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arXiv:2410.22861

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

Junyu Fan, Matthew Steinberg, A. Jahn, ChunJun Cao, Aritra Sarkar, Sebastian Feld

Published Oct 30, 2024• 1 citation
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2410.22861

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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.
We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation.More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion.Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network.The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as nonstabilizer codes over qubits and qudits.For a contractible tensor network, the sequence of contractions also constructs a decoding or encoding circuit.To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct nontrivial codes.These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a three-dimensional stabilizer code, and other stabilizer codes with interesting properties.Surprisingly, we find that the surface code is equivalent to the two-dimensional Bacon-Shor code after "dualizing" its tensor-network encoding map.
We re-visit the pentagon holographic quantum error correcting code from a ZX-calculus perspective. By expressing the underlying tensors as ZX-diagrams, we study the stabiliser structure of the code via Pauli webs. In addition, we obtain a diagrammatic understanding of its logical operators, encoding isometries, Rényi entropy and toy models of black holes/wormholes. Then, motivated by the pentagon holographic code's ZX-diagram, we introduce a family of codes constructed from ZX-diagrams on its dual hyperbolic tessellations and study their logical error rates using belief propagation decoders. Finally, we show how to construct spacetime ZX-diagrams that realise a fault-tolerant quantum channel in which every internal edge is a protected fault location.
We provide the first tensor-network method for computing quantum weight enumerator polynomials in the most general form. If a quantum code has a known tensor-network construction of its encoding map, our method is far more efficient, and in some cases exponentially faster than the existing approach. As a corollary, it produces decoders and an algorithm that computes the code distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can be substantially faster compared to the current methods. We also introduce novel weight enumerators and their applications. In particular, we show that these enumerators can be used to compute logical error rates exactly and thus construct (optimal) decoders for any independent and identically distributed single qubit or qudit error channels. The enumerators also provide a more efficient method for computing nonstabilizerness in quantum many-body states. As the power for these speedups rely on a quantum Lego decomposition of quantum codes, we further provide systematic methods for decomposing quantum codes and graph states into a modular construction for which our technique applies. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the two-dimensional Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force. Published by the American Physical Society 2024
We present a comprehensive mathematical framework establishing the foundations of holographic quantum computing, a novel paradigm that leverages holographic phenomena to achieve superior error correction and algorithmic efficiency. We rigorously demonstrate that quantum information can be encoded and processed using holographic principles, establishing fundamental theorems characterizing the error-correcting properties of holographic codes. We develop a complete set of universal quantum gates with explicit constructions and prove exponential speedups for specific classes of computational problems. Our framework demonstrates that holographic quantum codes achieve a code rate scaling as O( 1/ logn ) , superior to traditional quantum LDPC codes, while providing inherent protection against errors via geometric properties of the code structures. We prove a threshold theorem establishing that arbitrary quantum computations can be performed reliably when physical error rates fall below a constant threshold. Notably, our analysis suggests certain algorithms, including those involving high-dimensional state spaces and long-range interactions, achieve exponential speedups over both classical and conventional quantum approaches. This work establishes the theoretical foundations for a new approach to quantum computation that provides natural fault tolerance and scalability, directly addressing longstanding challenges of the field.
The study of holographic bulk-boundary dualities has led to the construction of novel quantum error correcting codes. Although these codes have shed new light on conceptual aspects of these dualities, they have widely been believed to lack a crucial feature of practical quantum error correction: The ability to support universal fault-tolerant quantum logic. In this work, we introduce a new class of holographic codes that realize this feature. These heterogeneous holographic codes are constructed by combining two seed codes in a tensor network on an alternating hyperbolic tiling. We show how this construction generalizes previous strategies for fault tolerance in tree-type concatenated codes, allowing one to implement non-Clifford gates fault-tolerantly on the holographic boundary. We also demonstrate that these codes allow for high erasure thresholds under a suitable heterogeneous combination of specific seed codes. Compared to previous concatenated codes, heterogeneous holographic codes achieve large overhead savings in physical qubits, e.g., a 21.8%21.8\%21.8% reduction for a two-layer Steane/quantum Reed-Muller combination. Unlike standard concatenated codes, we establish that the new codes can encode more than a single logical qubit per code block by applying ``black hole'' deformations with tunable rate and distance, while possessing fully addressable, universal fault-tolerant gate sets. Therefore, our work strengthens the case for the utility of holographic quantum codes for practical quantum computing.
Abstract We propose and analyze a hierarchical quantum error correction (QEC) scheme that concatenates hypergraph product (HGP) codes with rotated surface codes and that is compatible with quantum computers with only nearest-neighbor interactions. The outer code employs (3,4)-random HGP codes, known for their constant encoding rate and favorable distance scaling, while the inner code consists of a rotated surface code with distance 5, allowing hardware compatibility through lattice surgery. To address the decoding bottleneck, we utilize a soft-decision decoding strategy that combines belief propagation with ordered statistics decoding, enhanced by a syndrome-conditioned logical error probability computed via a tailored lookup table for the inner code. Numerical simulations under a code capacity noise model demonstrate that our hierarchical codes achieve logical error suppression below the threshold. Furthermore, we derive explicit conditions under which the proposed codes surpass surface codes as a QEC code in both qubit efficiency and error rate. In particular, for the size parameter s≥4s \ge 4s≥4 (which corresponds to 16 logical qubits) and the distance d≥25d\ge 25d≥25, our construction outperforms the rotated surface code in practical regimes with physical error rates around or less than 10−210^{-2}10−2. These results suggest that concatenated quantum low-density parity-check surface architectures can offer a scalable and resource-efficient path toward near-term fault-tolerant quantum computation.