QuantumComp
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- さあしゃ えいさか
- 5 years ago
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1 !! α!! /
2 ! PauliCliffordnon-Clifford, Solovay-KitaevCNOT!
3 i = 0i + 1i 1 0i = 0 0 1i = 1, 2 C =1,, + =( )/ 2, =( 0 1 )/ 2
4 i = 0i + 1i 1 0i = 0 0 1i = 1, 2 C =1,, + =( )/ 2, =( 0 1 )/ 2 y Bloch sphere z 2 x = cos, = e i sin
5 Pauli Pauli 0 1 X = 1 0 Z = Y = 0 i i 0 anticommute: XZ = iy ZX = XZ
6 Pauli Pauli 0 1 X = 1 0 Z = Y = 0 i i 0 anticommute: XZ = iy ZX = XZ qubit X 0 = 1 X 1 = 0 (bit-flip) Z 0 = 0 Z 1 = 1 (phase-flip) Y 0 = i 1 Y 1 = i 0 (bit&phase-flip + global phase)
7 Pauli Pauli 0 1 X = 1 0 Z = Y = 0 i i 0 anticommute: XZ = iy ZX = XZ qubit X 0 = 1 X 1 = 0 (bit-flip) Z 0 = 0 Z 1 = 1 (phase-flip) Y 0 = i 1 Y 1 = i 0 (bit&phase-flip + global phase) PauliPauli basis Z! 0i, 1i Z basis X! +i ( 0i + 1i)/ p 2, i ( 0i 1i)/ p 2 X basis
8 Bloch sphere ( 0i + i 1i)/ p 2 y 1i i z 2 x +i = cos, = e i sin 0i ( 0i i 1i)/ p 2
9 Bloch sphere ( 0i + i 1i)/ p 2 y e i Y 1i i e i Z z 2 x e i X +i = cos, = e i sin 0i ( 0i i 1i)/ p 2
10 Clifford CliffordPauli Pauli A = UBU Pauli H Hadamard = HXH = Z S Phase = 1 p i SXS = Y
11 Bloch sphere ( 0i + i 1i)/ p 2 y 1i i z 2 x +i = cos, = e i sin 0i ( 0i i 1i)/ p 2
12 non-clifford π y x 1! non-clifford
13 non-clifford π y x 1! non-clifford
14 non-clifford π y x 1! non-clifford
15 non-clifford π y x 1! non-clifford
16 non-clifford π y x 1! non-clifford
17 non-clifford π y x 1! non-clifford
18 non-clifford π y x 1! non-clifford
19 non-clifford π y x 1! non-clifford
20 non-clifford π y x 1! non-clifford
21 non-clifford π y x 1! non-clifford
22 non-clifford y x (3 p 5)
23 non-clifford y x (3 p 5)
24 non-clifford y x (3 p 5)
25 non-clifford y x (3 p 5)
26 non-clifford y x (3 p 5)
27 non-clifford y x (3 p 5)
28 non-clifford y x (3 p 5)
29 non-clifford y x (3 p 5)
30 non-clifford y x (3 p 5)
31 non-clifford y x (3 p 5)
32 non-clifford y x (3 p 5)
33 non-clifford y x (3 p 5)
34 non-clifford y x (3 p 5)
35 non-clifford y x (3 p 5)
36 non-clifford y x (3 p 5)
37 non-clifford y x (3 p 5)
38 non-clifford y x (3 p 5)
39 non-clifford y x (3 p 5)
40 non-clifford y x (3 p 5)
41 non-clifford y x (3 p 5)
42 non-clifford y x (3 p 5) θ
43 non-clifford y x (3 p 5) θ
44 non-clifford y x π (3-5)= (3 p 5) θ
45 non-clifford y x π (3-5)= (3 p 5) θ
46 non-clifford y π (3-5)= x (3 p 5) θ
47 non-clifford y π (3-5)= x (3 p 5) θ
48 non-clifford y π (3-5)= x (3 p 5) θ
49 non-clifford y π (3-5)= x (3 p 5) θ
50 non-clifford y π (3-5)= x (3 p 5) θ
51 non-clifford y (3 π (3-5)= x p 5) θ
52 non-clifford y (3 π (3-5)= x p 5) θ
53 non-clifford y (3 π (3-5)= x p 5) θ
54 non-clifford (3 y π (3-5)= x p 5) θ
55 non-clifford (3 y π (3-5)= x p 5) θ
56 non-clifford (3 y π (3-5)= x p 5) θ
57 non-clifford (3 y π (3-5)= x p 5) θ
58 non-clifford π/8 T = e i( /8)Z
59 non-clifford cos 8, sin 8, cos 8 y π/8 T = e i( /8)Z THTH = cos 2 8 I i h cos 8 (X + Z)+sin 8 Y i sin 8 z x
60 non-clifford π/8 T = e i( /8)Z THTH = cos 2 8 I i h cos 8 (X + Z)+sin 8 Y i sin 8 cos 8, sin 8, cos 8 y = 2 arccos[cos 2 ( /8)] z x {H, T} 1!
