It's often said that for any quantum circuit with intermediate measurements, there exists an equivalent circuit where all measurements are at the end of the circuit. Is anything ever argued about the complexity of the circuit with delayed measurements?
I'm imagining a circuit with $n$ qubits and $\mathrm{poly}(n)$ depth, where the first $n/2$ qubits are measured and depending on the out outcomes I apply one of the $2^{n/2}$ unitaries on the remaining qubits. If I was to naively delay measurements, I would add $2^{n/2}$ $C\mbox{-}U_i$ gates instead of measuring, but have now made the circuit $\exp(n)$ depth instead.
Is there a way to delay measurements to the end of the computation without changing the complexity of the circuit?
