Classical simulation of quantum circuits using a multiqubit Bloch vector representation of density matrices

In the Bloch sphere picture, one finds the coefficients for expanding a single-qubit density operator in terms of the identity and Pauli matrices. A generalization to n qubits via tensor products represents a density operator by a real vector of length 4n, conceptually similar to a state vector. Here, we study this approach for the purpose of quantum circuit simulation, including noise processes. The tensor structure leads to computationally efficient algorithms for applying circuit gates and performing few-qubit quantum operations. In view of variational circuit optimization, we study "backpropagation"through a quantum circuit and gradient computation based on this representation, and generalize our analysis to the Lindblad equation for modeling the (nonunitary) time evolution of a density operator.