Weighted coins and linear algebra

[Since the forum is in an early stage, I just throw all my considerations in one post here]

1. Is it possible to represent a real number by a qbit? I have encountered here an example of random number generator, equivalent to tossing some coins. As far as I understood it was possible by setting all the weights (energy coefficients?) to zero. Then the question arises, is it possible to toss a set of weighted coins? For example, we would like to toss a weighted coin, which gives 1 with probability p and 0 with probability 1-p. How could it be done on D-Wave?

2. If the answer to 1. is positive, then (at least in principle) we could represent a real number b (in the range [0,1], but that probably could be scaled to arbitrary interval) by a Constraint Satisfaction Problem x-b=0. Or in other words b would be represented as a weighted coin with probability corresponding to 1 equal to b. Then running the CSP many times, we would obtain an approximation for b based on statistics of the discrete outcomes x (equal 0 or 1). If that is correct we could approximately solve a linear equation using 1 qbit (in multiple runs).

3. If the statement above is correct, how one could represent linear algebra operations, e.g., matrix multiplication? Since the operation is still linear, I assume one could adjust the energy minimization problem to solve it, just using multiple qbits this time. This are already far reaching considerations, but solving linear systems is an important issue. Usually, the solution strategy and particular implementation is based on a prior knowledge about the system. As the number of qbits in available QPUs grow, at some point it might be possible to find a rough approximate solution of a linear system or an eigenvalue problem using QPU. This rough approximate solution might be used as a starting point for classical algorithms then. Are there already any successful attempts to use QPUs for linear algebra?




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