Quantum Error-Correcting Codes by Concatenation QEC11 Second International Conference on Quantum Error Correction University of Southern California, Los Angeles, USA December 5–9, 2011 Quantum Error-Correcting Codes by Concatenation Markus Grassl joint work with Bei Zeng Centre for Quantum Technologies National University of Singapore Singapore Markus Grassl – 1– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Why Bei isn’t here Jonathan, November 24, 2011 Markus Grassl – 2– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Overview Shor’s nine-qubit code revisited • The code [[25, 1, 9]] • Concatenated graph codes • Generalized concatenated quantum codes • Codes for the Amplitude Damping (AD) channel • Conclusions • Markus Grassl – 3– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Shor’s Nine-Qubit Code Revisited Bit-flip code: 0 000 , 1 111 . | → | | → | Phase-flip code: 0 + ++ , 1 . | → | | → | − −− Effect of single-qubit errors on the bit-flip code: X-errors change the basis states, but can be corrected • Z-errors at any of the three positions: • Z 000 = 000 | | “encoded” Z-operator Z 111 = 111 | −| = Bit-flip code & error correction convert the channel into a phase-error ⇒ channel = Concatenation of bit-flip code and phase-flip code yields [[9, 1, 3]] ⇒ Markus Grassl – 4– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 The Code [[25, 1, 9]] The best single-error correcting code is = [[5, 1, 3]] • C0 Re-encoding each of the 5 qubits with yields = [[52, 1, 32]] = [[25, 1, 9]] • C0 C The code is a subspace of five copies of [[5, 1, 3]] • C The stabilizer of is generated by five copies of the stabilizer of and an • C C0 encoded version of the stabilizer of C0 The code is degenerate • C m-fold self-concatenation of [[n, 1,d]] yields [[nm, 1,dm]] • Markus Grassl – 5– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Level-decoding of [[25, 1, 9]] The code corrects up to t = 4 errors (t<d/2) different error patterns: a) • •••• ••••• • •••• • •••• ••••• b) • • • • • ••••• • • • • • ••••• ••••• errror correction on both levels: • corrects a), but fails for b) errror detection on lowest level, error correction on higer level: • corrects b), but fails for a) = optimal decoding must pass information between the levels ⇒ Markus Grassl – 6– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Overview Shor’s nine-qubit code revisited • The code [[25, 1, 9]] • Concatenated graph codes ⇒ [Beigi, Chuang, Grassl, Shor & Zeng, Graph Concatenation for QECC, JMP 52 (2011), arXiv:0910.4129] Generalized concatenated quantum codes • Codes for the Amplitude Damping (AD) channel • Conclusions • Markus Grassl – 7– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Canonical Basis of a Stabilizer Code fix logical operators Xi and Zℓ • the stabilizer and the logical operators Zℓ mutually commute • S the logical state 00 ... 0 is a stabilizer state • | define the (logical) basis states as • i1 ik i i ...ik = X X 00 ... 0 | 1 2 1 ··· k | in terms of a classical code over a finite field: the logical state 00 ... 0 corresponds to a self-dual code • | C0 the basis states i i ...ik correspond to cosets of • | 1 2 C0 for a stabilizer code, the union of the cosets is an additive code ∗ • C Markus Grassl – 8– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Graphical Quantum Codes [D. Schlingemann & R. F. Werner: QECC associated with graphs, PRA 65 (2002), quant-ph/0012111] [Grassl, Klappenecker & R¨otteler: Graphs, Quadratic Forms, & QECC, ISIT 2002, quant-ph/0703112] Basic idea a classical symplectic self-dual code defines a single quantum state • = [[n, 0,d]]q C0 the standard form of the stabilizer matrix is (I A) • | the generators have exactly one tensor factor X • self-duality implies that A is symmetric • A can be considered as adjacency matrix of a graph with n vertices • logical X-operators give rise to more quantum states in the code • = [[n,k,d′]]q C use additionally k input vectices • Markus Grassl – 9– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Graphical Representation of [[6, 2, 3]]3 100000 000102 010000 001222 001000 010201 000100 122000 000010 020002 000001 221020 000000 101100 000000 100021 stabilizer & logical X-operators graphicalrepresentation Markus Grassl – 10– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Encoder based on Graphical Representation [M. Grassl, Variations on Encoding Circuits for Stabilizer Quantum Codes, LNCS 6639, pp. 142–158, 2011] F -1 × × × • φ ψ0 | in | F -1 × × × • 0 F Z Z2 | × × × × 0 F Z2 Z2 | × ××× 0 F Z2 Z | × × × × φ | enc 0 F X | × × × × 0 F | × × × 0 F X | × × × × × preparation of 0 ... 