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Classification of topologically protected gates for local stabilizer codes

Sergey Bravyi, Robert Koenig

TL;DR

The paper classifies the set of encoded gates that can be implemented topologically protected, i.e., by constant-depth, geometrically local circuits, in topological stabilizer codes. In 2D, any such gate must lie in the Clifford group, implying that universality requires breaking topological protection at some computation step (e.g., magic-state distillation). In 3D, a constant-depth implementable gate must map Pauli operators to Clifford operations, including certain non-Clifford gates like the pi/8 rotation, thereby broadening the allowed set but still prohibiting full universality for fixed logical-qubit number. The authors give a general theorem for D-dimensional TSCs that links depth-limited locality to the Dth level of the Gottesman-Chuang hierarchy, and provide a proof strategy using light-cone arguments and locality arguments, with corollaries about finite protected-gate sets for fixed k.

Abstract

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.

Classification of topologically protected gates for local stabilizer codes

TL;DR

The paper classifies the set of encoded gates that can be implemented topologically protected, i.e., by constant-depth, geometrically local circuits, in topological stabilizer codes. In 2D, any such gate must lie in the Clifford group, implying that universality requires breaking topological protection at some computation step (e.g., magic-state distillation). In 3D, a constant-depth implementable gate must map Pauli operators to Clifford operations, including certain non-Clifford gates like the pi/8 rotation, thereby broadening the allowed set but still prohibiting full universality for fixed logical-qubit number. The authors give a general theorem for D-dimensional TSCs that links depth-limited locality to the Dth level of the Gottesman-Chuang hierarchy, and provide a proof strategy using light-cone arguments and locality arguments, with corollaries about finite protected-gate sets for fixed k.

Abstract

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.

Paper Structure

This paper contains 1 section, 4 theorems, 14 equations, 2 figures.

Table of Contents

  1. Acknowledgments

Key Result

Theorem 1

Suppose a unitary operator $U$ implementable by a constant-depth quantum circuit preserves the codespace ${\cal C }$ of a topological stabilizer code on a $D$-dimensional lattice, $D\ge 2$. Then the restriction of $U$ onto ${\cal C }$ implements an encoded gate from the set ${\cal P }_D$.

Figures (2)

  • Figure 1: Non-contractible closed strips $\gamma_1,\gamma_2$ and $\delta_1,\delta_2$ on the torus.
  • Figure 2: Simplicial partition of the lattice $\Lambda=ABC$. Starting from a triangulation with regular triangles having sides of length $R$, let $C$ be the union of discs of radius $R/4$ centered on the vertices of the triangulation. Let $B\subset \Lambda\backslash C$ be union of the $R/8$-neighborhoods of each edge in the remaining surface. Finally, let $A=\Lambda\backslash (B\cup C)$ be the union of the remaining capped triangles. A similar but rectangular partition has been used in BPT10 to derive upper bounds on parameters of TSCs, but is less suitable for generalization to $D>2$.

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • Lemma 1: Cleaning Lemma BT08
  • Lemma 2: Union Lemma BPT10HP10