A refined controlled-Z bound for exchange-coupled qutrits: the trace lemma #

Physics context. Two transmons truncated to three levels are coupled by the exchange interaction H_c = J (a† ⊗ a + a ⊗ a†) (here J = 1). With ideal, instantaneous local control, the operator norm ‖H_c‖ gives only the crude controlled-Z time bound T ≥ π / (4 J ‖H_c‖)-type estimates. The sharper argument compares the true evolution U(t) with the evolution L(t) that has the same local controls but no H_c, on the reference-extended product state Ω₀ (a maximally entangled state between a rank-two code in ℂ³ and a two-level passive reference, on each side). The Fubini–Study speed of U†L Ω₀ is the spread of H_c in L(t) Ω₀, a state whose physical reduced state is (P_A ⊗ P_B) / 4 with P_A, P_B rank-two projectors of ℂ³. The key finite-dimensional fact is then

(1/4) Tr[(P_A ⊗ P_B) H_c²] ≤ 97/32,

with equality for P = 1 - |v⟩⟨v|, v = √(7/8) |0⟩ + √(1/8) |2⟩. Combined with the fact that L(T) Ω₀ is a product across the two sides while the CZ state is at Fubini–Study distance ≥ π/4 from every product, this gives T·√(97/32) ≥ π/4 - arccos(1 - e) for CZ error e.

The argument was suggested by an AI reviewer (ChatGPT, OpenAI); the lemma was checked numerically (research/qsl-hardware/cota_cz_refinada.py in the Curiosity workspace) and is formalized here.

Main results (all proved here, sorry-free) #

QSL.Exchange.exchC_sq_apply: the entries of H_c² (a 9 × 9 integer matrix, block diagonal by total excitation number; the only off-diagonal entries are ⟨20|H_c²|02⟩ = ⟨02|H_c²|20⟩ = 2).
QSL.Exchange.trace_expand: closed form of Tr[(P_A ⊗ P_B) H_c²] for P = 1 - |v⟩⟨v|, as a polynomial in the populations |v_i|², |w_i|² plus the cross term 2 (z + z̄), z = v₀ w₂ v̄₂ w̄₀. No normalization is used.
QSL.Exchange.refined_cz_lemma: for unit v, w ∈ ℂ³, (1/4) Re Tr[(P_v ⊗ P_w) H_c²] ≤ 97/32. Route: coordinate expansion, AM–GM on the cross term (re_crossZ_le), then the scalar inequality scalar_ineq (97/8 - G = 2 (p₂ + q₂ - 1/4)² + p₁(…) + q₁(…) ≥ 0, i.e. completing the square).
QSL.Exchange.refined_cz_equality: equality at v = w = √(7/8) |0⟩ + √(1/8) |2⟩.

Not formalized #

The time bound itself (P2): sAngle (U Ω₀) (L Ω₀) ≤ √(97/32) Σ τ_k and the corollary T √(97/32) ≥ π/4 - arccos(1 - e). Plan, reusing QSL.Qudit and QSL.Gate:

Show ‖(H_c ⊗ 1) (U_A ⊗ U_B ⊗ 1) Ω₀‖² = (1/4) Tr[(P_A ⊗ P_B) H_c²], where Ω₀ = Φ_{AA'} ⊗ Φ_{BB'}, Φ = 2^{-1/2} Σ_{a'} |c_{a'}⟩ ⊗ |a'⟩ (c_0, c_1 an orthonormal pair in ℂ³), U_A, U_B unitary, and P_A = Σ |U_A c_a'⟩⟨U_A c_a'|.
Show every rank-two projector of ℂ³ has the form 1 - |v⟩⟨v| with v a unit vector (the conjugate cross product of the code pair), so refined_cz_lemma applies.
Duhamel step: for s ↦ e^{-i(τ-s)(K+H)} H e^{-isK} ψ, an estimate in the style of QSL.Gate.sAngle_exp_le gives sAngle (e^{-iτ(K+H)} ψ) (e^{-iτK} ψ) ≤ τ · sup_s ‖H e^{-isK} ψ‖, with e^{-isK} Ω₀ of the form in step 1 by QSL.Qudit.exp_local; chain the steps with QSL.Gate.sAngle_le_sAngle_add_sAngle and unitary invariance.
Corollary: L Ω₀ is a product across AA' : BB'; the Choi state of CZ has Schmidt coefficients (1/√2, 1/√2) across it, so the existing product-overlap lemma (QSL.Qudit.norm_inner_bellPair_prodVec_le) gives π/4 and the triangle inequality with fsAngle_le_of_norm_sub_le/arccos_one_sub_le finishes.

References #

P. Wocjan, M. Rötteler, D. Janzing, T. Beth, Simulating Hamiltonians in quantum networks: efficient schemes and complexity bounds, Phys. Rev. A 65 (2002) 042309.