The Future. Currently state of the art chips have gates of length 35 nanometers.

Transcription

1 Quantum Computing

2 Moore s Law

3 The Future Currently state of the art chips have gates of length 35 nanometers.

4 The Future Currently state of the art chips have gates of length 35 nanometers. When gate lengths reach 5 nanometers there is a non-negligible probability of quantum tunneling --- an electron can pass through the gate even when it is closed.

5 The Future Currently state of the art chips have gates of length 35 nanometers. When gate lengths reach 5 nanometers there is a non-negligible probability of quantum tunneling --- an electron can pass through the gate even when it is closed.

6 The Future Currently state of the art chips have gates of length 35 nanometers. When gate lengths reach 5 nanometers there is a non-negligible probability of quantum tunneling --- an electron can pass through the gate even when it is closed. Can quantum effects be harnessed for our computational advantage?

7 A bit is 0 or 1. Bits

8 Bits A bit is 0 or 1. Current is flowing in a wire or not.

9 Bits A bit is 0 or 1. Current is flowing in a wire or not.

10 Bits A bit is 0 or 1. Current is flowing in a wire or not. n bits can be in states... 2 n possible

11 Bits A bit is 0 or 1. Current is flowing in a wire or not. n bits can be in states... 2 n possible...and can be described with n bits

12 Distributions A probabilistic bit is described by a non-negative real numbers p 0,p 1 satisfying p 0 + p 1 =1.

13 Distributions A probabilistic bit is described by a non-negative real numbers p 0,p 1 satisfying p 0 + p 1 =1. A coin flip where probability of heads is p 0.

14 Distributions A probabilistic bit is described by a non-negative real numbers satisfying p 0 + p 1 =1. p 0,p 1 A coin flip where probability of heads is p 0. A distribution on n-bits is described by a 2 n dimensional vector, giving the probability of each of the outcomes. 2 n possible

15 Distributions A probabilistic bit is described by a non-negative real numbers satisfying p 0 + p 1 =1. p 0,p 1 p 000 A coin flip where probability of heads is p 0. p 001 A distribution on n-bits is described by a 2 n dimensional vector, giving the probability of each of the 2 n possible outcomes. p 111

16 Qubits A qubit can be in a superposition of 0 and 1.

17 Qubits A qubit can be in a superposition of 0 and 1. Traditionally these states are represented in ket notation 0 and 1

18 Qubits A qubit can be in a superposition of 0 and 1. Traditionally these states are represented in ket notation 0 and 1 It is described by complex numbers α 0, α 1 satisfying α α 1 2 =1.

19 Qubits A qubit can be in a superposition of 0 and 1. It is described by complex Traditionally these states are represented in ket notation 0 and 1 That they are complex numbers α 0, α 1 satisfying is not so important... α α 1 2 =1. Important thing is that they can be negative!

20 Qubits A qubit can be in a superposition of 0 and 1. It is described by complex Traditionally these states are represented in ket notation 0 and 1 That they are complex numbers α 0, α 1 satisfying is not so important... α α 1 2 =1. 1 Important thing is that they can be negative! 0

21 Qubits A qubit can be in a superposition of 0 and 1. It is described by complex Traditionally these states are represented in ket notation 0 and 1 That they are complex numbers α 0, α 1 satisfying is not so important... α α 1 2 =1. 1 Important thing is that they can be negative!

22 Qubits An electron has a property called spin that can be up or down.

23 Qubits An electron has a property called spin that can be up or down. 1 spin up 1 spin down 2 2

24 Qubits An electron has a property called spin that can be up or down. 1 spin up 1 spin down 2 2 What does it mean to have an amplitude of 1 2?

25 Qubits An electron has a property called spin that can be up or down. 1 spin up 1 spin down 2 2 What does it mean to have an amplitude of 1 2? We have no access to the amplitudes of the state...

26 Qubits An electron has a property called spin that can be up or down. 1 spin up 1 spin down 2 2 What does it mean to have an amplitude of 1 2? We have no access to the amplitudes of the state... Our view is filtered through measurements.

