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1 22 Quantum error correction and its simulation
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3 Abstract Quantum error correction and its simulation Hiroko Dehare Researches in quantum information theory and technology, that mix quantum theory and informatics, have been very active in recent years. The quantum bits, that are used in quantum informatics, are susceptible to external noises and often error prone. The error-correction, therefore, is essential in the realization quantum information processing. The classical error correction schemes can not be used because they exacerbate errors in quantum bits with the disturbances caused by quantum observations. In this thesis, we develop simulation programs for four existing quantum errorcorrection algorithms, and test their performances and efficiencies in realistic settings. Based on the results of thqe simulations, we propose a new hybrid algorithm which balances the efficiency in single error-correction and the tolerance for multiple errors. key words quantum bits quantum error correction ii
4 iii
5 X Not Qbit Qbit Qbit Shor 5Qbit Shor 5Qbit Shor 5Qbit Shor 5Qbit Qbit Steane 7Qbit Shor 9Qbit Shor 9Qbit iv
6 39 A 40 A A A B 43 v
7 4.1 Not 3Qbit Not3Qbit Not3Qbit X Not3Qbit X Y Z Not5Qbit Not5Qbit Not5Qbit X Not5Qbit X Y Z Qbit Qbit Qbit X Qbit X Y Z Shor 5Qbit Shor 5Qbit Shor 5Qbit X Shor 5Qbit X Y Z Qbit Qbit X Qbit X Y Z :X :X Y Z ( ) :X Y Z ( ) vi
8 2.1 1Qbit Not3Qbit Not5Qbit Qbit Shor 5Qbit Steane 7Qbit Shor 9Qbit B.1 Shor 5Qbit 2Qbit B.2 5-3Qbit 2Qbit vii
9 Bennett Brassard BB84 Bennett B92 Ekert E91 BB84 1
10 1.2 B92 BB BB B92 E BB84 80km [5] Benioff Bennett Benioff 1985 Deutsch 1994 Shor 2
11 1.2 [7] ψ =α 0 + β 1 ψ α β ψ (1.1) (1.2) a A b B (1.1) (1.2) (1.3) ψ = α 0 A + β 1 A (1.1) φ AB = 1 2 ( 0 A 0 B + 1 A 1 B ) (1.2) ψ A φ AB = (α 0 A + β 1 A ) 1 2 ( 0 A 0 B + 1 A 1 B ) (1.3) (1.3) ψ 2.2 q ψ 00 ψ 00 ψ C Z 10 C X 11 C Z C X 3
12
