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 (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. © 2008 The American Physical Society.