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arXiv:2403.03237

Local Quantum Search Algorithm for Random $k$-SAT with $Ω(n^{1+ε})$ Clauses

Mingyou Wu

Published Mar 5, 2024• 1 citation
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Abstract

The random k-SAT instances undergo a "phase transition" from being generally satisfiable to unsatisfiable as the clause number m passes a critical threshold, $r_k n$. This causes a drastic reduction in the number of satisfying assignments, shifting the problem from being generally solvable on classical computers to typically insolvable. Beyond this threshold, it is challenging to comprehend the computational complexity of random k-SAT. In quantum computing, Grover's search still yields exponential time requirements due to the neglect of structural information. Leveraging the structure inherent in search problems, we propose the k-local quantum search algorithm, which extends quantum search to structured scenarios. Grover's search, by contrast, addresses the unstructured case where k=n. Given that the search algorithm necessitates the presence of a target, we specifically focus on the problem of searching the interpretation of satisfiable instances of k-SAT, denoted as max-k-SSAT. If this problem is solvable in polynomial time, then k-SAT can also be solved within the same complexity. We demonstrate that, for small $k \ge 3$, any small $ε>0$ and sufficiently large n: $\cdot$ k-local quantum search achieves general efficiency on random instances of max-k-SSAT with $m=Ω(n^{2+δ+ε})$ using $\mathcal{O}(n)$ iterations, and $\cdot$ k-local adiabatic quantum search enhances the bound to $m=Ω(n^{1+δ+ε})$ within an evolution time of $\mathcal{O}(n^2)$. In both cases, the circuit complexity of each iteration is $\mathcal{O}(n^k)$, and the efficiency is assured with overwhelming probability $1 - \mathcal{O}(\mathrm{erfc}(n^{δ/2}))$. By modifying this algorithm capable of solving all instances of max-k-SSAT, we further prove that max-k-SSAT is polynomial on average when $m=Ω(n^{2+ε})$ based on the average-case complexity theory.

DOI
10.48550/arxiv.2403.03237

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Abstract Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem k-SAT, Quantum k-SAT is QMA1\hbox {QMA}_1QMA1​ QMA 1 -complete (for k≥3k\ge 3k≥3 k ≥ 3 , and where QMA1\hbox {QMA}_1QMA1​ QMA 1 is a quantum generalization of NP with one-sided error), and thus likely intractable. But whereas k-SAT has been well-studied for special tractable cases, as well as from a “parameterized complexity” perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a “dimer covering” or “matching”; such systems are known to be frustration-free, but it remains open whether one can efficiently compute a ground state. Our results fall into three directions, all of which relate to the “dimer covering” setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two interaction terms or clauses. (2) We give a “parameterized algorithm” for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. An explicit family of hypergraphs, denoted Crash, for which such a speedup is obtained is introduced. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a “dimer covering”. We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.
The kkk-SAT problem is a canonical NP-complete problem for which no efficient algorithm is known. This paper proposes a hybrid quantum–classical algorithm that employs Grover's algorithm as the inner local search of a randomized outer loop. Each outer iteration measures a random assignment, prepares a problem-dependent superposition over the Hamming ball of radius rrr around it, and applies amplitude amplification to locate a satisfying assignment. Because the ball center is measured, the outer loop remains a classical repetition over independently sampled balls and contributes the inverse success probability rather than its square root; balancing this against the inner quadratic speedup shifts the optimal radius from n/(k+1)n/(k+1)n/(k+1) to n/(1+k)n/(1+\sqrt{k})n/(1+k​) and yields the running time (2−2/(1+k))n(2-2/(1+\sqrt{k}))^{n}(2−2/(1+k​))n, up to polynomial factors—1.268n1.268^{n}1.268n for k=3k=3k=3, below the best known classical 1.307n1.307^{n}1.307n. The contribution is not an asymptotic record—the best known quantum exponents for kkk-SAT, obtained by amplifying stronger classical solvers as black boxes, remain smaller—but a fully explicit, gate-level construction: the problem-dependent state over the Hamming ball and the kkk-SAT oracle are decomposed into elementary quantum gates, and the polynomial gate and qubit overheads are accounted for in full.
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The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.
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Satisfiability (SAT) is a central problem in computer science, and advances in SAT-solving algorithms have a far-reaching impact across many fields. Recent works have proposed quantum SAT solvers based on Grover's algorithm, a quantum search technique. However, Grover-based approaches face a key limitation: they typically require prior knowledge of the number of satisfying assignments of the target Boolean formula. This information is unavailable in most practical settings. Quantum counting can be used to estimate this quantity, but it incurs a computational overhead that is several orders of magnitude higher than Grover search. In this paper, we propose a novel quantum SAT solver based on entanglement and equivalence checking. Our method does not assume prior knowledge of the number of solutions and is computationally more efficient than quantum counting. Although the worst case time complexity is inevitably exponential, we prove that the expected time complexity of our approach is only constant time O(1) over random Boolean functions. Experimental results also support our theoretical claim.