LEGO_HQEC: Automating the Analysis, Construction, and Decoding of Holographic Quantum Codes

TL;DR

LEGO_HQEC addresses the challenge of studying holographic quantum codes by providing an open-source software tool that automates construction, stabilization/operator extraction, and decoding under common noise channels. It couples the quantum LEGO formalism with a modular Python implementation to enable code construction on regular hyperbolic tilings and evaluation with three decoders: erasure Gaussian-elimination, Pauli integer-optimization, and tensor-network decoding. The authors demonstrate the tool on canonical holographic codes (HaPPY and Steane) and present new numerical results for the holographic black-hole HaPPY code, establishing an erasure threshold and illustrating finite-size behavior. This framework enables systematic exploration and benchmarking of holographic code variants and supports extensions to broader tilings, seed codes, and fault-tolerant schemes.

Abstract

Quantum error correction (QEC) is a crucial prerequisite for future large-scale quantum computation. Finding and analyzing new QEC codes, along with efficient decoding and fault-tolerance protocols, is central to this effort. Holographic codes are a recent class of generalized concatenated codes derived from holographic bulk/boundary dualities. In addition to exploring the physics of such dualities, these codes possess useful QEC properties such as tunable encoding rates, distance scaling competitive with other well-studied code classes,and excellent recovery thresholds. To allow for a comprehensive analysis of holographic code constructions, we introduce LEGO_HQEC, a software package utilizing the quantum LEGO formalism. This package allows for the construction and analysis of holographic codes on regular hyperbolic tilings, computing their stabilizer generators and logical operators for a specified number of seed codes and layers. Three decoders are included: an erasure decoder based on Gaussian elimination; an integeroptimization decoder; and a tensor-network decoder. With these tools, LEGO_HQEC enables systematic studies of both previously known holographic codes and novel variants. As a demonstration, we provide new numerical results on the holographic blackhole pentagon code, establishing its threshold behavior under the erasure channel as a benchmark example.

Figures (9)

• Figure 1: Tensor-network diagrams depicting (a)-(b) the tensor $T_{i_{1} \cdots i_{m}}$ as it pertains to a quantum state, as well as an isometric map via the Choi-Jamiolkowski isomorphism wilde2013quantum. Here, red open legs represent logical input qubits, and black legs depict physical output qubits. (c)-(d) The action of tensor pushing an operator $\mathcal{O}$ through the input subsystems of the isometry in order to find an output-leg representation, which we name $\mathcal{O}'$.
• Figure 2: (a) One may define an operator $\mathcal{\bar{O}}$ on an input leg to a tensor by its representation (in the present case $\mathcal{O}_{1} \otimes \cdots \otimes \mathcal{O}_{6}$) on the output legs. (b) Contracting indices together yields a new, conjoined tensor network. (c) Deducing the new logical operator is possible by keeping the original logical operators that satisfy the matching condition $\mathcal{O}_{3} = \mathcal{O}'^{*}_{6}$ and $\mathcal{O}_{4} = \mathcal{O}'^{*}_{5}$quantumlegoFarrelly_ltnc.
• Figure 3: Examples of several holographic tensor-network codes that LEGO_HQEC can construct, decode, and analyze. (a) The maximum-rate Harlow-Preskill-Pastawski-Yoshida (HaPPY) code on a hyperbolic pentagon tiling, mapping from logical (red) to physical (black) legs (\ref{['section:happy_erasure']}). (b) A partially gauge-fixed version of the HaPPY code is created by projecting some logical legs onto an eigenstate $\ket{G}$ of a gauge operator $G$. (c) The asymptotically zero-rate holographic Steane code, which uses the same 8-leg tensor once as a logical-to-physical map (central heptagon tensor) and otherwise with only planar legs (octagon tensors); a test of this code is performed in \ref{['section:steane_depo']}.
• Figure 4: Methods by which one can perform holographic concatenation. Both (a) edge inflation and (b) vertex inflation of a tensor network on the $\{5,4\}$ face-centered hyperbolic tessellation conformal_quasicrystals_holographyjahn2022tensor_qcftjahn_topical. In both cases, we show the first three inflation steps around a central pentagon.
• Figure 5: Pictographic representation of the stabilizer / logical operator generator in LEGO_HQEC. TensorNetwork is a list containing all of the Tensor objects, whose attributes include the TensorLeg object. This list of tensors is then filtered into the Push function, which elementwise processes and pushes each operator from each tensor to the boundary, obtaining a boundary representation. Finally, a correctness checker (CorrectnessChecker) backtracks along the tensor network, checking to see that each contracted index from each tensor contains only the identity operator. The final boundary representation is read out with ReadOut.