Is "analog" quantum-computation not useful? I understand what a qubit-based quantum computer by the current definition is and how they are constructed. I read another thread where someone suggested encoding a computation into a dual-slit experiment and reading out the answer as interference vs non-interference, and this was mostly shot down since it's not "useful", because the system ceased to be a quantum computer when decoherence occurred.
But this reminded me of an old idea I had on other ways to encode computations in physics than the "classical" qubit-based schemes. As a striking example, consider one of the canonical uses for a quantum-computer - to efficiently simulate quantum-level systems. Well - to do this without a quantum-computer, you could actually assemble the system itself and let it "run", and then read out what you were looking for to simulate in the first place in many cases, in a vastly more simple way than digitally calculating it with qubits. Is this then not also a "quantum computation", really? The closest analogue I can come up with is the way analog computers were designed before digital computers took over, and that they were indeed much more efficient at a certain class of problems.
Even a system of 3 particles performs a calculation (of sorts) which is difficult to simulate traditionally.
More generally, if you could encode your problem into a hamiltonian which you subsequently map onto a physical structure wouldn't this be very useful? You might not be able to run algorithms on it, but you might be able to format the problems the algorithms were meant to solve in the first place. I'd be interested in hearing if this approach is pursued and I have missed it or if it is deemed to be useless for some reason, or if it's useful but "digital" quantum computation is simply more attractive in the long-run so most of the focus is on that. Maybe there is a measure on the computational complexity that can be mapped or performed by a general quantum structure (non-qubit-based)?
I seem to remember a Canadian company called D-Wave which I think do this more or less, and there was (is) much heated discussions on if it was "real quantum computation" or not.
EDIT: Nature published a special Nature Insight review on this in April 2012, especially about Quantum Simulators, where the issues I brought up in this question are discussed, including the "digital vs analog quantum computation" issue.
 A: Well, i am just not sure that it would be useless, and just meant as an example (and not as a full answer to the more general aspects of your question), you could have super-FFT if you could encode the inputs of your FFT as dark/transparent pixels in a 2D light filter and then recover the FFT output as an 2D interference pattern with CCD. If this is actually scalable or interesting from a cost-benefit perspective its another, engineering-related matter. 
This would of course be counted by many as a classical-analog computer, because it doesn't rely on entanglement of qubits, just good old wave huyghens interference
best FFT algorithms have complexities like $O( n \log(n^2) )$, so i don't know why no one has come up with an application for this yet. As i said, probably the reasons are entirely economical
A: The difference between a traditional analogue computer and a quantum computer is at least that the analogue computer is not stochastic (at least in concept; there's still noise, but the idea is to eliminate the noise as far as possible, not to use it). In particular, from the field theoretic perspective I usually work in, the mathematical structure of a classical field is of a lower order than the mathematical structure of a probability measure over values of a classical field.
If one goes to the higher order structure of probability measures over classical analogue structures instead of deterministic analogue structures, one is then close to the computational power of quantum mechanics. In particular, in a recent paper, "Equivalence of the Klein-Gordon random field and the complex Klein-Gordon quantum field", EPL 87 (2009) 31002, arXiv:0905.1263v2, I show that a quantized complex Klein-Gordon field is empirically equivalent to a real Klein-Gordon random field, which is essentially at the mathematical level of probability measures over classical fields. The traditional Klein-Gordon field, with no stochastic or probabilistic structure, however, is not of the same mathematical order.
More generally, a classical stochastic model can be presented in a Koopman-von Neumann formalism, which essentially just applies Hilbert space mathematics to classical probabilities. To return to the parenthesis of my first paragraph, for a classical analogue computer to approach the effectiveness of a quantum computer, my intuition (without enough justification) is that it would have to use some kind of noise in an essential way to improve the effectiveness of an algorithm. It may or may not be significant that Monte Carlo methods are a well-known example.
The real crux, however, is that a quantum mechanical way of thinking about the experimental apparatuses that are currently being used to push forward quantum computing as a technology, essentially in terms of qubits and closely similar finite-dimensional Hilbert space mathematics, is more productive than thinking in terms of classical stochastic signal processing, even though the concept of incompatible measurements is as natural in the classical mathematics of processing stochastic analogue signals as it is in quantum theory. [You go a long way towards providing this reason in your Question.] The particular advantage, as far as I can tell from outside the field, is that the mathematics of Hilbert spaces allows a quasi-digital way of algebraic thinking that bridges the gap between analogue and digital thinking more effectively than thinking just in terms of probability measures over analogue mathematical structures. Because almost no-one works in stochastic-analogue terms, there is a dearth of established results and a consequent lack of leverage for people who do work with these ideas.
It's a good Question. I look forward to seeing a more conventional Answer that I hope will be forthcoming from someone else.
A: It's been proposed, and there are a variety of possible experimental systems people are working on. I expect this to be a very useful way of simulating quantum systems, which will work well for many questions long before large-scale universal quantum computers can be built.
