Quantum Codes Suitable for Iterative Decoding

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25/26 Q-Tornado codes quantum codes Theorem 3. [Luby-Mitzenmacher-Shokrollahi-Spielman-1997] Classical Tornado codes attain the capacity of the erasure channel with linear encoding/decoding complexity. Theorem 4. [Tillich09] With a last level of size O(n 1 3) chosen at random and rate < 1 − 2p, quantum Tornado codes attain the capacity of the erasure channel of erasure probability p with linear decoding complexity. Drawback ( : size of the last level ⇒ probability that the decoding fails Ω e −αn1/3) .
Open problems quantum codes ◮ Increase the size of the last level in the quantum Tornado code construction. ◮ Obtaining Q-LDPC codes with linear minimum distance? ◮ Using Q-LDPC codes to improve the lower bounds known on the quantum channel capacity. ◮ Analyze rigorously iterative decoding of Q-LDPC codes. ◮ Find a modification of quantum serial turbo-code construction with unbounded minimum distance and where iterative decoding converges to the right solution <strong>for</strong> good enough channel conditions. ◮ Generalize other iterative decoding schemes. ◮ Use such codes to improve existing fault tolerant architectures. 26/26
Page 1 and 2: Quantum Codes Suitable for Iterativ
Page 3 and 4: Introduction Quantum LDPC codes ◮
Page 5 and 6: 4/26 Serial quantum turbo-codes Int
Page 7 and 8: 6/26 Another approach introduction
Page 9 and 10: 8/26 Quantum The Pauli group over n
Page 11 and 12: 10/26 Quantum code and encoding Qua
Page 13 and 14: 12/26 ◮ For E, F ∈ G n : The sy
Page 15 and 16: 14/26 3.Quantum LDPC code Q-LDPC
Page 17 and 18: 16/26 Iterative decoding Q-LDPC (0.
Page 19 and 20: 18/26 A construction Q-LDPC (v ,v )
Page 21 and 22: Q-LDPC X n = 2t 2 , k = 2, R = 1 t
Page 23 and 24: 22/26 Decoding the construction Q-L
Page 25: 24/26 Tornado ◮ Re-encode the las

Quantum Codes Suitable for Iterative Decoding

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