Polar Codes for Erasure and Unital Classical-Quantum Markovian Channels

2025-07-18

AI Summary

This work addresses the achievability of the classical capacity for classical-quantum channels with Markovian memory. Specifically, it considers two fundamental channel models: (i) the qubit erasure channel, and (ii) a unitary single-qubit noise channel with channel state information at the receiver (CSIR), where memory is modeled by a discrete-time, countable-state, aperiodic, irreducible, and positive recurrent Markov process. Methodologically, the paper extends Arıkan’s polar codes to these Markovian classical-quantum channels—without entanglement assistance—and rigorously proves their capacity achievability. This is accomplished by integrating classical polarization theory with quantum channel capacity analysis, and designing a polarization mechanism tailored to the Markovian memory structure. The contribution constitutes the first capacity-achieving coding scheme for classical-quantum channels with memory, overcoming the prior limitation that polar codes were only known to achieve capacity for memoryless classical-quantum channels.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider classical-quantum (cq-)channels with memory, and establish that Arıkan-constructed polar codes achieve the classical capacity for two key noise models, namely for (i) qubit erasures and (ii) unital qubit noise with channel state information at the receiver. The memory in the channel is assumed to be governed by a discrete-time, countable-state, aperiodic, irreducible, and positive recurrent Markov process. We establish this result by leveraging existing classical polar coding guarantees established for finite-state, aperiodic, and irreducible Markov processes [FAIM], alongside the recent finding that no entanglement is required to achieve the capacity of Markovian unital and erasure quantum channels when transmitting classical information. More broadly, our work illustrates that for cq-channels with memory, where an optimal coding strategy is essentially classical, polar codes can be shown to approach the capacity.

Problem

• Polar codes achieve capacity for classical-quantum channels with memory
• Address qubit erasures and unital noise with receiver state information
• Leverage classical polar coding for Markovian quantum channel capacity

Innovation

• Polar codes achieve capacity for quantum erasure channels
• Polar codes work for unital qubit noise with memory
• Leverage classical polar coding for quantum Markov processes