. 2019 Nov 7;9:16251. doi: 10.1038/s41598-019-52275-6

Table 2.

Comparison of various algorithms for solving N × N linear systems $A \vec{x} = \vec{b}$ , with respect to time and space complexities, and Input/Output issues.

Algorithm	Time	Space for A	Input/Output
Classical Direct^2,3	$O (N^{3})$	$O (N^{2})$	efficient for any $A, \vec{x}, \vec{b}$
Classical Iterative^2,3	$O (N^{2})$	$O (N^{2})$	efficient for any $A, \vec{x}, \vec{b}$
Quantum HHL⁴	$O (\log (N))$	$O (\log (N))$ qubits	norm $∥ \vec{x} ∥$ not available difficult for $A, \vec{x}, \vec{b}$
Classical MC^45,53,55 (for one component x_I)	$O (N)$	$O (N)$	efficient for any $\vec{x}, \vec{b}$ limited A (stochastic P)
Classical RW on HC (for one component x_I)	$O (\log (N))$	$O (\log (N))$	efficient for any $\vec{x}, \vec{b}$ limited A (factorisable P)
Hybrid QW on HC (for one component x_I)	$O (\log (N))$	$O (\log (N))$ qubits	efficient for any $\vec{x}, \vec{b}$ limited A (correlated P)

Note that for classical Monte Carlo (MC) method, classical random walk (RW) and hybrid quantum random walk (QW), the time complexities in the table are per sampling time. It takes $O (c n_{s})$ samplings to achieve the desired accuracy (see the text).