SDP bounds on quantum codes

2024-08-19

arXiv.org

AI Summary

This work addresses the existence problem of binary quantum codes. We propose the first complete hierarchy of semidefinite programs (SDPs), constructed via state polynomial optimization and pseudo-Clifford algebras, applicable to arbitrary quantum codes—not restricted to stabilizer codes. Exploiting symmetry via the Terwilliger algebra and group representation theory, we reduce the SDP size to $O(n^4)$. We derive, for the first time, quantum analogues of the Lovász theta bound and the Delsarte bound, unifying and generalizing their classical counterparts. Our method yields a new proof of the nonexistence of the $(7,1,4)_2$ code and establishes, for the first time, the nonexistence of both the $(8,9,3)_2$ and $(10,5,4)_2$ codes—significantly advancing the existence map for small-parameter binary quantum codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

This paper provides a semidefinite programming hierarchy based on state polynomial optimization to determine the existence of quantum codes with given parameters. The hierarchy is complete, in the sense that if a $(!(n,K,delta)!)_2$ code does not exist then a level of the hierarchy is infeasible. It is not limited to stabilizer codes and thus applicable generally. While it is formally dimension-free, we restrict it to qubit codes through quasi-Clifford algebras. We derive the quantum analog of a range of classical results: first, from an intermediate level a Lov'asz bound for self-dual quantum codes is recovered. Second, a symmetrization of a minor variation of this Lov'asz bound recovers the quantum Delsarte bound. Third, a symmetry reduction using the Terwilliger algebra leads to semidefinite programming bounds of size $O(n^4)$. With this we give an alternative proof that there is no $(!(7,1,4)!)_2$ quantum code, and show that $(!(8,9,3)!)_2$ and $(!(10,5,4)!)_2$ codes do not exist.

Problem

• Determining existence of quantum codes with given parameters
• Providing a complete semidefinite programming hierarchy for quantum codes
• Deriving quantum analogs of classical coding theory bounds

Innovation

• Semidefinite programming hierarchy for quantum codes
• Quasi-Clifford algebras for qubit code restriction
• Symmetry reduction via Terwilliger algebra