Inverse-polynomial OTOC fluctuations in the transition regime

Establish that, in the intermediate-depth regime between the commuting (low-depth) and scrambled (large-depth) limits for random quantum circuits U drawn from the specified 2D brickwork ensemble on n=ℓ×ℓ qubits with Haar-random two-qubit gates, the out-of-time-ordered correlator value ⟨0^n|C^2|0^n⟩, where C = U^† B U M with B equal to a Pauli X on qubit (ℓ,ℓ) and M equal to a Pauli Z on qubit (1,1), exhibits inverse-polynomial instance-to-instance fluctuations in the number of qubits n.

Background

The paper considers random quantum circuits U on a 2D grid and defines C = U^† B U M with B a Pauli X at (ℓ,ℓ) and M a Pauli Z at (1,1). For shallow depth, C^2 commutes with M and ⟨0^n|C^2|0^n⟩ = 1; for large depth, C^2 appears scrambled and ⟨0^n|C^2|0^n⟩ ≈ 0.

In the crossover between these regimes, the authors note an explicit conjecture (referencing prior work) that the OTOC exhibits inverse-polynomial instance-to-instance fluctuations. Pinning down these fluctuations would clarify the behavior of OTOC near the scrambling transition and underpin hardness claims tied to signal strength.