Thermodynamic implications of $^{29}$Si spin impurities on scalability of silicon-based quantum computing

It is anticipated that $^{31}$P donors in silicon have the potential for realizing scalable quantum processors in analogue to classical computer chips\footnote{B. E. Kane, {\em Nature} {\bf 393}, 133 (1998). }. In classical computing, removing excess heat is a challenge that sets practical limits on performance. Here we consider what fundamental thermodynamic limits exist for the P-donor quantum computer in isotopically enriched $^{28}$Si. Specifically, we consider the effect of $^{31}$P nuclear spin rotation on the nuclear spin dynamics of the remaining $^{29}$Si impurity atoms within a single-qubit gate volume. Our simulations show that a $\pi$ rotation of $^{31}$P nuclear spin induces $^{29}$Si nuclear spin flipping resulting in an average energy decrease of the $^{29}$Si nuclear spin bath. For a gate volume of 5 nm$^3$ and $^{29}$Si concentration of 800 PPM at 250$\mu K$, the average energy decrease per single qubit rotation is 4.74$\times10^{-12} eV$. This suggests that the scalability of $^{31}$P-donor quantum computer will not be limited by energy dissipation from single qubit control pulses into the $^{29}$Si nuclear spin bath. Moreover, randomized single qubit rotation promises to be useful for cooling the $^{29}$Si nuclear spin bath.