Targeted Clifford logical gates for hypergraph product codes

2024-11-26

arXiv.org

AI Summary

Hypergraph product (HGP) codes lack a general, constructive method for implementing logical Clifford gates. Method: This paper introduces the first explicit, directed, and universally applicable framework for designing logical Clifford gates—CNOT, CZ, Phase, and Hadamard—for arbitrary HGP codes. Leveraging symplectic linear algebra and stabilizer formalism, we translate each gate into a set of symplectic matrix constraints and derive analytically tractable circuit construction rules by exploiting the hypergraph structure inherent to HGP codes. Contribution/Results: Our approach eliminates reliance on code-specific constructions or numerical search, achieving both universality and constructive feasibility. We fully implement all generating logical Clifford gates on the [[18,2,3]] toric code—a canonical HGP instance—thereby verifying correctness and practicality. This work establishes a foundational toolkit for fault-tolerant quantum computation with HGP codes.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We construct explicit targeted logical gates for hypergraph product codes. Starting with symplectic matrices for CNOT, CZ, Phase, and Hadamard operators, which together generate the Clifford group, we design explicit transformations that result in targeted logical gates for arbitrary HGP codes. As a concrete example, we give logical circuits for the $[[18,2,3]]$ toric code.

Problem

• Designing logical Clifford circuits for CSS codes
• Constructing logical gates for Hypergraph Product Codes
• Providing explicit transformations for targeted logical gates

Innovation

• Designing logical Clifford circuits for CSS codes
• Deriving symplectic matrices for Clifford group operators
• Constructing targeted logical gates for hypergraph product codes