Do any single-cell organisms exist that approximate NP-hard problems within a factor better than $1/2$ $log$2?
In the realm of classical physics (where you don't need quantum mechanics to predict what is going on), we have no known method of computation that will be significantly faster than a standard classical computer.  Indeed, it is a standard belief that none exists.
The one exception is quantum mechanics.  Quantum computers are believed to be able to solve some problems significantly faster than classical computers -- though only a few special problems; for most, they offer no benefit (and quantum computers are not expected to offer any significant speedup for NP-complete problems).  It is a standard belief that no physical process that can be described by quantum mechanics offers any significant speedup over a quantum computer (in particular, a quantum computer is universal and can simulate any quantum mechanical process).
In short, the answer to your question is &quot;no, there is no known physical process that holds any promise for solving NP-hard problems in polynomial time&quot;.