A New Lower Bound Technique for Quantum Circuits without Ancillæ

We present a technique to derive depth lower bounds for quantum circuits. The technique is based on the observation that in circuits without ancillæ, only a few input states can set all the control qubits of a Toffoli gate to 1. This can be used to selectively remove large Toffoli gates from a quantum circuit while keeping the cumulative error low. We use the technique to give another proof that parity cannot be computed by constant depth quantum circuits without ancillæ.