Running multiple rounds of a BQP computation, without multiple measurements? BQP as usually defined is: the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances.

Just like BPP, the choice of 1/3 in the definition is arbitrary because we could just have a "higher level" algorithm that runs the "basic algorithm" many times and take a majority vote to achieve higher probability of correctness.

I can imagine this as:


initialize qbits based on the input of length n
apply polynomial(n) quantum gates
measure output qbit
repeat steps 1-3, K times and report the majority measurement from step 3


Essentially, having a classical computing loop call a quantum routine K times.

Is it possible instead to somehow do ALL the computation on the quantum computer, and only make a single measurement at the end which is equivalent to this?  

I'd hope for something like:


initialize qbits based on the input of length n
apply K * polynomial(n) quantum gates
measure just a single output qbit


Or is making multiple measurements absolutely necessary to the computational power here?
The principle of deferred measurement (see https://en.wikipedia.org/wiki/Deferred_Measurement_Principle) tells us that measurements in the middle of a quantum computation can be simulated by using additional quantum gates instead. In particular, instead of measuring a qubit, we can apply a CNOT gate to that qubit and another qubit (which is used only at this point of the computation).