Source-linked AI summary
Restrictions on Transversal Encoded Quantum Gate Sets
Bryan Eastin, Emanuel Knill
TL;DR
The paper addresses whether nontrivial quantum codes can implement all logical gates transversally, a long-standing question motivated by the desire to use transversal gates exclusively. Using a Lie-group proof, it shows that codes detecting arbitrary errors on any single subsystem cannot have universal transversal logical gates.
Problem
The paper asks whether nontrivial quantum codes can implement all logical gates transversally, a long-standing question motivated by the desire to use transversal gates exclusively.
Source: p. 1Method
The paper proves its result using the structure of the Lie group of transversal unitary operators and its action on local-error-detecting codes.
Source: p. 2Results
For any nontrivial local-error-detecting quantum code, the set of transversal logical unitary operators is not universal.
Source: p. 3Takeaways & Limitations
The paper identifies nonunitary operators, including measurements and classical feed-forward, as an alternative route to circumventing the impossibility.
Source: p. 3The paper leaves open the maximum size and computational power of finite transversal gate groups, which fall short of full universality.
Source: p. 4
Abstract
from arXiv · showhide
Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.