# Optimal dependency of HHL (or any QLSP) algorithm on condition number $\kappa$

This is conserning the optimal dependency on condition number for Quantum linear system problem (QLSP).

For solving QLSP, the HHL (algorithm) paper mentions any polylog($$\kappa$$) quantum algorihm would imply $$BPP=PSPACE$$. This means polylog($$\kappa$$) quantum algorithm is extremely unlikely. See page-3; paragraph 2 on the right side; Link.

I went through an improved algorithm by Child-Kothari-Somma (CKS). Which mentions any sublinear algorithm in $$\kappa$$ is unlikely. OR, linear dependency on $$\kappa$$ is optimal. See page-4; last paragraph; link. It cites HHL paper as reference.

I find CKS's statemnet quite stronger. Because, $$\kappa^{1/2}$$ dependency is also sublinear, thus excluded.

My query is:

What is the best known lower bound on condition number?

How to make sense of $$\Omega(\kappa)$$ mentioned in CKS paper?

Costa et. la.[1] use the discrete adiabatic theorem to develop a quantum algorithm for solving linear systems with complexity strictly linear in $$\kappa$$, matching the lower bound.
How to make sense of $$\Omega(\kappa)$$ mentioned in CKS paper?
CKS paper refers to HHL as the source of this fact. Actually, you can find it just after excluding the possibility of $$\mathrm{polylog}(\kappa)$$ quantum algorithm,
"Even improving our $$\kappa$$ dependence to $$\kappa^{1 - \delta}$$ for $$\delta \gt 0$$ would allow any time-$$T$$ quantum algorithm to be simulated in time $$o(T)$$; iterating this would again imply that $$\mathrm{BQP}=\mathrm{PSPACE}$$."