Finite-Bit Compilation into the VMW F/H Alphabet

For each fixed $U$, compute $z=\det U$ from its entries; $\mathrm{\mid }z\mathrm{\mid }=1$. Choose once, as part of the finite machine description, one of the constants $a\in \{1,-1,i,-i\}$ for which $\mathrm{Re}(az)>1\mathrm{/}2$. Such a constant exists because at least one of $\mathrm{\mid }\mathrm{Re}z\mathrm{\mid },\mathrm{\mid }\mathrm{Im}z\mathrm{\mid }$ is at least $1\mathrm{/}\sqrt{2}$. On the half-plane $\mathrm{Re}(az)>1\mathrm{/}2$, put $r=\sqrt{(1+\mathrm{Re}(az))\mathrm{/}2}$ and $v=r+i\mathrm{Im}(az)\mathrm{/}(2r)$. Then $r>\sqrt{3}\mathrm{/}2$, $\mathrm{\mid }v\mathrm{\mid }=1$ and $v^2=az$ by direct calculation. Fix one algebraic square root $b$ of $a$ and set $\lambda =b\mathrm{/}v$. Thus $\mathrm{\mid }\lambda \mathrm{\mid }=1$ and ${\lambda }^{2}z=1$, so $V=\lambda U\in SU(2)$. All denominators in this normalization are bounded below by a fixed positive constant. By [F3], its coordinates have polynomial-time Cauchy procedures. There is no exact-zero test or accuracy-dependent branch selection.

Fix, for example, ${\eta }_0=1\mathrm{/}8192$ and the gate package furnished by [F1]. Apply [F2] to $V$ with tolerance ${\epsilon }^{\mathrm{\prime }}=\min (\epsilon ,{\eta }_0\mathrm{/}2)$. The fixed algebraic gate descriptions and base-net constants satisfy every gate-package hypothesis. The normalized target's Cauchy query exponent is a fixed constant by step 1.1. Thus the compiler prints a word $S$ in the lifted alphabet with $\mathrm{\parallel }S-V\mathrm{\parallel }\le {\epsilon }^{\mathrm{\prime }}$, using polynomial bit time and polynomial expanded word length in the requested bit precision and tolerance-encoding length. Its work space is at most its bit time. This invocation uses the finite-bit theorem, not a unit-cost exact-real arithmetic count.

Erase scalar phases letter by letter using [F1]. The resulting physical word $W$ has the same length and differs from $S$ by one scalar of modulus one. Consequently its projective distance to $U$ is at most ${\epsilon }^{\mathrm{\prime }}\le \epsilon$. Printing the physical labels adds linear time in the expanded output length. All quantum letters act on the same one-qubit register. At tolerance $1\mathrm{/}(Ct)$ the requested precision and binary encoding length are $O(\log t)$, so every classical resource bound becomes $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}(t)$; a fixed family has one uniform maximum exponent and constant over its finitely many members.