27 Measurements α α 1 1

28 Measurements α α 1 1 When we measure this qubit, we observe state 0 with probability α0. 2

29 Measurements α α 1 1 When we measure this qubit, we observe state 0 with probability α0. 2 Remember we require α α 2 1 =1.

30 Measurements α α 1 1 When we measure this qubit, we observe state 0 with probability α0. 2 Remember we require α α 2 1 =1. After the measurement, the state collapses to either or 0 1

31 Multiple Qubits Two qubits can be in superposition over the 4 possible classical states.

32 Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α α α α 11 11

33 Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α α α α Again we require α α α α 2 11 =1

34 Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α α α α Again we require α α α α 2 11 =1 Like in probabilistic case, n qubits will be described by vector. 2 n dimensional

35 Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α 00 α α α α α 01 Again we require α α α α 2 11 =1 Like in probabilistic case, n qubits will be described by vector. 2 n dimensional α 10 α 11

36 Measurements Say we have a two qubit state. α α α α 11 11

37 Measurements Say we have a two qubit state. α α α α We could measure both qubits...

38 Measurements Say we have a two qubit state. α α α α We could measure both qubits... We could just measure the first qubit...

39 Measurements Say we have a two qubit state. α α α α We could measure both qubits... We could just measure the first qubit... Then probability we see 0 is α α 2 01

40 Measurements Say we have a two qubit state. α α α α We could measure both qubits... We could just measure the first qubit... Then probability we see 0 is α α 2 01 In this case, the state collapses to 1 α α11 2 (α α 01 01)

41 Spooky Action at a Distance

42 Spooky Action at a Distance

43 Spooky Action at a Distance

44 Spooky Action at a Distance

45 Spooky Action at a Distance

46 Spooky Action at a Distance

47 Spooky Action at a Distance

48 Spooky Action at a Distance

49 Spooky Action at a Distance

50 Spooky Action at a Distance Alice measures her qubit. This will collapse the system either to 00 or 11 each with 50% probability.

51 Spooky Action at a Distance Alice measures her qubit. This will collapse the system either to 00 or 11 each with 50% probability. This determines the state of Bob!

52 Entanglement What is special about the state Alice and Bob used?

53 Entanglement What is special about the state Alice and Bob used? Product states can be written in the form α 0 β α 0 β α 1 β α 1 β 1 11

54 Entanglement What is special about the state Alice and Bob used? Product states can be written in the form α 0 β α 0 β α 1 β α 1 β 1 11 These give statistics just like independent coin flips.

55 A simple game Remember the setting of communication complexity?

56 A simple game This is a game of no communication.

57 A simple game This is a game of no communication. s

58 A simple game This is a game of no communication. s t

59 A simple game This is a game of no communication. a s t

60 A simple game This is a game of no communication. a b s t

61 A simple game This is a game of no communication. a, b, s, t {0, 1} a b s t

62 A simple game This is a game of no communication. a, b, s, t {0, 1} a b s t Goal: a b = s t

63 A simple game This is a game of no communication. a, b, s, t {0, 1} a b s t Goal: a b = s t With what probability can Alice and Bob win when s,t chosen randomly?

64 XOR game Goal

65 XOR game Strategy: 0 1 Always answer 0! Goal

66 XOR game Strategy: 0 1 Always answer 0! Goal

67 XOR game Strategy: 0 1 Always answer 0! Goal

68 XOR game Strategy: 0 1 Always answer 0! Goal Win with prob. 3/4

69 No strategy is perfect Goal

70 No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 = Goal

71 No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 = b 0 + b 1 = Goal

72 No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 a 1 + b 0 =0 a 1 + b 1 = b 0 + b 1 = Goal

73 No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 a 1 + b 0 =0 a 1 + b 1 = b 0 + b 1 =0 b 0 + b 1 = Goal

74 No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 a 1 + b 0 =0 a 1 + b 1 = b 0 + b 1 =0 b 0 + b 1 = Maximum winning prob. is 3/4 Goal

75 Shared Randomness What if Alice and Bob share randomness? a b s t

76 Shared Randomness Even sharing randomness they cannot win with prob. greater than 3/4. This is known as a Bell Inequality a b s t

77 Shared Entanglement a b s t

78 Shared Entanglement a b s t

79 Shared Entanglement Using the state below, Alice and Bob can win this game with probability a b s t

80 Computation: Operations In the classical case we do operations like AND, OR, and NOT.

81 Computation: Operations In the classical case we do operations like AND, OR, and NOT. In the quantum case, we similarly build up computation from operations on a few qubits. The operations we can do are rotations and reflections.

