A Sum-of-Squares Hierarchy with Quadratic Convergence for Quantum Channel Coding

2026-09-08

AI Summary

本文通过构建一个Hermitian平方和层次结构，为任意数量的消息提供了一个具有二次收敛性的量子信道编码方法，解决了计算最优成功概率的NP难问题。

Abstract

Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.

Problem

• Quantum Channel Coding
• Optimal Success Probability
• NP-hard
• Semidefinite Programming Hierarchy
• Symmetric Extensions

Innovation

• Hermitian sum-of-squares hierarchy
• quadratic convergence
• state-discrimination duality
• positive polynomial kernels