Simulation of single-qubit open quantum systems

Abstract

A quantum algorithm is presented for the simulation of arbitrary Markovian dynamics of a qubit, described by a semigroup of single-qubit quantum channels $\left\{T_t\right\}$ specified by a generator $\mathcal{L}$. This algorithm requires only single-qubit and controlled-not gates and approximates the channel $T_t=e^{t\mathcal{L}}$ up to the chosen accuracy $\epsilon$, with a slightly superlinear cost ${O\left(\right(\parallel \mathcal{L}\parallel }_{(1\to 1)}t{)}^{1+1/2k}/{\epsilon }^{1/2k})$ for any integer $k$. Inspired by developments in Hamiltonian simulation, a decomposition and recombination technique is utilized which allows for the exploitation of recently developed methods for the approximation of arbitrary single-qubit channels. In particular, as a result of these methods the algorithm requires only a single ancilla qubit, the minimal possible dilation for a nonunitary single-qubit quantum channel.