![]() We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits $n$, the number of encoded qubits $k$, the code distance $d$, the accuracy parameter $\delta$ that quantifies how well the erasure channel can be reversed, and the locality parameter $\ell$ that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy $\delta$ that is exponentially small in $\ell$, which is the case for perturbations of local commuting projector codes, our bound reads $kd^\bigr)$ in operator norm. STEINSPRING QUANTUM ERROR CORRECTION CODEįinally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types. Terhal, ``Quantum error correction for quantum memories,'' Rev. The combination of symmetry and locality constraints influence our ability to perform quantum error correction in two counterposing ways. Gottesman, ``An introduction to quantum error correction and fault-tolerant quantum computation,'' in Quantum Information Science and Its Contributions to Mathematics, Vol. (American Mathematical Society, 2010) pp. Yamamoto, ``Approximate quantum error correction can lead to better codes,'' Phys. Smith, ``Approximate quantum error-correcting codes and secret sharing schemes,'' in Advances in Cryptology – EUROCRYPT 2005: 24th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Aarhus, Denmark, May 22-26, 2005. Cramer (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp. Read, ``Nonabelions in the fractional quantum hall effect,'' Nuclear Physics B 360, 362–396 (1991). Kitaev, ``Anyons in an exactly solved model and beyond,'' Annals of Physics 321, 2–111 (2006), cond-mat/0506438. Zwolak, ``Stability of frustration-free Hamiltonians,'' Communications in Mathematical Physics 322, 277–302 (2013), arXiv:1109.1588. Wen, ``Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance,'' Physical Review B 72, 045141 (2005), cond-mat/0503554. STEINSPRING QUANTUM ERROR CORRECTION CODE.
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