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ACM International Symposium on High-Performance Parallel and Distributed Computing (HPDC)

Quantum-Based SMT Solving for String Theory

Beatrice Casey, Joanna C. S. Santos, Andrew Hennessee

Published Jul 20, 2025• 0 citations
Publisher

Abstract

Satisfiability Modulo Theory (SMT) solvers are a useful tool that can be applied to a variety of problems, such as configuring relationships in distributed systems, detecting race conditions, and program analysis. String constraints are particularly difficult for SMT solvers to navigate, as the search space is generally large. Often times, classical SMT solvers will have to quit generating a solution for string constraints because it takes too long to find the solution. Quantum computing offers the advantages of quantum mechanics (e.g., superposition), which allows a system to explore a large search space much more efficiently. In this work, we explore creating a quantum-enabled SMT solver for string theory by using quantum annealing and Quadratic Unconstrained Binary Optimization (QUBO). Our preliminary results demonstrate that it is feasible to transform these string constraints to QUBO, and generate solutions for given constraints.

DOI
10.1145/3731545.3744151

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Satisfiability Modulo Theory (SMT) solvers are a useful tool that can be applied to a variety of problems, such as configuring relationships in distributed systems, detecting race conditions, and program analysis. String constraints are particularly difficult for SMT solvers to navigate, as the search space is generally large. Often times, classical SMT solvers will have to quit generating a solution for string constraints because it takes too long to find the solution. Quantum computing offers the advantages of quantum mechanics (e.g., superposition), which allows a system to explore a large search space much more efficiently. In this work, we explore creating a quantum-enabled SMT solver for string theory by using quantum annealing and Quadratic Unconstrained Binary Optimization (QUBO). Our preliminary results demonstrate that it is feasible to transform these string constraints to QUBO, and generate solutions for given constraints.
Satisfiability modulo theory (SMT), a problem of determining the satisfiability of a first-order formula with respect to a decidable first-order theory, has a wide range of applications because of its powerful ability of problem encoding, including formal verification and optimization workflows in both classical and quantum domains. Given an SMT formulaFFF, the classical lazy approach for SMT solving tries to (1) abstractFFFto a Boolean formulaFBF_{B}FB​, (2) find a Boolean solution toFBF_{B}FB​, and (3) check whether the Boolean solution is consistent with the theory. If the solution found in the Boolean domain is not consistent with that in the theory domain, the SMT solver needs to perform steps (2) and (3) back and forth until a consistent solution is found between the theory and Boolean domains. This classical approach becomes unscalable when the search space is huge. In this work, we develop the first Grover–based quantum SMT solver to solve formulas on the theory of bit-vectors. With the characteristic of superposition in quantum computing, the proposed SMT solver is able to consider all the possible inputs simultaneously and check their consistency between the theory and Boolean domains in one shot, which brings significant speed-up for SMT solving.
To date, conventional computers have never been able to efficiently handle certain tasks, where the number of computation steps is likely to blow up as the problem size increases. As an emerging technology and new computing paradigm, quantum computing has a great potential to tackle those hard tasks efficiently. Among all the existing quantum computation models, quantum annealing has drawn significant attention in recent years due to the realization of the commercialized quantum annealer, sparking research interests in developing applications to solve problems that are intractable for classical computers.However, designing and implementing algorithms that manage to harness the enormous computation power from quantum annealer remains a challenging task. Generally, it requires mapping of the given optimization problem into quadratic unconstrained binary optimization(QUBO) problem and embedding the subsequent QUBO onto the physical architecture of quantum annealer. Additionally, practical quantum annealers are susceptible to errors leading to low probability of the correct solution.In this study, we focus on solving Boolean satisfiability (SAT) problem using quantum annealer while addressing practical limitations. We have proposed a mapping technique that maps SAT problem to QUBO, and we have further devised a tool flow that embeds the QUBO onto the architecture of a quantum annealing device. Additionally, We have optimized the proposed embedding flow to reduce run-time in addition to shortening the qubit chain length, leading to robust quantum annealing. To further improve the reliability of quantum annealing, we have also developed a post processing embedding technique that enlarges the energy gap between ground state and the first excited state. To demonstrate the effectiveness of proposed methods, we have conducted experiments on real quantum annealing devices manufactured by D-Wave Systems, showing compelling result of using quantum annealer to solve SAT problem.
