1. Quantum Computing Lecture 1 Michele Mosca

2. l Course Outline http://cacr.math.uwaterloo.ca/~mmosca/outlinef00.htm http://cacr.math.uwaterloo.ca/~mmosca/outlinef00.htm l Web page http://cacr.math.uwaterloo.ca/~mmosca/quantumcoursef00.htm http://cacr.math.uwaterloo.ca/~mmosca/quantumcoursef00.htm l Course Notes: “Quantum Computation and Quantum Information” by Nielsen and Chuang (available at the UW Bookstore)

3. Overview l Chapter 1: General Introduction l Chapter 2: Intro to Quantum Mechanics l Chapter 3: Intro to Computer Science l Chapter 4: More on Quantum Circuits l Chapters 5,6: Quantum Algorithms l Chapter 7: Physical Realizations l Chapter 10: Quantum Error Correction l Chapter 8: General Quantum Operations

4. Reading for General Introduction l Chapter 1 of Text. Sections 1.1-1.5 l Introduction posted on web page I will cover parts of the introduction in Lecture 1. I will continue in Lecture 5 and subsequent lectures.

5. Introduction to Quantum Mechanics l Lectures 2,3 and 4 l Parts of Chapter 2, as instructed by Prof. Mann

6. General Introduction l Strong Church-Turing thesis states that a probabilistic Turing machine (ie a classical computer that can make fair coin flips) can efficiently simulate any realistic model of computing l Therefore if we are interested in which problems can be solved efficiently on a realistic model of computation, we can restrict attention to a probabilistic Turing machine (or an equivalent model)

7. Physics and Computation l Information is stored in a physical medium and manipulated by physical processes l Therefore the laws of physics dictate the capabilities and limitations of any information processor l The “classical” laws of physics are (in macroscopic systems moving relatively slowly) a good approximation to the laws of physics

8. Physics and Computation Realisations are getting smaller (and faster) and reaching a point where “classical” physics is not longer a sufficient model for the laws of physics

9. Physics and Computation l However the theory of quantum physics is a much better approximation to the laws of physics l The probabilistic Turing machine is implicitly a “classical” device and it is not known in general how to use it simulate quantum mechanical systems [Fey82] l A computer designed to exploit the quantum features of Nature (a quantum computer) seems to violate the Strong Church-Turing thesis

10. Physics and Computation l Is a quantum computer realistic? Answer seems to be YES (chapter 10) l If the quantum computers are a reasonable model of computation, and classical devices cannot efficiently simulate them, then the strong Church-Turing thesis needs to be modified to state that a quantum Turing machine can efficiently simulate any realistic model of computation

11. Quantum Communication and Cryptography l By exploiting the quantum mechanical behaviour of the communication medium, we can detect eavesdroppers (leading to quantum cryptography, section 12.6) and solve distributed computation tasks more efficiently. Unfortunately, we won’t be covering this in this course, but we will lay the foundation for further reading in quantum information theory.

12. A beam-splitter The simplest explanation is that the beam- splitter acts as a classical coin-flip, randomly sending each photon one way or the other.

13. Quantum Interference The simplest explanation must be wrong, since it would predict a 50-50 distribution.

14. More experimental data

15. A new theory The particle can exist in a linear combination or superposition of the two paths

16. Probability Amplitude and Measurement If the photon is measured when it is in the statethen we get with probability

17. Quantum Operations The operations are induced by the apparatus linearly, that is, if and then

18. Quantum Operations Any linear operation that takes states satisfying and maps them to states satisfying must be UNITARY

19. Linear Algebra corresponds to

20. Linear Algebra corresponds to

21. Linear Algebra corresponds to

22. Linear Algebra is unitary if and only if

23. Abstraction The two position states of a photon in a Mach-Zehnder apparatus is just one example of a quantum bit or qubit Except when addressing a particular physical implementation, we will simply talk about “basis” states and unitary operations like and

24. wherecorresponds to and corresponds to

25. An arrangement like is represented with a network like

26. More than one qubit If we concatenate two qubits we have a 2-qubit system with 4 basis states and we can also describe the state as or by the vector

27. More than one qubit In general we can have arbitrary superpositions where there is no factorization into the tensor product of two independent qubits. These states are called entangled.

28. Measuring multi-qubit systems If we measure both bits of we getwith probability