Topological quantum memory
2001/10/24 by Eric Dennis, Alexei Kitaev, Andrew Landahl +1 · 1,688 citations
Computer Science · Physics and Astronomy · #Quantum #Quantum Computing Algorithms and Architecture #Quantum algorithm #Quantum computer #Quantum error correction #Quantum gate #Quantum information #Quantum many-body systems #Quantum operation #Qubit #Topological Materials and Phenomena #Topology (electrical circuits) #Toric code #cond-mat.stat-mech #hep-lat #hep-th #quant-ph
paper · pdf · doi:10.1063/1.1499754
published in Journal of Mathematical Physics 43(9), 4452-4505 (American Institute of Physics) · 39 pages, 21 figures, REVTeX
arxiv created 2001/10/24 · openalex publication_date 2002/09/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
Abstract
We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z2 lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.
Citations
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