M. Grassl, Grundlagen der Quantenfehlerkorrektur, Universität Innsbruck, WS 2007/ , Folie 127

Transcription

1 M Grassl, Grundlagen der Quantenfehlerkorrektur, Universität Innsbruck, WS 2007/ , Folie 127

2 M Grassl, Grundlagen der Quantenfehlerkorrektur, Universität Innsbruck, WS 2007/ , Folie 128

3 M Grassl, Grundlagen der Quantenfehlerkorrektur, Universität Innsbruck, WS 2007/ , Folie 129

4 M Grassl, Grundlagen der Quantenfehlerkorrektur, Universität Innsbruck, WS 2007/ , Folie 130

5 References Markus Grassl, Encoding and decoding quantum error-correcting codes,, in: G Casati, D L Shepelyansky, and P Zoller, Proceedings of the International School of Physics E Fermi, Course CLXII on Quantum Computers, Algorithms and Chaos, IOS Press, 2006, pp Markus Grassl, Quantum Error Correction, International School of Physics Enrico Fermi, Course CLXII on Quantum Computers, Algorithms and Chaos, Varenna, Transparencies of the talk Markus Grassl, Martin Rötteler, and Thomas Beth, Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes, International Journal of Foundations of Computer Science (IJFCS), Vol 14, No 5 (2003), pp Preprint quant-ph/ , M Grassl, Grundlagen der Quantenfehlerkorrektur, Universität Innsbruck, WS 2007/ , Folie 131

6 International School of Physics Enrico Fermi Quantum Computers, Algorithms and Choas Varenna, 5 15 July 2005 Quantum Error Correction Markus Grassl Arbeitsgruppe Quantum Computing, Prof Beth Institut für Algorithmen und Kognitive Systeme Fakultät für Informatik, Universität Karlsruhe Germany

7 Outline general setting of quantum error correction discretizing errors encoding circuits References Markus Grassl, Algorithmic aspects of quantum error-correcting codes, in: Ranee K Brylinski and Goong Chen (Eds), Mathematics of Quantum Computation, Chapman & Hall/CRC, 2002, pp Markus Grassl, Martin Rötteler, and Thomas Beth, Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes, International Journal of Foundations of Computer Science (IJFCS), Vol 14, No 5 (2003), pp Preprint quant-ph/

8 Quantum Error Correction General scheme environment no access system φ interaction entanglement decoherence

9 Quantum Error Correction General scheme no access system φ encoding ancilla 0 environment encoding interaction three-party entanglement

10 Quantum Error Correction General scheme no access system φ encoding ancilla 0 environment encoding interaction syndrome ancilla 0 error correction code entanglement environment/ancilla

11 Quantum Error Correction General scheme no access system φ encoding ancilla 0 environment encoding interaction syndrome ancilla 0 error correction decoding φ 0

12 Quantum Error Correction General scheme no access system φ encoding ancilla 0 environment encoding interaction error correction syndrome ancilla 0 Basic requirement some knowledge about the interaction between system and environment Common assumptions no initial entanglement between system and environment local or uncorrelated errors, i e, only a few qubits are disturbed = CSS codes, stabilizer codes interaction with symmetry = decoherence free subspaces decoding φ 0

13 Interaction System/Environment Closed System environment ε system φ interaction } = U env/sys ( ε φ ) Channel Q: ρ in := φ φ ρ out := Q( φ φ ) := i E i ρ in E i with Kraus operators (error operators) E i Local/low correlated errors product channel Q n where Q is close to identity Q can be expressed (approximated) with error operators Ẽi such that each E i acts on few subsystems, e g quantum gates

14 Computer Science Approach: Discretize QECC Characterization [Knill & Laflamme, PRA 55, (1997)] A subspace C of H with orthonormal basis { c 1,, c K } is an error-correcting code for the error operators E = {E 1, E 2, }, if there exists constants α k,l such that for all c i, c j and for all E k, E l E: c i E k E l c j = δ i,j α k,l (1) It is sufficient that (1) holds for a vector space basis of E

