BK T-type — Stabilizer Rank Challenge

The state

| T ⟩ = \cos β | 0 ⟩ + e^{i π / 4} \sin β | 1 ⟩, \cos 2 β = \frac{1}{\sqrt{3}}

The qubit Bravyi-Kitaev T-type state, the face centre of the stabilizer octahedron. A different Clifford orbit from the H-type despite the overloaded name.

Rank 5 at six copies is conjectured impossible, with search terminating at exactly √(5/6) sin(π/12).

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| T ⟩}^{\otimes 1}) \geq 2$	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T ⟩}^{\otimes 1}) \leq 2$	1	2	1.0000	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| T ⟩}^{\otimes 2}) \geq 2$	2	2	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T ⟩}^{\otimes 2}) \leq 2$	2	2	0.5000	lean	@vprusso · stabrank annealer
$χ_{R} ({\| T ⟩}^{\otimes 3}) \geq 3$	3	3	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| T ⟩}^{\otimes 3}) \leq 3$	3	3	0.5283	lean	@vprusso · stabrank annealer
$χ_{R} ({\| T ⟩}^{\otimes 4}) \geq 3$	4	3	—	reproduced	@vprusso · stabrank PR #18
$χ_{R} ({\| T ⟩}^{\otimes 4}) \leq 3$	4	3	0.3962	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 3$	5	3	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 4$	5	4	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 5$	5	5	—	attested · 2/6 re-run	@vprusso · stabrank research/t5_rank4
$χ_{R} ({\| T ⟩}^{\otimes 5}) \geq 6$	5	6	—	attested · 2/160 re-run	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 5}) \leq 6$	5	6	0.5170	lean	@vprusso · product of board witnesses
$χ_{R} ({\| T ⟩}^{\otimes 6}) \geq 4$	6	4	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| T ⟩}^{\otimes 6}) \geq 5$	6	5	—	attested · 2/6 re-run	@vprusso · stabrank research/t5_rank4
$χ_{R} ({\| T ⟩}^{\otimes 6}) \geq 6$	6	6	—	attested · 2/160 re-run	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 6}) \leq 6$	6	6	0.4308	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| T ⟩}^{\otimes 7}) \geq 6$	7	6	—	attested · 2/160 re-run	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 7}) \leq 9$	7	9	0.4528	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T ⟩}^{\otimes 8}) \geq 6$	8	6	—	attested · 2/160 re-run	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 8}) \leq 9$	8	9	0.3962	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| T ⟩}^{\otimes 9}) \geq 6$	9	6	—	attested · 2/160 re-run	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 9}) \leq 15$	9	15	0.4341	verified	@vprusso · arxiv:2610.17500 (the m=10 identity) and stabrank bounds/qubit_T-m10-upper-15.json
$χ_{R} ({\| T ⟩}^{\otimes 10}) \geq 6$	10	6	—	attested · 2/160 re-run	@vprusso · stabrank research/t5_rank5
$χ_{R} ({\| T ⟩}^{\otimes 10}) \leq 15$	10	15	0.3907	verified	@sjbevins · arxiv:2610.17500
$χ_{R} ({\| T ⟩}^{\otimes 10}) \leq 18$	10	18	0.4170	verified	@vprusso · stabrank research/constructions/product_witness.py

What would move it

The published exponent is already beaten: χ ≤ 15 at m=10 gives γ = 0.3907. Lowering it further needs χ ≤ 6 at m=7→ 0.3693.