Quantum Computing
Moore s Law
The Future Currently state of the art chips have gates of length 35 nanometers.
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.
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.
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?
A bit is 0 or 1. Bits
Bits A bit is 0 or 1. Current is flowing in a wire or not.
Bits A bit is 0 or 1. Current is flowing in a wire or not.
Bits A bit is 0 or 1. Current is flowing in a wire or not. n bits can be in states... 2 n possible
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
Distributions A probabilistic bit is described by a non-negative real numbers p 0,p 1 satisfying p 0 + p 1 =1.
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.
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
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
Qubits A qubit can be in a superposition of 0 and 1.
Qubits A qubit can be in a superposition of 0 and 1. Traditionally these states are represented in ket notation 0 and 1
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 0 2 + α 1 2 =1.
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 0 2 + α 1 2 =1. Important thing is that they can be negative!
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 0 2 + α 1 2 =1. 1 Important thing is that they can be negative! 0
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 0 2 + α 1 2 =1. 1 Important thing is that they can be negative! 0 1 2 0 + 1 2 1
Qubits An electron has a property called spin that can be up or down.
Qubits An electron has a property called spin that can be up or down. 1 spin up 1 spin down 2 2
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?
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...
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.
Measurements α 0 0 + α 1 1
Measurements α 0 0 + α 1 1 When we measure this qubit, we observe state 0 with probability α0. 2
Measurements α 0 0 + α 1 1 When we measure this qubit, we observe state 0 with probability α0. 2 Remember we require α 2 0 + α 2 1 =1.
Measurements α 0 0 + α 1 1 When we measure this qubit, we observe state 0 with probability α0. 2 Remember we require α 2 0 + α 2 1 =1. After the measurement, the state collapses to either or 0 1
Multiple Qubits Two qubits can be in superposition over the 4 possible classical states.
Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α 00 00 + α 01 01 + α 10 10 + α 11 11
Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α 00 00 + α 01 01 + α 10 10 + α 11 11 Again we require α 2 00 + α 2 01 + α 2 10 + α 2 11 =1
Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α 00 00 + α 01 01 + α 10 10 + α 11 11 Again we require α 2 00 + α 2 01 + α 2 10 + α 2 11 =1 Like in probabilistic case, n qubits will be described by vector. 2 n dimensional
Multiple Qubits Two qubits can be in superposition over the 4 possible classical states. α 00 α 00 00 + α 01 01 + α 10 10 + α 11 11 α 01 Again we require α 2 00 + α 2 01 + α 2 10 + α 2 11 =1 Like in probabilistic case, n qubits will be described by vector. 2 n dimensional α 10 α 11
Measurements Say we have a two qubit state. α 00 00 + α 01 01 + α 10 10 + α 11 11
Measurements Say we have a two qubit state. α 00 00 + α 01 01 + α 10 10 + α 11 11 We could measure both qubits...
Measurements Say we have a two qubit state. α 00 00 + α 01 01 + α 10 10 + α 11 11 We could measure both qubits... We could just measure the first qubit...
Measurements Say we have a two qubit state. α 00 00 + α 01 01 + α 10 10 + α 11 11 We could measure both qubits... We could just measure the first qubit... Then probability we see 0 is α 2 00 + α 2 01
Measurements Say we have a two qubit state. α 00 00 + α 01 01 + α 10 10 + α 11 11 We could measure both qubits... We could just measure the first qubit... Then probability we see 0 is α 2 00 + α 2 01 In this case, the state collapses to 1 α 2 00 + α11 2 (α 00 00 + α 01 01)
Spooky Action at a Distance
Spooky Action at a Distance
Spooky Action at a Distance
Spooky Action at a Distance 1 1 + 2 2
Spooky Action at a Distance
Spooky Action at a Distance
Spooky Action at a Distance
Spooky Action at a Distance
Spooky Action at a Distance 1 1 + 2 2
Spooky Action at a Distance 1 1 + 2 2 Alice measures her qubit. This will collapse the system either to 00 or 11 each with 50% probability.
Spooky Action at a Distance 1 1 + 2 2 Alice measures her qubit. This will collapse the system either to 00 or 11 each with 50% probability. This determines the state of Bob!
Entanglement What is special about the state Alice and Bob used? 1 1 + 2 2
Entanglement What is special about the state Alice and Bob used? 1 1 + 2 2 Product states can be written in the form α 0 β 0 00 + α 0 β 1 01 + α 1 β 0 10 + α 1 β 1 11
Entanglement What is special about the state Alice and Bob used? 1 1 + 2 2 Product states can be written in the form α 0 β 0 00 + α 0 β 1 01 + α 1 β 0 10 + α 1 β 1 11 These give statistics just like independent coin flips.
A simple game Remember the setting of communication complexity? 0 1 0 0 0 1 0 1
A simple game This is a game of no communication.
A simple game This is a game of no communication. s
A simple game This is a game of no communication. s t
A simple game This is a game of no communication. a s t
A simple game This is a game of no communication. a b s t
A simple game This is a game of no communication. a, b, s, t {0, 1} a b s t
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
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?
