On the simulation of quantum circuits

We consider recent works on the simulation of quantum circuits using the formalism of matrix product states and the formalism of contracting tensor networks. We provide simplified direct proofs of many of these results, extending an explicit class of efficiently simulable circuits (log depth circuits with 2-qubit gates of limited range) to the following: let C be any poly sized quantum circuit (generally of poly depth too) on n qubits comprising 1and 2qubit gates and 1-qubit measurements (with 2-qubit gates acting on arbitrary pairs of qubit lines). For each qubit line i let Di be the number of 2-qubit gates that touch or cross the line i i.e. the number of 2-qubit gates that are applied to qubits j, k with j ≤ i ≤ k. Let D = maxiDi. Then the quantum process can be classically simulated in time npoly(2). Thus if D = O(log n) then C may be efficiently classically simulated.