61 Solovay-Kitaev Solovay-Kitaev(U,n)! if n=0, return basic approximation of U! else U n-1 =Solovay-Kitaev(U,n-1)! V,W s.t. VWV W =UU n-1! V n-1 = Solovay-Kitaev(V,n-1)! W n-1 = Solovay-Kitaev(V,n-1)! Return V n-1 W n-1 V n-1 W n-1 U n-1!! O(log c (1/ )) Dawson-Nielsen, QIC 6, 81 (2006)
62 a i b i 0i 1i = = C A Kronecker 10 11
63 a i b i 0i 1i = = C A Kronecker (X) CNOT (controlled NOT) = 0 0 I X (Z) CZ (controlled Z) e i /4(Z 1Z 2 Z 1 Z 2 I) = 0 0 I Z
64 Solovay-Kitaev {H, T} 1
65 Solovay-Kitaev {H, T} 1 CNOT + 1 n universal set { (X),H,T}
66 Solovay-Kitaev {H, T} 1 CNOT + 1 n Toffoli gate universal set { (X),H,T} e i 8 Z e i 8 Z e i 8 Z S H e i e i 8 Z 8 Z e i 8 Z e i 8 Z
67 Solovay-Kitaev {H, T} 1 CNOT + 1 n universal set { (X),H,T} { 000i, 111i} X X U X X X X X X.
68 ! PauliCliffordnon-Clifford, Solovay-KitaevCNOT! CliffordGottesman-Knill
69
70 n-: n = c s1 s 2...s n s 1 s 2...s n s 1,s 2,...,s n? s 1 s 2 s n
71 n-: n = Stabilizer s 1,s 2,...,s n c s1 s 2...s n s 1 s 2...s n? s 1 s 2 s n D. Gottesman, Ph.D. thesis, California Institute of Technology (1997); arxiv:quant- ph/
72 Pauli group, Stabilizer group n-qubit Pauli group: {±1, ±i} {I,X,Y,Z} n 2 P n
73 Pauli group, Stabilizer group n-qubit Pauli group: {±1, ±i} {I,X,Y,Z} n 2 P n Stabilizer group S = {S i } Pauli() S i 2 P, S i = S i, [S i,s j ]=0 ) hxx,zzi = {II,XX,ZZ, YY} stabilizer generator even overlap =!
74 Stabilizer states [Gottesman PhD thesis arxiv:quant- ph/ ] Stabilizer statestabilizer S S i = stabilizer group stabilizer generator S i
75 Stabilizer states [Gottesman PhD thesis arxiv:quant- ph/ ] Stabilizer statestabilizer S S i = stabilizer group stabilizer generator S i 1) S 1 = XX,ZZ Bell state ( )/ 2 2) S 2 = ZZ { 00, 11 } generatorqubit
76 Stabilizer state Graph state Z X i Z Z K i = X i Y j2v i Z j measurement-based quantum computation (MBQC)! [Raussendorf-Briegel PRL 01]
77 Stabilizer state Graph state Surface code (Toric code) state Z X i Z Z K i = X i Y j2v i Z j Z Z Af Z Z X X Bv A f = X X Z i i face f B v = X i i vertex v measurement-based quantum computation (MBQC)! [Raussendorf-Briegel PRL 01] Quantum error correction code/ ground state of topologically orderd system [Kitaev Ann Phys 03]
78 Stabilizer(Clifford) CliffordPauli product Pauli product! stabilizer state stabilizer state [Heisenberg] hs i i U S i i = i [Schrödinger] i U S 0 i = US i U hs 0 ii S 0 i 0 i = 0 i 0 i = U i
79 Stabilizer(Clifford) CliffordPauli product Pauli product! stabilizer state stabilizer state [Heisenberg] hs i i U S i i = i [Schrödinger] i U S 0 i = US i U hs 0 ii S 0 i 0 i = 0 i 0 i = U i CliffordU stabilizer S i! S 0 i = US i U
80 Clifford H Hadamard = HXH = Z
81 Clifford H Hadamard S Phase = = p i HXH = Z SXS = Y
82 Clifford H Hadamard S Phase = = p i HXH = Z SXS = Y (X) X X Z X Z Z CNOT (controlled NOT) = 0 0 I X (X)(X I) (X) =X X (X)(I Z) (X) =Z Z