15 Linearity of Conditions Assume that C can correct the errors E = {E 1, E 2, } New error-operators: A := k λ k E k and B := l µ l E l c i A B c j = k,l λ k µ l c i E k E l c j = k,l λ k µ l δ i,j α k,l = δ i,j α (A, B) It is sufficient to correct error operators that form a basis of the linear vector space spanned by the operators E = only a finite set of errors

16 Error Basis Pauli Matrices ( ) 0 1 ( ) 0 i ( 1 0 ) ( ) 1 0 σ x := 1 0, σ y := i 0, σ z := 0 1, I := 0 1 vector space basis of all 2 2 matrices unitary matrices which generate a finite group Error Basis for many Qubits/Qudits E error basis for subsystem of dimension d with I E = E n error basis with elements E := E 1 E n, E i E weight of E: number of factors E i I

17 Local Error Model Code Parameters C = [n, k, d] n: number of subsystems used in total k: number of (logical) subsystems encoded d: minimum distance correct all errors acting on at most (d 1)/2 subsystems detect all errors acting on less than d subsystems correct all errors acting on less than d subsystems at known positions

18 General Decoding Algorithm E 1 C E 2 C E k C V 0 E 1 c 0 E 2 c 0 E k c 0 V 1 E 1 c 1 E 2 c 1 E k c 1 V i E 1 c i E 2 c i E k c i c i E k E l c j = δ i,j α k,l (1)

19 General Decoding Algorithm E 1 C E 2 C E k C V 0 E 1 c 0 E 2 c 0 E k c 0 V 1 E 1 c 1 E 2 c 1 E k c 1 V i E 1 c i E 2 c i E k c i rows are orthogonal as c i E k E l c j = 0 for i j c i E k E l c j = δ i,j α k,l (1)

20 General Decoding Algorithm E 1 C E 2 C E k C V 0 E 1 c 0 E 2 c 0 E k c 0 V 1 E 1 c 1 E 2 c 1 E k c 1 V i E 1 c i E 2 c i E k c i inner product between columns is constant as c i E k E l c i = α k,l rows are orthogonal as c i E k E l c j = 0 for i j c i E k E l c j = δ i,j α k,l (1)

21 General Decoding Algorithm E 1 C E 2 C E k C V 0 E 1 c 0 E 2 c 0 E k c 0 V 1 E 1 c 1 E 2 c 1 E k c 1 V i E 1 c i E 2 c i E k c i inner product between columns is constant as c i E k E l c i = α k,l rows are orthogonal as c i E k E l c j = 0 for i j = simultaneous Gram-Schmidt orthogonalization within the spaces V i

22 Orthogonal Decomposition E 1C E 2C E k C V 0 E 1 c 0 E 2 c 0 E k c 0 V 1 E 1 c 1 E 2 c 1 E k c 1 V i E 1 c i E 2 c i E k c i columns are mutually orthogonal rows are mutually orthogonal new error operators E k are linear combinations of the E l projection onto E k C determines the error exponentially many orthogonal spaces E k C

23 Encoding Stabilizer Codes [Grassl, Rötteler, and Beth, IJFCS, 14 (2003), pp ] Basic idea: Use operations of the generalized Clifford group (or Jacobi group) to transform the stabilizer S into a trivial stabilizer S 0 := Z (1), Z (n k) row/column operations on the binary matrix (X Z) to obtain normal form (0 I0) operations on (X Z) correspond to elementary single-qubit gates CNOT-gate single qudit gate P := 1 0 = 0 e πi/ i

24 Action on Pauli Matrices Hadamard matrix H HXH = Z HY H = Y HZH = X (1, 0) (0, 1) (1, 1) (1, 1) (0, 1) (1, 0) exchange X and Z matrix P P XP = Y P Y P = X P ZP = Z (1, 0) (1, 1) (1, 1) (1, 0) (0, 1) (0, 1) multiply X by Z in mod2 operation on binary row vectors: (a, b) M = (a, b ) (arithmetic mod2) H ˆ= 0 1 and P ˆ= local operation on (X Z): multiplying column i in submatrix X and column i in submatrix Z by M

