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The University of BirminghamSchool of Computer Science

A Constant Complexity Algorithm for Solving the Boolean Satisfiability Problem on

Quantum Computers

(A Hybrid Quantum Search Engine)

Ahmed Younes – Julian Miller – Jon Rowe

October 2003

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Outline

n Intro. to Satisfiability Problem.n Representation of Boolean Functions on QC.

n The Algorithm.n Algorithm Performance.

n Iterating the Algorithm.n Performance of Iterating the Algorithm.

n Conclusion.n References.

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Satisfiability Problem-1

n Satisfiability problem is one of the famous NP-complete problems of special interest in theory of computation, artificial intelligence and the study of mathematical logic.

n It can be understood as follows: We have an nvariable Boolean expression and we need to know if there are any possible variable assignments within the 2n possible variable assignments that will make the expression evaluate to TRUE.

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Satisfiability Problem-2

n Solving an arbitrary Satisfiability Problem can be understood as a search problem within search space of size N=2n with one or more matching solutions within the search space.

n Classically, It was shown that this problem can be solved in where M is the number of solutions.

n Grover’s Algorithm was shown to solve this problem in on quantum computers.

n It was shown that number of iterations required by Grover’s Algorithm increases for . 2/NM ≥

)/( MNO

)/( MNO

5Grover's Probability Distribution after one iteration.

Satisfiability Problem-3

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n There are two well developed paradigms of Boolean logic. The first uses the {AND,OR,NOT} operations and called Canonical Boolean Logic. The second uses the {AND,XOR,NOT} operations and called Reed-Muller logic (RM). It means that any arbitrary Boolean expression (SAT or K-SAT) can be represented in RM form.

Representation of Boolean Functions on QC -1

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Representation of Boolean Functions on QC -2

n Consider the following Boolean expression:

n In previous work, we showed that any Boolean function represented in RM can be implemented as a reversible Boolean quantum circuit, so RM expression for the above Boolean expression takes this form:

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n The Boolean Quantum Circuit that represents this expression for Positive Polarity RM is as follows :

Representation of Boolean Functions on QC -3

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Representation of Boolean Functions on QC -4

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n The meaning of circuit optimality [5] has many constraints, for e.g.:

1.The interaction between qubits in the circuit. It may be difficult to involve certain qubits on the same CNOT gates.

2.The number of control qubits per CNOT gate should be minimum.

3.The total number of CNOT gates should be kept to a minimum so it is possible to maintain coherence during the operation of the circuit.

n Optimization algorithms is required/can be found to optimize quantum circuits represented as RM similar to that found on classical computers

Representation of Boolean Functions on QC -5

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The Algorithm -1

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The Algorithm -21-Register Preparation:

2-Register Initialization:

3- Boolean Function Evaluation:

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The Algorithm -34- Completing Superposition and Changing Sign :

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n Let M be the number of possible variable assignments that will make the Boolean expression evaluate to TRUE.

n : no solution: f(i) = 0n : there exist at least one

solution within the superposition.

The Algorithm -4

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Case 1:

(M) states with amplitude :

(P-M) states with amplitude:

(2M) states represent the solutions.(P-2M) states represent non-solutions.

The Algorithm -5

P1−

P1

∑'

∑ ''

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The Algorithm -6

Case 2:

P states with amplitude: P1

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The Algorithm -7

5- Inversion About the Mean: Apply Diffusion Operator D on the n+1 qubit system:

n Applying D on general state gives:

n In the case we have:

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The Algorithm -8

n For the case where:Applying D will transform amplitudes as follows:

The state of the system after applying D can be written as follows:

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The Algorithm -9

n For the case where:Applying D will not change the amplitudes:

6- Measurement: Apply measurement on the first n qubits.

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Algorithm Performance - 1

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Algorithm Performance - 2

n To verify this results, the average probability that the algorithm can find a solution can be calculated as follows:

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Algorithm Performance - 3

n The average probability that the algorithm will find non-solution can be calculated as follows:

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Algorithm Performance - 4

n Classically if we tried to guess the solution, we may succeed with probability M/N, so the average probability by guessing can be calculated as follows:

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Algorithm Performance - 5

n Similarly, The average probability that Grover’s algorithm can find a solution after any arbitrary t of iterations can be calculated as follows:

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Algorithm Performance - 6

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One Iteration Probability Distribution of Grover’s Algorithmvs.Proposed Algorithm .

