Abstract
Given a linear system of equations Ax=b, quantum linear system solvers (QLSSs) approximately prepare a quantum state |x for which the amplitudes are proportional to the solution vector x. Asymptotically optimal QLSSs have query complexity O(κ log(1/ε)), where κ is the condition number of A, and ε is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm x is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is (1+O(ε))κ ln(2√2/ε). If the norm is unknown, our method allows it to be estimated up to a constant factor using O(loglog(κ)) applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal O(κ loglog(κ)logloglog(κ)+κlog(1/ε)) total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that O(κ) complexity can be achieved for norm estimation, yielding an optimal QLSS with O(κlog(1/ε)) complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of 56κ+1.05κ ln(1/ε)+o(κ) for the complexity of our optimal QLSS.