Introduction to Quantum Computing

This talk is aimed at software engineers and computer scientists with no knowledge of quantum mechanics. Check out the code at https://github.com/ericmarcos/pyqu

Transcript

1. A (light) introduction to
 Quantum Computing Eric Marcos barcelonaqbit.com
2. What QC is?
3. Quantum cryptography Quantum simulation Quantum computing
4. Quantum cryptography Quantum simulation Quantum computing
5. What QC is NOT?
6. “QC can solve classically unsolvable problems”
7. “QC is exponentially faster than classical computation”
9. A little bit of history
10. 1900 Max Plank E = ħv 1905 Albert Einstein Annus mirabilis 1924 Louis de Broglie λ = ħ / p 1927 Werner Heisenberg σxσp ≥ ħ / 2 1928 Paul Dirac (i - ∂ - m) ψ = 0 Quantum Mechanics
11. 1931 Kurt Gödel Incompleteness Theorems 1936 Alan Turing Turing Machines 1952 Stephen Kleene Church-Turing Thesis 1956 Noam Chomsky Chomsky Hierarchy 1971 Stephen Cook Leonid Levin NP-Completeness Theory of Computation
12. 1982 Yuri Manin Richard Feynmann Quantum Computing ideas… 1985 David Deutsch Quantum Turing Machines 1992 Deutsch-Josza Algorithm 1994 Peter Shor Shor’s Algorithm 1996 Lov Grover Grover’s Algorithm Quantum Computing 2001 IBM Stanford First physical implementation of Shor’s Algorithm
13. So… What QC is?
14. First… some thoughts on classical computing
15. What is a BIT?
16. BIT x ∈ { false , true } OPERATIONS NOT(x) = ¬x AND(x, y) = x ∧ y TRUTH VALUES
17. BIT x ∈ { 0 , 1 } OPERATIONS NOT(x) = 1 - x AND(x, y) = x · y INTEGER VALUES
18. VECTOR VALUES (0,1) (1,0)
19. ( ) VECTOR VALUES (0,1) = True = |1> (1,0) = False = |0> x ∈ { , } 1 0 ( ) 0 1
20. ( ) VECTOR VALUES x ∈ { , } 1 0 ( ) 0 1 OPERATIONS ?
21. ( ) NOT( ) = 1 0 ( ) 0 1 NOT(x ̅) = x ̅ ( ) 0 1 1 0 VECTOR VALUES
22. In CQ bits are just vectors and operations are just matrices
23. In CQ bits are just unit vectors and operations are just unitary matrices
24. “I am a Quantum Engineer, but on Sundays I Have Principles.” John Stewart Bell
25. “Oh oh, mamma mia, here I go again, my my, how can I resist you” Benny Andersson (ABBA)
26. AND(x ̅, y ̅) = ?? VECTOR VALUES OPERATIONS
27. AND(x ̅, y ̅) = ?? We need to combine x ̅ and y ̅ into a joint state VECTOR VALUES OPERATIONS
28. COMBINING VECTOR BITS x ̅ ⊗ y ̅ = z ̅ In CQ the joint state of 2 vector bits is its tensor product
29. COMBINING VECTOR BITS w ̅ ⊗ x ̅ ⊗ y ̅ = z ̅ In CQ the joint state of N vector bits is its tensor product
30. COMBINING VECTOR BITS x ̅ ⊗ y ̅ = ( ) x0 · y0 x0 · y1 x1 · y0 x1 · y1
31. COMBINING VECTOR BITS |0> ⊗ |0>
32. COMBINING VECTOR BITS |0>|0>
33. COMBINING VECTOR BITS |00>
34. COMBINING VECTOR BITS ( ) 1 0 ( ) 1 0 ⊗
35. COMBINING VECTOR BITS ( ) 1 0 0 0 |00> =
36. COMBINING VECTOR BITS ( ) 0 1 0 0 |01> =
37. COMBINING VECTOR BITS ( ) 0 0 1 0 |10> =
38. COMBINING VECTOR BITS ( ) 0 0 0 1 |11> =
39. In CQ a register of N bits is represented by a vector space of 2N dimensions