Algorithm Performance - 7

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Iterating the Algorithm - 1

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Iterating the Algorithm - Performance-1

Probability Distribution for 1,2,..6 iterations.

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Iterating the Algorithm – Performance-2

Five Iterations Probability Distribution of Grover’s Algorithmvs.Proposed Algorithm .

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Iterating the Algorithm – Performance-3

n The Probability of finding a solution after q steps can be calculated as follows:

n The number of iterations q required to get probability at least one-half:

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Iterating the Algorithm – Performance-4

n It means the algorithm has the following behaviour:

* O(1) :*O(N/M) :

n It means there is no big gain from iterating the algorithm.

8N

M N≤ ≤

81

NM <≤

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Conclusion-1

n We propose a hybrid quantum search engine which works as follows:n If the number of solutions M is known in

advance, so we can pick which algorithm to use as follows:• If : We use the proposed

algorithm in .• If : We use Grover’s algorithm

in .

/8N M N≤ ≤

8/1 NM <≤)1(O

)/( MNO

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Conclusion - 2

n If the number of solutions M is unknown, so the proposed algorithm may succeed to solve the problem with probability 87.5% after one iteration.

n Special cases: n If : The proposed algorithm

with find a solution with certainty.n If : Grover’s algorithm with find

a solution with certainty after one iteration.

2/NM =

4/NM =

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Conclusion - 3

n It would be better to have a single quantum algorithm which combines both behaviours (Grover’s for few number of solutions and the proposed algorithm here):i.e.n If : It runs in .n If : It runs in .

(current research)

/8N M N≤ ≤ )1(O8/1 NM <≤ )/( MNO

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References

[1] L. K. Grover (1996). [1] L. K. Grover (1996). A Fast Quantum Mechanical Algorithm for Database SearchA Fast Quantum Mechanical Algorithm for Database Search. . Proceedings of the TwentyProceedings of the Twenty--Eighth Annual ACM Symposium on the Theory of Computing, Eighth Annual ACM Symposium on the Theory of Computing, 212212--219.219.

[2] M.Boyer, G. Brassard, P. [2] M.Boyer, G. Brassard, P. HHøøyeryer and A. Tapp(1998), and A. Tapp(1998), Tight Bounds on Quantum SearchingTight Bounds on Quantum Searching. . FortschritteFortschritte derder PhysikPhysik, vol. 46(4, vol. 46(4--5), pp. 4935), pp. 493--505. 505.

[3] M. Nielsen and I. [3] M. Nielsen and I. ChuangChuang (2000), (2000), Quantum Computation and Quantum InformationQuantum Computation and Quantum Information. . Cambridge University Press, Cambridge, United Kingdom.Cambridge University Press, Cambridge, United Kingdom.

[4] A. Younes, J. Miller (2003), [4] A. Younes, J. Miller (2003), Automated Method for Building CNOT Based Quantum Circuits Automated Method for Building CNOT Based Quantum Circuits for Boolean Functionfor Boolean Function. Los Alamos Physics preprint archive, quant. Los Alamos Physics preprint archive, quant--ph/0304099.ph/0304099.

[5] A. Younes, and J. Miller (2003), [5] A. Younes, and J. Miller (2003), Representation of Boolean Quantum Circuits as ReedRepresentation of Boolean Quantum Circuits as Reed--Muller ExpansionsMuller Expansions. Los Alamos Physics preprint archive,quant. Los Alamos Physics preprint archive,quant--ph/0305134.ph/0305134.

[6] A. Younes, and J. Miller (2003), [6] A. Younes, and J. Miller (2003), A Constant Complexity Algorithm for Solving the Boolean A Constant Complexity Algorithm for Solving the Boolean Satisfiability Problem on Quantum ComputersSatisfiability Problem on Quantum Computers. Los Alamos Physics preprint archive,quant. Los Alamos Physics preprint archive,quant--ph/0305134.ph/0305134.