Efficient quantum algorithms for solving quantum linear system problems

We transform the problem of solving linear system of equations $A\mathbf{x}=\mathbf{b}$ to a problem of finding the right singular vector with singular value zero of an augmented matrix $C$, and present two quantum algorithms for solving this problem. The first algorithm solves the problem directly by applying the quantum eigenstate filtering algorithm with query complexity of $O\left( sκ\log \left( 1/ε\right) \right)$ for a $s$-sparse matrix $C$, where $κ$ is the condition number of the matrix $A$, and $ε$ is the desired precision. The second algorithm uses the quantum resonant transition approach, the query complexity scales as $O\left[sκ+ \log\left( 1/ε\right)/\log \log \left( 1/ε\right) \right]$. Both algorithms meet the optimal query complexity in $κ$, and are simpler than previous algorithms.

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