Accurate quantum Z rotations with less magic

Andrew J. Landahl, Chris Cesare

The quantum Fourier transform Ubiquitous in quantum algorithms If native gates have error ε, how many instructions (gates, preparations, measurements) does it take to create a synthetic Z k gate whose error is ε' << ε? N.B.: Z 2 = T, Z 1 = S.Accurate quantum Z rotations with less magic.

Andrew J. Landahl, Chris Cesare

Otto C. W. Kong

Andrew J. Landahl, Chris Cesare

We present quantum protocols for executing arbitrarily accurate $π/2^k$ 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 $T = Z(π/4)$ 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 $Z(π/2^k)$ 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 $k \leq 6$.

James Mc Connell