Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation

We present a quantum algorithm to solve systems of linear equations of the form $A\mathbf{x}=\mathbf{b}$, where $A$ is a tridiagonal Toeplitz matrix and $\mathbf{b}$ results from discretizing an analytic function, with a circuit complexity of $poly(\log(\kappa), 1/\sqrt{\epsilon}, \log(N))$, where $N$ denotes the number of equations, $\epsilon$ is the accuracy, and $\kappa$ the condition number. The \emph{repeat-until-success} algorithm has to be run $\mathcal{O}\left(\kappa/(1-\epsilon)\right)$ times to succeed, leveraging amplitude amplification, and sampled $\mathcal{O}(1/\epsilon^2)$ times. Thus, the algorithm achieves an exponential improvement with respect to $N$ over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with $1/\sqrt{\epsilon}$ circuit complexity instead of $1/\epsilon$ and which can be parallelized. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although, our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.