Quantum Search Heuristics: Quantum Search Heuristics: Tad Hogg’s PerspectiveTad Hogg’s Perspective

George ViamontesGeorge Viamontes

February 4, 2002February 4, 2002

OutlineOutline

General StructureGeneral Structurek-SAT Examplek-SAT ExampleComparisons to TrugenbergerComparisons to TrugenbergerConclusionsConclusions

What are we trying to solve?What are we trying to solve?

Quantum Heuristics may be most useful Quantum Heuristics may be most useful for NP problemsfor NP problems

NP problem structure:NP problem structure:Exponential number of candidate solutions as Exponential number of candidate solutions as

problem size increasesproblem size increasesQuick test for any given candidate solution to Quick test for any given candidate solution to

see if it is indeed a correct solutionsee if it is indeed a correct solution

Quantum Heuristic vs. NPQuantum Heuristic vs. NP

Quantum algorithms can represent all Quantum algorithms can represent all candidate solutions simultaneously in a candidate solutions simultaneously in a superpositionsuperposition

Tests of candidate solutions can be done Tests of candidate solutions can be done on all candidates at once with a single on all candidates at once with a single operationoperationTest is often in the form of a cost functionTest is often in the form of a cost function

Generic Quantum HeuristicGeneric Quantum Heuristic

C

H U H P …

Implementation-defined interatcion with Psi …

Generic Quantum HeuristicGeneric Quantum Heuristic

Hadamards put Psi into a superposition of Hadamards put Psi into a superposition of candidate solutionscandidate solutions

U modifies the probability amplitudes of U modifies the probability amplitudes of Psi to favor better candidate solutionsPsi to favor better candidate solutions

P does phase adjustments on PsiP does phase adjustments on PsiC is a control or work qubitC is a control or work qubitQuantum Heuristics vary a lotQuantum Heuristics vary a lot

P is optionalP is optionalC can have different rolesC can have different roles

High-Level BreakdownHigh-Level Breakdown

Put data qubits (Psi) into a superposition of all Put data qubits (Psi) into a superposition of all possible solutionspossible solutions

Do stuff to the probability amplitudes in order to Do stuff to the probability amplitudes in order to increase the chance of measuring a good increase the chance of measuring a good solution and decrease the chance of measuring solution and decrease the chance of measuring a bad onea bad one

““Un-superposition” the data qubitsUn-superposition” the data qubits Do optional other stuff to the data (like changing Do optional other stuff to the data (like changing

phases)phases) Use extra control/work qubit(s) as necessaryUse extra control/work qubit(s) as necessary

The GoalThe Goal

Iterate the previous circuit until there is a Iterate the previous circuit until there is a good probability of measuring good good probability of measuring good candidate solutionscandidate solutions

Hopefully the number of iterations will be Hopefully the number of iterations will be kept to a minimumkept to a minimumThis is the arena of competition with classical This is the arena of competition with classical

heuristicsheuristics

Generic Quantum HeuristicGeneric Quantum Heuristic

C

H U H P …

Implementation-defined interatcion with Psi …

G

G

B

OutlineOutline

General StructureGeneral Structurek-SAT Examplek-SAT ExampleComparisons to TrugenbergerComparisons to TrugenbergerConclusionsConclusions

What is k-SAT?What is k-SAT?

k-SAT is the problem of finding a satisfying k-SAT is the problem of finding a satisfying truth assignment for a boolean function in truth assignment for a boolean function in CNF (i.e. an assignment that causes the CNF (i.e. an assignment that causes the whole function to be a 1)whole function to be a 1)

The “k” represents the number of variables The “k” represents the number of variables per clauseper clause

E.G. A 3-SAT instance:E.G. A 3-SAT instance:

))(( dcbcba

One Way to Solve k-SATOne Way to Solve k-SAT

The GSAT (“Greedy SAT”) algorithm:The GSAT (“Greedy SAT”) algorithm: First produce a random set of variable assignments First produce a random set of variable assignments

(select a random set of variables and negate each (select a random set of variables and negate each one with probability ½)one with probability ½)

Flip (negate) variables whose new value will result in Flip (negate) variables whose new value will result in the satisfying of more clausesthe satisfying of more clauses

The flipping is essentially a cost function in which The flipping is essentially a cost function in which unsatisfied clauses result in a higher costunsatisfied clauses result in a higher cost

GSAT runs until an overall minimum cost is reached GSAT runs until an overall minimum cost is reached or it has run for a prespecified number of stepsor it has run for a prespecified number of steps