40. We can store 2N bits of information with just N qbits!
41. AND(x ̅, y ̅) VECTOR VALUES OPERATIONS
42. AND(x ̅ ⊗ y ̅) VECTOR VALUES OPERATIONS
43. VECTOR VALUES OPERATIONS quantum gates AND x ̅ y ̅ x ̅ (output) y ̅ (trash)
44. VECTOR VALUES OPERATIONS quantum circuits AND x ̅ y ̅ w ̅ OR
45. VECTOR VALUES OPERATIONS quantum circuits AND x ̅ y ̅ w ̅ OR I I
46. VECTOR VALUES OPERATIONS quantum circuits AND x ̅ y ̅ w ̅ OR I I I ⊗ AND = Uand OR ⊗ I = Uor
47. (OR ⊗ I) × (I ⊗ AND) × (w ̅ ⊗ x ̅ ⊗ y ̅)
48. Uor × Uand × z ̅0 = z ̅1
49. Reversible computation
51. (reminder) In CQ bits are just unit vectors and operations are just unitary matrices
52. UNIT VECTOR ||v|| = <v,v> = √( ∑ vi 2 ) = 1
53. ( ) 1 0 0 0 |00> =
54. ( ) 0.5 0.5 0.5 0.5 |??> =
55. ( ) 1 0 0 0 0.5 × ( ) 0 1 0 0 + 0.5 × ( ) 0 0 1 0 + 0.5 × ( ) 0 0 0 1 + 0.5 ×
56. 0.5 × |00> + 0.5 × |01> + 0.5 × |10> + 0.5 × |11>
57. |Ψ> = |0> + |1> = THE QUBIT , ∈ ℂ ( )
58. |Ψ> = |0> + |1> = THE QUBIT , ∈ ℂ ( ) Quantum superposition
60. Operating on a qubit in a superposition state means operating on all the values at once THE GOOD NEWS
61. THE GOOD NEWS ( ) 0.5 0.5 0.5 0.5 Uf × This is like applying Uf to |00>, |01>, |10> and |11> in just one computation step!
62. Quantum parallelism THE GOOD NEWS
63. Superposition only works if the quantum system is isolated THE BAD NEWS
64. THE BAD NEWS ( ) 0.5 0.5 0.5 0.5 Uf × = ( ) ?? ?? ?? ??
65. Reading the qubit destroys superposition and makes it collapse to one of the base states randomly THE BAD NEWS
66. THE BAD NEWS ( ) 0.5 0.5 0.5 0.5 Uf × = |Ψ> read(|Ψ>) p(|00>) = ?? p(|01>) = ?? p(|10>) = ?? p(|11>) = ?? p = ||||2
68. How can we build a superposition state?
69. HADAMARD GATE ( ) 1 1 1 -1 H = 1/√2
70. HADAMARD GATE H |0> = ( ) 1/√2 1/√2 |0> H |0> + |1> √2 H |0> H |1> = ( ) 1/√2 -1/√2
71. CNOT GATE ( ) 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 CNOT =
72. CNOT GATE |0> |0> H |0> + |1> √2 CNOT |00> + |11> √2
73. Quantum entanglement
75. COMPUTING FUNCTIONS :|> → |()> Generally not possible because of reversible computation : {0,1} → {0,1}
76. COMPUTING FUNCTIONS :|>|0> → |x>|()> : {0,1} → {0,1}
77. COMPUTING FUNCTIONS |0> |0> H |0> + |1> √2 U |(0)> + |(1)> √2
78. COMPUTING FUNCTIONS |0> H |0> + |1> √2 U |0> - |1> √2 |Ψ> = 1/2[(-1)f(0)+(-1)f(1)]|0> + 1/2[(-1)f(0)-(-1)f(1)]|1> H |Ψ>
79. The Deutsch algorithm : {0,1} → {0,1} Given , tell me whether is constant (all outputs are the same for whatever input) or balanced (same number of 1s and 0s)
80. Uses the phase kick-back technique and quantum interference to determine whether a function is constant or balanced in just one call The Deutsch algorithm
81. The Deutsch-Josza algorithm : {0,1}n → {0,1} Given , tell me whether is constant (all outputs are the same for whatever input) or balanced (same number of 1s and 0s)
82. “I think I can safely say that nobody understands quantum mechanics.” Richard Feynman
83. @ericmarcosp ericmarcos.p@gmail.com github.com/ericmarcos/pyqu