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The realization of quantum algorithms relies on specific quantum compilations according to the underlying quantum processors. However, there are various ways to physically implement qubits in different physical devices and manipulate those qubits. These differences lead to different communication methods and connection topologies, with each vendor implementing its own set of primitive gates. Therefore, quantum circuits have to be rewritten or transformed in order to be transplanted from one platform to another. We propose a pattern matching-based framework for rewriting quantum circuits, called QRewriting. It takes advantage of a new representation of quantum circuits using symbol sequences. Unlike the traditional way of using directed acyclic graphs, the new representation allows us to easily identify the patterns that appear non-consecutively but reducible. Then, we convert the problem of pattern matching into that of finding distinct subsequences and propose a polynomial-time dynamic programming-based pattern matching and replacement algorithm. We develop a rule library for basic optimizations and use it to rewrite the Arithmetic and Toffoli benchmarks from the $G_{IBM}$ gate set to the $G_{Sur}$ gate set. Compared with the existing tool PaF, QRewriting obtains an improvement of reducing depths (resp. gate counts) by 29\% (resp. 14\%).