Source-linked AI summary
Classification of topologically protected gates for local stabilizer codes
Sergey Bravyi, Robert Koenig
TL;DR
The paper asks which fault-tolerant encoded gates constant-depth circuits can realize in topological stabilizer codes. It classifies these gates within the hierarchy P_D, showing Clifford restrictions in two dimensions and broader but still constrained classes in higher dimensions.
Problem
The paper studies which encoded operations can be implemented fault-tolerantly by constant-depth circuits in topological stabilizer codes.
Method
The authors analyze constant-depth, geometrically local circuits preserving topological stabilizer-code spaces and classify the induced encoded gates using the hierarchy P_j.
Results
The induced encoded gate belongs to P_D; in two dimensions this is the Clifford group, while the result applies under sufficiently large-distance and locality conditions.
Takeaways & Limitations
For fixed logical-qubit count, the finite set of topologically protected gates cannot provide computational universality, and two-dimensional computation must use steps without topological protection.
Takeaways & Limitations
The stated circuit extension requires ξ, hr ≪ d^(1/D), with d the code distance and ξ the maximum parity-check range.
Abstract
from arXiv · showhide
Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault-tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be "turned off" for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate U is implementable by a constant-depth circuit only if the image of any Pauli operator under conjugation by U belongs to the Clifford group. This class of gates includes some non-Clifford gates such as the π/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance.