Deutsch s Algorithm. Dimitrios I. Myrisiotis. MPLA CoReLab. April 26, / 70
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1 Deutsch s Algorithm Dimitrios I. Myrisiotis MPLA CoReLab April 6, / 70
2 Contents Introduction Preliminaries The Algorithm Misc. / 70
3 Introduction 3 / 70
4 Introduction Figure : David Deutsch (1953 ). 4 / 70
5 Introduction Deutsch s Problem. Input : A black-box U f for f : {0, 1} {0, 1}. Output : The value of XOR (f (0), f (1)) = f (1) f (0). 5 / 70
6 Introduction Deutsch s Problem. Input : A black-box U f for f : {0, 1} {0, 1}. Output : The value of XOR (f (0), f (1)) = f (1) f (0). Classically, we need two queries to U f. Quantumly, one query to U f suffices! 5 / 70
7 Preliminaries 6 / 70
8 Preliminaries }{{} 16 : {0, 1} Figure : Our 16-bit computer, with 16 configurations. 7 / 70
9 Preliminaries p 1 p... p 16 i { 1,,..., 16} : p i {0, 1} p i = 1!k : p k = 1 16 i=1 Figure : Communicating a configuration of a deterministic computer. 8 / 70
10 Preliminaries p 1 p... p 16 i { 1,,..., 16} : p i [0, 1] p i = 1 16 i=1 Figure : Communicating a configuration of a probabilistic computer. 9 / 70
11 Preliminaries c 1 c... c 16 i { 1,,..., 16} : c i C c i = 1 (*) 16 i=1 Figure : Communicating a configuration of a quantum computer. 10 / 70
12 Preliminaries Quantum States 11 / 70
13 Preliminaries: Quantum States B = {v i } 16 i=1 v 1 = v = v 16 = }{{} 16 1 / 70
14 Preliminaries: Quantum States B = {v i } 16 i=1 v 1 = } 0..{{. 000} = 16. v 16 = } 1..{{. 111} = / 70
15 Preliminaries: Quantum States B = {v i } 16 i=1 16 H = span (B, C) = c i v i i : c i C and v i B = C16 H = i=1 { } q C 16 q = 1 C / 70
16 Preliminaries: Quantum States H = { } q C 16 q = 1 = where 16-register quantum computers live C / 70
17 Preliminaries: Quantum States H = { q C n q = 1 } = where n-register quantum computers live C n 16 / 70
18 Preliminaries: Quantum States H = { q C n q = 1 } 16 n q = c i v i H q = 1 c i = 1 (*) i=1 i=1 17 / 70
19 Preliminaries: Quantum States ψ 18 / 70
20 Preliminaries: Quantum States ψ H C n 19 / 70
21 Preliminaries: Quantum States ψ = a ket = a column vector in H ψ = a bra = a row vector in H = the dual of ψ = ψ = ( ψ ) T = ( ψ ) T 0 / 70
22 Preliminaries: Quantum States B = {v i } 16 i=1 v 1 = v = v 16 = / 70
23 Preliminaries: Quantum States B = { v i } 16 i=1 v 1 = v = v 16 = / 70
24 Preliminaries: Quantum States B = { v i } 16 i=1 v 1 = = 1 v = =.. v 16 = = 16 3 / 70
25 Preliminaries: Quantum States I ( ψ, φ ) = inner product = ψ φ = ψ φ C O ( ψ, φ ) = outer product = ψ φ = ψ φ C n n 4 / 70
26 Preliminaries Unitary Evolution 5 / 70
27 Preliminaries: Unitary Evolution U q old = q new U 1 = U = (U ) T = (U ) T 6 / 70
28 Preliminaries: Unitary Evolution q initial H 7 / 70
29 Preliminaries: Unitary Evolution U 1 q initial H 8 / 70
30 Preliminaries: Unitary Evolution U U 1 q initial H 9 / 70
31 Preliminaries: Unitary Evolution m N U m U U 1 q initial H 30 / 70
32 Preliminaries: Unitary Evolution m N q final = U m U U 1 q initial H 31 / 70
33 Preliminaries: Unitary Evolution m N q final = U m U U 1 q initial H Figure : Our first quantum algorithm. 3 / 70
34 Preliminaries Measurements 33 / 70
35 Preliminaries: Measurements ψ = c i v i H n i=1 c i = 1 (*) n i=1 34 / 70
36 Preliminaries: Measurements ψ = c i v i Measurement j : ψ = v j n i=1 Pr [The outcome is j.] = c j 35 / 70
37 Preliminaries: Measurements ψ = c i v i Measurement j : ψ = v j n i=1 Measurement ψ = v j Pr [The outcome is j.] = 1 = 1 36 / 70
38 Preliminaries: Measurements 37 / 70
39 Preliminaries Composition 38 / 70
40 Preliminaries: Composition H 1 H H 1 H 39 / 70
41 Preliminaries: Composition H qubit H qubit H qubit H qubit 40 / 70
42 Preliminaries: Composition C C C C 41 / 70
43 Preliminaries: Composition C C C 4 / 70
44 Preliminaries: Composition C C C 4 43 / 70
45 Preliminaries: Composition C C C 4 = H two qubits 44 / 70
46 Preliminaries: Composition A useful property: (U 1 U ) ( x y ) = U }{{} 1 x U y. U 45 / 70
47 Preliminaries A Comparison 46 / 70
48 Preliminaries: A Comparison Table : Quantum mechanics and probability theory. Probability Theory Real numbers in [0, 1] Real numbers that sum to 1 Quantum Mechanics Complex numbers Complex numbers that the squares of their magnitudes sum to 1 The sum is equal to 1 The Euclidean norm is equal to 1 The sum is preserved The L 1 -norm is preserved Use of stochastic matrices The Euclidean norm is preserved The L -norm is preserved Use of unitary matrices 47 / 70
49 The Algorithm 48 / 70
50 The Algorithm Deutsch s Problem. Input : A black-box U f for f : {0, 1} {0, 1}. Output : The value of XOR (f (0), f (1)) = f (1) f (0). Classically, we need two queries to U f. Quantumly, one query to U f suffices! 49 / 70
51 The Algorithm two qubits qubit 1 H U f qubit I I H two qubits = qubit 1 qubit 50 / 70
52 The Algorithm 0 H U f I I H 51 / 70
53 The Algorithm 0 H U f I I H 5 / 70
54 The Algorithm 0 H U f I I ( 1 ψ 0 = ) 1 ψ 1 = U 1 ψ 0 = (H I ) ψ 0 ψ = U ψ 1 = U f ψ 1 ψ 3 = U 3 ψ = (H I ) ψ ψ 4 = the state after we measure the first qubit of ψ 3 H 53 / 70
55 The Algorithm 0 H U f I I H H 0 = U f x y = x y f (x) H 1 = H 1 = H I x = x 54 / 70
56 The Algorithm H U f H H 0 = U f x y = x y f (x) H 1 = H 1 = H I x = x 55 / 70
57 The Algorithm H U f H ( 1 ψ 0 = ) 1 ( 1 ψ 1 = (H I ) ψ 0 = = 1 ( ) ) ( ) 1 ψ = U f ψ 1 = 1 ( 0 0 f (0) 0 1 f (0) +... ) ( ) ( ) (0) f (1) 1 ( 1)f ( 1) f (0) (0) ( 1)f = / 70
58 The Algorithm H U f H ψ = ( ) ( ) (0) f (1) ( 1)f ( 1) f (0) (0) ( 1)f / 70
59 The Algorithm H U f H ( ) 1 ψ = 0 + ( 1)1 1 ( ) ( 1) f (0) (0) ( 1)f / 70
60 The Algorithm H U f H ψ = ( ) ( ) ( 1) f (0) (0) ( 1)f / 70
61 The Algorithm H U f H ψ = ( ) ( ) ( 1) f (0) (0) ( 1)f ψ 3 = (H I ) ψ = 1 ( ) ( 1) f (0) (0) ( 1)f / 70
62 The Algorithm H U f H ψ = ( ) ( ) ( 1) f (0) (0) ( 1)f ψ 3 = (H I ) ψ = 1 = f (0) f (1) ( ) ( 1) f (0) (0) ( 1)f 0 1 ( ) ( 1) f (0) (0) ( 1)f / 70
63 The Algorithm H U f H ψ = ( ) ( ) (0) f (1) ( 1)f ( 1) f (0) (0) ( 1)f / 70
64 The Algorithm H U f H ( ) 1 ψ = 0 + ( 1)0 1 ( ) ( 1) f (0) (0) ( 1)f / 70
65 The Algorithm H U f H ψ = ( ) ( ) ( 1) f (0) (0) ( 1)f / 70
66 The Algorithm H U f H ψ = ( ) ( ) ( 1) f (0) (0) ( 1)f ψ 3 = (H I ) ψ = 0 ( ) ( 1) f (0) (0) ( 1)f / 70
67 The Algorithm H U f H ψ = ( ) ( ) ( 1) f (0) (0) ( 1)f ψ 3 = (H I ) ψ = 0 = f (0) f (1) ( ) ( 1) f (0) (0) ( 1)f 0 1 ( ) ( 1) f (0) (0) ( 1)f / 70
68 The Algorithm :D 67 / 70
69 Misc. 68 / 70
70 Misc.: Where to start?/references An Introduction to Quantum Computing (009), by Kaye, Laflamme, and Mosca. Quantum Computing Since Democritus (013), by Aaronson. 69 / 70
71 Thank you! 70 / 70
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