H-type — Stabilizer Rank Challenge

The state

| H ⟩ = \cos (π / 8) | 0 ⟩ + \sin (π / 8) | 1 ⟩

The qubit H-type state, the edge centre of the stabilizer octahedron. Clifford-equivalent to the phase state (|0⟩ + e^iπ/4|1⟩)/√2, which is what the literature usually means by 'the T state'.

The most studied cell in the field: χ ≤ 7 at six copies underpins the standard 2^0.47n figure, and the cat-state rank 6 there (exponent 0.431, Qassim, Pashayan, and Gosset, arXiv:2106.07740) refuted the conjecture that 7 was optimal. The published exponent log₂3/4 comes from their asymptotic contracted-cat-state family rather than from any single m, so every finite-m entry here sits above it. The ZX restatement of Kissinger, van de Wetering, and Vilmart (arXiv:2202.09202) adds a partial decomposition of |T⟩^⊗5 into three terms that each keep one |T⟩, so χ(T^t) ≤ 3χ(T^t−4); it gives 9 at m=7 and 12 at m=8, and the glued |cat₁₀⟩ gives 18 at m=10 (exponent 0.417).

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| H ⟩}^{\otimes 1}) \geq 2$	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| H ⟩}^{\otimes 1}) \leq 2$	1	2	1.0000	verified	@vprusso · stabrank research/constructions/data
$χ_{R} ({\| H ⟩}^{\otimes 2}) \geq 2$	2	2	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| H ⟩}^{\otimes 2}) \leq 2$	2	2	0.5000	lean	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 3}) \geq 3$	3	3	—	reproduced	@vprusso · stabrank PR #18
$χ_{R} ({\| H ⟩}^{\otimes 3}) \leq 3$	3	3	0.5283	lean	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 4}) \geq 3$	4	3	—	reproduced	@vprusso · stabrank PR #18
$χ_{R} ({\| H ⟩}^{\otimes 4}) \geq 4$	4	4	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| H ⟩}^{\otimes 4}) \leq 4$	4	4	0.5000	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H ⟩}^{\otimes 5}) \geq 4$	5	4	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| H ⟩}^{\otimes 5}) \geq 5$	5	5	—	reproduced · 3600 s	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| H ⟩}^{\otimes 5}) \geq 6$	5	6	—	attested · 2/323 re-run	@vprusso · stabrank research/h5_rank5
$χ_{R} ({\| H ⟩}^{\otimes 5}) \leq 6$	5	6	0.5170	lean	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 6}) \geq 4$	6	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| H ⟩}^{\otimes 6}) \geq 6$	6	6	—	attested · 2/180 re-run	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 6}) \leq 6$	6	6	0.4308	lean	Qassim, Pashayan, and Gosset · arXiv:2106.07740
$χ_{R} ({\| H ⟩}^{\otimes 6}) \leq 7$	6	7	0.4679	verified	Bravyi, Smith, and Smolin · arXiv:1506.01396
$χ_{R} ({\| H ⟩}^{\otimes 7}) \geq 6$	7	6	—	attested · 2/180 re-run	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 7}) \leq 9$	7	9	0.4528	lean	Kissinger, van de Wetering, and Vilmart · arXiv:2202.09202
$χ_{R} ({\| H ⟩}^{\otimes 8}) \geq 6$	8	6	—	attested · 2/180 re-run	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 8}) \leq 12$	8	12	0.4481	lean	Kissinger, van de Wetering, and Vilmart · arXiv:2202.09202
$χ_{R} ({\| H ⟩}^{\otimes 9}) \geq 6$	9	6	—	attested · 2/180 re-run	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 9}) \leq 18$	9	18	0.4633	verified	@vprusso · arXiv:2106.07740, arXiv:2202.09202
$χ_{R} ({\| H ⟩}^{\otimes 10}) \geq 6$	10	6	—	attested · 2/180 re-run	@vprusso · stabrank research/h6_rank5
$χ_{R} ({\| H ⟩}^{\otimes 10}) \leq 18$	10	18	0.4170	lean	Kissinger, van de Wetering, and Vilmart · arXiv:2202.09202

What would move it

Beating the published exponent needs χ ≤ 6 at m=7→ 0.3693.