  We have investigated the origin of the magnetic dipole correlations $\langle
\sigma_{\boldsymbol{Q}}^z \sigma_{\boldsymbol{-Q}}^z \rangle$ characterized by
the modulation wave vector $\boldsymbol{k} \sim
(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$ observed in the frustrated pyrochlore
magnet Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$. This magnetic short-range order cannot be
accounted for by adding further-neighbor exchange interactions to the
nearest-neighbor pseudospin-$\tfrac{1}{2}$ Hamiltonian for quantum pyrochlore
magnets. Using classical Monte Carlo simulation and quantum simulation based on
thermally pure quantum (TPQ) states we have shown that the spin correlations
with $\boldsymbol{k} \sim (\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$ are induced
at low temperatures by a three-spin interaction of a form
$\sigma_{\boldsymbol{r}}^{\pm} \sigma_{\boldsymbol{r}^{\prime}}^z
\sigma_{\boldsymbol{r}^{\prime \prime}}^z $, which is a correction to the
Hamiltonian due to the low crystal-field excitation. Simulations using TPQ
states have shown that the spin correlations coexist with electric quadrupole
correlations $\langle \sigma_{\boldsymbol{Q}}^{\alpha}
\sigma_{\boldsymbol{-Q}}^{\beta} \rangle$ ($\alpha, \beta = x, y$) with
$\boldsymbol{k} \sim \boldsymbol{0}$. These results suggest that the putative
quantum spin liquid state of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ is located close to
phase boundaries of the spin-ice, quadrupole-ordered, and magnetic-ordered
states in the classical approximation, and that the three-spin interaction
brings about a quantum disordered ground state with both spin and quadrupole
correlations.
