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

TL;DR

This paper proves that subsystem stabilizer codes in any dimension cannot have a universal set of transversal gates for even one encoded qudit, implying the necessity of alternative methods like quantum teleportation for universal fault-tolerant quantum computation.

Contribution

It generalizes previous results to show the non-existence of universal transversal gates for subsystem stabilizer codes in any dimension, including nonprime dimensions.

Findings

Universal transversal gates do not exist for subsystem stabilizer codes in any dimension.

Supports the need for alternative primitives like quantum teleportation.

Implications for fault-tolerance noise thresholds.

Abstract

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.