Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit

Abstract

A long-standing open problem in fault-tolerant quantum computation has been to find a universal set of transversal gates. As shown by Zeng et al. (e-print arXiv:0706.1382), such a set does not exist for binary stabilizer codes. Here we generalize our work to show that for subsystem stabilizer codes in $d$-dimensional Hilbert space, such a universal set of transversal gates cannot exist for even one encoded qudit, for any dimension $d$, prime or nonprime. This result strongly supports the idea that other primitives, such as quantum teleportation, are necessary for universal fault-tolerant quantum computation, and may be an important factor for fault-tolerance noise thresholds.