.55%
Monomial - No Tensor	0	0%
Non-Mono + Approx Tensor	300	85.47%
Non-Mono - Obstructed	0	0%
Total	351	100%

Diagonal Verification

Success Rate by (d, Level)

Verification Details

d	Level	Operators	Success	Rate
2	1	2	2	1.0
2	2	4	4	1.0
2	3	8	8	1.0
3	1	3	3	1.0
3	2	9	9	1.0
3	3	10	10	1.0
4	1	4	4	1.0
4	2	10	10	1.0
4	3	10	10	1.0
5	1	5	5	1.0
5	2	10	10	1.0
5	3	10	10	1.0

Perturbation Analysis

Monomial Residual vs Perturbation

Diagonal Weight vs Perturbation

Conjugation Stability

Preservation Rate

Conjugation Test Results

d	Level	Tests	Preserved	Rate
2	1	16	16	1.0
2	2	24	24	1.0
3	1	24	24	1.0
3	2	24	24	1.0
4	1	24	24	1.0
4	2	24	24	1.0
Total		136	136	1.0