Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 324 / 1 / a / i

A strong classical simulation of a quantum circuit computes any requested output probabilityin polynomial time, to the prescribed inverse-polynomial accuracy. A weak classical simulation of a quantum circuit instead produces classical samples from the circuit's output distribution, with exact or suitably small total-variation error.

The Extended Gottesman--Knill theorem states that a unitary Clifford circuit with an arbitrary product state input and final computational-basis measurements is weakly classically simulable. It is strongly simulable when only $O(\log n)$ output qubits are measured. Indeed, each joint output projector expands into $2^k$ Pauli operators, and Clifford conjugation maps every such operator to another Pauli operator whose expectation factors over the input qubits. The factor $2^k$ is polynomial precisely for $k=O(\log n)$.