1. Intro to Quantum Algorithms SUNY Polytechnic Institute Chen-Fu Chiang Fall 2015

2. Foreword … There are 2 parts in this lecture  Part I: Scientists’ vision & current status  Part II: The basics & simple algorithms showing exponential speedup

3. Part I : Scientists’ vision and current status See @ https://www.youtube.com/watch?v=CMdHDHEuOUE https://www.youtube.com/watch?v=CMdHDHEuOUE

4. Part II: Content Quantum computing & the basics  Quantum bits & superposition  Quantum circuit & data manipulation Elementary quantum algorithms  Deutsch’s algorithm  Deutsch-Jozsa algorithm

5. Quantum Computing A quantum computer is a machine that performs calculations based on the laws of quantum mechanics, which is the behavior of particles at the sub-atomic level.

6. Quantum Computing A quantum computer is a machine that performs calculations based on the laws of quantum mechanics, which is the behavior of particles at the sub-atomic level. Quantum computing uses the quantum mechanical properties in order to build computers and algorithms that have a better performance than current computer technology.

7. Quantum Computing Before we continue, I assume that you have the knowledge of the following  Circuit models (computer organization) Gates (NOT, Controlled-NOT, XOR, etc.) Wires  Linear algebra Matrix operations (multiplication, eigenvector, eigenvalue) Tensor product Vector space

8. Representation of Data - Qubits Excited State Ground State Nucleu s Light pulse of frequency for time interval t Electro n State |0>State |1> A bit of data is represented by a single atom that is in one of two states denoted by |0> and |1>. A single bit of this form is known as a qubit A physical implementation of a qubit could use the two energy levels of an atom. An excited state representing |1> and a ground state representing |0>.

9. Represent Data – Superposition A quantum state can be described by a vector state. Thus a qubit maybe written as  |ψ ˃ = α|0> + β|1> where and |α| 2 +|β| 2 = 1

10. Represent Data – Superposition A quantum state can be described by a vector state. Thus a qubit maybe written as  |ψ ˃ = α|0> + β|1> where and |α| 2 +|β| 2 = 1  Hadamard Transform

11. Simple Illustration – Deutsch’s Problem Problem:  Given a black box function  Task: determine whether is constant or balanced

12. Simple Illustration – Deutsch’s Problem Problem:  Given a black box function  Task: determine whether is constant or balanced  BalancedConstant Q: How many queries do you need classically? xf 1 (x) f 2 (x) 00 1 11 0 xf 3 (x)f 4 (x) 010 110

13. Simple Illustration – Deutsch’s Problem Problem:  Given a black box function  Task: determine whether is constant or balanced  BalancedConstant Q: How many queries do you need classically?  2 Q: Quantumly ? xf 1 (x)f 2 (x) 001 110 xf 3 (x)f 4 (x) 010 110

14. A quantum algorithm for Deutsch’s Problem Simple quantum circuit

15. A quantum algorithm for Deutsch’s Problem Simple quantum circuit

16. A quantum algorithm for Deutsch’s Problem Simple quantum circuit

17. A quantum algorithm for Deutsch’s Problem Simple quantum circuit

18. A quantum algorithm for Deutsch’s Problem Simple quantum circuit

19. A quantum algorithm for Deutsch’s Problem Simple quantum circuit recall that

20. A quantum algorithm for Deutsch’s Problem Simple quantum circuit

21. A quantum algorithm for Deutsch’s Problem Simple quantum circuit

22. A Deutsch-Jozsa Problem Problem:  Given a black box function  Promise: is either constant or balanced Balanced : for exactly half values of x, we have Constant : is independent of x  Determine whether if is constant or balanced

23. A Deutsch-Jozsa Problem Problem:  Given a black box function  Promise: is either constant or balanced Balanced : for exactly half values of x, we have Constant : is independent of x  Determine whether if is constant or balanced Q: How many queries are needed?

24. A Deutsch-Jozsa Problem Problem:  Given a black box function  Promise: is either constant or balanced Balanced : for exactly half values of x, we have Constant : is independent of x  Determine whether if is constant or balanced Q: How many queries are needed? Classically, queries with certainty (at least) Quantumly ?

25. A Deutsch-Jozsa Problem Simple quantum Circuit

26. A Deutsch-Jozsa Problem Simple quantum Circuit

27. A Deutsch-Jozsa Problem Simple quantum Circuit

28. Look at Hadamard Again

31. A Deutsch-Jozsa Problem – finishing up Recall the state before the last Hadamard gates The operation of n Hadamard could have the effect Hence, we obtain the state (let us ignore the last qubit)

32. A Deutsch-Jozsa Problem – finishing up The final state before measurement is If is constant, the amplitude of is?

33. A Deutsch-Jozsa Problem – finishing up The final state before measurement is If is constant, the amplitude of is?  Since the amplitude is, we will only see after the measurement. So, we are sure the function is a constant function.

34. A Deutsch-Jozsa Problem – finishing up The final state before measurement is If is balanced, the amplitude of is?

35. A Deutsch-Jozsa Problem – finishing up The final state before measurement is If is balanced, the amplitude of is?  Since the amplitude is 0, that means if we measure and obtain a non state, then the function must be balanced.

36. Opportunity – For Undergraduates Quantum summer school @ IQC Canada  USEQIP