Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding

TL;DR

This work analyzes asymptotically zero-rate holographic quantum codes under biased Pauli noise, employing tensor-network maximum-likelihood decoding to assess thresholds. Using seed tensors such as HaPPY, SCF, Steane, and a tailored 7,1,3 code (including a Clifford-deformed Steane variant), the authors demonstrate that these holographic codes can reach or closely approach the hashing bound $R = 1 - H(\bar p)$ across a range of biases, with $\bar p = p\bar r$ and $\eta = r_Z/(r_X+r_Y)$ controlling the bias. Notably, pure 1-Pauli biases allow HaPPY and tailored 7,1,3 to attain the hashing bound, while several codes approach or surpass it for 2-Pauli noise; Clifford deformations further enhance thresholds for certain channels. The results establish holographic codes as a competitive, efficiently decodable class under biased noise, offering practical decoding advantages and guiding future explorations of gauge fixing, finite-rate variants, and hardware-aware noise tailoring.

Abstract

A crucial insight for practical quantum error correction is that different types of errors, such as single-qubit Pauli operators, typically occur with different probabilities. Finding an optimal quantum code under such biased noise is a challenging problem, related to the (generally unknown) maximum capacity of the corresponding noisy channel. A benchmark for this capacity is given by the hashing bound, which describes the performance of random stabilizer codes and leads to the matter of identifying codes that come close to the bound while also being efficiently decodable. In this work, we perform the first comprehensive analysis of asymptotically zero-rate holographic codes under biased noise. We show that many representatives from such models of this code class fulfill both the channel optimality and efficient decoding guarantees for tensor-network codes. In fact, all holographic codes tested were found to reach the hashing bound in some bias regime, while several built from the $\codepar{5,1,2}$ surface code and $\codepar{6,1,3}$ code exceed state-of-the-art code performance in the 2-Pauli noise regime. Furthermore, we consider Clifford deformations which allow all considered codes to reach the hashing bound for 1-Pauli noise as well. Our results establish that holographic codes, which were previously shown to possess efficient tensor-network decoders, also exhibit competitive thresholds under biased noise.

Figures (6)

• Figure 1: The holographic tensor-network codes considered in this work: (a) The Harlow-Preskill-Pastawski-Yoshida (HaPPY) code happy_paper whose pentagon-hexagon geometry also underlies the holographic surface-code fragment (SCF) model harris_iod, (b) the holographic $\llbracket 6,1,3 \rrbracket$ code farrelly_tn_codes, and (c) the holographic Steane code harris_css. All are defined as tensor network contractions of copies of a fixed $q$-leg tensor on a hyperbolic tiling, with the central tensor used as a seed tensor for the encoding isometry of a $\llbracket q-1,1,d \rrbracket$ code with some distance $d$. The remaining tensors have all $q$ legs contracted in the plane, leading to a larger encoding map between one logical qubit (central red leg) and the boundary physical qubits (open black legs). Here we depict the codes with $R=2$ layers of edge inflation. In the $R\to\infty$ limit, the rate of each code goes to zero.
• Figure 2: Threshold data points surveyed for the holographic codes tested in this work. All data points were ascertained using four threshold crossing points at each corresponding marker in the diagram. In blue, we highlight the pure Pauli biased points, and in red, we have accentuated the 2-Pauli bias points considered in this work. The depolarizing noise point is denoted in violet.
• Figure 3: Ternary plots for all holographic codes investigated in this study under biased noise, in the code-capacity setting. (a)-(d) depicts the zero-rate HaPPY, Steane, $\llbracket 6,1,3 \rrbracket$, and SCF codes; thresholds are color-coded from dark blue ($p_{th} = 10\%$) to dark red ($p_{th} = 50\%$). Additionally, we tested the tailored$\llbracket 7,1,3 \rrbracket$ code from tailored_713tailored_513_xzzx_codes, which achieved an identical threshold profile to that of the HaPPY code.
• Figure 4: Threshold curves for depolarizing, pure $X$, $Y$, and $Z$ noise, as studied using the tensor-network decoder, for the zero-rate HaPPY code at up to $R=3$ layers of edge inflation.
• Figure 5: Hashing bound plots for all of the codes tested in this work; we denote the hashing bound values in black solid line, while holographic codes are shown in varying colors and markers, together with calculated uncertainties. For all of the plots, biases ranged from $\eta = \{0, +\infty\}$. We list particular points of interest in \ref{['table:hb_extremal_points']}.