Correcting Quantum Errors with Entanglement

Abstract

We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error correcting codes, thus allowing us to ``quantize'' all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement, these codes can also be used as catalytic codes, in which a small amount of initial entanglement enables quantum communication.

17 pages, no figure. To appear in Science

References in corpus (11)

• Mixed State Entanglement and Quantum Error Correction
• Good Quantum Error-Correcting Codes Exist
• A Theory of Fault-Tolerant Quantum Computation
• Quantum Error Correction and Orthogonal Geometry
• On the role of entanglement in quantum computational speed-up
• Sparse Graph Codes for Quantum Error-Correction
• A family of quantum protocols
• Entanglement required in achieving entanglement-assisted channel capacities
• Entanglement in the stabilizer formalism
• Climbing Mount Scalable: Physical-Resource Requirements for a Scalable Quantum Computer
• GHZ extraction yield for multipartite stabilizer states

Cited by in corpus (150)

• Quantum channels and memory effects
• Optimal Entanglement Formulas for Entanglement-Assisted Quantum Coding
• General entanglement-assisted quantum error-correcting codes
• Power law violation of the area law in quantum spin chains
• Catalytic quantum error correction
• Distributed Quantum Computation Architecture Using Semiconductor Nanophotonics
• Entanglement-assisted quantum error-correcting codes over arbitrary finite fields
• Entanglement-assisted quantum turbo codes
• Entanglement in a 20-Qubit Superconducting Quantum Computer
• Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators
• Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs
• Entanglement Increases the Error-Correcting Ability of Quantum Error-Correcting Codes
• Entanglement-Assisted Quantum Quasi-Cyclic Low-Density Parity-Check Codes
• Multiplicativity of completely bounded $p$-norms implies a strong converse for entanglement-assisted capacity
• Trade-off capacities of the quantum Hadamard channels
• Duality in Entanglement-Assisted Quantum Error Correction
• Perfect state transfer and efficient quantum routing: a discrete-time quantum walk approach
• Graphical Nonbinary Quantum Error-Correcting Codes
• High performance entanglement-assisted quantum LDPC codes need little entanglement
• NP-hardness of decoding quantum error-correction codes
• Entanglement-assisted quantum low-density parity-check codes
• Entropic proofs of Singleton bounds for quantum error-correcting codes
• Trading classical communication, quantum communication, and entanglement in quantum Shannon theory
• Quantifying resource in catalytic resource theory
• Entanglement-Assisted Quantum Communication Beating the Quantum Singleton Bound
• Entanglement-Assisted Quantum Error-Correcting Codes with Imperfect Ebits
• Many-body quantum teleportation via operator spreading in the traversable wormhole protocol
• Entanglement-Assisted Quantum Convolutional Coding
• Quantum rate distortion, reverse Shannon theorems, and source-channel separation
• New entanglement-assisted MDS quantum codes from constacyclic codes
• Energetic cost of Hamiltonian quantum gates
• Communication Links for Distributed Quantum Computation
• Decoding algorithms for surface codes
• Polar codes for private and quantum communication over arbitrary channels
• Polar Codes and Their Quantum-Domain Counterparts
• Maximal Success Probabilities of Linear-Optical Quantum Gates
• Enhanced Feedback Iterative Decoding of Sparse Quantum Codes
• On the logical operators of quantum codes
• The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
• Encoding One Logical Qubit Into Six Physical Qubits
• Bell nonlocality and fully-entangled fraction measured in an entanglement-swapping device without quantum state tomography
• Classical Enhancement of Quantum Error-Correcting Codes
• Dualities and Identities for Entanglement-Assisted Quantum Codes
• How Much Entanglement Does a Quantum Code Need?
• Generalized Semi-Quantum Secret Sharing Schemes
• Linear-Optical Hyperentanglement-Assisted Quantum Error-Correcting Code
• Channel Capacities of an Exactly Solvable Spin-Star System
• Asymmetric entanglement-assisted quantum error-correcting codes and BCH codes
• Approximate simulation of quantum channels
• Quantum Coding with Entanglement
• Entanglement-assisted codeword stabilized quantum codes
• Entanglement-Assisted Quantum Error Correction with Linear Optics
• Quantum Information Encoding, Protection, and Correction from Trace-Norm Isometries
• Quantum polar codes for arbitrary channels
• Polar codes for degradable quantum channels
• Coherent Communication with Continuous Quantum Variables
• Characterization of entanglement on superconducting quantum computers of up to 414 qubits
• Logical Clifford Synthesis for Stabilizer Codes
• A characterization of entanglement-assisted quantum low-density parity-check codes
• Efficient Quantum Polar Codes Requiring No Preshared Entanglement
• On the MacWilliams Identity for Classical and Quantum Convolutional Codes
• Constant-Overhead Fault-Tolerant Bell-Pair Distillation using High-Rate Codes
• One-shot quantum error correction of classical and quantum information
• Nonlocal quantum information in bipartite quantum error correction
• Efficiently implementable codes for quantum key expansion
• Entanglement-Assisted Quantum Error Correcting Codes From RS Codes and BCH Codes with Extension Degree 2
• Coding for quantum channels with side information at the transmitter
• Perfect many-to-one teleportation with stabilizer states
• Minimal-memory, non-catastrophic, polynomial-depth quantum convolutional encoders
