Rational encoding preserves a promised local-Hamiltonian gap

Request each of the at most $D^2$ entries of each $H_j$ to absolute complex error at most $\tau \mathrm{/}(2mD)$. Replace each entry by its rational approximation and symmetrize the matrix by $(A+A^{\ast })\mathrm{/}2$; this preserves rationality and Hermiticity, and does not increase the Frobenius error beyond the entrywise bound. The Frobenius norm of a $D\times D$ error matrix is at most $D$ times its largest entry error, so [F3] gives $\mathrm{\parallel }{H}_j-{\tilde{H}}_j\mathrm{\parallel }\le \tau \mathrm{/}(2m)$. The precision request has $O(\log (mD\mathrm{/}\tau ))$ bits beyond the input description and is computed in polynomial time by [F1]. The term list and common embedding are those of [F4].

Identity embedding preserves operator norm, so $\mathrm{\parallel }H-\tilde{H}\mathrm{\parallel }\le {\sum }_j\mathrm{\parallel }{H}_j-{\tilde{H}}_j\mathrm{\parallel }\le \tau \mathrm{/}2\le \tau$. For each unit $v$, $\mathrm{\mid }⟨Hv,v⟩-⟨\tilde{H}v,v⟩\mathrm{\mid }\le \tau$; taking minima over the same unit sphere in both directions and using [F2] yields $\mathrm{\mid }{\lambda }_{\min }(H)-{\lambda }_{\min }(\tilde{H})\mathrm{\mid }\le \tau$. If ${\lambda }_{\min }(H)\le a$, then ${\lambda }_{\min }(\tilde{H})\le a+\tau$; if ${\lambda }_{\min }(H)\ge b$, then ${\lambda }_{\min }(\tilde{H})\ge b-\tau$. Their new gap is $\mathrm{\Delta }-2\tau \ge 3\mathrm{\Delta }\mathrm{/}4$.