Why isn't Montgomery modular exponentiation considered for use in quantum factoring? It is well known that modular exponentiation (the main part of an RSA operation) is computationally expensive, and as far as I understand things the technique of Montgomery modular exponentiation is the preferred method. Modular exponentiation is also prominently featured in the quantum factoring algorithm, and it is expensive there as well. 

So: why isn't Montgomery modular exponentiation apparently present in current detailed subroutines for quantum factoring?

The only thing I can imagine is that there's a high qubit overhead for some non-obvious reason.

Running montgomery quantum "modular exponentiation" through Google Scholar yields no useful results. I am aware of work by Van Meter and others on quantum addition and modular exponentiation, but examining their references (I have yet to read this work) shows no indication that Montgomery methods are considered there.

The single reference I have found that appears to discuss this is in Japanese, which lamentably I cannot read, though apparently it is from a 2002 conference proceedings. A machine translation yields nuggets appended below that indicate there might be something useful. However, I can't find any indication that this has been followed up, which makes me think that the idea has been a) considered and then b) discarded.


  Quantum circuit in performing
  arithmetic Noboru Kunihiro
  
  ...In this study, but requires
  relatively large qubit, we propose a
  modular exponentiation circuit quantum
  computation time is short. Montgomery
  Reduction [8] and right binary method
  [9] Combined, they constitute a
  circuit Ru. Reduction Montgomery is, m
  randomly chosen as a natural number,
  mod 2m by the operation, perform the
  remainder operation If, mod n
  operations in eliminating. This will
  lead to reduction of computation
  time...
  
  Application of 3.2 Montgomery
  Reduction Montgomery Reduction [8] is
  formulated as follows...This algorithm
  can return the correct values can be
  easily confirmed. M R (Y) he asks for
  a law 2m Polynomials with 2m points
  are important and only requires
  division by. In addition, Montgomery
  Reduction in, there are different
  calculation methods....In general,
  Reduction Montgomery is not one-to-one
  function...
  
  ...The proposed method uses a right
  binary method, Montgomery Reducton has
  a feature that is adopted. Than the
  conventional method, characterized by
  a small component of the circuit Have.
  qubit fault that is required to have a
  lot of expectations can be computed in
  less computational time Be. The
  future, Montgomery Reduction and
  control circuitry specifically NOT
  described by the qubit really needed
  Evaluate the number is expected to
  evaluate the computation time. In
  addition, each taking advantage of
  research findings, more than modular
  exponentiation Non-arithmetic (Euclid
  mutual division, etc.) also with
  respect to the planned configuration
  of an efficient quantum circuit.
  
  ...[8] PL Montgomery, "Modular
  Multiplication Without Trial
  Division," Mathematics of Computation, 44, 170, pp. 519-521, 1985...
Could you post the original Japanese title/reference?
Also, you might consider just writing to the author - assuming it's the same guy he is a professor at the University of Tokyo:
http://www.it.k.u-tokyo.ac.jp/~kunihiro/
and almost certainly would reply.
Sorry to post this as an answer, it should be a comment but I don't have the rep for that yet apparently...
EDIT:  So, I took a look at the original Japanese.  As a preface, I am currently a PhD student in the EE department at U. Tokyo, originally from the U.S., and I do technical JA-&gt;EN translation as a part-time job. However, this topic area is well outside my comfort zone so please take my opinion with a grain of salt!
Basically the conclusion (4) says:

べき乗剰余演算を行う量子回路を提案した。提案方式は、右向きbinary methodを採用し、Montgomery Reductionを採用するという特徴を持っている。従来の方式よりも、回路の構成要素が少ないという特徴を持っている。qubitが多く必要となるという欠点は持つが、より少ない計算時間で計算ができると期待される。
[In this paper] We proposed a new quantum circuit for computing modular exponentiation. The proposed method utilizes an LR binary method, and is also characterized by use of the Montgomery Reduction.  Compared with previous methods, the proposed method requires fewer components to construct the circuit.  The proposed method does however, have the drawback of requiring a large number of qubits, but we are confident that it will be computationally efficient ( lit: require very little computation time ).

I tried searching for related follow-up papers in both English and Japanese but was unsuccessful.   My guess is that the approach proved unsuccessful, or the professor got busy with something else (looks like this was around when he switched universities).
I think that your best bet at this point, assuming you want to follow up the rest of the way and get a concrete answer, is to write professor Kunihiro directly (in English!)