13 2 2.1 Quantum bit 1bit 1Qbit 1Qbit x(x = 0, 1) x (2.1) (2.1) (2.2) x = α 0 + β 1 ( α 2 + β 2 = 1) (2.1) ( ) ( ) 1 0 x = α + β 0 1 (2.2) 0 1 (2.2) (2.3) ( 1 ) ( 0 ) = 0 = 1 (2.3) θ (2.1) (2.2) (2.3) (2.4) θ = sin θ cos θ 2 1 = ( sin θ 2 cos θ 2 ) (2.4) 5
14 2.1 (2.4) (2.5) ( ) ( = = ) (2.5) θ θ + π θ θ + π (2.6) = { 50% 50% (2.6) BB84 B W.K.Heisenberg ( ) [4] ( ) ( ) 6
15 2.1 (2.7) (2.8) { : 50% 50% : 50% 50% (2.7) { : 50% 50% : 50% 50% (2.8) [1] ψ φ ( ψ φ ) ψ φ ( ψ φ = 0) 0 (2.9) ψ 0 φ 0 (2.9) (2.9) U (2.10) { U( ψ 0 ) = ψ ψ U( φ 0 ) = φ φ (2.10) (2.10) (2.11) ψ φ = ( ψ φ ) 2 (2.11) (2.11) ψ φ = 0 ψ φ = 1 ψ = φ ψ φ (2.11) 7
16 θ R (2.12) R(θ) = ( cos θ 2 sin θ 2 sin θ 2 cos θ 2 ) (2.12) C X C Y C Z I H( ) 5 C X C Y C Z A.1 C X C Y C Z (2.13) C X = σ x C Y = iσ y (2.13) C Z = σ z 1Qbit (2.14) 1Qbit C X C Y C Z 8
17 2.2 ( ) ( ) ( ) C X = C Y = C Z = ( ) ( ) (2.14) I = H = H C X C Z H (2.15) H = 1 2 (C X + C Z ) (2.15) C Y C Z C X C Y C X C Z (2.16) { CZ C X = C Y C X C Z = C Y (2.16) (2.14) (2.16) (2.17) (2.18) (2.19) { CX C Y = C Z C Y C Z = C X (2.17) C X C Y = C Y C X C X C Z = C Z C X (2.18) C Y C Z = C Z C Y C 2 X = C 2 Y = C 2 Z = I (2.19) 0 1 1Qbit (2.1) (2.1) A.2 (2.1) α β I C X C Y C Z (2.12) R θ I C Y (2.20) R θ = I cos θ 2 C Y sin θ 2 (2.20) 9
18 Qbit C X C Y C Z I H ( ) ( 0 1 ) Not 2Qbit Not (A.3) Not XOR Not 3Qbit And Not Cont X Cont Y ContZ 2Qbit (A.8) (A.10) (A.11) (A.12) (2.19) C Y C Y (2.16) C Y = C Z C X C X C Z C Y 10
19 2.3 U α 0 + β 1 (2.21) U{U(α 0 + β 1 )} = α 0 + β 1 (2.21) Not 2.4 ( ) [3] ψ = a b 2Qbit (2.22) 2Qbit (2.23) a b = 1 2 ( ) (2.22) a b = 1 2 ( + (2.23) 11
20 2.4 a a = b (2.23) b a b [1] b a b ( ) b = 1 2 ( ± ) Belle B [3] 12
21 X Y Z 3 X Y Z X 0 1 Z Y (2.16) Z X I C X C Y C Z a X θ (3.1) e iθx a = (I cos θ + ix sin θ) a ( ) cos θ i sin θ = a (3.1) i sin θ cos θ 13
22 α 0 + β 1 5Qbit F (3.2) F = α β (3.2) α β
23 C X C Y C Z Y C Y
24 4 4.1 nqbit F = f 0 f n 1 E i 2 100,000Qbit X X Y Z 1 3 X X Y Z (2.1) Y Z X Y Z i (i=0,,4) C X 16