82 Computation: Operations In the classical case we do operations like AND, OR, and NOT. In the quantum case, we similarly build up computation from operations on a few qubits. The operations we can do are rotations and reflections

83 Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice?

84 Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice? Of course, this sends 0 to 1

85 Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice? Of course, this sends 0 to 1 Viewed another way,

86 Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice? Of course, this sends 0 to 1 Viewed another way,

87 Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice? Of course, this sends 0 to 1 Viewed another way, 0 The 0 paths cancel out!

88 Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice? Of course, this sends 0 to 1 Viewed another way, 0 Quantum mechanics is linear! The 0 paths cancel out!

89 Double-Slit Experiment

90 Double-Slit Experiment

91 Double-Slit Experiment

92 Double-Slit Experiment Interference pattern persists even when intensity is such that only one photon passes through slits at a time!

93 Double Slit Experiment

94 Big Picture input: classical n bit string Picture modeled after Algorithms, Dasgupta et al.

95 Big Picture input: classical n bit string Picture modeled after Algorithms, Dasgupta et al.

96 Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

97 Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

98 Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

99 Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

100 Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

101 Big Picture input: classical n bit string good answers operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

102 Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

103 Big Picture input: classical n bit string measure: n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.

104 Algorithmic Highlights Shor s factoring algorithm 1994: polynomial time algorithm to factor integers.

105 Algorithmic Highlights Shor s factoring algorithm 1994: polynomial time algorithm to factor integers. Grover s search algorithm 1996: desired item can be found in unstructured database of N items with N probes.

106 Algorithmic Highlights Shor s factoring algorithm 1994: polynomial time algorithm to factor integers. Grover s search algorithm 1996: desired item can be found in unstructured database of N items with N probes. Fault tolerance: Even if gates fail with some small probability, reliable computation still possible.

107 Practical Highlights

108 Practical Highlights With high probability...

109 Practical Highlights With high probability... 15=5*3

110 Quantum and NP-hard problems

111 Quantum and NP-hard problems The search algorithm of Grover gives a way to solve 3-SAT in time 2 n/2.

112 Quantum and NP-hard problems The search algorithm of Grover gives a way to solve 3-SAT in time 2 n/2. Any faster algorithm will have to exploit specific details about the structure of 3-SAT solutions.

113 Quantum and NP-hard problems The search algorithm of Grover gives a way to solve 3-SAT in time 2 n/2. Any faster algorithm will have to exploit specific details about the structure of 3-SAT solutions. In the black box---or query---model we know that searching a list of N items requires quantum queries. N

114 NP and physics The study of quantum computing opens up a wider question...

115 NP and physics The study of quantum computing opens up a wider question... Why stop there? What do other physical theories say about computation?

116 NP and physics The study of quantum computing opens up a wider question... Why stop there? What do other physical theories say about computation? Do the laws of physics allow solution of NP-hard problems efficiently?

117 NP and physics The study of quantum computing opens up a wider question... Why stop there? What do other physical theories say about computation? Do the laws of physics allow solution of NP-hard problems efficiently? Can nature solve NP-hard problems?

118 Acme Private Line Salt Lake City New York Houston

119 Acme Private Line Salt Lake City New York Cost of line = MST Houston

120 Acme Private Line Salt Lake City New York Cost of line = MST Houston

121 Acme Private Line Salt Lake City New York Houston

122 Acme Private Line Salt Lake City Open office in Dallas. New York Houston

123 Acme Private Line Salt Lake City Open office in Dallas. New York Dallas Houston

124 Acme Private Line Salt Lake City New York Dallas Houston

125 Acme Private Line Salt Lake City New York Dallas Houston

126 Acme Private Line Salt Lake City New York Dallas Cost of line Houston decreases!

127 Steiner Tree Problem Telephone company faces a new problem: given a collection of cities, what is length of shortest network connecting them, possibly adding new nodes?

