CSS codes Shyam Sundhar R November 12, 2017 Shyam Sundhar R CSS - PowerPoint PPT Presentation

CSS codes Shyam Sundhar R November 12, 2017 Shyam Sundhar R CSS codes November 12, 2017 1 / 10

Outline Codes as a group 1 Code Construction 2 Hadamard Transform 3 Error Modelling 4 Error Correction 5 Shyam Sundhar R CSS codes November 12, 2017 2 / 10

Codes as a group W.K.T sum of any two code-words is also a code-word. closed under addition And 0 is also a codeword. identity element exists So any classical code C can be seen as a sub-group of Z n 2 Shyam Sundhar R CSS codes November 12, 2017 3 / 10

Let C 1 and C 2 be two linear codes. C 2 ⊂ C 1 C 2 is a subgroup of C 1 C 1 can be partitioned into cosets of C 2 Shyam Sundhar R CSS codes November 12, 2017 4 / 10

Code construction C 1 and C 2 are [n,k,d] and [ n , k 1 , d 1 ] codes respectively. C 2 ⊂ C 1 C 1 and C ⊥ 2 are both t-error correcting codes. C 1 = C 2 ∪ ( c 1 + C 2 ) ∪ ( c 2 + C 2 ) . . . ∪ ( c n + C 2 ) where n is the number of cosets. N = 2 k 2 k 1 We associate each basis state with a unique coset. 1 2 k Σ c ′ ∈ C 2 | c i + c ′ � | s i � = √ Each basis state is in a superposition of all the codewords in a coset. c i ∈ C 1 so If c 1 i and c 2 i belong to the same coset , they define the same basis codeword. Shyam Sundhar R CSS codes November 12, 2017 5 / 10

Application of Hadamard Phase flips become bit flips after hadamard application. Hadamard transforms a codes to its dual W.K.T | i � H ⊗ n → Σ j ( − 1) ( i . j ) | j � − − ′ . j ) | j � Σ c ′ ∈ C 2 | c ′ � H ⊗ n → Σ j Σ c ′ ∈ C 2 ( − 1) ( c − − ′ . j ) Σ c ′ ∈ C 2 ( − 1) ( c non-zero only if j ∈ C ⊥ 2 Σ c ′ ∈ C 2 | c ′ � H ⊗ n − − → Σ j ∈ C ⊥ 2 | j � Shyam Sundhar R CSS codes November 12, 2017 6 / 10

1 2 k Σ c ′ ∈ C 2 | c i + c ′ � H ⊗ n ′ ) . j | j � → Σ j Σ c ′ ∈ C 2 ( − 1) ( c i + c | s i � = √ − − ′ . j | j � → Σ j ( − 1) ( c i . j ) Σ c ′ ∈ C 2 ( − 1) c − By previous result ′ . j | j � = | w i � | s i � H ⊗ n 2 ( − 1) c − − → Σ j ∈ C ⊥ In | s i � basis the states are in the superposition of codewords of C 1 , bit flips can be corrected In | w i � basis the states are in the superposition of codewords of C ⊥ 2 , phase flips can be corrected Shyam Sundhar R CSS codes November 12, 2017 7 / 10

Error Modelling We model bit flips as an addition of a vector with same length | e � bit flip | c i + c ′ + e � | c i + c ′ � − − − − → We model phase flips as the multiplication by ( − 1) m where m is the number of phase flipped qubits and state as ’1’. | e � phase flip ( − 1) ( c i + c ′ ) . e | c i + c ′ � | c i + c ′ � − − − − − → Shyam Sundhar R CSS codes November 12, 2017 8 / 10

Error Correction Let e 1 and e 2 be 2 vectors of hamming weight atmost ’t’ → ( − 1) ( c i + c ′ ) . e 1 | c i + c ′ + e 2 � | c i + c ′ � error − − ′ ∈ C 2 where c i ∈ C 1 and c use C 1 → ( − 1) ( c i + c ′ ) . e 1 | c i + c ′ � − − − − Apply hadamard ′ ) . ( j + e 1 ) | j � H ⊗ n → Σ j Σ c ′ ∈ C 2 ( − 1) ( c i + c − − j iterates over all possible strings, so replacing j with j+e Shyam Sundhar R CSS codes November 12, 2017 9 / 10

′ ) . ( j ) | j + e 1 � → Σ j Σ c ′ ∈ C 2 ( − 1) ( c i + c − ′ . j | j + e 1 � → Σ j ( − 1) c i . j Σ c ′ ∈ C 2 ( − 1) c − Each term is non zero only if j ∈ C ⊥ 2 ′ . j | j + e 1 � 2 ( − 1) c → Σ j ∈ C ⊥ − W.K.T C ⊥ 2 is t-error correcting use C ⊥ ′ . j | j � 2 ( − 1) c 2 − − − − → Σ j ∈ C ⊥ apply hadamard again H ⊗ n → | c i + c ′ � − − Shyam Sundhar R CSS codes November 12, 2017 10 / 10