Classical hardness of estimating ⟨0^n|C^4|0^n⟩ (the quantum (2) problem)

Prove that, for large n, no classical algorithm running in time polynomial in n, circuit depth d, and 1/ε can estimate to additive error ε the quantity ⟨0^n|C^4|0^n⟩ for C = U^† B U M, where U is drawn from the described random circuit ensemble on a 2D grid with B a Pauli X on qubit (ℓ,ℓ) and M a Pauli Z on qubit (1,1); more specifically, establish hardness for n=ℓ×ℓ, some depth d in Θ(ℓ), and ε = 1/poly(n).

Background

The paper introduces the (2) problem: given a random unitary U from a specified distribution on an n-qubit 2D grid and ε>0, output ⟨0^n|C^4|0^n⟩ to additive error ε, where C = U^† B U M with fixed single-qubit Paulis B and M at opposite corners. The authors explain how a quantum computer can estimate this efficiently, while numerical and heuristic evidence suggests classical algorithms struggle.

They explicitly conjecture a general classical hardness result for this estimation task, including concrete parameter scaling (d ∈ Θ(ℓ), ε = 1/poly(n)) for n = ℓ×ℓ. Proving this would provide a rigorous average-case hardness foundation for beyond-classical demonstrations using this observable.