25 4.2 Not X i C Y C Z Y i Z i i X Y Z C X C Y C Z X i Y i Z i i (i=0,,4) X X α 0 + β 1 α β X α β Y Z 4.2 Not Not 2.3 f i = 1 Not X Y Y Z Qbit Not 3Qbit 2Qbit P = p 0 p 1 p i (i = 0, 1) f i f i+1 17
26 4.2 Not 4.1 Not 3Qbit (4.1) P (4.1) X 4.1 Not3Qbit I X 0 X 1 X 2 p p Not3Qbit (4.1) (4.2) (4.2) X X Y Z 2 (4.3) (4.4) 4.3 Not3Qbit 4.4 Not3Qbit X X Y Z 18
27 4.2 Not Qbit 4Qbit P = p 0 p 1 p 2 p 3 p i f i f i Not5Qbit (4.5) P (4.2) 2Qbit X 4.2 Not5Qbit I X 0 X 1 X 2 X 3 X 4 X 0 X 1 X 0 X 2 X 0 X 3 X 0 X 4 X 1 X 2 X 1 X 3 X 1 X 4 X 2 X 3 X 2 X 4 X 3 X 4 p p p p
28 Not5Qbit (4.2) (4.6) (4.6) (4.7) (4.8) 4.7 Not5Qbit 4.8 Not5Qbit X X Y Z 4.3 Cont X Cont Y Cont X H nbit M M (4.1) (4.2) M 2 = I (4.1) MM F = F (4.2) M M i M n
29 E i M (2.18) X i Z i (2.16) C Y C Z C X X i Y j Y i Z j (i j) (4.3) (4.4) (5.1) (5.2) Qbit 2Qbit p 0 p 1 i X E i F = X i F { M0 = Z 0 Z 1 M 1 = Z 1 Z 2 (4.3) i = 0 X 0 F (4.3) M 0 M 1 (4.2) (4.4) X 0 F M 0 M 1 { M0 X 0 F = X 0 F M 1 X 0 F = X 0 F (4.4) 21
30 Qbit (4.9) P E i M 0 M 1 (4.3) p i = 1 { M0 E i F = ( 1) p 0 E i F M 1 E i F = ( 1) p 1 E i F (4.5) (4.1) Not3Qbit (4.9) 3Qbit A Qbit X 0 X 1 X 2 M 0 + M Qbit (4.9) P (4.5) M 0 M 1 E i P (4.3) X (4.10) 3Qbit (4.11) (4.12) 22
31 Qbit Qbit X X Y Z Shor 5Qbit Shor 5Qbit [2] 4Qbit P = p 0 p 1 p 2 p 3 M (4.6) (4.7) (4.8) M = (1 + M 0 )(1 + M 1 )(1 + M 2 )(1 + M 3 ) (4.6) M 0 = Z 4 X 3 X 2 Z 1 M 1 = X 4 X 3 Z 2 Z 0 (4.7) M 2 = X 4 Z 3 Z 1 X 0 M 3 = Z 4 Z 2 X 1 X 0 M i (1 + M i ) = 1 + M i (4.8) G = M F (4.9) (4.7) M i G M i (4.8) (4.10) M i G = G (4.10) (4.7) Shor 5Qbit (4.13) 23
32 Shor 5Qbit M 0 E i G = ( 1) p 0 E i G M 1 E i G = ( 1) p 1 E i G M 2 E i G = ( 1) p 2 E i G M 3 E i G = ( 1) p 3 E i G (4.11) (4.13) P (4.11) E i M i (4.4) G 1Qbit 4.4 Shor 5Qbit X 0 Y 0 Z 0 X 1 Y 1 Z 1 X 2 Y 2 Z 2 X 3 Y 3 Z 3 X 4 Y 4 Z 4 1 M M M M (4.4) (4.14) Y i Y i 24
33 Shor 5Qbit (4.14) (4.2) (4.12) M G F M G = F (4.12) (4.16) (4.12) F 4.15 Shor 5Qbit X 4.16 Shor 5Qbit X Y Z Shor 5Qbit Shor 5Qbit 1Qbit 2Qbit
