Efficient quantum circuit simulation using a multi-qubit Bloch vector representation of density matrices

TL;DR

This paper introduces a multi-qubit Bloch vector representation for density matrices to improve quantum circuit simulation efficiency, including noise modeling and gradient computation for variational algorithms.

Contribution

It generalizes the Bloch sphere to multi-qubit systems, enabling efficient simulation and gradient calculation for quantum circuits with noise.

Findings

Efficient algorithms for applying quantum gates using the tensor-structured Bloch vector.

Effective simulation of noise processes and non-unitary evolution.

Facilitates gradient-based optimization in variational quantum algorithms.

Abstract

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 $4^n$, conceptually similar to a statevector. 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 (non-unitary) time evolution of a density operator.