Far from Perfect: Quantum Error Correction with (Hyperinvariant) Evenbly Codes

Abstract

We introduce a new class of qubit codes that we call Evenbly codes, building on a previous proposal of hyperinvariant tensor networks. Its tensor network description consists of local, non-perfect tensors describing CSS codes interspersed with Hadamard gates, placed on a hyperbolic $\{p,q\}$ geometry with even $q\ge 4$, yielding an infinitely large class of subsystem codes. We construct an example for a $\{5,4\}$ manifold and describe strategies of logical gauge fixing that lead to different rates $k/n$ and distances $d$, which we calculate analytically, finding distances which range from $d=2$ to $d\sim n^{2/3}$. Investigating threshold performance under erasure, depolarizing, and pure Pauli noise channels, we find that the code exhibits a depolarizing noise threshold of about 19.1% in the code-capacity model and 50% for pure Pauli and erasure channels under suitable gauges. We also test a constant-rate version with $k/n=0.125$, finding excellent error resilience (about 40%) under the erasure channel. Recovery rates for these and other settings are studied both under an optimal decoder as well as a more efficient but non-optimal greedy decoder. We also consider generalizations beyond the CSS tensor construction, compute error rates and thresholds for other hyperbolic geometries, and discuss the relationship to holographic bulk/boundary dualities. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.

30 pages, 13 figures

References in corpus (56)

• Quantum Computing in the NISQ era and beyond
• Photonic quantum technologies
• Improved Simulation of Stabilizer Circuits
• Trapped-Ion Quantum Computing: Progress and Challenges
• A class of quantum many-body states that can be efficiently simulated
• Entanglement Renormalization and Holography
• Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
• Bulk Locality and Quantum Error Correction in AdS/CFT
• Photonic quantum information processing: a concise review
• Quantum computing with neutral atoms
• Repeated Quantum Error Detection in a Surface Code
• Stabilizer Formalism for Operator Quantum Error Correction
• Quantum Low-Density Parity-Check Codes
• Hyperbolic Lattices in Circuit Quantum Electrodynamics
• The XZZX Surface Code
• Universal fault-tolerant quantum computation with only transversal gates and error correction
• Maximally multipartite entangled states
• Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
• Genuinely multipartite entangled states and orthogonal arrays
• Strong Resilience of Topological Codes to Depolarization
• Flat entanglement spectra in fixed-area states of quantum gravity
• Flag fault-tolerant error correction with arbitrary distance codes
• Holographic Renyi Entropy from Quantum Error Correction
• Constructions and Noise Threshold of Hyperbolic Surface Codes
• Tensor Networks from Kinematic Space
• Consistency Conditions for an AdS/MERA Correspondence
• Learning the Alpha-bits of Black Holes
• Quantum minimal surfaces from quantum error correction
• Hyper-invariant tensor networks and holography
• Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation
• Code properties from holographic geometries
• Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes
• Quantum circuits for maximally entangled states
• Conformal Quasicrystals and Holography
• Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks
• Central charges of aperiodic holographic tensor network models
• Majorana dimers and holographic quantum error-correcting codes
• Constructions of k-uniform and absolutely maximally entangled states beyond maximum distance codes
• Tensor-network codes
• Entanglement renormalization, quantum error correction, and bulk causality
• Approximate Bacon-Shor Code and Holography
• Holographic Codes from Hyperinvariant Tensor Networks
• Decoding Holographic Codes with an Integer Optimisation Decoder
• Planar Maximally Entangled States
• Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computing
• $k$-uniform mixed states
• Conformal Properties of Hyperinvariant Tensor Networks
• Fault-tolerant logical gates in holographic stabilizer codes are severely restricted
• Modifying method of constructing quantum codes from highly entangled states
• Bulk Locality and Entanglement Swapping in AdS/CFT
• Hyper-Invariant MERA: Approximate Holographic Error Correction Codes with Power-Law Correlations
• Deconfinement and Error Thresholds in Holography
• Near-Term Spin-Qubit Architecture Design via Multipartite Maximally-Entangled States
• Multipartite entanglement and quantum error identification in $D$-dimensional cluster states
• Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes
• Bulk-boundary correspondence from hyper-invariant tensor networks

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