We present experiments in solving combinatorial optimization and constraint satisfaction problems by means of Quantum Annealing. We describe how to model classic constraint problems such as N-queens and Magic Square as well as hard combinatorial problems such as the Costas Array Problem or the Quadratic Assignment Problem in terms of QUBO (Quadratic Unconstrained Binary Optimization). QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the ”quantum-inspired” but classical devices such as Fujitsu’s Digital Annealing Unit or Hitachi’s CMOS Annealing Machine. We present preliminary results for solving these combinatorial optimization and constraint satisfaction problems by implementation on the D-Wave quantum computer.
Quantum annealing device has shown a great potential in solving discrete problems that are theoretically and empirically hard. Boolean Satisfiability (SAT) problem, determining if there is an assignment of variables that satisfies a given Boolean function, is the first proven NP-complete problem widely used in various domains. Here, we present a novel mapping of the SAT problem to the quadratic unconstrained binary optimization problem (QUBO), and further develop a tool flow embedding the proposed QUBO to the architecture of the commercialized quantum computer D-Wave. By leveraging electronic design automation techniques including synthesis, placement and routing, this is not only the first work providing the detail flow that embeds the QUBO, but also a technique scalable for real world applications and some hard SAT problems with over 6000 variables in QUBO. Based on our results, we discuss the challenges in solving SAT using the current generation of annealing device, and explore the problem solving capability of future quantum annealing computers.
Abstract Boolean satisfiability (SAT) is a propositional logic problem of determining whether an assignment of variables satisfies a Boolean formula. Many combinatorial optimization problems can be formulated in Boolean SAT logic—either as k-SAT decision problems or Max k-SAT optimization problems, with conflict-driven clause learning (CDCL) solvers being the most prominent. Despite their ability to handle large instances, CDCL-based solvers have fundamental scalability limitations. In light of this, we propose recently-developed quadratic unconstrained binary optimization (QUBO) solvers as an alternative platform for 3-SAT problems. To utilize them, we implement a 2-step [3-SAT]-[Max 2-SAT]-[QUBO] conversion procedure and present a novel approach using linear systems and Diophantine equation to calculate the number of both satisfied and violated clauses of the original 3-SAT instance from the transformed Max 2-SAT formulation. We then demonstrate, through numerical simulations on several benchmark instances, that digital QUBO solvers can achieve state-of-the-art accuracy on 78-variable 3-SAT benchmark problems. Our work facilitates the broader use of quantum annealers on noisy intermediate-scale quantum devices, as well as their quantum-inspired digital counterparts, for solving 3-SAT problems.
Optimization is a crucial challenge across various domains, including finance, resource allocation, and mobility. Quantum computing has the potential to redefine the way we handle complex problems by reducing computational complexity and enhancing solution quality. Optimization, particularly of objective functions, stands to benefit significantly from quantum solvers, which leverage principles of quantum mechanics like superposition, entanglement, and tunneling. The Ising and Quadratic Unconstrained Binary Optimization (QUBO) models are the most suitable formulations for these solvers, involving binary variables and constraints treated as penalties within the overall objective function. To harness quantum approaches for optimization, two primary strategies are employed: exploiting quantum annealers—special-purpose optimization devices—and designing algorithms based on quantum circuits. This review provides a comprehensive overview of quantum optimization methods, examining their advantages, challenges, and limitations. It demonstrates their application to real-world scenarios and outlines the steps to convert generic optimization problems into quantum-compliant models. Lastly, it discusses available tools and frameworks that facilitate the exploration of quantum solutions for optimization tasks.