83 Clifford H Hadamard S Phase = = p i HXH = Z SXS = Y (X) X X Z X Z Z CNOT (controlled NOT) = 0 0 I X (X)(X I) (X) =X X (X)(I Z) (X) =Z Z (Z) X X Z Z X X CZ (controlled Z) e i /4(Z 1Z 2 Z 1 Z 2 I) = 0 0 I Z (Z)(X I) (Z) =X Z (Z)(I X) (Z) =Z X
84 Clifford +i H X +i X H Z +i H +i H 0i +i H H 0i +i X H H Z 0i 0i H H 0i 0i X H H Z 0i H 0i X H Z
85 Gottesman-Knill (Clifford classically simulatable) Input!!! Pauli.! Operation!! Clifford! Measurement!Pauli n qubit stabilizer state n [Heisenberg] [Schrödinger] hs i i S i i = i U i U S 0 i = US i U hs 0 ii S 0 i 0 i = 0 i 0 i = U i
86 Gottesman-Knill (Clifford classically simulatable) Input!!! Pauli.! Operation!! Clifford! Measurement!Pauli Classically simulatable! n qubit stabilizer state n [Heisenberg] [Schrödinger] hs i i S i i = i U i U S 0 i = US i U hs 0 ii S 0 i 0 i = 0 i 0 i = U i
87 Gottesman-Knill (Clifford classically simulatable) Input!!! Pauli.! Operation!! Clifford! Measurement!Pauli Classically simulatable! n qubit stabilizer state n [Heisenberg] [Schrödinger] hs i i S i i = i U i U S 0 i = US i U hs 0 ii S 0 i 0 i = 0 i 0 i = U i Paulistabilizer! Gottesman-Knill
88 Magic state (noisy magic state is enough for universal QC) INPUT!!!! Pauli! OPERATION!! Clifford! MEASUREMENT! Pauli Pauli classically simulatable Bloch y z x (Tr[ X], Tr[ Y ], Tr[ Z])
89 Magic state (noisy magic state is enough for universal QC) INPUT!!!! Pauli! OPERATION!! Clifford! MEASUREMENT! Pauli Pauli classically simulatable Bloch y z Hi = cos( /8) 0i + sin( /8) 1i magic state H =(1 p) HihH + p H? ihh? p = 1 2 (1 1/p 2) z x (Tr[ X], Tr[ Y ], Tr[ Z]) x Clifford pure magic state Bravyi-Kitaev, PRA 71, (2005); Reichardt, QIP 4, 251 (2005)
90 ! PauliCliffordnon-Clifford, Solovay-KitaevCNOT! CliffordGottesman-Knill
91 Graph state (cluster state) Graph state stabilizer generators: K i = X i Y j2v i Z j i K i Gi = Gi for all i 2 V graph G=(V,E) V:E: Z X Z Z
92 Graph state (cluster state) Graph state stabilizer generators: K i = X i Y j2v i Z j i K i Gi = Gi for all i 2 V graph G=(V,E) V:E: Z X Z Z Gi = Y e2e e (Z) +i V
93 Graph state (cluster state) Graph state stabilizer generators: K i = X i Y j2v i Z j i K i Gi = Gi for all i 2 V graph G=(V,E) V:E: Z X Z Z Gi = Y e2e e (Z) +i V XI XZ IX ZX X X Z Z X X CZ (controlled Z) = 0 0 I Z e i /4(Z 1Z 2 Z 1 Z 2 I)
94 Cluster state computation Raussendorf-Briegel PRL (2001); Raussendorf-Browne-Briegel PRA (2003). 2D cluster state projective measurement Z i +w Z i +n X Z i i +e Z i +s 2D resource state K i = X i Z i+n Z i+e Z i+s Z i+w
95 Cluster state computation Raussendorf-Briegel PRL (2001); Raussendorf-Browne-Briegel PRA (2003). 2D cluster state projective measurement Z i +w Z i +n X Z i i +e Z i +s 2D resource state K i = X i Z i+n Z i+e Z i+s Z i+w
96 Cluster state computation Raussendorf-Briegel PRL (2001); Raussendorf-Browne-Briegel PRA (2003). 2D cluster state projective measurement Z i +n Z i +w X Z i i +e Z i +s 2D resource state! O h i Gi i K i = X i Z i+n Z i+e Z i+s Z i+w
97 Cluster state computation Raussendorf-Briegel PRL (2001); Raussendorf-Browne-Briegel PRA (2003). 2D cluster state projective measurement Z i +n Z i +w X Z i i +e Z i +s 2D resource state =! O h i Gi i K i = X i Z i+n Z i+e Z i+s Z i+w space U1 U2 U4 time U3 U5 h0 N U n U 1 0i N
98 Cluster state computation Raussendorf-Briegel PRL (2001); Raussendorf-Browne-Briegel PRA (2003). 2D cluster state projective measurement Z i +n Z i +w X Z i i +e Z i +s 2D resource state =! O h i Gi i K i = X i Z i+n Z i+e Z i+s Z i+w space U1 U2 U4!! time U3 U5 h0 N U n U 1 0i N
99 [Bennet et al., PRL 93] X X, Z Z Bell measurement output input Bell state! (maximally entangled state) i 00i + 11i p 2 = identity gate
100 [Bennet et al., PRL 93] X X, Z Z Bell measurement UX m 1 Z m 2 i output U input Bell state! (maximally entangled state) i 00i + 11i p 2 = U identity unitary gate
101 [Bennet et al., PRL 93] X X, Z Z Bell measurement UX m 1 Z m 2 i output U input Bell state! (maximally entangled state) i 00i + 11i p 2 = U identity unitary gate =
102 One-bit teleportation one-bit teleportation Zhou-Leung-Chuang, Phys. Rev. A 62, (2000). input + X X m H i output! Pauli byproduct
103 One-bit teleportation one-bit teleportation Zhou-Leung-Chuang, Phys. Rev. A 62, (2000). input + X X m H i output! Pauli byproduct input state X i X m H i H X m H i = X m circuit diagram
104 One-bit teleportation Z( ) =e i Z/2 + Z( ) X
105 One-bit teleportation + Z( ) =e i Z/2 Z( ) X Z( ) i = + X
106 One-bit teleportation + Z( ) =e i Z/2 Z( ) X Z( ) i = + X X m HZ( ) i
107 One-bit teleportation + Z( ) =e i Z/2 Z( ) X Z( ) i = + X X m HZ( ) i = input state = Z( ) H X m X m HZ( ) i X m HZ( ) i
108 One-bit teleportation + Z( ) =e i Z/2 Z( ) X Z( ) i = + X X m HZ( ) i = input state = Z( ) H X m X m HZ( ) i X m HZ( ) i 1D cluster state Hadamard gate, Z-rotation gate
109 Gate teleportation Gate teleportationd. Gottesman and I. L. Chuang, Nature (London) 402, 390 (1999). input input 2 X output 1 output 2 X
110 Gate teleportation Gate teleportationd. Gottesman and I. L. Chuang, Nature (London) 402, 390 (1999). input 1 + X output 1 input 1 circuit diagram H output 1 + output 2 = input 2 H output 2 input 2 X
111 Gate teleportation Gate teleportationd. Gottesman and I. L. Chuang, Nature (London) 402, 390 (1999). input 1 + X output 1 input 1 circuit diagram H output 1 + output 2 = input 2 H output 2 input 2 X = input 1 + X + input 2 X
112 Gate teleportation Gate teleportationd. Gottesman and I. L. Chuang, Nature (London) 402, 390 (1999). input 1 + X output 1 input 1 circuit diagram H output 1 + output 2 = input 2 H output 2 input 2 X = input X = X input 2 X X
113 Gate teleportation Gate teleportationd. Gottesman and I. L. Chuang, Nature (London) 402, 390 (1999). input 1 + X output 1 input 1 circuit diagram H output 1 + output 2 = input 2 H output 2 input 2 X = input input 2 X X = X X 2D cluster state! two-qubit gate
