√T — Stabilizer Rank Challenge

The state

| \sqrt{T} ⟩ = \frac{| 0 ⟩ + e^{i π / 8} | 1 ⟩}{\sqrt{2}} = Z (π / 8) | + ⟩

The √T state, Z(π/8)|+⟩: the magic state of the square root of the T gate, at the fourth level of the Clifford hierarchy. Its Bloch vector (cos(π/8), sin(π/8), 0) is neither an edge nor a face center of the stabilizer octahedron, and the Clifford group acts on Bloch vectors by signed coordinate permutations, so this is a third single-qubit orbit beside the H-type and T-type states. Its amplitudes lie in Q(ζ₁₆), a degree-four extension of the Q(i) the qubit stabilizer states live in, and that extra degree is what the Galois argument uses: a stabilizer span containing |√T⟩^⊗m contains its four conjugates, so χ ≥ min(4, m+1) exactly and with no search.

The published exponent is 1/2, Theorem 2 of Qassim, Pashayan, and Gosset (arXiv:2106.07740): χ(R_θ^⊗m) = O(2^m/2) for every angle θ. The small cells take the generic single-qubit values: χ = 2, 3, 4 at one to three copies (one more than the T state at two and three) and 4 ≤ χ ≤ 5 at four, the upper bounds from product stabilizer states spanning the symmetric subspace. The question the orbit asks is whether a finer phase is dearer per copy than T's log₂3/4; the cells that would beat 1/2 are χ ≤ 5 at m = 5 and χ ≤ 7 at m = 6.

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 1}) \geq 2$	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 1}) \leq 2$	1	2	1.0000	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 2}) \geq 3$	2	3	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 2}) \leq 3$	2	3	0.7925	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 3}) \geq 4$	3	4	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 3}) \leq 4$	3	4	0.6667	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 4}) \geq 4$	4	4	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 4}) \geq 5$	4	5	—	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 4}) \leq 5$	4	5	0.5805	verified	@vprusso · this repository (stabrank-research, round 31 new-orbit survey)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 5}) \geq 6$	5	6	—	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 5}) \leq 6$	5	6	0.5170	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 6}) \geq 6$	6	6	—	verified	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 6}) \geq 7$	6	7	—	attested · 2/859 re-run	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 6}) \leq 12$	6	12	0.5975	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 7}) \geq 7$	7	7	—	attested · 2/859 re-run	@vprusso · this repository (stabrank-research, round 33)
$χ_{R} ({\| \sqrt{T} ⟩}^{\otimes 7}) \leq 12$	7	12	0.5121	verified	@vprusso · stabrank autoresearch/run.py on Perlmutter (round 33 of stabrank-research, job 60161156)

What would move it

Beating the published exponent needs χ ≤ 7 at m=6→ 0.4679.