Quantum Error Correcting Codes Using Qudit Graph States

Abstract

Graph states are generalized from qubits to collections of $n$ qudits of arbitrary dimension $D$, and simple graphical methods are used to construct both additive and nonadditive quantum error correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large $n$ and $D$ are constructed using simple graphs, except when $n$ is odd and $D$ is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general $D$, and shown to provide a dual representation of an additive graph code.

Version 4 is almost exactly the same as the published version in Phys. Rev. A

References in corpus (6)

• Quantum error-correcting codes associated with graphs
• On optimal quantum codes
• Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic
• A nonadditive quantum code
• Graphical Nonbinary Quantum Error-Correcting Codes
• A simple family of nonadditive quantum codes

Cited by in corpus (66)

• Quantum secret sharing with qudit graph states
• Pair-cat codes: autonomous error-correction with low-order nonlinearity
• Standard Form of Qudit Stabilizer Groups
• Encoding Hypergraphs into Quantum States
• Efficient entanglement purification protocols for d-level systems
• Approximate quantum error correction for generalized amplitude damping errors
• Quantum circuits for maximally entangled states
• Generalized Concatenated Quantum Codes
• Quantum capacity analysis of multi-level amplitude damping channels
• Securing quantum networking tasks with multipartite Einstein-Podolsky-Rosen steering
• Generalized Semi-Quantum Secret Sharing Schemes
• A Non-Commuting Stabilizer Formalism
• Nonbinary Codeword Stabilized Quantum Codes
• Verifying Genuine High-Order Entanglement
• Low-complexity quantum codes designed via codeword-stabilized framework
• Multi-Error-Correcting Amplitude Damping Codes
• Experimental violation of Bell inequalities for multi-dimensional systems
• Universal quantum computing using $({\mathbb{Z}}_d{)}^3$ symmetry-protected topologically ordered states
• Tripartite Entanglement in Qudit Stabilizer States and Application in Quantum Error Correction
• Graph-state formalism for mutually unbiased bases
• Deterministic Fast Scrambling with Neutral Atom Arrays
• Encoding graphs into quantum states: an axiomatic approach
• Qudit Hypergraph States
• Location of quantum information in additive graph codes
• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• Examining the dimensionality of genuine multipartite entanglement
• Relationship among locally maximally entanglable states, W states and hypergraph states under local unitary transformations
• A Consistent Quantum Ontology
• Quantum Error-Correcting Codes over Mixed Alphabets
• Qudit hypergraph states and their properties
• Loss-tolerant architecture for quantum computing with quantum emitters
• Multipartite Entanglement and Hypergraph states of three qubits
• Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics
• Analysis of quantum error correction with symmetric hypergraph states
• Partially Coherent Direct Sum Channels
• Calculating Concentratable Entanglement in Graph States
• Symmetric hypergraph states: Entanglement quantification and robust Bell nonlocality
• Quantum LOSR networks cannot generate graph states with high fidelity
• Graph states in phase space
• Error-correcting one-way quantum computation with global entangling gates
• Parafermion stabilizer codes
• Investigation of the Quantum Correlations for a S=1/2 Ising- Heisenberg Model on a Symmetrical Diamond Chain
• Codeword Stabilized Quantum Codes for Asymmetric Channels
• Construction of Equientangled Bases in Arbitrary Dimensions via Quadratic Gauss Sums and Graph States
• Entropic measure and hypergraph states
• Resource optimization for fault-tolerant quantum computing
• Symmetries of Codeword Stabilized Quantum Codes
• Scalable noncontextuality inequalities and certification of multiqubit quantum systems
• GraphiQ: Quantum circuit design for photonic graph states
• Heuristic construction of codeword stabilized codes
• Transforming graph states via Bell state measurements
• Generating graph states with a single quantum emitter and the minimum number of fusions
• Finite-Function-Encoding Quantum States
• Measurement-based quantum computation using two-component BECs
• Tensor Rank and Other Multipartite Entanglement Measures of Graph States
• Almost device-independent certification of GME states with minimal measurements
• MUB-like structures and tomographic reconstruction for N-ququart systems
• Robustness of Bell Violation of Graph States to Qubit Loss
• Maximal Entanglement: Applications in Quantum Information and Particle Physics
• Accurate Leakage Speculation for Quantum Error Correction
• Fault-tolerant Quantum Error Correction Using a Linear Array of Emitters
• Multipartite Entanglement Distribution in Quantum Networks using Subgraph Complementations
• A new approach to codeword stabilized quantum codes using the algebraic structure of modules
• A Combinatorial Approach to Quantum Error Correcting Codes
• Measurement-based quantum computation utilizing the graph states of Bose-Einstein condensates and continuous variables
• Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation