An Introduction to Quantum Error Correction

Abstract

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements.

15 pages, AMSLaTeX, talk given at AMS Short Course on Quantum Computation

References in corpus (8)

• Mixed State Entanglement and Quantum Error Correction
• Good Quantum Error-Correcting Codes Exist
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• Reliable Quantum Computers
• A Theory of Fault-Tolerant Quantum Computation
• Quantum Error Correction and Orthogonal Geometry
• Codes for the Quantum Erasure Channel
• Threshold Accuracy for Quantum Computation

Cited by in corpus (28)

• Quantum Error Correction for Beginners
• Longer-Baseline Telescopes Using Quantum Repeaters
• Introduction to Quantum Algorithms
• Constructing optimal quantum error correcting codes from absolute maximally entangled states
• Permutation-invariant constant-excitation quantum codes for amplitude damping
• On Self-Dual Quantum Codes, Graphs, and Boolean Functions
• Introduction to Quantum Error Correction and Fault Tolerance
• Lower Bounds on the Quantum Capacity and Highest Error Exponent of General Memoryless Channels
• The multiplicative domain in quantum error correction
• The Minimum Distance Problem for Two-Way Entanglement Purification
• Is Fault-Tolerant Quantum Computation Really Possible?
• Continuous quantum error correction for evolution under time-dependent Hamiltonians
• Channel-Adapted Quantum Error Correction
• Fault-Tolerant Computing with Single Qudit Encoding
• Generalized Multiplicative Domains and Quantum Error Correction
• The equivalence of two approaches to the design of entanglement distillation protocols
• A Unified Picture of Decoherence Control
• Interference in discrete Wigner functions
• On Quantum and Classical Error Control Codes: Constructions and Applications
• Fault-Tolerant Quantum Computation
• Nonbinary stabilizer codes over finite fields
• Enhancing quantum models of stochastic processes with error mitigation
• Thoughts on Noise and Quantum Computation
• Quantum Communication Complexity of Quantum Authentication Protocols
• Robust quantum memory via quantum control
• New non-binary quantum codes from skew constacyclic codes over the ring ${\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+v^2{\mathbb{F}}_{p^m}$
• Quantum correction with three codes
• Recovery Operator for Local Errors Generated by Gauge Multiplets