Exploring Quantum Heuristics: A Comparative Analysis of Hogg's Approach to k-SAT Solutions

1. Quantum Search Heuristics: Tad Hogg’s Perspective George Viamontes February 4, 2002

2. Outline • General Structure • k-SAT Example • Comparisons to Trugenberger • Conclusions

3. What are we trying to solve? • Quantum Heuristics may be most useful for NP problems • NP problem structure: • Exponential number of candidate solutions as problem size increases • Quick test for any given candidate solution to see if it is indeed a correct solution

4. Quantum Heuristic vs. NP • Quantum algorithms can represent all candidate solutions simultaneously in a superposition • Tests of candidate solutions can be done on all candidates at once with a single operation • Test is often in the form of a cost function

5. Generic Quantum Heuristic … H U H P Implementation-defined interatcion with Psi …

6. Generic Quantum Heuristic • Hadamards put Psi into a superposition of candidate solutions • U modifies the probability amplitudes of Psi to favor better candidate solutions • P does phase adjustments on Psi • C is a control or work qubit • Quantum Heuristics vary a lot • P is optional • C can have different roles

7. High-Level Breakdown • Put data qubits (Psi) into a superposition of all possible solutions • Do stuff to the probability amplitudes in order to increase the chance of measuring a good solution and decrease the chance of measuring a bad one • “Un-superposition” the data qubits • Do optional other stuff to the data (like changing phases) • Use extra control/work qubit(s) as necessary

8. The Goal • Iterate the previous circuit until there is a good probability of measuring good candidate solutions • Hopefully the number of iterations will be kept to a minimum • This is the arena of competition with classical heuristics

9. Generic Quantum Heuristic G … H U H P B G Implementation-defined interatcion with Psi …

10. Outline • General Structure • k-SAT Example • Comparisons to Trugenberger • Conclusions

11. What is k-SAT? • k-SAT is the problem of finding a satisfying truth assignment for a boolean function in CNF (i.e. an assignment that causes the whole function to be a 1) • The “k” represents the number of variables per clause • E.G. A 3-SAT instance:

12. One Way to Solve k-SAT • The GSAT (“Greedy SAT”) algorithm: • First produce a random set of variable assignments (select a random set of variables and negate each one with probability ½) • Flip (negate) variables whose new value will result in the satisfying of more clauses • The flipping is essentially a cost function in which unsatisfied clauses result in a higher cost • GSAT runs until an overall minimum cost is reached or it has run for a prespecified number of steps

13. Not the Best Solution • It turns out that GSAT isn’t the best heuristic for solving k-SAT • Walk-SAT on average performs better • Difference is that Walk-SAT doesn’t always rely on the cost function • It will randomly choose between minimizing cost and flipping a random variable in an unsatisfied clause

14. However… • Hogg introduces a quantum heuristic for solving k-SAT and chooses to compare it with GSAT rather than Walk-SAT • Though not very useful, it makes sense to compare with GSAT since quantum heuristics, like GSAT, generally rely exclusively on a cost function

15. Limitations of Hogg’s Decision • Overlooks an unexplored avenue of research which involves introducing random walks into quantum heuristics • Hogg’s heuristic on average has about the same performance as GSAT • Evidence that quantum heuristics may not be better than classical heuristics since Walk-SAT is better than GSAT

16. One Possible Benefit • Portfolios involve running different heuristics concurrently on the same problem instances • Halt when one of the heuristics has a solution • The problem instances that GSAT performs well on are different than the instances Hogg’s quantum heuristic performs well on • Perhaps quantum heuristics could be used to create more powerful heuristic portfolios

17. Mathematical View • Hogg’s implementation of the U operator: • Diagonal matrix with as the elements • s is the number of 1-bits in the overall superposition (tau is explained in the next slide) • And the P operator • Diagonal matrix with as the elements • c(s) is the number of unsatisfied clauses introduced by a particular solution in the superposition s (rho is explained in the next slide)

18. Mathematical View

19. Other Details • Hogg’s heuristic uses only a single work qubit in addition to the data qubits (Psi) • As the h term indicates, the heuristic is applied iteratively

20. More Limitations • Phase Parameters seem to be determined experimentally (Hogg does not indicate where he gets particular values from) • Since an iteration counter is used directly, the quantum circuit requires a counter of some sort (Hogg does not mention this at all)

21. Recap of Hogg’s Heuristic • On average, performs as well as GSAT but has different behavior for different problem instances • Not as good as the best classical heuristic • Has certain non-trivial implementation details that aren’t discussed

22. Outline • General Structure • k-SAT Example • Comparisons to Trugenberger • Conclusions

23. Recall… • Carlo Trugenberger has also presented a quantum heuristic • Bears some similarities to Hogg’s heuristic but also has fundamental differences

24. Similarities • Trugenberger uses a U operator that is also a diagonal matrix with terms • Seems to indicate that such terms would be prevalent in any quantum heuristic due to their property of using phase to cancel out bad solutions

25. Similarities • Trugenberger’s heuristic also follows the Hadamard – U – Hadamard pattern • A cost function is also used

26. Differences • Trugenberger’s heuristic is far more general and robust (possible advantage) • The cost function is user-defined • Multiple control qubits are used rather than the single work qubit used by Hogg (possible drawback) • No dependence on iterations is explicitly defined (possible advantage)

27. Differences • Trugenberger does not utilize the extra P operator to modify phases • Instead, Trugenberger’s U gate is enhanced to take care of the cost function and phase modification in a single operator • He does this by expanding the U gate to also include U inverse • By controlling this beefed up U gate with a control bit, the phase modifications can be combined with cost • The U inverse functionality helps to cancel out bad solutions and beef up good solutions

28. The Winner? • Hard to say without simulation • Probably boils down to three factors: • Will quantum counting be worse than using multiple control qubits? • Is it harder to implement the beefed up U gate or the “simpler” U gate/P gate combination • Will Hogg’s heuristic suffer significantly from the delay of transforming any NP problem to SAT (Trugenberger is not bound to SAT)

29. Outline • General Structure • k-SAT Example • Comparisons to Trugenberger • Conclusions

30. Hope for Quantum Heuristics? • Hogg’s heuristic doesn’t show a benefit in doing things “quantumly” rather than classically • However, from the theory of portfolios, we can already see that there is some benefit to combining the quantum and the classical • Perhaps a good cost function definition in Trugenberger’s heuristic would save the day

31. Smoke and Mirrors • There seems to be a communication gap between quantum heuristic researchers • Despite the striking similarities, Hogg does not cite Trugenberger and Trugenberger only cites one of Hogg’s earlier works • Hogg’s experimental results are not encouraging, and Trugenberger presents no experimental results

32. Future Avenues • On the bright side, since quantum heuristics have not been widely explored or applied, there is still hope • Introduction of randomness into quantum heuristics may allow them to surpass classical heuristics which exploit randomness • Problems whose cost functions are more expensive to compute would give quantum heuristics the edge • Exploration of quantum-classical portfolios • Perhaps restructuring of the major gates would lead to further improvement