Not the Best SolutionNot the Best Solution

It turns out that GSAT isn’t the best It turns out that GSAT isn’t the best heuristic for solving k-SATheuristic for solving k-SAT

Walk-SAT on average performs betterWalk-SAT on average performs betterDifference is that Walk-SAT doesn’t Difference is that Walk-SAT doesn’t alwaysalways

rely on the cost functionrely on the cost function It will randomly choose between minimizing It will randomly choose between minimizing

cost and flipping a random variable in an cost and flipping a random variable in an unsatisfied clauseunsatisfied clause

However…However…

Hogg introduces a quantum heuristic for Hogg introduces a quantum heuristic for solving k-SAT and chooses to compare it solving k-SAT and chooses to compare it with GSAT rather than Walk-SATwith GSAT rather than Walk-SAT

Though not very useful, it makes sense to Though not very useful, it makes sense to compare with GSAT since quantum compare with GSAT since quantum heuristics, like GSAT, generally rely heuristics, like GSAT, generally rely exclusively on a cost functionexclusively on a cost function

Limitations of Hogg’s DecisionLimitations of Hogg’s Decision

Overlooks an unexplored avenue of Overlooks an unexplored avenue of research which involves introducing research which involves introducing random walks into quantum heuristicsrandom walks into quantum heuristics

Hogg’s heuristic on average has about the Hogg’s heuristic on average has about the same performance as GSATsame performance as GSATEvidence that quantum heuristics may not be Evidence that quantum heuristics may not be

better than classical heuristics since Walk-better than classical heuristics since Walk-SAT is better than GSATSAT is better than GSAT

One Possible BenefitOne Possible Benefit

PortfoliosPortfolios involve running different heuristics involve running different heuristics concurrently on the same problem instancesconcurrently on the same problem instances Halt when one of the heuristics has a solutionHalt when one of the heuristics has a solution

The problem instances that GSAT performs well The problem instances that GSAT performs well on are on are differentdifferent than the instances Hogg’s than the instances Hogg’s quantum heuristic performs well onquantum heuristic performs well on

Perhaps quantum heuristics could be used to Perhaps quantum heuristics could be used to create more powerful heuristic portfolioscreate more powerful heuristic portfolios

Mathematical ViewMathematical View

Hogg’s implementation of the U operator:Hogg’s implementation of the U operator:Diagonal matrix with as the elementsDiagonal matrix with as the elementss is the number of 1-bits in the overall s is the number of 1-bits in the overall

superposition (tau is explained in the next superposition (tau is explained in the next slide)slide)

And the P operatorAnd the P operator Diagonal matrix with as the elementsDiagonal matrix with as the elements c(s) is the number of unsatisfied clauses introduced c(s) is the number of unsatisfied clauses introduced

by a particular solution in the superposition s (rho is by a particular solution in the superposition s (rho is explained in the next slide)explained in the next slide)

si he

)(sci he

Mathematical ViewMathematical View

j

hRjh

11

j

hTjh

11

1

,1,...,1

10

10

TTT

RRRwithjh

"",,, 1010 parametersphaseareTTRR

iterationsofnumbertheish

Other DetailsOther Details

Hogg’s heuristic uses only a single work Hogg’s heuristic uses only a single work qubit in addition to the data qubits (Psi)qubit in addition to the data qubits (Psi)

As the h term indicates, the heuristic is As the h term indicates, the heuristic is applied iterativelyapplied iteratively

More LimitationsMore Limitations

Phase Parameters seem to be determined Phase Parameters seem to be determined experimentally (Hogg does not indicate experimentally (Hogg does not indicate where he gets particular values from)where he gets particular values from)

Since an iteration counter is used directly, Since an iteration counter is used directly, the quantum circuit requires a counter of the quantum circuit requires a counter of some sort (Hogg does not mention this at some sort (Hogg does not mention this at all)all)

Recap of Hogg’s HeuristicRecap of Hogg’s Heuristic

On average, performs as well as GSAT On average, performs as well as GSAT but has different behavior for different but has different behavior for different problem instancesproblem instances

Not as good as the best classical heuristicNot as good as the best classical heuristicHas certain non-trivial implementation Has certain non-trivial implementation

details that aren’t discusseddetails that aren’t discussed

OutlineOutline

General StructureGeneral Structurek-SAT Examplek-SAT ExampleComparisons to TrugenbergerComparisons to TrugenbergerConclusionsConclusions

Recall…Recall…

Carlo Trugenberger has also presented a Carlo Trugenberger has also presented a quantum heuristicquantum heuristic

Bears some similarities to Hogg’s heuristic Bears some similarities to Hogg’s heuristic but also has fundamental differencesbut also has fundamental differences

