Turning error-reducing quantum turbo codes into error-correcting - PowerPoint PPT Presentation

Turning error-reducing quantum turbo codes into error-correcting codes Mamdouh Abbara (MEc), Iryna Andriyanova (ENSEA), Jean-Pierre Tillich (INRIA) QEC14 December the 17th, 2014

introduction The 5 − qubit code Probability of error after decoding the 5 -qubit code 1/34

introduction An alternative strategy for concatenation inner convolutional interleaver code of rate 1 0> 0> C 0> 0> 0> 0> C 0> 0> 0> 0> C 0> 0> 0> 0> C 0> 0> 0> 0> C 0> 0> 0> 0> C 0> 0> 0> 0> C 0> 0> 2/34

introduction The message of this talk ◮ It is possible to concatenate with a rate 1 code (so no protection against errors at all...) and still achieve something nontrivial when the rate 1 code is a convolutional code. 3/34

introduction Improving the 5 − qubit code Figure 1: Probability of error after decoding – complexity of encoding ≈ complexity of encoding a 5 -qubit code – Rate 1 5 → 1 8 – same complexity of decoding as the 5 -qubit code – modified quantum turbo-code construction 4/34

Introduction serial quantum turbo-codes ◮ as for quantum LDPC codes it is possible to build such codes and decode them with iterative decoding algorithms. ◮ freedom to introduce randomness in the construction what we do not have for quantum LDPC codes. ◮ much simpler to construct. ◮ but there are also some problems related to encoding issues... 5/34

serial concatenation 2. Concatenation of codes – P n Pauli group over n quits – Clifford transformation U : U † P n U ∈ P n P L 1 1 P L 2 2 P 3 ... ... φ> k ... −1 U U L k ... 0> S 1 0> S 2 ... n−k ... 0> P S n n−k 6/34

serial concatenation ◮ Physical error P = P 1 P 2 . . . P n = U † PU ◮ Logical error,syndrome LS = L 1 L 2 . . . L k S 1 . . . S n − k � �� � � �� � logical error syndrome k L n P U n−k S 7/34

serial concatenation Stabilizer, Normalizer ◮ Stabilizer set S corresponds to L = I . . . I , S ∈ { I, Z } n − k : � � S ) U † , S ∈ { I, Z } n − k S = U ( I . . . I � �� � k ◮ Normalizer set N corresponds to S ∈ { I, Z } n − k . � U ( L, S ) U † , S ∈ { I, Z } n − k � N = ◮ Quantum minimum distance = min {| P | ∈ N \ S } d quantum = min {| P | ∈ N \ { I . . . I }} d classical 8/34

introduction Serial concatenation of codes L’ L E 1 1 1 E L 2 2 information symbols ... ... ... ... out U Π L k S 1 syndrome ... ... ... U inner of the outer L’ n code S n−k S 1 S 2 syndrome ... ... of the inner code E S N N−n 9/34

minimum distance 3. Minimum Distance Properties When the inner code is a juxtaposition of small codes L 0 P U 0 S 0 L 1 U P 1 S 1 ..... L N U P N S N U Clifford transformation on n qubits, D in ≤ n , D cont ≤ D out n. 10/34

minimum distance D out L out U out Sout S in D con ≤ D out n. 11/34

minimum distance The problem (I) U out S ′ 1 ... S ′ L ′ 1 , ... , L ′ k 1 + r 1 S ′ 1 , ... , S ′ → L 1 ... L k 1 S 1 ... S r 1 r 2 r 2 � �� � � �� � � �� � S out S in S in Π L ′ π (1) , ... , L ′ π ( k 1 + r 1 ) , S ′ 1 , ... , S ′ → r 2 U in → P 1 , ... , P k 1 + r 1 + r 2 12/34

minimum distance The problem (II) Assume that there exists for the inner code a bound D such that for each i ∈ { 1 , . . . , k 1 + r 1 } and every P ∈ { X, Y, Z } there exists a choice for the S ′ j ’s in { I, Z } such that � � k 1 + r 1 − i times � � i − 1 times � � � �� � � �� � r 2 ) U † S ′ 1 , . . . S ′ U in ( ≤ D I . . . I P I . . . I � � in � � � � then if the minimum distance of the outer code is D out the minimum distance of the concatenated code is upper bounded by D out D 13/34

stabilizer The problem(III) U out S ′ 1 ... S ′ L ′ 1 , ... , L ′ k 1 + r 1 S ′ 1 , ... , S ′ → L 1 ... L k 1 S 1 ... S r 1 r 2 r 2 � �� � � �� � � �� � S out S in S in � � � L ′ 1 , ... , L ′ with = D out � k 1 + r 1 Π L ′ π (1) , ... , L ′ π ( k 1 + r 1 ) , S ′ 1 , ... , S ′ → r 2 for each of the L ′ the corresponding S ′ i 1 . . . S ′ i π ( i ) � = I consider r 2 S ′ 1 , . . . , S ′ and mutiply them to obtain r 2 U in → P 1 , ... , P k 1 + r 1 + r 2 with | P 1 . . . P k 1 + r 1 + r 2 | ≤ D out D 14/34