34 4.3 2Qbit 1Qbit (B.1) 3Qbit Shor 5Qbit Shor 5Qbit Shor 5Qbit 5Qbit Shor 5Qbit 4Q 16 Y i X i Z i i = 0 X 0 = (+ ++) Z 0 = (+ + ) Y 0 = (+ ) X 0 Z 1 X 0 Z 1 = (+ + ) (B.1) X 2 X 0 Z 1 X 2 Shor 5Qbit X 0 Z 1 X 2 = (+ + ++) X 0 Z 1 X 2 (4.13) (4.13) M 0 X 0 Z 1 X 2 G = X 0 Z 1 X 2 G M 1 X 0 Z 1 X 2 G = X 1 Z 1 X 2 G M 2 X 0 Z 1 X 2 G = X 2 Z 1 X 2 G M 3 X 0 Z 1 X 2 G = X 3 Z 1 X 2 G (4.13) 1Qbit 26
35 4.3 3Qbit 2Qbit Shor 5Qbit Shor 5Qbit 2Qbit 1Qbit 2Qbit 2Qbit X i Y i Z i X i X j Y i Y j Z i Z j X i Y j X i Z j Y i Z j (i j) 105 (B.1) 2Qbit 1Qbit 1 2Qbit Qbit 2Qbit Shor 5Qbit Shor 5Qbit 2Qbit 2Qbit Shor 5Qbit 7 3Qbit Shor 5Qbit F 3Qbit 2 27
36 4.3 3Qbit 2 Not 5Qbit Shor 5Qbit 3Qbit 1Qbit 3Qbit X 2Qbit (B.1) X Y Shor 5Qbit X i Y i 3Qbit Shor 5Qbit 3Qbit (B.2) 2Qbit (4.17) 3Qbit 3Qbit Qbit (4.17) Shor 5Qbit X i Y i Z i 5-3Qbit 2Qbit (4.18) (4.19) 28
37 Qbit X Qbit X Y Z 29
38 X (5.1) X Y Z (5.2) (5.3) 5.1 :X Not 2 3Qbit 3 X Shor 5Qbit 5-3Qbit X Y Z X Shor 5Qbit X Y Z
39 :X Y Z 5.3 :X Y Z ( ) ( ) Not 3Qbit 5-3Qbit X Y Z Shor 5Qbit 3Qbit Not5Qbit Not Y Z X Shor 5Qbit 5-3Qbit Not 3Qbit X Y Z 100% (4.4) (4.8) (4.12) 100% 3Qbit 2Qbit 0 Z (2.1) Shor 5Qbit 5-3Qbit
40 Qbit 5Qbit [1] Steane 7Qbit A.M.Steane 7Qbit [9] [1] 1Qbit Shor 5Qbit Steane 7Qbit 2Qbit Shor 5Qbit X Z Steane 7Qbit 6Qbit 3Qbit X 3Qbit Z Y Y i = Z i X i X i Z i X Z 1Qbit 2Qbit 2Qbit X i Z j (i j) X i X j Z i Z j (2.16) X i Y j Y i Z j Y i Y j (i j) Steane 7Qbit X i Y i Z i 7 X i Z j Steane 7Qbit Z M i X N i (5.1) Shor 5Qbit 32
41 5.2 (5.1) M 0 = X 0 X 4 X 5 X 6 M 1 = X 1 X 3 X 5 X 6 M 2 = X 2 X 3 X 4 X 6 N 0 = Z 0 Z 4 Z 5 Z 6 N 1 = Z 1 Z 3 Z 5 Z 6 N 2 = Z 2 Z 3 Z 4 Z 6 (5.1) 5.1 Steane 7Qbit I Z 0 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 M M M I X 0 X 1 X 2 X 3 X 4 X 5 X 6 N N N Shor 9Qbit Shor 9Qbit [10] 3Qbit 3Qbit Z Y Z Shor 9Qbit Z 3Qbit F (5.2) Z 0 F = Z 1 F = Z 2 F Z 3 F = Z 4 F = Z 5 F Z 6 F = Z 7 F = Z 8 F (5.2) 33
