Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 324 / 4 / ii / Solution

For the first output qubit,A Clifford circuit maps the Pauli observable $Z_1$ under conjugation to a tensor-product Pauli operator $\pm P_1\otimes \cdots \otimes P_n$, found by propagating it backward through the circuit in polynomial time. Since the input is a product state,Each factor is efficiently computable, so both one-bit output probabilities are strongly simulatable. This is Heisenberg propagation of a Pauli observable through a Clifford circuit.