114 MBQC on 2D cluster state Z X Z Z Z Z X Z Z Z Z X Z Z Z Z X Z Z Z Z Z Z X Z Z Z Z Z Z Z X Z Z Z Z X X Z Z Z Z X Z Z X Z X Z Z Z Z X Z Z X Z X Z Z
115 MBQC on 2D cluster state
116 MBQC on 2D cluster state
117 MBQC on 2D cluster state
118 MBQC! ()
119 MBQC Bell pair! micro-cluster! ~divide and conquer ~! 1/2 () brute force MBQC! Measurement-based quantum computation! Nielsen Phys. Rev. Lett. 93, (2004) micro-cluster! Yoran-Reznik Phys. Rev. Lett. 91, (2003)! Browne-Rudolph Phys. Rev. Lett. 95, (2005) fusion gate! Duan-Raussendorf Phys. Rev. Lett. 95, (2005) cross-strategy! K. Kieling, T. Rudolph, and J. Eisert Phys. Rev. Lett. 99, (2007) percolation
120 MBQC snowflake! by Matsuzaki-Benjamin-Fitzsimons! Phys. Rev. Lett. 104, (2010) star-cluster! by KF & Tokunaga! Phys. Rev. Lett. 105, (2010)! () Li et al.! Phys. Rev. Lett. 105, (2010) ps = 0.1, pu = 10-4 ps = 0.1, pu = 10-4 fault-tolerant!
121 MBQC! ()
122 MBQC Valence-bond solid (VBS):! edge state () singlet = maximally entangled state Matrix product state (MPS): i = X single site measurement site edge state i 1,,i N hr A[i N ] A[i 1 ] Li i 1 i N i correlation space Gross-Eisert, PRL (2007)! Brennen-Miyake PRL (2008)! Miyake, Ann. Phys. (2011)! Wei et al., PRL (2011)! Li et al., PRL (2011)! KF-Morimae PRA (2012), KF et al., PRL (2013). Affleck-Kennedy-Lieb-Tasaki : H = J i Affleck-Kennedy-Lieb-Tasaki, Comm.! Math. Phys. 115, 477 (1988). [S i S i+1 1/3(S i S i+1 ) 2 ]
123 MBQC! ()
124 MBQC! c (t) 1 c 2 c (t0 ) 1 () c 0(t0 ) c 2 Raussendorf-Harrington-Goyal, Annals Phys. 321, 2242 (2006); NJP 9, 199 (2007).
125 MBQC! ()
126 MBQC! () Broadbent-Fitzsimons-Kashefi, FOCS 2009! Morimae-Fujii, Nature Comm. 2012; PRL 2013; PRA 2013
127 MBQC! ()
128 MBQC! () Matsuo-KF-Imoto, PRA (2014)
129 MBQC! ()
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1: Pauli 2 Heisenberg [3] 3 r 1, r 2 V (r 1, r 2 )=V (r 2, r 1 ) V (r 1, r 2 ) 5 ϕ(r 1, r 2 ) Schrödinger } { h2 2m ( 1 + 2 )+V (r 1, r 2 ) ϕ(r 1, r 2
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[I486S] 2018 5 1 (JAIST) 2018 5 1 1 / 22 : I486S I URL:https://wwwjaistacjp/~fujisaki/i486S (Tuesdays) 5 17:10 18:50 4/17, 4/24, 5/1, 5/15, 5/22, 5/29, 6/5, 6/19, 6/26, 7/3, 7/10, 7/17, 7/24, 7/31 (JAIST)
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量子スプレマシー セキュアクラウド 量子計算 量子計算の検証 森前智行 (京都大学基礎物理学研究所) 大阪市立大コロキウム 内容 量子スプレマシー セキュアクラウド量子計算 量子計算の検証 小柴 藤井 森前 コロナ社 森前 森北出版 量子スプレマシー ( 量子超越性 ) 弱い量子計算機が古典より強いことを強い基盤に基づいて証明する 量子計算の歴史は約 36 年 おもいついた 1982 年 36 年間の間に非常に多くの理論的
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量子暗号の安全性と量子系の基本的性質 小芦雅斗 阪大基礎工 Introduction Constraint on disturbance-free operations - Quantifying quantum information Monogamy of entanglement - Quantifying entanglement in unit of bits Unconditional
