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

Understand

A long-standing open problem in fault-tolerant quantum computation has been to find a universal set of transversal gates.

• As three of us proved in 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.