Concatenating Algebraic Codes over High-Rate Quantum LDPC Codes

TL;DR

This paper introduces a novel concatenation scheme using algebraic outer codes over high-rate quantum LDPC codes, enabling efficient fault-tolerant quantum memory with reduced overhead and improved error correction capabilities.

Contribution

It develops a new approach to quantum code concatenation with algebraic outer codes and Galois qudits, enhancing fault tolerance and decoding strategies for high-rate quantum codes.

Findings

Achieves teraquop regime at 10^{-3} physical noise with lower space overhead.

Develops a Galois qudit Shor scheme for fault-tolerant syndrome extraction.

Demonstrates advantages of large-alphabet qudits in fault tolerance.

Abstract

Different quantum error correction schemes trade off overhead, error suppression, and hardware connectivity. Code concatenation can relax these tradeoffs by using an outer code whose non-local connectivity is supplied by logical operations of an inner code rather than directly by hardware. Prior works showed that this can reduce memory overhead for local low-rate inner codes such as the surface code. Here, we study concatenation over non-local, high-rate inner codes. Such inner codes experience correlated errors among the many logical qubits in a single codeblock. We handle this by treating each block as a single logical Galois qudit, enabling concatenation with algebraic outer codes with excellent parameters and, crucially, list decoders. In particular, we consider a memory system formed by concatenating quantum Reed-Solomon outer codes over the gross code. For fault-tolerant syndrome extraction, we develop a Galois qudit Shor scheme using "time-like" Reed-Solomon protection against measurement errors. Interestingly, a lightweight fault tolerance scheme, that would fail for qubits, works well for large-alphabet qudits, suggesting a very different theory of fault tolerance for such qudits. The whole protocol is optimised via improved bicycle instruction logical error rates, novel compilation strategies, and recent decoder post-selection rules. At uniform $10^{-3}$ physical noise, the concatenated gross code reaches the teraquop regime, which it previously could not access, with a lower space overhead than the $288$-qubit two-gross code, while offering several advantages from the engineering standpoint. Beyond our main case study, we believe the core ideas of Galois qudits, quantum Reed-Solomon outer codes, and list decoding, will prove generically powerful and highly transferable ideas across high-rate quantum architectures.