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Classification of topologically protected gates for local stabilizer codes.

Sergey Bravyi¹, Robert König

¹IBM Watson Research Center, Yorktown Heights, New York 10598, USA.

Physical Review Letters

|May 18, 2013

Summary

Topological stabilizer codes offer error protection for quantum computations. In 2D, only Clifford gates are efficiently implementable, while 3D codes allow some non-Clifford gates for universal quantum computation.

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Area of Science:

Quantum Information Science
Quantum Computing
Error Correction Codes

Background:

Efficient and fault-tolerant implementation of encoded operations is crucial for quantum error correcting codes.
Topological stabilizer codes offer inherent protection against errors through their structure.
Constant-depth quantum circuits provide limited error propagation, enhancing gate protection.

Purpose of the Study:

To investigate the limitations and possibilities of implementing encoded unitary gates using constant-depth quantum circuits within topological stabilizer codes.
To classify which encoded gates can be implemented fault-tolerantly in 2D and 3D topological stabilizer codes.
To determine the conditions for achieving universal quantum computation with topological protection.

Main Methods:

Analysis of encoded unitary gates implementable by constant-depth quantum circuits.
Classification of gates based on their behavior under conjugation by Pauli operators.
Examination of topological stabilizer codes in 2D and 3D geometries.

Main Results:

In 2D topological stabilizer codes, only Clifford group gates can be implemented by constant-depth circuits.
In 3D topological stabilizer codes, gates U satisfying UPU(†) in the Clifford group for all Pauli P are implementable.
This class of gates in 3D includes some non-Clifford gates, like the π/8 rotation, enabling universality.

Conclusions:

Achieving universal quantum computation with topological protection requires temporarily "turning off" some protection, particularly in 2D.
3D topological stabilizer codes offer a pathway to implement a broader set of gates, including some non-Clifford gates, within constant-depth circuits.
The findings provide a classification for fault-tolerant gate implementation in stabilizer codes with local stabilizers and sufficient code distance.