Analog computing is quite old. The issue with going analog is that when you have errors and noise as you do calculations, you can't do long chains of calculations without ending up with just the noise.
OP link isn't analog, it's digital. (There are analog approaches with large speedups, of course.)
Here is presentation from a researcher at MIT on a novel way of designing computer processors. It relies on performing approximate, rather than exact, mathematical operations (like the meat-based processor in our heads!). Claimed benefits are a 10,000-fold improvement in speed, while the errors introduced by the approximations are postulated to be insignificant in many applications.
http://web.media.mit.edu/~bates/Summary_files/BatesTalk.pdf
Slide #2 of the presentation offers a fascinating insight: We currently work around the limitations of the processing substrate to implement a precise computation, and it is becoming increasingly difficult:
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THE MOTIVATING PROBLEM:
Computations specified by programmers are implemented as behavior in physical material
• Hardware designer’s job:
efficiently implement Math (what sw wants) using Physics (what silicon offers)
(near) perfect arith noisy, approximate
uniform mem delay delay ~ distance
• Increasingly difficult as decades passed and transistor counts exploded
• Now each instruction (increment, load register, occasionally multiply) invokes >10M transistor operations, even though a single transistor can perform, for instance, an approximate exponentiate or logarithm
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The parallels and contrasts with our own brain are what interested me the most. Perhaps one day the most powerful computers will be running on "corrupted hardware" of sorts.