42 5.2 X i (i = 0 8) Z j Z j+1 Z j+2 (j = 0 3 6) Shor 9Qbit (5.3) (5.2) M 0 = X 0 X 1 X 2 X 3 X 4 X 5 M 1 = X 3 X 4 X 5 X 6 X 7 X 8 N 0 = Z 0 Z 1 N 1 = Z 1 Z 2 N 2 = Z 3 Z 4 N 3 = Z 4 Z 5 N 4 = Z 6 Z 7 N 5 = Z 7 Z 8 (5.3) 5.2 Shor 9Qbit I Z 0 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 q X 0 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 M M N N N N N N Shor 9Qbit (5.2) 1Qbit Qbit X 3Qbit 3Qbit 34
43 5.2 2Qbit Shor 9Qbit 3Qbit Z 2 Z 3 Z 5 Z 6 9Qbit M 0 M 1 Shor 9Qbit Shor 9Qbit 1Qbit Shor 9Qbit 3Qbit 3Qbit 1Qbit X X i X j X k (i = 0 2 j = 3 5 k = 6 8) Z X X 35
44 6 X X Y Z 2 A.2 X X Y Z 5 Shor 5Qbit 5-3Qbit 2 5-3Qbit 2Qbit Steane 7Qbit X i X j Y i Y j Z i Z j X i Y j Y i Z j (i j) 2bit 5Qbit 8Qbit Shor 5Qbit Steane 7Qbit 36
45 5-3Qbit 2.4 [6] 9Qbit Shor exponential 37
46 38
47 [1] N.David Mermin 2009 [2] D P DiVincenzo P W Shor Fault-Tolerant Error Correction with Efficient Quantum Codes Phys Rev Lett 77(15): [3] Belle B News@KEK [4] BP [5] 80km JST [6] T.Aoki G.Takahashi T.Kajiya J.Yoshikawa S.L.Braunstein P.v.Loock A.Furusawa Quantum eroor correction beyong qubits Nature Physics 5: [7] Colin P.Williams Scott H.Chearwater [8] K.Fujii Y.Tokunaga Fault-Tolerant Topological One-Way Quantum Compution with Probabilistic Two-Qubit Gates Phys.Rev.Lett [9] A.M.Steane Error Correcting Codes in Quantum Theory Phys. Rev.Lett. 77(5): [10] P.W.Shor Scheme for reducing decoherence in quantum conputer memory Phys.Rev.A 52(4):
48 A A ( ) ( ) ( i 1 0 σ x = σ y = σ z = 1 0 i ) (A.1) (A.1) σ x σ y σ z (A.2) (A.3) (A.4) σx 2 = σy 2 = σz 2 = I σ x σ y = iσ z σ y σ z = iσ x σ z σ x = iσ y σ x σ y = σ y σ x σ y σ z = σ z σ y σ z σ x = σ x σ z (A.2) (A.3) (A.4) A.2 a a (2.1) 1 a a (A.5) a 40
49 A.3 { = = 0 1 (A.5) A.3 ( 0 1 ) ( 1 1 ) ( 1 0 ) ( 1 1 ) 1 0 = (A.6) ( 1 0 ) ( 1 1 ) ( 0 1 ) ( 1 1 ) 0 1 = (A.7) (A.8) = (A.9) 41
50 A (A.10) (A.11) = (A.12) = (A.13) 42
51 B B.1 Shor 5Qbit 2Qbit I XX YY ZZ XY YZ ZX X 0 Y 4 Y 1 Z 3 Z 2 X 4 Y 2 X 1 Y 3 Z 4 X 3 Z 1 X 2 Y 0 X 3 X 2 Z 4 Z 1 X 4 Y 3 X 1 Y 2 Y 4 Z 2 Y 1 Z 3 Z 0 X 4 X 1 Y 3 Y 2 Y 4 Z 3 Y 1 Z 2 Z 1 X 3 Z 4 X 2 X 1 Y 2 Y 0 Z 4 Z 3 X 2 Y 4 X 0 Y 3 Z 0 X 4 Z 2 X 3 Y 1 X 4 X 3 Z 2 Z 0 X 0 Y 4 X 2 Y 3 Y 2 Z 4 Y 0 Z 3 Z 1 X 2 X 0 Y 4 Y 3 