Quantum Low-Density Parity-Check Codes

Abstract

Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of quantum codes called low-density parity-check (LDPC) quantum codes. The codes we discuss are alternatives to the surface code, which is the currently leading candidate to implement quantum fault-tolerance. We introduce the zoo of LDPC quantum codes and discuss their potential for making quantum computers robust against noise. In particular, we explain recent advances in the theory of LDPC quantum codes related to certain product constructions and discuss open problems in the field.

17 pages, 11 figures

References in corpus (60)

• Fault-tolerant quantum computation by anyons
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Surface codes: Towards practical large-scale quantum computation
• Good Quantum Error-Correcting Codes Exist
• Quantum computational advantage using photons
• Topological quantum memory
• Quantum Error Correction for Quantum Memories
• Quantum Computing with Very Noisy Devices
• Stabilizer Codes and Quantum Error Correction
• Local stabilizer codes in three dimensions without string logical operators
• Roads towards fault-tolerant universal quantum computation
• Quantum Glassiness
• Sparse Graph Codes for Quantum Error-Correction
• Resilient Quantum Computation: Error Models and Thresholds
• A Survey on Quantum Channel Capacities
• Hyperbolic Lattices in Circuit Quantum Electrodynamics
• Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}
• Fast Decoders for Topological Quantum Codes
• Degenerate Quantum LDPC Codes With Good Finite Length Performance
• Topological order in an exactly solvable 3D spin model
• Quantum memories at finite temperature
• Tradeoffs for reliable quantum information storage in 2D systems
• Quantum LDPC Codes with Almost Linear Minimum Distance
• Single-shot fault-tolerant quantum error correction
• Classification of topologically protected gates for local stabilizer codes
• Quantum Graphical Models and Belief Propagation
• Constructions and Noise Threshold of Hyperbolic Surface Codes
• Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes
• Constant overhead quantum fault-tolerance with quantum expander codes
• Fault-tolerant logical gates in quantum error-correcting codes
• Majorana Fermion Codes
• Asymptotically Good Quantum Codes
• Homological Error Correction: Classical and Quantum Codes
• Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage
• Subsystem codes with spatially local generators
• Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds
• Algebraic Methods for Quantum Codes on Lattices
• Minimally complex ion traps as modules for quantum communication and computing
• Combining hard and soft decoders for hypergraph product codes
• Improved quantum hypergraph-product LDPC codes
• Refined Belief Propagation Decoding of Sparse-Graph Quantum Codes
• Enhanced Feedback Iterative Decoding of Sparse Quantum Codes
• Higher-dimensional quantum hypergraph-product codes
• Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms
• Space-Time Circuit-to-Hamiltonian Construction and Its Applications
• Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs
• Numerical and analytical bounds on threshold error rates for hypergraph-product codes
• The Small Stellated Dodecahedron Code and Friends
• Numerical study of hypergraph product codes
• Fault-tolerant gates via homological product codes
• Freely Scalable Quantum Technologies using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links
• Quantum Codes from High-Dimensional Manifolds
• Fractalizing quantum codes
• Optimal quantum subsystem codes in 2-dimensions
• On Quantum Weight Reduction
• Limitations on transversal gates for hypergraph product codes
• Weight Reduction for Quantum Codes
• Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders
• Hyperbolic quantum color codes
• New Cosystolic Expanders from Tensors Imply Explicit Quantum LDPC Codes with $\Omega (\sqrt{n}{\log }^{k}n)$ Distance

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• Multireference error mitigation for quantum computation of chemistry
• Approximate Quantum Codes From Long Wormholes
• Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms
• Heralding entangled optical photons from a microwave quantum processor
• Emergent unitary designs for encoded qubits from coherent errors and syndrome measurements
• Quantum Scar States in Coupled Random Graph Models
• Logical entanglement distribution between distant 2D array qubits
• Enhanced Lieb-Robinson bounds for commuting long-range interactions
• Flying-cat parity checks for quantum error correction
• Low-depth quantum error correction via three-qubit gates in Rydberg atom arrays
• Single-shot preparation of hypergraph product codes via dimension jump
• Fundamental thresholds for computational and erasure errors via the coherent information
• Solvable Quantum Circuits in Tree+1 Dimensions
• Distributed quantum error correction based on hyperbolic Floquet codes
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