Great Ideas in Theoretical Computer Science. Lecture 28: A Gentle Introduction to Quantum Computation

Transcription

1 5-25 Great Ideas in Theoretical Computer Science Lecture 28: A Gentle Introduction to Quantum Computation May st, 208

2 Announcements Please fill out the Faculty Course Evaluations (FCEs).

3 Announcements You can vote to eliminte 2 topics from the final exam: Stable Matchings Boolean Circuits NP and Logic (Descriptive Complexity) Transducers Presburger Arithmetic

4 Announcements The Last Lecture on Thursday Daniel Sleator Mor Harchol-Balter Ariel Procaccia Rashmi Vinayak Anupam Gupta Ryan O Donnell

5 Announcements The Last Lecture on Thursday

6 Quantum Computation

7 The plan Classical computers and classical theory of computation Quantum physics (what the fuss is all about) Quantum computation (practical, scientific, and philosophical perspectives)

8 The plan Classical computers and classical theory of computation

9 What is computer/computation? A device that manipulates data (information) Usually Input Device Output

10 Theory of computation Mathematical model of a computer: Turing Machines ~ Boolean (uniform) Circuits

11 Theory of computation Turing Machines

12 Theory of computation Boolean Circuits AND NOT OR AND OR AND OR AND OR OR 0 gates AND OR NOT

13 Theory of computation Boolean Circuits INPUT n bits AND NOT OR AND OR AND OR AND OR OR OUTPUT 0 bit n bits Circuit bit (or m bits)

14 Physical Realization Circuits implement basic operations / instructions. Everything follows classical laws of physics!

15 (Physical) Church-Turing Thesis Turing Machines ~ (uniform) Boolean Circuits universally capture all of computation. Python Java, C, TMs All types of computation

16 (Physical) Church-Turing Thesis Turing Machines ~ (uniform) Boolean Circuits universally capture all of computation. (Physical) Church Turing Thesis Any computational problem that can be solved by a physical device, can be solved by a Turing Machine. Strong version Any computational problem that can be solved efficiently by a physical device, can be solved efficiently by a TM.

17 The plan Classical computers and classical theory of computation Quantum physics (what the fuss is all about) Quantum computation (practical, scientific, and philosophical perspectives)

18 The plan Quantum physics (what the fuss is all about)

19 One slide course on physics Classical Physics General Theory of Relativity Quantum Physics

20 One slide course on physics Classical Physics General Theory of Relativity String Theory (?) Quantum Physics

21 Video: Double slit experiment Nature has no obligation to conform to your intuitions.

22 Video: Double slit experiment

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24 2 interesting aspects of quantum physics. Having multiple states simultaneously e.g.: electrons can have states spin up or spin down : upi or downi In reality, they can be in a superposition of two states. 2. Measurement Quantum property is very sensitive/fragile! If you measure it (interfere with it), it collapses. So you either see upi or downi.

25 It must be just our ignorance - There is no such thing as superposition. - We don t know the state, so we say it is in a superposition. - In reality, it is always in one of the two states. - This is why when we measure/observe the state, we find it in one state. God does not play dice with the world. - Albert Einstein Einstein, don t tell God what to do. - Niels Bohr

26 How should we fix our intuitions to put it in line with experimental results?

27 Removing physics from quantum physics mathematics underlying quantum physics = generalization/extension of probability theory (allow negative probabilities )

28 Probabilistic states and evolution vs Quantum states and evolution

29 Probabilistic states Suppose an object can have n possible states: i, 2i,, ni At each time step, the state can change probabilistically. What happens if we start at state i and evolve? Initial state: i 2i 3i ni i 2i ni i

30 Probabilistic states Suppose an object can have n possible states: i, 2i,, ni At each time step, the state can change probabilistically. What happens if we start at state i and evolve? After one time step: i 2i 3i ni Transition 7 Matrix 5 = / /2 2 i 2i ni i

31 Probabilistic states i 2i 3i ni Transition 7 Matrix 5 = / /2 the new state (probabilistic) A general probabilistic state: 2 3 p p 2 p i = the probability of being in state i p + p p n = p n (` norm is )

32 Probabilistic states A general probabilistic state: p p 2. p n p i + p 2 2i + + p n ni = i 2i 3i ni Transition Matrix = /2 0. / the new state (probabilistic)

33 Probabilistic states 2 Evolution of probabilistic states 3 6Transition 7 4 Matrix 5 Any matrix that maps probabilistic states to probabilistic states. We won t restrict ourselves to just one transition matrix. K 0! K! 2 K 2! 3

34 Quantum states 2 p 3 p p n p i s can be negative.

35 Quantum states n = Unitary Matrix i s can be negative. i + 2 2i + + n ni n = n = n ( i s are called amplitudes.) (`2 norm is ) ( i can be a complex number) n = any matrix that preserves quantumness

36 Quantum states 2 Evolution of quantum states Unitary Matrix 7 5 Any matrix that maps quantum states to quantum states. We won t restrict ourselves to just one unitary matrix. U 0! U! 2 U 2! 3

37 Quantum states n Measuring quantum states = i + 2 2i + + n ni n = When you measure the state, you see state i with probability i 2.