minimum distance When the inner encoder is convolutional S 0 P 1 L 1 U S 1 P 2 U L 2 S 2 . . . P N L N U S N P N +1 U { S N +1 . . . P N + t − 1 U S N + t − 1 { m M i 1 n − P i } k U U L i { P N + t S N + t n − k m S i M i D in = O (1) D con ≤ ? 15/34

minimum distance L U out S 16/34

minimum distance Classical setting ◮ Choose U out and U in as (classical) convolutional encoders. ◮ [Kahale-Urbanke-ISIT 1998] In the classical case, by an averaging argument, if the free distance of C out is d out and if U in is a non- catastrophic and recursive encoder, then the minimum distance of � � d out − 2 the resulting code is typically of order Θ . N d out ◮ Generalizes easily to the quantum setting ? 17/34

minimum distance A first problem Theorem 1. [Poulin-Tillich-Ollivier-ISIT 2008] There are no quantum convolutional encoders which are at the same time non- catastrophic and recursive. 18/34

minimum distance Catastrophic/recursive conv. encoder ( S 0 , L 1 , S 1 , . . . , L i , S i , . . . ) − → P = ( P 1 , P 2 , . . . , ) with S 0 ∈ { I, Z } m , S i ∈ { I, Z } n − k for i ≥ 1 def = L L 1 , L 2 , . . . , ◮ Non-catastrophic encoder : supp ( P ) finite ⇒ supp ( L ) finite. ◮ Recursive encoder : | L | = 1 ⇒ supp ( P ) infinite. 19/34

minimum distance A crucial argument used in the classical setting Consider convolutional encoders for which | L | ≤ | P | 20/34

minimum distance A quantum convolutional encoder that does the job 21/34

minimum distance A theorem Theorem 2. [Abbara-Tillich - ITW 2011] If the inner code is the aforementioned convolutional code of rate 1 and the outer code is a juxtaposition of copies of a quantum code of classical minimum distance d classical and quantum minimum distance d quantum , then with probability → 1 as the length N of the inner code → ∞ the minimum distance D con of the concatenated scheme satisfies � � d classical − 2 – D con = Ω if d classical > 2 N d classical � � log N – D con = Ω if d classical = 2 and d quantum ≥ 3 . log log N 22/34

construction The construction A first attempt inner convolutional interleaver code of rate 1 0> C 0> 0> 0> C 0> 0> 0> C 0> 0> 0> C 0> 0> 0> C 0> 0> 0> C 0> 0> 0> C 0> 0> 23/34

construction Decoding inner convolutional interleaver code of rate 1 L 1 L’ P 1 1 0> S L 1 P C 1 0> 2 0> 0> C L n 0> 0> 0> C 0> 0> 0> C 0> 0> 0> C 0> 0> 0> C 0> 0> L’ n 0> S C n’ 0> P 0> n L n decoding the outer code deinterleaving decoding the inner code 24/34

construction The problem starting point 2.0 = 1.8 endpoint entropy of the output of the inner code entropy of the input of the inner code 1.6 1.4 inner code 1.2 outer code 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 entropy of the output of the outer code entropy of the input of the inner code 25/34

construction The modified construction inner convolutional interleaver code of rate 1 0> C 0> 0> 0> 0> 0> C 0> 0> 0> 0> 0> C 0> 0> 0> 0> 0> C 0> 0> 0> 0> 26/34

Results QuBit-error probability after decoding 1 K = 17143 K = 1143 K = 143 0.1 Bitwise error probability 0.01 0.001 0.0001 1e-05 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 Depolarizing channel strength 27/34

Results Probability of error per block 1 Decoding error probability 0.1 N = 60000 N = 4000 N = 500 0.01 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 Depolarizing channel strength 28/34

Results Entropy evolution during decoding (P1) + 1st position of the outer code sent directly to the channel starting point 2 1.8 1.6 inner code entropy of the output of the inner code entropy of the input of the outer code 1.4 outer code 1.2 decoding trajectory 1 0.8 0.6 0.4 0.2 endpoint 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 entropy of the output of the outer code entropy of the input of the inner code 29/34

further 5. Going further : a multilevel construction inner convolutional interleaver code of rate 1 inner convolutional interleaver 0> code of rate 1 C 0> 0> 0> C 0> 0> 0> 0> 0> 0> 0> C 0> 0> 0> C 0> 0> 0> 0> channel 0> 0> 0> C 0> channel 0> 0> C 0> 0> 0> 0> 0> 0> 0> C 0> 0> 0> C 0> 0> 0> 0> 0> 0> 30/34