128 Steiner Tree Problem Telephone company faces a new problem: given a collection of cities, what is length of shortest network connecting them, possibly adding new nodes? The new nodes added are known as Steiner points.

129 Steiner Tree Problem Telephone company faces a new problem: given a collection of cities, what is length of shortest network connecting them, possibly adding new nodes? The new nodes added are known as Steiner points. This problem was studied at Bell labs, and shown to be NP-hard!

130 Soap Bubble Computation

131 Soap Bubble Computation But soap bubbles naturally minimize surface area...

132 Soap Bubble Computation But soap bubbles naturally minimize surface area... Can we use soap bubbles to compute minimum Steiner trees?

133 Soap Bubble Computation But soap bubbles naturally minimize surface area... Can we use soap bubbles to compute minimum Steiner trees? Try It!

134 Soap Bubble Computation

135 Soap Bubble Computation Please wait while the computer boots up...

136 Soap Bubble Computation Please wait while the computer boots up... This may take a minute...

137 First Problem How about corners of a rectangle?

138 First Problem How about corners of a rectangle?

139 Now for something harder...

140 Basic Steiner Questions

141 Basic Steiner Questions How many Steiner points might you have to add?

142 Basic Steiner Questions How many Steiner points might you have to add? Never more than n-2.

143 Basic Steiner Questions How many Steiner points might you have to add? Never more than n-2. What is the maximum degree of a Steiner node?

144 Basic Steiner Questions How many Steiner points might you have to add? Never more than n-2. What is the maximum degree of a Steiner node? Degree of Steiner nodes will be 3.

145 Basic Steiner Questions How many Steiner points might you have to add? Never more than n-2. What is the maximum degree of a Steiner node? Degree of Steiner nodes will be 3. Outgoing edges form 120 degree angles.

146 Now for something harder...

147 Now for something harder...

148 Relativity Computing

149 Relativity Computing If you can t speed up the computation, speed up yourself!

150 Relativity Computing If you can t speed up the computation, speed up yourself!

151 Relativity Computing If you can t speed up the computation, speed up yourself!

152 Relativity Computing If you can t speed up the computation, speed up yourself! Time on the ship will run slower by a factor 1 v 2 /c 2

153 Time Travel Computing Say that time travel is possible...can we use it to do computation?

154 Time Travel Computing Say that time travel is possible...can we use it to do computation? Knowledge creation paradox: travelers from the future can reveal technologies to people in the past...

155 Time Travel Computing Say that time travel is possible...can we use it to do computation? Knowledge creation paradox: travelers from the future can reveal technologies to people in the past... When was the work of creation done?

156 Grandfather Paradox Have you heard of the grandfather paradox...

157 Grandfather Paradox Have you heard of the grandfather paradox... How can this be resolved with consistency?

158 Grandfather Paradox Have you heard of the grandfather paradox... How can this be resolved with consistency? Elegant solution of David Deutsch: you are born with probability 1/2. If you are born, you go back and time and kill your grandfather.

159 Deutsch s Model Nature chooses a probability distribution on events so that your behavior will leave this probability distribution invariant.

160 Deutsch s Model Nature chooses a probability distribution on events so that your behavior will leave this probability distribution invariant. But finding this distribution might be quite hard...can we use this for computation?

161 Deutsch s Model Nature chooses a probability distribution on events so that your behavior will leave this probability distribution invariant. But finding this distribution might be quite hard...can we use this for computation? Yes! Computational models have been built on this and one can solve NP-hard problems and beyond!

162 Intractability as physical law

163 Intractability as physical law Scott Aaronson has suggested taking the intractability of NP-hard problems as a physical law.

164 Intractability as physical law Scott Aaronson has suggested taking the intractability of NP-hard problems as a physical law. As we have seen, such a law has implications for physics---for example that time travel is not possible.

165 Intractability as physical law Scott Aaronson has suggested taking the intractability of NP-hard problems as a physical law. As we have seen, such a law has implications for physics---for example that time travel is not possible. For much more, see Scott s survey NP-complete problems and physical reality.

166 Questions?

167 Questions? Thanks! You have been a great class!