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Beyond the Hashing Limit of Quantum Pauli Channels

Jihao Fan

Published Jul 7, 2026• 0 citations
Publisher

Abstract

Abstract The classical noisy-channel coding theorem developed by Shannon establishes a fundamental limit on reliable information transmission. In quantum regime, this framework is extended by results such as the Lloyd-Shor-Devetak theorem, which gives the hashing limit on quantum channel capacity. In this work, we beat the hashing limit for low rate coding schemes over quantum Pauli channels. Our approach is based on a re-examination of Bacon-Shor codes and their generalizations, which we show are not merely simple constructions but instances of a broader mechanism that fundamentally alters achievable rate limits. Over depolarizing channels with depolarizing probability of p, these codes exceed the hashing bound with 0.1145 ≤ p ≤ 0.75, and further surpass the asymptotic quantum Singleton bound under bounded minimum-distance decoding for 0.3301 ≤ p ≤ 0.75. We further present explicit constructions based on quantum low-density parity-check codes, achieving frame error rates below 10−4 while exceeding the hashing bound. The advantage persists and is pronouncedly enhanced in the presence of biased noise, demonstrating robustness across realistic quantum Pauli channels. Our results challenge the prevailing understanding of fundamental limits in quantum coding and information theory.

DOI
10.21203/rs.3.rs-10243494/v1

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Abstract The classical noisy-channel coding theorem developed by Shannon establishes a fundamental limit on reliable information transmission. In quantum regime, this framework is extended by results such as the Lloyd-Shor-Devetak theorem, which gives the hashing limit on quantum channel capacity. In this work, we beat the hashing limit for low rate coding schemes over quantum Pauli channels. Our approach is based on a re-examination of Bacon-Shor codes and their generalizations, which we show are not merely simple constructions but instances of a broader mechanism that fundamentally alters achievable rate limits. Over depolarizing channels with depolarizing probability of p, these codes exceed the hashing bound with 0.1145 ≤ p ≤ 0.75, and further surpass the asymptotic quantum Singleton bound under bounded minimum-distance decoding for 0.3301 ≤ p ≤ 0.75. We further present explicit constructions based on quantum low-density parity-check codes, achieving frame error rates below 10−4 while exceeding the hashing bound. The advantage persists and is pronouncedly enhanced in the presence of biased noise, demonstrating robustness across realistic quantum Pauli channels. Our results challenge the prevailing understanding of fundamental limits in quantum coding and information theory.
Abstract Recent progress in quantum computing has enabled systems with tens of reliable logical qubits, built from thousands of noisy physical qubits1. However, many impactful applications demand quantum computations with millions of logical qubits2, necessitating highly scalable quantum error correction. In classical information theory, low-density parity-check (LDPC) codes3 can approach channel capacity efficiently4. Yet, no quantum error-correcting codes with efficient decoding have been shown to approach the hashing bound—a fundamental limit on quantum capacity—despite decades of research5–7. Here, we present quantum LDPC codes that not only approach the hashing bound but also allow decoding with computational cost linear in the number of physical qubits. This breakthrough paves the way for large-scale, fault-tolerant quantum computation. Combined with emerging hardware that manages many qubits, our approach brings quantum solutions to important real-world problems significantly closer to reality.
Quantum dense coding enables the transmission of two classical bits of information through the exchange of a single qubit, provided that an entangled state is shared between the communicating parties. While this protocol achieves a fundamental increase in channel capacity, its practical realization is strongly influenced by the presence of noise in realistic quantum channels. This work presents a simulation-based study of dense coding performance under two representative noise models: the depolarizing channel and the amplitude damping channel. Using Qiskit Aer, the bit error rate (BER), frame error rate (FER), and effective channel capacity are evaluated across a range of noise parameters. In addition, a classical threefold repetition strategy is examined as a lightweight error-mitigation technique. Results indicate that depolarizing noise produces more rapid degradation in both BER and channel capacity compared to amplitude damping, while repetition coding provides partial improvement in reliability at the expense of resource overhead. These findings highlight the trade-offs between efficiency and robustness in noisy implementations of dense coding and motivate further exploration of more advanced quantum error correction methods.
The quantum hashing bound guarantees that rates up to 1−H(pI,pX,pY,pZ)1-H(p_I, p_X, p_Y, p_Z)1−H(pI​,pX​,pY​,pZ​) are achievable for memoryless Pauli channels, but it is not generally tight. A known way to improve achievable rates for certain asymmetric Pauli channels is to apply a small inner stabilizer code to a few channel uses, decode, and treat the resulting logical noise as an induced Pauli channel; reapplying the hashing argument to this induced channel can beat the baseline hashing bound. We generalize this induced-channel viewpoint to arbitrary stabilizer codes used purely as channel transforms. Given any $ [\![ n, k ]\!] $ stabilizer generator set, we construct a full symplectic tableau, compute the induced joint distribution of logical Pauli errors and syndromes under the physical Pauli channel, and obtain an achievable rate via a hashing bound with decoder side information. We perform a structured search over small transforms and report instances that improve the baseline hashing bound for a family of Pauli channels with skewed and independent errors studied in prior work.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.
A key appeal of quantum low-density parity check (qLDPC) codes is their ability to suppress stochastic Pauli noise below nonzero thresholds. Coherent errors are fundamentally different: they produce superpositions of error patterns whose amplitudes can interfere even after syndrome measurement. Rigorous understanding of coherent errors remains limited. Here we show that general qLDPC codes admit a nonzero code capacity threshold against local coherent noise and more generally local channel noise. For any family of qLDPC codes with distance d=Ω(log⁡n)d=Ω(\log n)d=Ω(logn), we show that there is a constant noise strength below which the logical recovery error in diamond distance decays exponentially with the code distance. The result is established for optimal recovery as well as the minimum-weight decoder. The key technical ingredient is what we call a \emph{cluster resummation}: rather than bounding superposed error configurations one by one, we isolate a large connected error cluster in the channel expansion and exactly resum all errors disconnected from it before taking norms. Standard cluster counting then yields exponential suppression. This work resolves a longstanding challenge in fault tolerance theory, providing general robustness guarantees for qLDPC codes against coherent noise and laying a rigorous foundation for future studies of fault-tolerant quantum technologies.
Quantum error correction is essential for realizing scalable quantum computation. Among various approaches, low-density parity-check codes over higher-order Galois fields have shown promising performance due to their structured sparsity and compatibility with iterative decoding algorithms whose computational complexity scales linearly with the number of physical qubits. In this work, we demonstrate that explicitly exploiting the degeneracy of quantum errors can significantly enhance the decoding performance. Simulation results over the depolarizing channel indicate that the proposed method, at a coding rate of 1/3, achieves a frame error rate as low as 10−410^{-4}10−4 at a physical error rate of 9.45% for a code with 104,000 logical qubits and 312,000 physical qubits, approaching the quantum hashing bound. These findings highlight the critical role of degeneracy in closing the gap to the fundamental limits of quantum error correction.