SimilaritiesSimilarities

Trugenberger uses a U operator that is Trugenberger uses a U operator that is also a diagonal matrix with termsalso a diagonal matrix with termsSeems to indicate that such terms would be Seems to indicate that such terms would be

prevalent in any quantum heuristic due to prevalent in any quantum heuristic due to their property of using phase to cancel out their property of using phase to cancel out bad solutionsbad solutions

ie

SimilaritiesSimilarities

Trugenberger’s heuristic also follows the Trugenberger’s heuristic also follows the Hadamard – U – Hadamard patternHadamard – U – Hadamard pattern

A cost function is also usedA cost function is also used

DifferencesDifferences

Trugenberger’s heuristic is far more Trugenberger’s heuristic is far more general and robust (possible advantage)general and robust (possible advantage)The cost function is user-definedThe cost function is user-defined

Multiple control qubits are used rather than Multiple control qubits are used rather than the single work qubit used by Hogg the single work qubit used by Hogg (possible drawback)(possible drawback)

No dependence on iterations is explicitly No dependence on iterations is explicitly defined (possible advantage)defined (possible advantage)

DifferencesDifferences

Trugenberger does not utilize the extra P Trugenberger does not utilize the extra P operator to modify phasesoperator to modify phases

Instead, Trugenberger’s U gate is enhanced to Instead, Trugenberger’s U gate is enhanced to take care of the cost function take care of the cost function andand phase phase modification in a single operatormodification in a single operator He does this by expanding the U gate to also include He does this by expanding the U gate to also include

U inverseU inverse By controlling this beefed up U gate with a control bit, By controlling this beefed up U gate with a control bit,

the phase modifications can be combined with costthe phase modifications can be combined with cost The U inverse functionality helps to cancel out bad The U inverse functionality helps to cancel out bad

solutions and beef up good solutionssolutions and beef up good solutions

The Winner?The Winner?

Hard to say without simulationHard to say without simulationProbably boils down to three factors:Probably boils down to three factors:

Will quantum counting be worse than using Will quantum counting be worse than using multiple control qubits?multiple control qubits?

Is it harder to implement the beefed up U gate Is it harder to implement the beefed up U gate or the “simpler” U gate/P gate combinationor the “simpler” U gate/P gate combination

Will Hogg’s heuristic suffer significantly from Will Hogg’s heuristic suffer significantly from the delay of transforming any NP problem to the delay of transforming any NP problem to SAT (Trugenberger is not bound to SAT)SAT (Trugenberger is not bound to SAT)

OutlineOutline

General StructureGeneral Structurek-SAT Examplek-SAT ExampleComparisons to TrugenbergerComparisons to TrugenbergerConclusionsConclusions

Hope for Quantum Heuristics?Hope for Quantum Heuristics?

Hogg’s heuristic doesn’t show a benefit in Hogg’s heuristic doesn’t show a benefit in doing things “quantumly” rather than doing things “quantumly” rather than classicallyclassically

However, from the theory of portfolios, we can However, from the theory of portfolios, we can already see that there is some benefit to already see that there is some benefit to combiningcombining the quantum and the classical the quantum and the classical

Perhaps a good cost function definition in Perhaps a good cost function definition in Trugenberger’s heuristic would save the dayTrugenberger’s heuristic would save the day

Smoke and MirrorsSmoke and Mirrors

There seems to be a communication gap There seems to be a communication gap between quantum heuristic researchersbetween quantum heuristic researchersDespite the striking similarities, Hogg does not Despite the striking similarities, Hogg does not

cite Trugenberger and Trugenberger only cite Trugenberger and Trugenberger only cites one of Hogg’s earlier workscites one of Hogg’s earlier works

Hogg’s experimental results are not Hogg’s experimental results are not encouraging, and Trugenberger presents encouraging, and Trugenberger presents no experimental resultsno experimental results

Future AvenuesFuture Avenues On the bright side, since quantum heuristics On the bright side, since quantum heuristics

have not been widely explored or applied, there have not been widely explored or applied, there is still hopeis still hope

Introduction of randomness into quantum Introduction of randomness into quantum heuristics may allow them to surpass classical heuristics may allow them to surpass classical heuristics which exploit randomnessheuristics which exploit randomness

Problems whose cost functions are more Problems whose cost functions are more expensive to compute would give quantum expensive to compute would give quantum heuristics the edgeheuristics the edge

Exploration of quantum-classical portfoliosExploration of quantum-classical portfolios Perhaps restructuring of the major gates would Perhaps restructuring of the major gates would

lead to further improvementlead to further improvement