• Entanglement-assisted concatenated quantum codes
• High-rate quantum low-density parity-check codes assisted by reliable qubits
• Quantum error correction via less noisy qubits
• Operator quantum error correction for continuous dynamics
• Two families of Entanglement-assisted quantum MDS codes from constacyclic codes
• A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices
• Quantum Convolutional Coding with Shared Entanglement: General Structure
• Quantum Circuits for Stabilizer Error Correcting Codes: A Tutorial
• Quantum Error Correction from Complexity in Brownian SYK
• Entanglement-assisted Coding Theory
• Extra Shared Entanglement Reduces Memory Demand in Quantum Convolutional Coding
• Asymmetric Quantum Concatenated and Tensor Product Codes with Large Z-Distances
• Systematic Design and Optimization of Quantum Circuits for Stabilizer Codes
• Convolutional Entanglement Distillation
• Optimized Entanglement-Assisted Quantum Error Correction
• On the probabilistic quantum error correction
• Coherent Communication with Linear Optics
• Entanglement generation with a quantum channel and a shared state
• Some optimal entanglement-assisted quantum codes constructed from quaternary Hermitian linear complementary dual codes
• The Encoding and Decoding Complexities of Entanglement-Assisted Quantum Stabilizer Codes
• Constructions of $\ell$-Adic $t$-Deletion-Correcting Quantum Codes
• Stabilizer Formalism for Operator Algebra Quantum Error Correction
• Quantum resource theory of coding for error correction
• The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems
• Entanglement-assisted quantum codes from Galois LCD codes
• Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance
• Degenerate Local-dimension-invariant Stabilizer Codes and an Alternative Bound for the Distance Preservation Condition
• Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed-Solomon codes
• Protecting Quantum Information with Entanglement and Noisy Optical Modes
• Improved Gilbert-Varshamov Bound for Entanglement-Assisted Asymmetric Quantum Error Correction by Symplectic Orthogonality
• Experimental diagnostics of entanglement swapping by a collective entanglement test
• Stabilizer codes for Open Quantum Systems
• Hulls of projective Reed-Muller codes over the projective plane
• Entanglement-assisted quantum error-correcting codes from units
• Quantum codes from classical graphical models
• Quantum Error Correction Code in the Hamiltonian Formulation
• Family of Finite Geometry Low-Density Parity-Check Codes for Quantum Key Expansion
• Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes
• Quantum error-correcting codes from projective Reed-Muller codes and their hull variation problem
• Two families of Entanglement-assisted Quantum MDS Codes from cyclic Codes
• Decoherence and Quantum Error Correction for Quantum Computing and Communications
• Galois hulls of MDS codes and their quantum error correction
• Advance sharing of quantum shares for quantum secrets
• Error Correction for Reliable Quantum Computing
• Parsing a sequence of qubits
• Transforming Collections of Pauli Operators into Equivalent Collections of Pauli Operators over Minimal Registers
• Codes for entanglement-assisted classical communication
• Entanglement-assisted Quantum Codes from Algebraic Geometry Codes
• An Algorithmic Approach to Entanglement-Assisted Quantum Error-Correcting Codes from the Hermitian Curve
• The Learnability of Unknown Quantum Measurements
• Approximate quantum error correction, covariance symmetry, and their relation
• Semidefinite programming bounds on the size of entanglement-assisted codeword stabilized quantum codes
• Harnessing Coding Theory for Reliable Network Quantum Communication
• An Optimized Nearest Neighbor Compliant Quantum Circuit for 5-qubit Code
• Contextuality of Quantum Error-Correcting Codes
• Classical Coding Approaches to Quantum Applications
• Rapid creation of distant entanglement by multiphoton resonant fluorescence
• A new method for constructing EAQEC MDS codes
• Entanglement-assisted Quantum Reed-Muller Tensor Product Codes
• On Quantum Tensor Product Codes
• Stabilization of symmetry-protected long-range entanglement in stochastic quantum circuits
• Entangling capabilities and unitary quantum games
• Censorship of quantum resources against catalytic account sharing
• Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q + 1
• Quantum steganographic protocols using degenerate and entanglement-assisted quantum codes
• Matrix-product entanglement characterizing the optimality of state-preparation quantum circuits
• Entanglement-assisted Quantum MDS Codes from generalized Reed-Solomon Codes
• Construction and Performance of Quantum Burst Error Correction Codes for Correlated Errors
• Explicit constructions of optimal linear codes with Hermitian hulls and their application to quantum codes
• On pure MDS asymmetric entanglement-assisted quantum error-correcting codes
• Steane enlargement of Entanglement-Assisted Quantum Error-Correcting Codes
• New Galois Hulls of GRS Codes and Application to EAQECCs
• Changing the Local-Dimension of an Entanglement-Assisted Stabilizer Code Removes Entanglement Need
• Multivariate Goppa codes
• On Euclidean Hulls of MDS Codes
• Extended quasi-cyclic constructions of quantum codes and entanglement-assisted quantum codes
• Cyclic codes and some new entanglement-assisted quantum MDS codes
• Codeword Stabilized Codes from m-Uniform Graph States
• New entanglement-assisted quantum MDS codes with length $n=\frac{q^2+1}{5}$
• Bounds on concatenated entanglement-assisted quantum error-correcting codes
• Computational Perspectives on Bell Inequalities and Many-body Quantum Correlations