Unfolding the color code (1503.02065v1)

Abstract: The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of the $d$-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for $d=2$, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the $d$-dimensional color code with $d+1$ boundaries of $d+1$ distinct colors, we find that the code is equivalent to multiple copies of the $d$-dimensional toric code which are attached along a $(d-1)$-dimensional boundary. In particular, for $d=2$, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the $d$-dimensional toric code admits logical non-Pauli gates from the $d$-th level of the Clifford hierarchy, and thus saturates the bound by Bravyi and K\"{o}nig. In particular, we show that the $d$-qubit control-$Z$ logical gate can be fault-tolerantly implemented on the stack of $d$ copies of the toric code by a local unitary transformation.

• The paper demonstrates that d-dimensional color codes can be decomposed into multiple toric codes using local unitary operations.
• It extends code equivalence from 2D to higher dimensions and systems with boundaries by leveraging lattice colorability.
• The work presents fault-tolerant implementations of non-Clifford logical gates, bridging transversal operations between the two codes.

Overview of Quantum Error-Correcting Codes: Color Code and Toric Code Equivalence

In the pursuit of fault-tolerant quantum computation, topological quantum codes have emerged as pivotal structures that safeguard quantum information against errors through geometrically local generators. Among these, the topological color code and the toric code are prominent candidates that facilitate fault-tolerant quantum computation by leveraging topological properties to protect information.

The paper "Unfolding the Color Code" by Kubica, Yoshida, and Pastawski explores the nuanced relationship between the color code and the toric code. The authors establish that the color code defined on a $d$-dimensional manifold can be decomposed into multiple independent toric codes through local unitary transformations—transitioning from a fundamentally interconnected structure to decoupled components, thus consolidating our understanding of topological phase equivalence across dimensions.

Detailed Findings and Methodology

1. Equivalence on Closed Manifolds: The authors extend previously known results by showing the equivalence of the color code and toric code beyond two dimensions. For a $d$-dimensional topological color code defined on a closed manifold, this code is equivalent to multiple decoupled copies of the $d$-dimensional toric code. The transformation relies on local unitary operations and the strategic addition or removal of ancilla qubits, bringing to light the significance of a lattice's colorability in constructing such equivalences.
2. Equivalence in the Presence of Boundaries: The extension of these equivalence results to systems with boundaries reveals a rich structure. In $d$ dimensions, a color code with $d+1$ boundaries, each of a distinct color, is shown to correspond to $d$ decoupled toric codes that share a common boundary. Specifically, for $d=2$, the color code with triangular boundary conditions translates into a toric code with boundary conditions amenable to folding into a surface code format.
3. Implications for Logical Gates: The authors discuss the implementation of non-Clifford logical gates, particularly focusing on the $d$-qubit control-Z logical gate within the framework of the $d$-dimensional toric code. They establish the fault-tolerant implementation of such a gate through local unitary transformations, bridging the capabilities of the color code in transversal gate execution to the toric code framework.

Implications and Future Directions

The results have substantial implications in classifying topological quantum phases and improving quantum computation techniques. The equivalence between color codes and toric codes across dimensions underscores their topological similarity and enhances the toolkit available for manipulating quantum information.

Furthermore, the ability to transform between these structures aids in code deformation methodologies and lattice surgery, potentially minimizing qubit overhead and enhancing error suppression. The insights into boundary conditions and fault-tolerant gate implementation pave the way for more resource-efficient quantum computing models and strengthen the link between topological quantum codes and physical realizability in quantum computers.

In conclusion, this research offers a deeper understanding of the interplay between topological structures in quantum error correction and opens avenues for practical developments in quantum computation architectures, particularly in constructing robust, resource-efficient quantum codes capable of supporting a comprehensive suite of logical operations.