H₃ — Stabilizer Rank Challenge

The state

| H_{3} ⟩ = \sqrt{\frac{3 + \sqrt{3}}{6}} | 0 ⟩ + \sqrt{\frac{3 - \sqrt{3}}{12}} (| 1 ⟩ + | 2 ⟩)

The eigenvector of the qutrit Fourier transform with eigenvalue 1. The qutrit analogue of the qubit H-type state, and like it the natural target of distillation.

Its m=4 cell sits at 8; rank 6 terminates at exactly √(70−37√3)/12, identified to fifteen digits across five runs on two independent engines.

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| H_{3} ⟩}^{\otimes 1}) \geq 2$	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| H_{3} ⟩}^{\otimes 1}) \leq 2$	1	2	0.6309	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| H_{3} ⟩}^{\otimes 2}) \geq 3$	2	3	—	lean	@vprusso · stabrank PR #18
$χ_{R} ({\| H_{3} ⟩}^{\otimes 2}) \leq 3$	2	3	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 3}) \geq 4$	3	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 3}) \leq 4$	3	4	0.4206	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 3$	4	3	—	reproduced	@vprusso · stabrank rank_exclusion_codes.py
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 5$	4	5	—	reproduced · 3600 s	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 6$	4	6	—	attested · 2/57 re-run	@vprusso · stabrank research/qutrit_m4_rank5
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \geq 7$	4	7	—	attested · 2/1021 re-run	@vprusso · stabrank research/h3_m4_rank6
$χ_{R} ({\| H_{3} ⟩}^{\otimes 4}) \leq 8$	4	8	0.4732	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H_{3} ⟩}^{\otimes 5}) \geq 7$	5	7	—	attested · 2/1021 re-run	@vprusso · stabrank research/h3_m4_rank6
$χ_{R} ({\| H_{3} ⟩}^{\otimes 5}) \leq 12$	5	12	0.4524	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| H_{3} ⟩}^{\otimes 6}) \geq 7$	6	7	—	attested · 2/1021 re-run	@vprusso · stabrank research/h3_m4_rank6
$χ_{R} ({\| H_{3} ⟩}^{\otimes 6}) \leq 16$	6	16	0.4206	verified	@vprusso · stabrank research/constructions/product_witness.py

What would move it

Beating the published exponent needs χ ≤ 15 at m=6→ 0.4108.