[Paper Review] Topological Quantum Computation with Gapped Boundaries and Boundary Defects

Iris Cong, Zhenghan Wang|arXiv (Cornell University)|Oct 19, 2017

Quantum Computing Algorithms and Architecture27 references3 citations

TL;DR

This paper proposes a framework for universal topological quantum computation using gapped boundaries and boundary defects in Dijkgraaf-Witten topological quantum field theories, particularly for finite groups like $\mathbb{Z}_3$ and $S_3$. By combining braiding operations with generalized topological charge measurements, the authors achieve universality in quantum computation, extending the reach of topological quantum computation beyond non-abelian anyons to include boundary-based anyonic-like degrees of freedom.

ABSTRACT

We survey some recent work on topological quantum computation with gapped boundaries and boundary defects and list some open problems.

Motivation & Objective

• To extend topological quantum computation beyond non-abelian anyons to include gapped boundaries and boundary defects as computational resources.
• To establish a systematic method for encoding qudits and performing topologically protected operations using gapped boundaries in unitary modular categories.
• To demonstrate universality in quantum computation by combining braiding with generalized topological charge measurements in $\mathfrak{D}(\mathbb{Z}_3)$.
• To generalize Hamiltonian constructions for gapped boundaries to non-trivial cocycles and more general TQFTs beyond finite group models.
• To explore the computational power of boundary defects and domain walls in topological phases, including their role in universal gate sets.

Proposed method

• Construct a corrected and generalized local commuting projector Hamiltonian for Dijkgraaf-Witten theories $\mathfrak{D}(G)$ with gapped boundaries labeled by subgroups $K \subseteq G$ and trivial 2-cocycles $\omega$.
• Use Lagrangian algebras $\mathcal{A}$ in the doubled category $\mathcal{B} = \mathcal{Z}(\mathcal{C})$ to classify stable gapped boundaries and model them as condensates of bulk anyons.
• Implement generalized topological charge measurements as a computational primitive to supplement braiding, enabling universality in $\mathfrak{D}(\mathbb{Z}_3)$.
• Leverage the correspondence between Lagrangian algebras and indecomposable module categories over $\mathcal{C}$ to label and classify gapped boundaries and boundary defects.
• Extend the Kitaev toric code model to include boundary terms that realize gapped boundaries via local Hamiltonian terms on the boundary edges.
• Use the folding trick to treat gapped boundaries as special cases of domain walls, enabling a unified mathematical and physical description of boundary and defect structures.

Experimental results

Research questions

• RQ1 Can gapped boundaries and boundary defects in Dijkgraaf-Witten theories support universal topological quantum computation when combined with braiding and generalized charge measurements?
• RQ2 How can the Hamiltonian framework for gapped boundaries be generalized to include non-trivial 2-cocycles $\omega \in H^2(K, \mathbb{C}^\times)$ and more general TQFTs?
• RQ3 What is the computational power of boundary defects in abelian theories like the $\mathbb{Z}_2$ toric code, and can they support universal gate sets?
• RQ4 Can a purely topological implementation of charge measurement be achieved in abelian theories, avoiding symmetry protection and enabling universal topological gate sets?
• RQ5 How do gapped domain walls and boundary defects in $\mathfrak{D}(S_3)$ support universal quantum computation via braid group representations and tunneling operators?

Key findings

• Universal quantum computation is achieved in $\mathfrak{D}(\mathbb{Z}_3)$ by combining braiding of gapped boundaries with generalized topological charge measurements, overcoming the limitation of non-universality via braiding alone.
• Gapped boundaries in $\mathfrak{D}(G)$ are classified by subgroups $K \subseteq G$ and trivial 2-cocycles $\omega$, with the boundary anyon having quantum dimension $d_{\mathcal{A}}$ corresponding to the Lagrangian algebra $\mathcal{A}$.
• The generalized topological charge measurement is shown to be a necessary and sufficient primitive to achieve universality when braiding is insufficient, particularly in weakly integral anyon theories.
• The corrected Hamiltonian construction for gapped boundaries in $\mathfrak{D}(G)$ extends Kitaev's original toric code model to include boundary terms that stabilize the topological degeneracy.
• Boundary defects in $\mathfrak{D}(S_3)$ support non-unitary tunneling operators and full braid group representations, suggesting potential for universal quantum computation in non-abelian theories.
• A systematic framework for encoding qudits and performing topologically protected operations on boundary defects is proposed, with potential for physical realization in fractional quantum Hall systems with superconducting coupling.