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International Conference on Machine Intelligence and Smart Innovation (ICMISI)

Quantum Answer Set Programming Solver Using Amplitude Amplification

Esraa Ebdelrehime, Ahmed Younes, Islam Elkabani

Published May 12, 2024• 1 citation
Publisher

Abstract

In this paper, a Quantum Answer set programming solver (QASP) will be proposed to solve NP-hard combinatorial search problems. The paper shows that problems such as Hamiltonian cycle problem and N-queen problem encoded as ASP can be reduced to a MAX-3-SAT problem in a 3-CNF form. The 3-CNF Boolean formula will be solved using the proposed quantum solver with$O(n+ m)$qubits where$n$is the input variables number,$m$is the clauses number in$O(\sqrt{2^{n/m}})$steps.

DOI
10.1109/icmisi61517.2024.10580520

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In this paper, a Quantum Answer set programming solver (QASP) will be proposed to solve NP-hard combinatorial search problems. The paper shows that problems such as Hamiltonian cycle problem and N-queen problem encoded as ASP can be reduced to a MAX-3-SAT problem in a 3-CNF form. The 3-CNF Boolean formula will be solved using the proposed quantum solver withO(n+m)O(n+ m)O(n+m)qubits wherennnis the input variables number,mmmis the clauses number inO(2n/m)O(\sqrt{2^{n/m}})O(2n/m​)steps.
The relation between answer set programming (ASP) and combinatorial problems, especially the satisfiability problem (SAT) has attracted significant interest in academic literature. The aim of this paper is to introduce an efficient quantum answer set programming solver (QSAT) that can handle NP-hard combinatorial search problems by reducing the problem to MAX-3-SAT problem. The paper proposes a quantum algorithm that solves the MAX-3-SAT problem using amplification techniques that exploits entanglement and partial diffusion operator to find a solution with high probability inO(√2n/l), where n is the number of variables and l denotes the number of literals. The proposed algorithm shows a speed-up in solving ASP problems compared to classical approaches.
The aim of the paper is to answer a long-standing open problem on the relationship between NP and BQP. The paper shows that BQP contains NP by proposing a BQP quantum algorithm for the MAX-E3-SAT problem which is a fundamental NP-hard problem. Given an E3-CNF Boolean formula, the aim of the MAX-E3-SAT problem is to find the variable assignment that maximizes the number of satisfied clauses. The proposed algorithm runs in O(m2)O(m^2)O(m2) for an E3-CNF Boolean formula with mmm clauses and in the worst case runs in O(n6)O(n^6)O(n6) for an E3-CNF Boolean formula with nnn inputs. The proposed algorithm maximizes the set of satisfied clauses using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of 1−ε1-ε1−ε for small ε>0ε>0ε>0 using a polynomial resources. In addition to solving the MAX-E3-SAT problem, the proposed algorithm can also be used to decide if an E3-CNF Boolean formula is satisfiable or not, which is an NP-complete problem, based on the maximum number of satisfied clauses.
To demonstrate the advantage of quantum computation, many attempts have been made to attack classically intractable problems, such as the satisfiability problem (SAT), with quantum computer. To use quantum algorithms to solve these NP-hard problems, a quantum oracle with quantum circuit implementation is usually required. In this manuscript, we first introduce a novel algorithm to synthesize quantum logic in the Conjunctive Normal Form (CNF) model. Compared with linear ancillary qubits in the implementation of Qiskit open-source framework, our algorithm can synthesize an m clauses n variables k-CNF with O(k2m2/n)O(k^2 m^2/n)O(k2m2/n) quantum gates by only using three ancillary qubits. Both the size and depth of the circuit can be further compressed with the increase in the number of ancillary qubits. When the number of ancillary qubits is Ω(mϵ)\Omega(m^\epsilon)Ω(mϵ) (for any ϵ>0\epsilon > 0ϵ>0), the size of the quantum circuit given by the algorithm is O(km), which is asymptotically optimal. Furthermore, we design another algorithm to optimize the depth of the quantum circuit with only a small increase in the size of the quantum circuit. Experiments show that our algorithms have great improvement in size and depth compared with the previous algorithms.
Quantum computing has emerged as a promising method for solving combinatorial optimization problems. These problems are often encoded as the Satisfiability (SAT) problem. In this study, we explore the quantum annealing based approach to solving the SAT problems. We conduct experiments on uniformly random 3-CNF instances that have 400 propositional variables and 1,000 to 2,499 clauses. We measure the time required to solve them using Fixstars Amplify Annealing Engine. The results show that the monotonic increase in solving time as the number of clauses increases, except for a discontinuity point at approximately 1,800 clauses. In addition, we observe some cases of false UNSAT reports. Our findings demonstrate both the characteristics and the limitations of this quantum annealing based solver. We plan to reproduce the solver on a local machine to inspect the performance when varying the parameters of the quantum annealing and the environment.
In this paper, we propose a construction method of Grover's algorithm to solve the N-Queen problem. Quantum permutation state was designed and applied to the initialization and amplitude amplification process in Grover's algorithm. An oracle-level quantum circuit was constructed using Boolean algebraic expressions. We reduced the number of iterations of the Grover's algorithm by decreasing the number of superposed inputs in the initialize step using quantum permutation state. We show that our algorithm has less time complexity compared to previous study that solved the N -Queen problem using Grover's algorithm with W state as a input.
The use of Boolean Satisfiability (SAT) solver for hardware verification incurs exponential runtime in several instances. In this work, we have proposed an efficient quantum SAT (qSAT) solver for equivalence checking of Boolean circuits employing Grover’s algorithm. The Exclusive-Sum-of-Product (ESOP)-based generation of the Conjunctive Normal Form (CNF) equivalent clauses demands less qubits and minimizes the gates and depth of quantum circuit interpretation. The consideration of reference circuits for verification affecting Grover’s iterations and quantum resources are also presented as a case study. Experimental results are presented assessing the benefits of the proposed verification approach using open source Qiskit platform and IBM quantum computer.