1. KEG PARTY!!!!!  Keg Party tomorrow night  Prof. Markov will give out extra credit to anyone who attends* *Note: This statement is a lie

2. Trugenberger’s Quantum Optimization Algorithm Overview and Application

3. Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

4. Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

5. Two Main Sources of Inspiration  Exploiting Quantum Parallelism  Analogy of Simulated Annealing

6. What is quantum parallelism?  What is quantum parallelism?  We can represent super-positions of specific instances of data in a single quantum state  We can then apply a single operator to this quantum state and thereby change all instances of data in a single step

7. What is Simulated Annealing?  Comes from physical annealing  Iteratively heat and cool a material until there’s a high probability of obtaining a crystalline structure  Can be represented as a computational algorithm  Iteratively make changes to your data until there is a high probability of ending up with the data you want

8. Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

9. Basic Idea  Use this inspiration to come up with a more generalized quantum searching algorithm  Trugenberger’s algorithm does a heuristic search on the entire data set by applying a cost function to each element in the data set  Goal is to find a minimal cost solution

10. The high-level algorithm  Use quantum parallelism to apply the cost function to all elements of the data set simultaneously in one step  Iteratively apply this cost function to the data set  Number of iterations is analogous to an instance of simulated annealing

11. Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

12. Representing the Problem: Graph Coloring  Super-position of the data elements  N instances  Use n qubits to represent the N instances  Each instance encoded as a binary number I^k whose value is between 0 and 2^n

13. Cost Functions in General  should return a cost for that data element  In this algorithm we will want to minimize cost  Data elements with lower cost are better solutions

14. Skeleton of the U operator  The imaginary exponential of the cost function is the main engine of the quantum optimization

15. What is Cnor?  We know in general that exp(i*theta) = cos(theta) + i*sin(theta)  Since U will need the imaginary exponential of the cost function, we want to normalize the cost function  By normalizing, we ensure that the cost function result is between 0 and pi/2

16. What is Cnor?  C(I^k) can at most be Cmax and is at least Cmin  Cnor is always between 1 and 0

17. And Cmin and Cmax?  Simple to determine for graph coloring  Cmin = 0 (no pair connected vertices shares the same color)  Cmax = # of edges (every pair of connected vertices shares the same color)  More general method for determining Cmin and Cmax will be introduced later

18. Fleshing out U for Graph Coloring

19. Still don’t quite have our magic operator  As written, U by itself will not lower the probability amplitude of bad states and increase the amplitude of good states  If we apply U now, the probability amplitudes of both the best and worst data elements will be the same and differ only in phase

20. Take Advantage of Phase Differences  We can accomplish the proper amplitude modifications by using a controlled form of the U gate  Can’t be an ordinary controlled gate though

21. Ucs: The Answer to our Problems  Ucs is a controlled gate that applies U to the data elements when the control bit is |0> and applies the inverse of U when the control bit is |1>

22. Control Bits also need some modification  Control bit always starts out in |0> state  Before applying Ucs, we run the control bit through a Hadamard gate  After applying Ucs, we run it through another Hadamard gate  This gives us a nice super-position of minimal and maximal cost elements

23. Matlab results for Graph Coloring Data element Probability amplitude 0000 0010.25 0100.3536 0110.25 1000.25 1010.3536 1110 ----------------------------------------------------------------- 000i*0.3536 001i*0.25 0100 011i*0.25 100i*0.25 1010 110i*0.25 111i*0.3536

24. Measurement  If we were to measure the control bit now and get a |0>, we’d know that the data will get the “first half” of the super-position: Data element Probability amplitude 0000 0010.25 0100.3536 0110.25 1000.25 1010.3536 1110

25. Measurement  However if we got a |1> instead, we’d know that the data will get the “second half” of the super-position: Data element Probability amplitude 000i*0.3536 001i*0.25 0100 011i*0.25 100i*0.25 1010 110i*0.25 111i*0.3536

26. Measurement  A control qubit measurement of |0> means we have a better chance of getting a lower cost state (a good solution)  A control qubit measurement of |1> means we have a better chance of getting a higher cost state (a bad solution)

27. Measurement  Assume the world is perfect and we always get a |0> when we measure the control qubit  We can effectively increase our probability of getting good solutions and decrease the probability of getting bad solutions by iterating the H,Ucs,H operations  We iterate by duplicating the circuit and adding more control qubits

28. Matlab Results after 26 “Ideal” Iterations Data element Probability amplitude 0000 0010 0100.3536 0110 1000 1010.3536 1110 ----------------------------------------------------------------- 0000 0010 0100 0110 1000 1010 1100 1110

29. Life Isn’t Fair  We don’t always get a |0> for all the control qubits when we measure  Some of the qubits are bound to be measured in the |1> state  Upon measuring the control qubits we can at least know the quality of our computation

30. The Tradeoff  If we increase the number of control qubits (b), then we have a chance of bumping up the probability amplitudes of the lower cost solutions and canceling out the probability amplitudes of the higher cost solutions

31. The Tradeoff  However, if we increase the number of control qubits (b), we ALSO lower our chances of getting more control qubits in the |0> state

32. Some good news  As mentioned earlier, the measurement of the control qubits will tell us how good our bad a particular run was  Trugenberger gives an equation for the expected number of runs needed for a good result:

33. Analogy to Simulated Annealing  Can view b, the number of control qubits, as a sort of temperature parameter  Trugenberger gives some energy distributions based on the “effective temperature” being equal to 1/b  Simply an analogy to the number of iterations needed for a probabilistically good solution

34. A Whole New Meaning for k  k can be seen as a certain subset of the |S> super-position of data elements  For the graph coloring problem, k=3  More generally for other problems, k can vary from 1 to K where K > 1

35. Equations affected by generalization  Cnor changes:

36. Equations effected by generalization  U changes (this in turn changes Ucs which utilizes U):

37. Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

38. U operator  Constructing the U operator may itself be exponential in the number of qubits  Perhaps some physical process to get around this

39. Cost Function Oracle?  Trugenberger glosses over the implementation of the cost function (in fact no implementation is suggested)  Some problems may still be intractable if cost function is too complicated

40. Only a Heuristic  Trugenberger’s algorithm may not get the exact minimal solution  Although, keeping in mind the tradeoff, more control qubits can be added to increase the odds of a good solution

41. Overview  Inspiration  Basic Idea  Mathematical and Circuit Realizations  Limitations  Future Work

42. Future Work  Look into physical feasibility of cost function and construction of Ucs  Run more simulations on various problems and compare against classical heuristics  Compare with Grover’s algorithm

43. Reference  Quantum Optimization by C.A. Trugenberger, July 22, 2001 (can be found on LANL archive)