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We present two quantum algorithms based on evolution randomization, a simplevariant of adiabatic quantum computing, to prepare a quantum state $\vert x\rangle$ that is proportional to the solution of the system of linear equations$A \vec{x}=\vec{b}$. The time complexities of our algorithms are $O(\kappa^2\log(\kappa)/\epsilon)$ and $O(\kappa \log(\kappa)/\epsilon)$, where $\kappa$is the condition number of $A$ and $\epsilon$ is the precision. Both algorithmsare constructed using families of Hamiltonians that are linear combinations ofproducts of $A$, the projector onto the initial state $\vert b \rangle$, andsingle-qubit Pauli operators. The algorithms are conceptually simple and easyto implement. They are not obtained from equivalences between the gate modeland adiabatic quantum computing. They do not use phase estimation orvariable-time amplitude amplification, and do not require large ancillarysystems. We discuss a gate-based implementation via Hamiltonian simulation andprove that our second algorithm is almost optimal in terms of $\kappa$. Likeprevious methods, our techniques yield an exponential quantum speedup undersome assumptions. Our results emphasize the role of Hamiltonian-based models ofquantum computing for the discovery of important algorithms.

Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing