Stabilizer quantum codes via the CWS framework Leonid Pryadko University of California, Riverside Y. Li, I. Dumer, & LPP, PRL (2010) Y. Li, I. Dumer, M. Grassl, & LPP, PRA (2010) A. Kovalev, I. Dumer, & LPP, PRA (accepted) NSF ARO 1-1
Stabilizer quantum codes via the CWS framework Leonid Pryadko University of California, Riverside • Intro: Stabilizer codes, graph states, and CWS codes • Upper bounds on generic CWS codes • GF(4) representation of an additive CWS code & lower bound for codes from a given graph • Cyclic CWS & related codes Y. Li, I. Dumer, & LPP, PRL (2010) Y. Li, I. Dumer, M. Grassl, & LPP, PRA (2010) A. Kovalev, I. Dumer, & LPP, PRA (accepted) NSF ARO 1-2
Stabilizer codes General quantum code is a subspace Q of n -qubit Hilbert space H ⊗ n 2 . Stabilizer code Q is determined by an Abelian stabilizer group S of Pauli operators Q ≡ {| ψ � : S | ψ � = | ψ � , ∀ S ∈ S } If S = � G 1 , . . . , G n − k � , with ( n − k ) generators, the code encodes k logical qubits. There are k logical operators X i , Z i , i = 1 , . . . , k which commute with every element in S . The code is denoted [[ n, k, d ]] , where d is the distance of the code. Errors are detected by measuring the generators G i of the stabilizer S The group � G 1 , . . . , G n − k , Z 1 , . . . , Z k � stabilizes a unique stabilizer state | s � ≡ | ¯ 0 . . . ¯ 0 � ; the basis of the code is α 1 α k | α 1 , . . . α k � ≡ X 1 . . . X k | s � , α j = { 0 , 1 } , j = 1 , . . . , k . 2-1
Example: [[5,1,3]] stabilizer code Q ≡ {| ψ � : G i | ψ � = | ψ � , i = 1 , . . . , 4 } with generators G 1 = XZZXI, G 2 = IXZZX, G 3 = XIXZZ, G 4 = ZXIXZ A basis of the code space is (up to normalization) 4 � | ¯ | ¯ 1 � = X | ¯ 0 � = ( 1 + G i ) | 00000 � , 0 � . i =1 The logical operators can be taken as X = ZZZZZ, Z = XXXXX. Measure generators of the stabilizer to find the error, e.g., � � � ˜ = X 1 ( A | ¯ 0 � + B | ¯ ψ 1 � ) gives unique syndrome � � G 1 � = 1 , � G 2 � = 1 , � G 3 � = 1 , � G 4 � = − 1 . For this code, there are total of 15 single-qubit errors, and exactly 15 distinct syndromes (apart from � G i � = 1 for any | ψ � ∈ Q ). 3-1
Graph states For a simple graph G = ( V, E ) with adjacency matrix R ≡ γ ij , n � Z γ ij the generators S i ≡ X i j =1 These define the Abelian graph stabilizer group S G ≡ � S 1 , . . . , S n � and the graph state | s � : S i | s � = | s � , a [[ n, 0 , d ]] stabilizer code Distance of a graph state is defined as the minimum weight of an element of the graph stabilizer S G . 4-1
Graph states For a simple graph G = ( V, E ) with adjacency matrix R ≡ γ ij , n � Z γ ij the generators S i ≡ X i j =1 These define the Abelian graph stabilizer group S G ≡ � S 1 , . . . , S n � and the graph state | s � : S i | s � = | s � , a [[ n, 0 , d ]] stabilizer code Distance of a graph state is defined as the minimum weight of an element of the graph stabilizer S G . Example: Ring graph for n = 3 ; S 1 = XZZ , S 2 = ZXZ , S 3 = ZZX . | s � is an equal superposition of all 2 3 states, taken with positive or negative signs depending on the number of pairs of ones at positions connected by the edges of the graph. | s � = | 000 � + | 001 � + | 010 � − | 011 � + | 100 � − | 101 � − | 110 � − | 111 � = S 2 | s � = | 010 � − | 011 � + | 000 � + | 001 � − | 110 � − | 111 � + | 100 � − | 101 � 4-2
Code-Word Stabilized codes Invented by Cross, Smith, Smolin & Zeng (2007). Generally non-additive, but include all stabilizer codes as a subclass. Standard form: Q = ( G , C ) . Graph G → graph state | s � Classical binary code ( n, K, d ) = C = { c i } K i =1 , with n -bit c i . Quantum basis vectors | i � ≡ W i | s � codeword operators W i ≡ Z c i, 1 . . . Z c i,n . Error E = Z u X v maps to binary vector [Cl G ( E )] j ≡ u j + γ ij v i Example: Non-additive CWS code ((5 , 6 , 2)) . The n = 5 ring graph generated by S 2 = ZXZII and cyclic permutations. Classical codewords c 0 = 00000 , c 1 = 01101 , c 2 = 10110 , c 3 = 01011 , c 4 = 10101 , c 5 = 11010 . 5-1
Code-Word Stabilized codes Invented by Cross, Smith, Smolin & Zeng (2007). Generally non-additive, but include all stabilizer codes as a subclass. Standard form: Q = ( G , C ) . Graph G → graph state | s � Classical binary code ( n, K, d ) = C = { c i } K i =1 , with n -bit c i . Quantum basis vectors | i � ≡ W i | s � codeword operators W i ≡ Z c i, 1 . . . Z c i,n . Error E = Z u X v maps to binary vector [Cl G ( E )] j ≡ u j + γ ij v i Example: Non-additive CWS code ((5 , 6 , 2)) . The n = 5 ring graph generated by S 2 = ZXZII and cyclic permutations. Classical codewords c 0 = 00000 , c 1 = 01101 , c 2 = 10110 , c 3 = 01011 , c 4 = 10101 , c 5 = 11010 . Unfortunately, no known efficient algorithm to decode non-additive CWS codes. 5-2
Error correction for CWS codes Error detection condition � i | E | j � = C E δ ij (a) Non-degenerate case C E = 0 : 0 = � i | E | j � = � s | W † i EW j S | s � = ± � s | W † i W j ( ES ) | s � . If E = X v Z u , get rid of all X operators with S i : v i � = 0 Error mapping to binary vector [Cl G ( E )] j ≡ u j + γ ij v i Power of Z : c i ⊕ c j ⊕ Cl G ( E ) If this is non-zero, classical and quantum error detection conditions coinside (b) Degenerate case C E � = 0 . Nothing to do in classical case. Quantum: E must commute with every W i ⇒ � c i , v � = 0 6-1
Upper bounds for a CWS code • Distance of a CWS code Q = ( G , C ) does not exceed that of the binary code C , d Q ≤ d C . • Distance of a non-degenerate CWS code Q = ( G , C ) does not exceed that of the graph state induced by G [Grassl et al., 2009], d non − deg ≤ d ′ G Q • If a bit j is involved in the code C [ ∃ c ∈ C : c j � = 0 ], then the distance of the CWS code Q = ( G , C ) does not exceed the weight of S j , d Q ≤ wgt S j . 7-1
Upper bounds for a CWS code • Distance of a CWS code Q = ( G , C ) does not exceed that of the binary code C , d Q ≤ d C . • Distance of a non-degenerate CWS code Q = ( G , C ) does not exceed that of the graph state induced by G [Grassl et al., 2009], d non − deg ≤ d ′ G Q • If a bit j is involved in the code C [ ∃ c ∈ C : c j � = 0 ], then the distance of the CWS code Q = ( G , C ) does not exceed the weight of S j , d Q ≤ wgt S j . Consequences • For a graph G with all vertices of the same degree r , the distance of a CWS code Q = ( G , C ) cannot exceed r + 1 . • The distance of a CWS code Q = ( G , C ) where the binary code C involves all bits cannot be bigger than the minimum weight of S i . 7-2
Examples: Simple stabilizer codes via CWS framework 2 1 • The [[5 , 1 , 3]] code is generated by the bi- nary code C = � (11111) � and the 5-ring graph G . Graph stabilizer generators S 2 = 3 Z 1 X 2 Z 3 and cyclic shifts. Stabilizer gener- ators S 2 S 3 = Z 1 Y 2 Y 3 Z 4 and cyclic shifts. 5 4 8-1
Examples: Simple stabilizer codes via CWS framework 2 1 • The [[5 , 1 , 3]] code is generated by the bi- nary code C = � (11111) � and the 5-ring graph G . Graph stabilizer generators S 2 = 3 Z 1 X 2 Z 3 and cyclic shifts. Stabilizer gener- ators S 2 S 3 = Z 1 Y 2 Y 3 Z 4 and cyclic shifts. 5 4 • The second [[6 , 1 , 3]] code is generated by 1 2 6 the binary code C = � (011100) � and the graph shown. Code degenerate above the 3 graph distance d ′ ( G ) = 2 : S 1 S 2 = X 1 X 2 . 5 8-2
Examples: Simple stabilizer codes via CWS framework 2 1 • The [[5 , 1 , 3]] code is generated by the bi- nary code C = � (11111) � and the 5-ring graph G . Graph stabilizer generators S 2 = 3 Z 1 X 2 Z 3 and cyclic shifts. Stabilizer gener- ators S 2 S 3 = Z 1 Y 2 Y 3 Z 4 and cyclic shifts. 5 4 • The second [[6 , 1 , 3]] code is generated by 1 2 6 the binary code C = � (011100) � and the graph shown. Code degenerate above the 3 graph distance d ′ ( G ) = 2 : S 1 S 2 = X 1 X 2 . 5 • Steane’s [[7 , 1 , 3]] code with the stabilizer 6 1 generators X 1 X 2 X 3 X 4 , Z 1 Z 2 Z 3 Z 4 and 3 their cyclic shifts. It is generated by the code 4 7 C = � (1110000) � and the graph shown. No graph with explicit circulant symmetry ex- 2 5 ists. 8-3
GF (4) representation of the stabilizer Background: GF (4) map for the stabilizer of an additive code: U ≡ i m ′ Z u X v → ( v , u ) → u + ω v , where ω is the generator of GF (4) : ω ≡ ω 2 = ω + 1 , ωω = 1 . Operators U 1 and U 2 commute iff e 1 ∗ e 2 ≡ e 1 · e 2 + e 1 · e 2 = v 1 · u 2 + u 1 · v 2 . Generator matrix for the additive GF (4) code ⇔ CWS code G = P ( ω 1 + R ) Parity check matrix Graph adjacency for the binary code matrix 9-1
GF (4) representation of the stabilizer Background: GF (4) map for the stabilizer of an additive code: U ≡ i m ′ Z u X v → ( v , u ) → u + ω v , where ω is the generator of GF (4) : ω ≡ ω 2 = ω + 1 , ωω = 1 . Operators U 1 and U 2 commute iff e 1 ∗ e 2 ≡ e 1 · e 2 + e 1 · e 2 = v 1 · u 2 + u 1 · v 2 . Generator matrix for the additive GF (4) code ⇔ CWS code G = P ( ω 1 + R ) Parity check matrix Graph adjacency for the binary code matrix G is automatically self-orthogonal as long as R = R t t = P ( ω 1 + R ) ∗ ( ω 1 + R ) P t = 0 G ∗ G 9-2
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