  In the framework of the chiral quark model along with complex scaling range,
we perform a dynamical study on the low-lying $S$-wave doubly-heavy tetraquark
states ($QQ\bar{q}\bar{q}$, $Q=c, b$ and $q=u, d$) with an accurate computing
approach, Gaussian expansion method. The meson-meson and diquark-antidiquark
configurations within all possible color structures for spin-parity quantum
numbers $J^P=0^+$, $1^+$ and $2^+$, and in the $0$ and $1$ isospin sectors are
considered. Possible tightly bound and narrow resonance states are obtained for
doubly-charm and doubly-bottom tetraquarks with $IJ^P=01^+$, and these exotic
states are also obtained in charm-bottom tetraquarks with $00^+$ and $01^+$
quantum numbers. Only loosely bound state is found in charm-bottom tetraquarks
of $02^+$ states. All of these bound states within meson-meson configurations
are loosely bound whether in color-singlet channels or coupling to hidden-color
ones. However compact structures are available in diquark-antidiquark channels
except for charm-bottom tetraquarks in $02^+$ states.
