Complex instruction set computing architecture for performing accurate quantum $Z$ rotations with less magic

We present quantum protocols for executing arbitrarily accurate $\pi \mathrm{/}{2}^k$ rotations of a qubit about its $Z$ axis. Reduced instruction set computing ($\text{textsc}risc$) architectures typically restrict the instruction set to stabilizer operations and a single non-stabilizer operation, such as preparation of a “magic” state from which $T=Z(\pi \mathrm{/}4)$ gates can be teleported. Although the overhead required to distill high-fidelity copies of this magic state is high, the subsequent quantum compiling overhead to realize $Z$ rotations in a $\text{textsc}risc$ architecture can be much greater. We develop a complex instruction set computing ($\text{textsc}cisc$) architecture whose instruction set includes stabilizer operations and preparation of magic states from which $Z(\pi \mathrm{/}{2}^k)$ gates can be teleported, for $2 \leq k \leq k_{$\text{max}$}$. This results in a substantial overall reduction in the number of gates required to achieve a desired gate accuracy for $Z$ rotations. The key to our construction is a family of shortened quantum Reed-Muller codes of length $2^{k+2}-1$, whose magic-state distillation threshold shrinks with $k$ but is greater than 0.85% for $k\le 6$.