QUANTUM ERROR CORRECTING CODES FREE

This theorem seems to present an obstacle to formulating a theory of quantum error correction. Similar to classical error correction, QEC codes do not always correctly decode logical qubits, but their use reduces the effect of noise.Ĭopying quantum information is not possible due to the no-cloning theorem. In this example, the logical information was a single bit in the one state, the physical information are the three copied bits, and determining what logical state is encoded in the physical state is called decoding. It is possible that a double-bit error occurs and the transmitted message is equal to three zeros, but this outcome is less likely than the above outcome. Assuming that noisy errors are independent and occur with some sufficiently low probability p, it is most likely that the error is a single-bit error and the transmitted message is three ones. Suppose further that a noisy error corrupts the three-bit state so that one of the copied bits is equal to zero but the other two are equal to one. suppose we copy a bit in the one state three times. The idea is to store the information multiple times, and-if these copies are later found to disagree-take a majority vote e.g. The simplest albeit inefficient approach is the repetition code. Quantum error correction is theorised as essential to achieve fault-tolerant quantum computation that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum preparation, and faulty measurements.Ĭlassical error correction employs redundancy. 259– 273.Quantum error correction ( QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Zémor, “ On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), Lecture Notes in Computer Science Vol. Zémor, “ Quantum LDPC codes with positive rate and minimum distance proportional to n 1/2,” in Proceedings of the 2009 IEEE International Conference on Symposium on Information Theory (ISIT), 2009. Shastri, Basic Algebraic Topology ( CRC Press, 2013). Presskil, Quantum computation, see Google ScholarĪ. Chuang, Quantum Computation and Quantum Information ( Cambridge University Press, 2000). Congr., Queens Univ., Kingston, Ont., 1977). 48 ( Queen's University, Kingston, Ontario, 1978), pp. Mostow, “ Discrete subgroups of Lie groups,” Lie Theories and Their Applications, Queen's Papers in Pure and Applied Mathematics Vol. Millson, “ On the first Betti number of a constant negatively curved manifold,” Ann. Luo, “ Z 2-systolic freedom and quantum codes,” Mathematics of Quantum Computation ( Chapman and Hall/CRC, 2002), pp. Lubotzky, “ Normal subgroup growth of linear groups: The(G2,F4,E8)-theorem,” Algebraic Groups and Arithmetic ( Tata Inst.

QUANTUM ERROR CORRECTING CODES FREE

Lubotzky, “ Free quotients and the first Betti number of some hyperbolic manifolds,” Transform.

Lubotzky, “ Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem,” Ann. Vishne, “ Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups,” J. Gromov, “ Systoles and intersystolic inequalities,” Actes de la Table Ronde de Gomtrie Diffrentielle (Luminy, 1992), Smin. Gallager, Low-Density Parity-Check Codes ( MIT Press, 1963). Freedman, “ Z 2-systolic freedom,” in Proceedings of the Kirbyfest, Geometry and Topology Monograph Vol. Fetaya, “ Bounding the distance of quantum surface codes,” J. Fetaya, “ Homological error correcting codes and systolic geometry,” e-print arXiv:1108.2886. Delfosse, “ Tradeoffs for reliable quantum information storage in surface codes and color codes,” e-print arXiv:1301.6588. Chern, “ A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds,” Ann. Sarnak, “ On the period matrix of a Riemann surface of large genus,” Invent. 16 ( Mathematical Society of Japan, Tokyo, 2006). Bowditch, A Course on Geometric Group Theory, Mathematical Society of Japan Memoirs Vol.

Belolipetsky, “ On 2-systoles of hyperbolic 3-manifolds,” Geom. Anderson, “ Complete minimal varieties in hyperbolic space,” Invent.

Quantum error correcting codes