Counting amplitude moduli: χ(|H⟩^(⊗ m)) ≥ 4 and χ(|T⟩^(⊗ m)) ≥ 4 for every m ≥ 31
Abstract
The stabilizer rank χ(psi) of an n-qubit state psi is the least number of stabilizer states whose complex span contains psi. For the two Clifford-inequivalent single-qubit magic-state orbits — the octahedron-edge state |H⟩=cos(π)/(8)|0⟩+sin(π)/(8)|1⟩ and the octahedron-face state |T⟩=cosβ|0⟩+e^(iπ/4)sinβ|1⟩, cos2β=1/sqrt3 — we prove that χ(|H⟩^(⊗ m)) ≥ 4 and χ(|T⟩^(⊗ m)) ≥ 4 for every m ≥ 31. The proof counts moduli. Every amplitude of a stabilizer state is 0 or γ iᵉ for one fixed scale γ, so an amplitude of a state in the span of three stabilizer states is determined by a type in a 5³=125-element set; multiplying all three symbols of a type by i multiplies the amplitude by i and leaves its modulus alone, so at most (125-1)/4=31 distinct nonzero moduli can occur. The m+1 Hamming-weight classes of a magic state carry m+1 pairwise distinct moduli, and 32 of them do not fit. Both statements quantify over every finite set of stabilizer states: no dictionary of stabilizer states is enumerated, and no floating-point search enters. For comparison, the published lower bounds carrying explicit constants first reach 4 at m=40 for the H orbit and at m=148 for the T orbit; asymptotically they all dominate the count used here, and the improvement is in the constant, not the rate. Every statement below is machine-checked in Lean 4 by kernel reduction alone.
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Source snapshot 2026-08-30 15:34 UTC
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Claim ledger
Stated results
S1routine2026-08-23
Z[i,sqrt q] as a computable ring with an injective evaluation into C for prime q
S2routine2026-08-23
qubit stabilizer states in canonical form, the stabilizer rank, and the two certificate shapes
S3routine2026-08-23
|T>ᵐ is a nonzero scalar multiple of an exact Z[i,sqrt3] vector; cos(2 beta) = 1/sqrt3 and unitarity
S4known data2026-08-23
the source's m = 4 amplitude table derived from cos(beta), sin(beta) rather than assumed
S5known data2026-08-23
chi(|T>⁴) <= 3 – the source's own decomposition, re-checked against a concrete stabilizer predicate
S6known data2026-08-23
chi(|T>ᵐ) <= 2, 2, 3 for m = 1, 2, 3
S7known2026-08-23
chi(|T>) = 2 and chi(|T>²) = 2, with the lower half complete over all stabilizer states
S8known data2026-08-23
chi(|T>⁵) <= 6 and chi(|T>⁶) <= 6 as explicit six-state witnesses
S9known data2026-08-23
the stabrank library's conjecturally optimal rank-5 span at m = 6 does not contain |T>⁶
S10routine2026-08-23
negative controls
S11routine2026-08-23
the support of a canonical-form stabilizer state: 2ᵏ elements, exactly the nonzero amplitudes, and full at three qubits once at most three vanish
S12routine2026-08-23
χ ≥ 3 on three qubits from four weight-class amplitudes, over all stabilizer states
S13known2026-08-23
χ(| T⟩^(⊗3)) = 3, both halves kernel-checked
S14known data2026-08-23
the H-type magic state exactly in ℤ[i,√2], and an explicit rank-3 witness at three copies
S15known2026-08-23
χ(| H⟩^(⊗3)) = 3, both halves kernel-checked
S16routine2026-08-23
negative controls for the lower-bound route
S17routine2026-08-30
the computational basis as canonical-form data, and non-emptiness of the decomposition set for every bounded amplitude vector
S18routine2026-08-30
the five-valued type of an amplitude, and the 5ᵏ value bound for a rank-k state
S19routine2026-08-30
a rank-two state has at most six distinct nonzero moduli; seven separated moduli force χ ≥ 3
S20routine2026-08-30
full support forced on both rays by small modulus classes, cutting six moduli to four; five separated moduli force χ ≥ 3
S21routine2026-08-30
Hamming-weight machinery for product states: pcnt, the strictly decreasing profile cᵐ⁻ʷsʷ, the class representatives 2ʷ-1, and the three ready-made bounds
S22known2026-08-30
χ(|H⟩^(⊗m)) ≥ 3 and χ(|T⟩^(⊗m)) ≥ 3 for every m ≥ 4
S23routine2026-08-30
the value ladder: χ > k for the magic states whenever 5ᵏ < m+1
S24known2026-08-30
χ(|T⟩^(⊗4)) = 3 with both halves kernel-checked, and the two open cells bracketed 3 ≤ χ(|T⟩^(⊗5)), χ(|T⟩^(⊗6)) ≤ 6
S25routine2026-08-30
negative controls for the counting layer
S26known data2026-08-30
χ(|H⟩^(⊗4)) ≤ 4 as an explicit exact four-state witness in ℤ[√2]
S27known data2026-08-30
χ(|H⟩^(⊗5)) ≤ 6 as an explicit exact six-state witness
S28known data2026-08-30
χ(|H⟩^(⊗6)) ≤ 6 as an explicit exact six-state witness, by the cat-state route
S29known2026-08-30
the three H cells m = 4, 5, 6 bracketed
S30routine2026-08-30
negative controls for the H witnesses
S31routine2026-08-30
five-valued symbols, the ℤ₄ rotation on the 125 type codes of a rank-three decomposition, and its 31 nonzero orbits
S32candidate2026-08-30
a state in the span of three stabilizer rays has at most 31 distinct nonzero moduli; thirty-two separated moduli force χ ≥ 4
S33candidate2026-08-30
χ(|H⟩^(⊗m)) ≥ 4 and χ(|T⟩^(⊗m)) ≥ 4 for every m ≥ 31
S34routine2026-08-30
negative controls for the rank-three exclusion
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- For an n-qubit state ψ, the stabilizer rank χ(ψ) is the least k such that ψ lies in the complex span of k n-qubit stabilizer states. For the two Clifford-inequivalent qubit magic-state orbits — H-type (octahedron *edges*, |H⟩ = cos(π/8)|0⟩ + sin(π/8)|1⟩) and T-type (octahedron *faces*, |T⟩ = cos β|0⟩ + e^(iπ/4) sin β|1⟩, cos 2β = 1/√3) — the state of knowledge is arXiv:2605.28586 Table (tab:qubit-ub):
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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