Parallelizing quantum circuits

We present a novel automated technique for parallelizing quantum circuits via forward and backward translation to measurement-based quantum computing patterns and analyze the trade oo in terms of depth and space complexity. As a result we distinguish a class of polynomial depth circuits that can be parallelized to logarithmic depth while adding only polynomial many auxiliary qubits. In particular, we provide for the rst time a full characterization of patterns with ow of arbitrary depth, based on the notion of innuencing paths and a simple rewriting system on the angles of the measurement. Our method leads to insightful knowledge for constructing parallel circuits and as applications, we demonstrate several constant and logarithmic depth circuits. Furthermore, we prove a logarithmic separation in terms of quantum depth between the quantum circuit model and the measurement-based model.