38 Probabilistic states vs Quantum states Suppose we have just 2 possible states: apple apple apple /2 /2 /2 = /2 /2 0 /2 apple apple apple /2 /2 0 /2 = /2 /2 /2 0i and i randomize a random state random state 0i! 2 0i + i 2 2 0i + 2 0i i 2 2 i 4 0i + 4 i + 4 0i + 4 i

39 Probabilistic states vs Quantum states Suppose we have just 2 possible states: apple / p 2 / p 2 apple 0 = 0i and i / p 2 / p 2 apple p p / 2 / 2 / p 2 / p 2 apple 0 = apple / p 2 / p 2 apple p / 2 / p 2 0i! p 0i + p i 2 2 p 2 p2 0i + p i 2 p2 0i + p p i i + 2 0i + 2 i 2 i = i

40 Probabilistic states vs Quantum states Classical Probability To find the probability of an event: add the probabilities of every possible way it can happen

41 Probabilistic states vs Quantum states Quantum To find the probability of an event: add the amplitudes of every possible way it can happen, then square the value to get the probability. one way has positive amplitude the other way has equal negative amplitude event never happens!

42 Probabilistic states vs Quantum states A final remark Quantum states are an upgrade to: 2-norm (Euclidean norm) and algebraically closed fields. Nature seems to be choosing the mathematically more elegant option.

43 The plan Classical computers and classical theory of computation Quantum physics (what the fuss is all about) Quantum computation (practical, scientific, and philosophical perspectives)

44 The plan Quantum computation (practical, scientific, and philosophical perspectives)

45 Two beautiful theories Theory of computation Quantum physics

46 Quantum Computation: Information processing using laws of quantum physics.

47 It would be super nice to be able to simulate quantum systems. With a classical computer this is extremely inefficient. n-state quantum system complexity exponential in n Richard Feynman (98-988) Why not view the quantum particles as a computer simulating themselves? Why not do computation using quantum particles/physics?

48 Representing data/information An electron can be in spin up or spin down state. upi or downi 0i or i ~ A quantum bit: (qubit) 0 0i + i, = 0i i A superposition of and. When you measure: With probability 0 2 it is 0i. With probability 2 it is i.

49 upi Representing data/information An electron can be in spin up or spin down state. or downi 0i or i ~ A quantum bit: (qubit) 0 0i + i, = 2 qubits: 00 00i + 0 0i + 0 0i + i =

50 upi Representing data/information An electron can be in spin up or spin down state. or downi 0i or i ~ A quantum bit: (qubit) 0 0i + i, = 3 qubits: i i i + 0 0i i + 0 0i + 0 0i + i =

51 upi Representing data/information An electron can be in spin up or spin down state. or downi 0i or i ~ A quantum bit: (qubit) 0 0i + i, = For n qubits, how many amplitudes are there?

52 Processing data What will be our model? In the classical setting, we had: - Turing Machines - Boolean circuits In the quantum setting, more convenient to use the circuit model.

53 Processing data: quantum gates One non-trivial classical gate for a single classical bit: 0 NOT NOT 0 There are many non-trivial quantum gates for a single qubit. One famous example: Hadamard gate 0i H i H p 0i + p i 2 2 p 2 0i p 2 i transition matrix: apple / p 2 / p 2 / p 2 / p 2

54 Processing data: quantum gates Examples of classical gates on 2 classical bits: AND OR A famous example of a quantum gate on 2 qubits: For x, y 2 {0, } controlled NOT xi yi xi x yi transition matrix:

55 Processing data: quantum circuits INPUT A classical circuit OUTPUT n bits bit n bits Classical Circuit bit (or m bits)

56 Processing data: quantum circuits INPUT A quantum circuit OUTPUT n qubits n qubits quantum gates (acts on qubit) (acts on 2 qubits)

57 Processing data: quantum circuits INPUT A quantum circuit OUTPUT n qubits n qubits n qubits Quantum Circuit n qubits

58 Processing data: quantum circuits INPUT A quantum circuit OUTPUT n qubits n qubits 000i Quantum Circuit

59 Processing data: quantum circuits INPUT A quantum circuit OUTPUT n qubits n qubits 000i Quantum Circuit i i i+ i

60 Processing data: quantum circuits INPUT A quantum circuit OUTPUT n qubits n qubits n qubits Quantum Circuit superposition of possible states. 2 n ( 2 n amplitudes)

61 Processing data: quantum circuits INPUT A quantum circuit OUTPUT n qubits n qubits How do we get classical information from the circuit? We measure the output qubit(s). e.g. we measure: i i+ + i

62 Processing data: quantum circuits INPUT A quantum circuit OUTPUT n qubits n qubits Complexity? number of gates ~ computation time

63 Physical Realization?

64 Practical, Scientific and Philosophical Perspectives

65 Practical perspective What useful things can we do with a quantum computer? We can factor large numbers efficiently! So what? Can break RSA! Can we solve every problem efficiently? No!

66 Practical perspective What useful things can we do with a quantum computer? Can simulate quantum systems efficiently! Better understand behavior of atoms and moleculues. Applications: - nanotechnology - microbiology - pharmaceuticals - superconductors....

67 Scientific perspective To know the limits of efficient computation: Incorporate actual facts about physics.

68 Scientific perspective (Physical) Church Turing Thesis Any computational problem that can be solved by a physical device, can be solved by a Turing Machine. Strong version Any computational problem that can be solved efficiently by a physical device, can be solved efficiently by a TM. Strong version doesn t seem to be true!

69 Philosophical perspective Is the universe deterministic? How does nature keep track of all the numbers? 000 qubits! amplitudes How should we interpret quantum measurement? (the measurement problem) Does quantum physics have anything to say about the human mind? Quantum AI?

70 Where are we at building quantum computers? When can I expect a quantum computer on my desk? After about 20 years and billion dollars of funding : Can factor 2 into 3 x 7. (with high probability) Challenge: Interference with the outside world. quantum decoherence

71 A whole new exciting world of computation. Potential to fundamentally change how we view computation.