Parallel Quantum Computation and Quantum Codes

We propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum errorcorrecting codes, or more generally any circuit consisting of controllednot gates, controlled π-shifts, and Hadamard gates. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we conjecture that an even simpler ‘staircase’ circuit cannot be parallelized to less than linear depth, and might be used to prove that QNC < QP.