Maximum-Entropy Inference with a Programmable Annealer.

Nicholas Chancellor,Szilard Szoke,Walter Vinci,Paul A Warburton,Gabriel Aeppli

doi:10.1038/srep22318

Abstract

Optimisation problems typically involve finding the ground state (i.e. the minimum energy configuration) of a cost function with respect to many variables. If the variables are corrupted by noise then this maximises the likelihood that the solution is correct. The maximum entropy solution on the other hand takes the form of a Boltzmann distribution over the ground and excited states of the cost function to correct for noise. Here we use a programmable annealer for the information decoding problem which we simulate as a random Ising model in a field. We show experimentally that finite temperature maximum entropy decoding can give slightly better bit-error-rates than the maximum likelihood approach, confirming that useful information can be extracted from the excited states of the annealer. Furthermore we introduce a bit-by-bit analytical method which is agnostic to the specific application and use it to show that the annealer samples from a highly Boltzmann-like distribution. Machines of this kind are therefore candidates for use in a variety of machine learning applications which exploit maximum entropy inference, including language processing and image recognition.

Highlights

Optimisation problems typically involve finding the ground state of a cost function with respect to many variables
We show experimentally that finite temperature maximum entropy decoding can give slightly better bit-error-rates than the maximum likelihood approach, confirming that useful information can be extracted from the excited states of the annealer
It has been long recognised[1,2,3] that there are two generic approaches for doing this: 1. maximum a priori (MAP) estimation, which results in a unique conclusion which maximises the likelihood of being correct; 2. marginal posterior maximisation (MPM), which results in a probabilistic conclusion whose distribution maximises entropy

Summary

Experimental Methods

To demonstrate that the maximum entropy approach is effective for larger systems, let us consider decoding for a 4 × 4 array of Chimera unit cells While the macroscopic investigation demonstrates that the chip can be used for finite temperature decoding, it does not provide strong evidence on how suitable the chip may be for other maximum entropy tasks To answer this question we need to examine whether or not the individual spin orientations on the annealing machine look similar to those expected from a Boltzmann distribution.

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Conclusion

Author Contributions

Findings

Additional Information