On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number

Quantum algorithms for solving the Quantum Linear System (QLS) problem are among the most investigated quantum algorithms of recent times, with potential applications including the solution of computationally intractable differential equations and speed-ups in machine learning. A fundamental parameter governing the efficiency of QLS solvers is $\kappa$, the condition number of the coefficient matrix $A$, as it has been known since the inception of the QLS problem that for worst-case instances the runtime scales at least linearly in $\kappa$ [Harrow, Hassidim and Lloyd, PRL 103, 150502 (2009)]. However, for the case of positive-definite matrices classical algorithms can solve linear systems with a runtime scaling as $\sqrt{\kappa}$, a quadratic improvement compared to the the indefinite case. It is then natural to ask whether QLS solvers may hold an analogous improvement. In this work we answer the question in the negative, showing that solving a QLS entails a runtime linear in $\kappa$ also when $A$ is positive definite. We then identify broad classes of positive-definite QLS where this lower bound can be circumvented and present two new quantum algorithms featuring a quadratic speed-up in $\kappa$: the first is based on efficiently implementing a matrix-block-encoding of $A^{-1}$, the second constructs a decomposition of the form $A = L L^\dagger$ to precondition the system. These methods are widely applicable and both allow to efficiently solve BQP-complete problems.