25 Action of CNOT Error propagation X ˆ= X X Z ˆ= Z X ˆ= ˆ= X Z Z Z

26 Action of CNOT Modifying stabilizers X ˆ= X X Z ˆ= Z X ˆ= X Z ˆ= Z Z

27 Action of CNOT Modifying stabilizers X ˆ= X X Z ˆ= Z X ˆ= X Z ˆ= Z Z add X from source to target add Z from target to source

28 Example: 5 Qubit Code [5, 1, 3] Generators of stabilizer XXZIZ ZXXZI ˆ= IZXXZ ZIZXX

29 Step I: X-Only Generator Local operations ˆ= operations on corresponding X and Z columns T 1 := I I I I I

30 Step I: X-Only Generator Local operations ˆ= operations on corresponding X and Z columns T 1 := I I H I I

31 Step I: X-Only Generator Local operations ˆ= operations on corresponding X and Z columns T 1 := I I H I H

32 Step II: X-Generator of Weight One CNOT ˆ= operations on pairs of corresponding X and Z columns T 1 := I I H I H T 2 := I

33 Step II: X-Generator of Weight One CNOT ˆ= operations on pairs of corresponding X and Z columns T 1 := I I H I H T 2 := I

34 Step II: X-Generator of Weight One CNOT ˆ= operations on pairs of corresponding X and Z columns T 1 := I I H I H T 2 := CNOT (1,2)

35 Step II: X-Generator of Weight One CNOT ˆ= operations on pairs of corresponding X and Z columns T 1 := I I H I H T 2 := CNOT (1,2)

36 Step II: X-Generator of Weight One CNOT ˆ= operations on pairs of corresponding X and Z columns T 1 := I I H I H T 2 := CNOT (1,2) CNOT (1,3)

37 Step II: X-Generator of Weight One CNOT ˆ= operations on pairs of corresponding X and Z columns T 1 := I I H I H T 2 := CNOT (1,2) CNOT (1,3)

38 Step II: X-Generator of Weight One CNOT ˆ= operations on pairs of corresponding X and Z columns T 1 := I I H I H T 2 := CNOT (1,2) CNOT (1,3) CNOT (1,5)

39 Step III: Row Operations multiplying generators ˆ= adding/permuting rows T 1 := I I H I H T 2 := CNOT (1,2) CNOT (1,3) CNOT (1,5)

40 Step III: Row Operations multiplying generators ˆ= adding/permuting rows T 1 := I I H I H T 2 := CNOT (1,2) CNOT (1,3) CNOT (1,5)

41 Step IV: Next Row/Generator T 3 := I I I I I

42 Step IV: Next Row/Generator T 3 := I I H H I

43 Step IV: Next Row/Generator T 3 := I I H H I T 4 := CNOT (2,3) CNOT (2,4)

44 Step IV: Next Row/Generator T 3 := I I H H I T 4 := CNOT (2,3) CNOT (2,4)

45 Pre-Final Result: Only X-Generators

46 Final Result: Only Z-Generators T final := H H H H I

47 Final Result: Only Z-Generators T final := H H H H I combining all transformations: quantum circuit that maps encoded state ψ L to the un-encoded state 0 ψ

48 Final Result: Only Z-Generators T final := H H H H I combining all transformations: quantum circuit that maps encoded state ψ L to the un-encoded state 0 ψ Homework : derive the complete circuit

49 Decoding Circuit for Ternary QECC C = [9, 5, 3] 3 F 0 F 0 M 2 F P 1 M 2 M 2 F 0 P 1 M 2 P 1 P 1 M 2 P 2 F 0 φ enc P 1 P 2 M 2 F P 2 M 2 F P 2 P 2 M 2 M 2 F P 1 M 2 F P 2 M 2 P 1 M 2 M 2 P 1 F P 1 M 2 P 1 M 2 φ in