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

Outline General Structure General Structure k-SAT Example k-SAT Example Comparisons to Trugenberger Comparisons to Trugenberger Conclusions Conclusions

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

Quantum Heuristic vs. NP Quantum algorithms can represent all candidate solutions simultaneously in a superposition 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 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 Test is often in the form of a cost function

Generic Quantum Heuristic HUHP … Implementation-defined interatcion with Psi …

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

High-Level Breakdown Put data qubits (Psi) into a superposition of all possible solutions 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 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 “Un-superposition” the data qubits Do optional other stuff to the data (like changing phases) Do optional other stuff to the data (like changing phases) Use extra control/work qubit(s) as necessary Use extra control/work qubit(s) as necessary

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

Generic Quantum Heuristic HUHP … Implementation-defined interatcion with Psi … G G B

Outline General Structure General Structure k-SAT Example k-SAT Example Comparisons to Trugenberger Comparisons to Trugenberger Conclusions Conclusions

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) 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 The “k” represents the number of variables per clause E.G. A 3-SAT instance: E.G. A 3-SAT instance:

One Way to Solve k-SAT The GSAT (“Greedy SAT”) algorithm: 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) 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 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 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 GSAT runs until an overall minimum cost is reached or it has run for a prespecified number of steps

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

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

Limitations of Hogg’s Decision Overlooks an unexplored avenue of research which involves introducing random walks into quantum heuristics 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 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 Evidence that quantum heuristics may not be better than classical heuristics since Walk- SAT is better than GSAT

One Possible Benefit Portfolios involve running different heuristics concurrently on the same problem instances Portfolios involve running different heuristics concurrently on the same problem instances Halt when one of the heuristics has a solution 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 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 Perhaps quantum heuristics could be used to create more powerful heuristic portfolios

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

Mathematical View

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

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

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

Outline General Structure General Structure k-SAT Example k-SAT Example Comparisons to Trugenberger Comparisons to Trugenberger Conclusions Conclusions

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

Similarities Trugenberger uses a U operator that is also a diagonal matrix with terms 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 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

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

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

Differences Trugenberger does not utilize the extra P operator to modify phases 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 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 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 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 The U inverse functionality helps to cancel out bad solutions and beef up good solutions

The Winner? Hard to say without simulation Hard to say without simulation Probably boils down to three factors: Probably boils down to three factors: Will quantum counting be worse than using multiple control qubits? 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 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) Will Hogg’s heuristic suffer significantly from the delay of transforming any NP problem to SAT (Trugenberger is not bound to SAT)

Outline General Structure General Structure k-SAT Example k-SAT Example Comparisons to Trugenberger Comparisons to Trugenberger Conclusions Conclusions

Hope for Quantum Heuristics? Hogg’s heuristic doesn’t show a benefit in doing things “quantumly” rather than classically 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 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 Perhaps a good cost function definition in Trugenberger’s heuristic would save the day

Smoke and Mirrors There seems to be a communication gap between quantum heuristic researchers 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 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 Hogg’s experimental results are not encouraging, and Trugenberger presents no experimental results

Future Avenues On the bright side, since quantum heuristics have not been widely explored or applied, there is still hope 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 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 Problems whose cost functions are more expensive to compute would give quantum heuristics the edge Exploration of quantum-classical portfolios Exploration of quantum-classical portfolios Perhaps restructuring of the major gates would lead to further improvement Perhaps restructuring of the major gates would lead to further improvement