0 operators X operators Z | Markus Grassl – 11– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Encoder based on Graphical Representation F -1 × × × • φ ψ0 | in | F -1 × × × • 0 F Z Z2 | × × × × 0 F Z2 Z2 | × ××× 0 F Z2 Z | × × × × φ | enc 0 F X | × × × × 0 F | × × × 0 F X | × × × × × preparation of 0 ... 0 operators X operators Z | Markus Grassl – 11– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Concatenation of Graph Codes [Beigi, Chuang, Grassl, Shor & Zeng, Graph Concatenation for QECC, JMP 52 (2011), arXiv:0910.4129] self-concatenation of [[5, 1, 3]] • = ⇒ measure the five auxillary nodes in X-bases • • X-measurement corresponds to sequence of local complementations • = many different choices ⇒ Markus Grassl – 12– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 [[5, 1, 3]] = [[25, 1, 9]] ⇒ Markus Grassl – 13– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 [[5, 1, 3]] = [[25, 1, 9]] ⇒ Markus Grassl – 13– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 [[7, 1, 3]] = [[49, 1, 9]] ⇒ Markus Grassl – 14– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 [[7, 1, 3]] = [[49, 1, 9]] ⇒ Markus Grassl – 14– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 General Concatenation Rule (for qubit codes; see paper for qudit codes) Any edge connecting an input vertex with an auxiilary vertex is replaced by • a set of edges connecting the input vertex with all neighbors of the auxillary vertex. Any edge between two auxiliary vertices A and B is replaced by a complete • bipartite graph connecting any neighbor of A with all neighbors of B. = ⇒ Markus Grassl – 15– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Overview Shor’s nine-qubit code revisited • The code [[25, 1, 9]] • Concatenated graph codes • Generalized concatenated quantum codes ⇒ [Grassl, Shor, Smith, Smolin & Zeng, PRA 79 (2009), arXiv:0901.1319] [Grassl, Shor & Zeng, ISIT 2009, arXiv:0905.0428] Codes for the Amplitude Damping (AD) channel • Conclusions • Markus Grassl – 16– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Stabilizer Codes stabilizer group = S1,...,Sn k generated by n k mutually • S − − commuting tensor products of (generalized) Pauli matrices C = [[n,k,d]] is a common eigenspace of the Si • n orthogonal decomposition of the vector space ⊗ into joint eigenspaces • H E n-k−1 q n E Cq i E1 C labelling of the spaces by the eigenvalues of the Si • errors that change the eigenvalues can be detected • Markus Grassl – 17– 07.12.2011 Markus Grassl Quantum Error-Correcting Codes by Concatenation quantum codes decomposition of new basis: B 0 (1) 0; 0 0; 1 B | | 1 (1) 1; 0 1; 1 B {| | | 2 (1) i ; 2; 0 2; 1 j B | | B 3 : (1) i ( (1) i 3; 0 3; 1 C B = 0 | | 2 = ((5 ) 4 (1) Variations on 4; 0 4; 1 ⊗ ,..., 5 B | | = , 5 (1) 5; 0 5; 1 2 B , 15; B | | 3)) (0) 6 (1) 6; 0 6; 1 2 j = ((5 B | | = 0 7 (1) 7; 0 7; 1 – 18– B , , | | 2 1 8 (1) 5 } 8; 0 8; 1 , 1)) B | | [[5 9 (1) 9; 0 9; 1 2 , into B | | 1 10 (1) 10; 0 10; 1 , 3]] 16 B | | 11 (1) 2 11; 0 11; 1 mutually orthogonal B | | 12 (1) 12; 0 12; 1 B | | 13 (1) 13; 0 13; 1 B | | 14 (1) 14; 0 14; 1 B | | 15 (1) 15; 0 15; 1 | | 07.12.2011 | | 0 1 QEC11 L L Quantum Error-Correcting Codes by Concatenation QEC11 7 Construction of ((15, 2 , 3))2 basis i; j : i = 0,..., 15; j = 0, 1 of B(0) = ((5, 25, 1)) • {| } 2 – states i; 0 and i; 1 are in the code B(1) = ((5, 2, 3)) | | i 2 – for i = i′, some states i; j and i′; j′ have distance < 3 | | protect the quantum number i • a classical code of distance three suffices for this purpose • generalized concatenated QECC ((3 5, 16 23, 3)) with basis • × × i; j i; j i; j : i = 0,..., 15; j = 0, 1; j = 0, 1; j = 0, 1 {| 1| 2| 2 1 2 3 } normalizer code is a generalized concatenated code with • – inner codes (0) = (5, 210, 1) and (1) = (5, 26, 3) B 4 B 4 – outer codes = [3, 1, 3] and = [3, 3, 1] 6 A1 16 A2 2 Markus Grassl – 19– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 7 Encoding of ((15, 2 , 3))2 encoder for the nested codes ((5, 2, 3)) ((5, 25, 1)) 2 ≤ 2 i i i i | 1| 2| 3| 4 ? ? ? ? j - B(1) | i ? ? ? ? ? i , i , i , i ; j |1 2 3 4 Markus Grassl – 20– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 7 Encoding of ((15, 2 , 3))2 generalized concatenated encoder i i i i | 1 | 2 | 3 | 4 ? ? ? ? outer code A @ @ @ @ @ @ @ @ @ @ @ @ © © © © ? ? ? ? @R @R @R @R - (1) - (1) - (1) j1 B j2 B j3 B | i | i | i ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Markus Grassl – 21– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 A New Qubit Non-Stabilizer Code [Grassl, Shor, Smith,