Abstract
We demonstrate a direct mapping of max k-SAT problems (and weighted max k-SAT) to a Chimera graph, which is the non-planar hardware graph of the devices built by D-Wave Systems Inc. We further show that this mapping can be used to map a similar class of maximum satisfiability problems where the clauses are replaced by parity checks over potentially large numbers of bits. The latter is of specific interest for applications in decoding for communication. We discuss an example in which the decoding of a turbo code, which has been demonstrated to perform near the Shannon limit, can be mapped to a Chimera graph. The weighted max k-SAT problem is the most general class of satisfiability problems, so our result effectively demonstrates how any satisfiability problem may be directly mapped to a Chimera graph. Our methods faithfully reproduce the low energy spectrum of the target problems, so therefore may also be used for maximum entropy inference.
Highlights
Many interesting computer science problems have been shown to be directly mappable to finding the ground state of a Ising spin model on the hardware graph of the devices by D-Wave systems Inc. [1], the Chimera graph
We demonstrate a direct mapping of max k-SAT problems to a Chimera graph, which is the non-planar hardware graph of the devices built by D-Wave Systems Inc
It is for this reason that we are interested in direct mappings of interesting problems onto the Chimera graph, and why a direct mapping of a very general problem such as max k-SAT is of interest
Summary
Any Boolean clause can always be written out as logical AND operations performed on strings of logical OR operators performed on bit values or the logical negation of bit values, e.g. (a1 OR a2...)AND(NOT a1 OR a5...). Any Boolean clause can always be written out as logical AND operations performed on strings of logical OR operators performed on bit values or the logical negation of bit values, e.g. A general clause is of the form (a(11) ∨ a(21)...) ∧ (a(12) ∨ a(22)...) ∧ ...,. To implement a SAT problem in terms of energy computation, we could construct such a term by enforcing a penalty of the form,. One can construct a SAT problem by summing many such penalties and obtaining an energy E = l P en({a(l)}). If one or more bit-strings exist where. In the case where the clauses are satisfiable, the bit-strings which yield E = 0 are the ones which satisfy hi hj i
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