Y 0 Z 4 Y 2 Z 3 Z 2 X 4 Z 0 X 3 X 2 Y 3 Y 1 Z 4 Z 0 X 3 Y 0 X 1 Y 4 Z 3 X 4 Z 1 X 0 Y 2 X 4 X 0 Z 3 Z 1 X 1 Y 0 X 3 Y 4 Y 3 Z 0 Y 1 Z 4 Z 2 X 3 X 1 Y 4 Y 0 Y 1 Z 0 Y 3 Z 4 Z 1 X 4 Z 3 X 0 X 3 Y 4 Y 2 Z 1 Z 0 X 4 Y 3 X 2 Y 0 Z 4 X 0 Z 2 X 1 Y 3 X 1 X 0 Z 4 Z 2 X 4 Y 0 X 2 Y 1 Y 4 Z 1 Y 2 Z 0 Z 3 X 3 X 1 Y 4 Y 0 Y 4 Z 0 Y 2 Z 1 Z 4 X 1 Z 2 X 0 X 4 Y 3 Y 0 Z 2 Z 1 X 3 Y 1 X 0 Y 2 Z 3 X 2 Z 0 X 1 Y 4 X 2 X 1 Z 3 Z 0 X 3 Y 2 X 0 Y 1 Y 3 Z 1 Y 0 Z 2 Z 4 X 3 X 0 Y 2 Y 1 Y 3 Z 2 Y 0 Z 1 Z 0 X 2 Z 3 X 1 43
52 B.2 5-3Qbit 2Qbit X 0 X 0 Z 3 Z 2 Z 1 X 2 Y 0 Y 0 Z 4 Z 1 Y 1 Z 3 X 1 Y 2 X 4 Y 3 Z 0 Z 0 Y 1 Z 2 Z 4 X 2 X 1 X 1 Z 4 Z 3 Y 2 Y 0 Y 1 Y 1 Y 0 Z 3 Z 2 Z 0 Y 2 Z 4 X 4 X 3 Z 1 Z 1 Y 0 Z 4 Y 4 Y 3 Y 2 Z 3 X 2 X 0 X 2 X 2 Z 4 Z 0 Z 1 X 0 Y 2 Y 2 X 3 Y 4 X 1 Y 0 Y 1 Z 4 Z 3 Z 1 Z 2 Z 2 Z 3 X 0 Y 1 Z 0 X 3 X 3 Y 4 Y 2 X 4 Y 1 Y 3 Y 3 Y 4 Z 1 X 4 Y 0 Z 3 Z 3 Z 2 X 0 Y 1 Y 0 Z 4 X 1 Y 2 Z 1 X 4 X 4 Y 3 Y 0 X 3 Y 1 Y 4 Y 4 X 3 Y 2 Y 3 Z 1 Z 4 Z 4 Y 0 Z 1 Z 3 X 1 Y 2 Y 1 Z 0 X X 0 Y 4 Y 1 Z 4 X 3 X 1 Y 3 X 4 Y 2 Y 0 Y 4 Z 2 X 3 X 2 Z 0 Y 4 Z 3 Y 3 Y 2 X 4 X 1 Z 1 X 3 X 1 Z 0 X 4 X 2 Y 4 X 0 Y 3 Z 2 X 3 Y 1 X 2 Y 3 X 0 Y 4 Z 1 Z 2 X 4 Y 3 Y 2 X 4 X 1 Z 1 X 3 X 2 Y 3 Y 1 X 3 Y 0 X 1 Y 4 Z 3 X 4 Y 2 Y 3 Z 0 X 4 X 0 Z 2 Z 1 X 4 Y 4 Y 0 Y 3 Z 4 X 3 X 1 X 3 X 2 Y 0 Z 1 Z 0 Z 4 X 0 Z 2 X 1 Y 3 X 2 Y 1 Y 2 Z 0 Z 4 Z 2 X 1 X 0 Z 3 Y 4 Z 0 X 4 X 2 X 4 Z 2 Z 1 Z 0 X 1 Z 3 X 2 X 0 Y 2 Y 4 Z 3 Z 0 Y 0 Z 2 X 0 Y 1 X 2 X 1 Z 4 Y 3 Z 2 X 3 X 0 44
21 Quantum calculator simulator based on reversible operation
21 Quantum calculator simulator based on reversible operation 1100366 2010 3 1 i Abstract Quantum calculator simulator based on reversible operation Ryota Yoshimura Quantum computation is the novel computational
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32 2008 pp. 121 133 1 Received November 4, 2008 The aim of this paper is to clarify some profound changes in the language used in the visual media, especially in TV news programs in Japan, and show what
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