XOR game 0 1 0 0 0 1 0 1 Goal
XOR game Strategy: 0 1 Always answer 0! 0 0 0 1 0 1 Goal
XOR game Strategy: 0 1 Always answer 0! 0 0 0 1 0 1 Goal
XOR game Strategy: 0 1 Always answer 0! 0 0 0 0 0 1 0 1 0 0 Goal
XOR game Strategy: 0 1 Always answer 0! 0 0 0 0 0 1 0 1 0 0 Goal Win with prob. 3/4
No strategy is perfect... 0 0 0 1 0 1 Goal
No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 0 0 0 1 0 1 Goal
No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 0 0 0 b 0 + b 1 =0 1 0 1 Goal
No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 a 1 + b 0 =0 a 1 + b 1 =1 0 0 0 b 0 + b 1 =0 1 0 1 Goal
No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 a 1 + b 0 =0 a 1 + b 1 =1 0 0 0 b 0 + b 1 =0 b 0 + b 1 =1 1 0 1 Goal
No strategy is perfect... a 0 + b 0 =0 a 0 + b 1 =0 a 1 + b 0 =0 a 1 + b 1 =1 0 0 0 b 0 + b 1 =0 b 0 + b 1 =1 1 0 1 Maximum winning prob. is 3/4 Goal
Shared Randomness What if Alice and Bob share randomness? 011 010 001 110 a b s t
Shared Randomness Even sharing randomness they cannot win with prob. greater than 3/4. This is known as a Bell Inequality. 011 010 001 110 a b s t
Shared Entanglement a b s t
Shared Entanglement 1 1 + 2 2 a b s t
Shared Entanglement Using the state below, Alice and Bob can win this game with probability 1 2 + 1 2 2 0.85 1 1 + 2 2 a b s t
Computation: Operations In the classical case we do operations like AND, OR, and NOT.
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.
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. 1 1 0 0
Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice?
Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice? Of course, this sends 0 to 1
Interference What happens when we apply the operation rotate counterclockwise by 45 degrees twice? Of course, this sends 0 to 1 Viewed another way,
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 1 2 0 1 2 1 1 2 0 1 2 1 1 2 1 1 2 0
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 1 2 0 1 2 1 cancel out! 1 2 0 1 2 1 1 2 1 1 2 0
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 1 2 0 1 2 1 cancel out! 1 2 0 1 2 1 1 2 1 1 2 0
Double-Slit Experiment
Double-Slit Experiment
Double-Slit Experiment
Double-Slit Experiment Interference pattern persists even when intensity is such that only one photon passes through slits at a time!
Double Slit Experiment
Big Picture input: classical n bit string Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string good answers operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Big Picture input: classical n bit string measure: n bit string operations on 2 n dimensional vectors Picture modeled after Algorithms, Dasgupta et al.
Algorithmic Highlights Shor s factoring algorithm 1994: polynomial time algorithm to factor integers.
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.
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.
Practical Highlights
Practical Highlights With high probability...
Practical Highlights With high probability... 15=5*3
Quantum and NP-hard problems
Quantum and NP-hard problems The search algorithm of Grover gives a way to solve 3-SAT in time 2 n/2.
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.
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
NP and physics The study of quantum computing opens up a wider question...
NP and physics The study of quantum computing opens up a wider question... Why stop there? What do other physical theories say about computation?
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?
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?
Acme Private Line Salt Lake City New York Houston
Acme Private Line Salt Lake City New York Cost of line = MST Houston
Acme Private Line Salt Lake City New York Cost of line = MST Houston
Acme Private Line Salt Lake City New York Houston
Acme Private Line Salt Lake City Open office in Dallas. New York Houston
Acme Private Line Salt Lake City Open office in Dallas. New York Dallas Houston
Acme Private Line Salt Lake City New York Dallas Houston
Acme Private Line Salt Lake City New York Dallas Houston
Acme Private Line Salt Lake City New York Dallas Cost of line Houston decreases!
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?
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.
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!
Soap Bubble Computation
Soap Bubble Computation But soap bubbles naturally minimize surface area...
Soap Bubble Computation But soap bubbles naturally minimize surface area... Can we use soap bubbles to compute minimum Steiner trees?
Soap Bubble Computation But soap bubbles naturally minimize surface area... Can we use soap bubbles to compute minimum Steiner trees? Try It!
Soap Bubble Computation
Soap Bubble Computation Please wait while the computer boots up...
Soap Bubble Computation Please wait while the computer boots up... This may take a minute...
First Problem How about corners of a rectangle?
First Problem How about corners of a rectangle?
Now for something harder...
Basic Steiner Questions
Basic Steiner Questions How many Steiner points might you have to add?
Basic Steiner Questions How many Steiner points might you have to add? Never more than n-2.
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?
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.
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.
Now for something harder...
Now for something harder...
Relativity Computing
Relativity Computing If you can t speed up the computation, speed up yourself!
Relativity Computing If you can t speed up the computation, speed up yourself!
Relativity Computing If you can t speed up the computation, speed up yourself!
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
Time Travel Computing Say that time travel is possible...can we use it to do computation?
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...
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?
Grandfather Paradox Have you heard of the grandfather paradox...
Grandfather Paradox Have you heard of the grandfather paradox... How can this be resolved with consistency?
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.
Deutsch s Model Nature chooses a probability distribution on events so that your behavior will leave this probability distribution invariant.
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?
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!
Intractability as physical law
Intractability as physical law Scott Aaronson has suggested taking the intractability of NP-hard problems as a physical law.
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.
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.
Questions?
Questions? Thanks! You have been a great class!