Low-dimensional or frustrated quantum magnets were revealed to exhibit exotic ground states, magnetic excitations, and quantum phase transitions (QPTs). The external magnetic field can always induce magnetic transitions. In long-range-ordered antiferromagnetic state, the magnetic field can induce the re-orientation of the magnetic structure and the spin polarization. In the spin-gapped antiferromagnets, the magnetic field can close the gap, which results in a QPT between a low-field disordered phase and a high-field long-range ordered one. An intriguing finding is that this ordered phase can be approximately described as a Bose-Einstein condensation (BEC) of magnons. In this talk, we show the low-temperature thermal conductivity () of several low-dimensional quantum magnets to study the magnetic heat transport associated with the field-induced transitions. In particular, we pay attention to a fundamental issue, whether the ballistic magnon heat transport (T3), which has actually never been observed in antiferromagnetic materials, can be realized.