Compilation with a witness / certificate
In theory, what you are asking about has been studied, under the name "translation validation".  See, e.g., the following classic paper:

George C. Necula. "Translation validation for an optimizing compiler." PLDI 2000.

In practice, compilation is such a complicated messy process that I doubt there will be any easy-to-verify certificate that proves the entire compilation process was done correctly (including lexing, parsing, front-end, optimization, back-end, assembly, linking, etc.).  Academic work typically focuses on just one or two phases of the process (e.g., some of the stages of optimization).

One should separate two separate issues: can the verifier be simpler than the compiler?  can the verifier be faster than the compiler?

Simpler is interesting, because it means the verifier can potentially be more trustworthy (e.g., less likely to have bugs).  Classic work on translation validation focuses on that case.

Faster is a different question.  Some optimizations are deterministic and won't benefit, but there are certainly plenty of optimizations that involve a search over some space of possibilities.  An extreme version of that is the concept of "superoptimization", which optimizes a short code sequence by searching over all possible optimized versions and finding the one that is the fastest while remaining semantically equivalent to the original.  See, e.g., the following papers:

R. Sausnauskas, Y. Chen, P. Collingbourne, J. Ketema, G. Lup, J. Taneja, J. Regehr.  "A Synthesizing Superoptimizer."

Sorav Bansal, Alex Aiken. "Automatic Generation of Peephole Superoptimizers."  ASPLOS 2006.