KEG PARTY!!!!!

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 000 0 001 0.25 010 0.3536 011 0.25 100 0.25 101 0.3536 111 0 ----------------------------------------------------------------- 000 i*0.3536 001 i*0.25 010 0 011 i*0.25 100 i*0.25 101 0 110 i*0.25 111 i*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 000 0 001 0.25 010 0.3536 011 0.25 100 0.25 101 0.3536 111 0

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 000 i*0.3536 001 i*0.25 010 0 011 i*0.25 100 i*0.25 101 0 110 i*0.25 111 i*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 000 0 001 0 010 0.3536 011 0 100 0 101 0.3536 111 0 ----------------------------------------------------------------- 000 0 001 0 010 0 011 0 100 0 101 0 110 0 111 0

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)