  We quantize the problem considered by Bott-Samelson who applied Morse theory
to any compact symmetric space $G/K$ and the associated real flag manifold
$G_{\mathbb{R}}/B$ which is a real locus of a complex partial flag variety
$G_{\mathbb{C}}/P_{\sigma}$. We prove that the Pontryagin ring
$H_{-*}(\Omega(G/K))$ of the based loop space $\Omega(G/K)$ is isomorphic to
the Floer cohomology ring $HF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$ after
localization. When $G/K$ is a Lie group, this is a conjecture of Peterson,
proved combinatorially by Lam-Shimozono, in the context of quantum cohomologies
of complex flag varieties.
  Our approach is geometric in nature: we construct a Lagrangian correspondence
from $T^*(G/K)$ to $G_{\mathbb{C}}/P_{\sigma}$ which geometrically composes
with a cotangent fiber to $G_{\mathbb{R}}/B$, and compute the linear part of
the associated Ma'u-Wehrheim-Woodward's $A_{\infty}$ homomorphism from a Floer
model of $\Omega(G/K)$ to $CF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$. The crux
is to make use of the geometry of $G/K$ to construct specific perturbation data
which enables us to reduce the computations to the case when $G/K$ is a torus.
