Except that in our result, the spectrum is either continuous, or the spectral gap is guaranteed to be &gt;= 1 (in natural units). It does not become arbitrarily small in the gapped case.<p>The reason you can&#x27;t use this to compute the uncomputable is that real systems are finite, and the spectral gap is always computable in principle (maybe with a lot of effort) for any finite system. A real (possibly very large but still finite) system will either have a gap or not, and you&#x27;ll be able to measure it. This definitely doesn&#x27;t solve an undecidable problem.<p>However, the undecidability in the idealised infinite lattice limit &quot;shows through&quot; to the experimentally accessible finite-size case, in the form of some rather unusual finite-size physics. This is discussed in more detail in the paper itself (the relevant section is quoted verbatim here <a href="http:&#x2F;&#x2F;mathoverflow.net&#x2F;a&#x2F;225905" rel="nofollow">http:&#x2F;&#x2F;mathoverflow.net&#x2F;a&#x2F;225905</a>) and in the comments on Scott Aaronson&#x27;s blog.