Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 4

The process admits a strong classical simulation of a quantum circuit if a classical algorithm, given its circuit and input descriptions, computes either output probability to the requested polynomial precision in time polynomial in the input size and precision parameter. This asks for the probabilities themselves, rather than only efficient samples from their distribution.

Suppose output qubit $j$ is measured. Its probability of outcome zero isBecause $C$ is a Clifford operation, the Pauli group is preserved under conjugation. Propagating $Z_j$ backward through the $N=\mathrm{poly}(n)$ gates therefore produces, including its sign, a tensor-product PauliThe input is a product state, so the expectation factors:Each one-qubit factor follows directly from the classical description of $|{\alpha }_k⟩$, and conjugating a Pauli through each Clifford gate takes constant classical work. Hence $p_0$, and $p_1=1-p_0$, are computable in polynomial time. This is Heisenberg propagation of a Pauli observable through a Clifford circuit.

Write $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ and $t=e^{i\theta }$. Before measurement, the three controlled-NOT gates map a computational-basis component $|x,y,0⟩$ toThe measured bit is therefore $b=y\oplus x$. For $b=0$, the unnormalized state of the first qubit isFor $b=1$, it isEach branch has probability $1/2$; after normalization and removal of the irrelevant global phase $t$, the two outputs arewith equal probability. This is probabilistic phase-gate injection.