We present a new method for the verification of quantum circuits based on a novel symbolic representation of sets of quantum states using \emph{level-synchronized tree automata} LSTAs.
LSTAs extend classical tree automata by labeling each transition with a set of \emph{choices}, which are then used to synchronize subtrees of an accepted tree. Compared to the traditional tree automata, LSTAs have an incomparable expressive power while maintaining important properties, such as closure under union and intersection and decidable language emptiness and inclusion. We have developed an efficient and fully automated symbolic verification algorithm for quantum circuits based on LSTAs. The complexity of supported gate operations is at most quadratic, dramatically improving the exponential worst-case complexity of an earlier tree automata-based approach. Furthermore, we show that LSTAs are a promising model for \emph{parameterized verification}, i.e., verifying the correctness of families of circuits with the same structure for any number of qubits involved, which principally lies beyond the capabilities of previous automated approaches. We implemented this method as a C++ tool and compared it with three symbolic quantum circuit verifiers and two simulators on several benchmark examples. The results show that our approach can solve problems with size orders of magnitude larger than the state of the art.