Complex instruction set computing architecture for performing accurate quantum \(Z\) rotations with less magic

Abstract

We present quantum protocols for executing arbitrarily accurate \(\pi/2^k\) rotations of a qubit about its \(Z\) axis. Reduced instruction set computing (\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 \(T = Z(\pi/4)\) 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 \textsc{risc} architecture can be much greater. We develop a complex instruction set computing (\textsc{cisc}) architecture whose instruction set includes stabilizer operations and preparation of magic states from which \(Z(\pi/2^k)\) 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 \(k \leq 6\).