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 $\{T_t\}$ specified by a generator $\mathcal{L}$. This algorithm requires only $\mathcal{O}((||\mathcal{L}|{|}_{(1\to 1)}t{)}^{3/2}/{ϵ}^{1/2})$ single qubit and CNOT gates and approximates the channel $T_t=e^{t\mathcal{L}}$ up to chosen accuracy $ϵ$. Inspired by developments in Hamiltonian simulation, a decomposition and recombination technique is utilised 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 non-unitary single-qubit quantum channel.