Abstract The Boolean satisfiability problem (SAT), in particular 3SAT with its bounded clause size, is a well-studied problem since a wide range of decision problems can be reduced to it. The Quantum Approximate Optimization Algorithm (QAOA) is a promising candidate for solving 3SAT for Noisy Intermediate-Scale Quantum devices in the near future due to its simple quantum ansatz. However, although QAOA generally exhibits a high approximation ratio, there are 3SAT problem instances where the algorithm’s success probability when obtaining a satisfying variable assignment from the approximated solution drops sharply compared to the approximation ratio. To address this problem, in this paper, we present variants of the algorithm that are inspired by the amplitude amplification algorithm to improve the success probability for 3SAT. For this, (i) three amplitude amplification-inspired QAOA variants are introduced and implemented, (ii) the variants are experimentally compared with a standard QAOA implementation, and (iii) the impact on the success probability and ansatz complexity is analyzed. The experiment results show that an improvement in the success probability can be achieved with only a moderate increase in circuit complexity.
Abstract The Boolean satisfiability problem (SAT), in particular 3SAT with its bounded clause size, is a well-studied problem since a wide range of decision problems can be reduced to it. The Quantum Approximate Optimization Algorithm (QAOA) is a promising candidate for solving 3SAT for Noisy Intermediate-Scale Quantum devices in the near future due to its simple quantum ansatz. However, although QAOA generally exhibits a high approximation ratio, there are 3SAT problem instances where the algorithm’s success probability when obtaining a satisfying variable assignment from the approximated solution drops sharply compared to the approximation ratio. To address this problem, in this paper, we present variants of the algorithm that are inspired by the amplitude amplification algorithm to improve the success probability for 3SAT. For this, (i) three amplitude amplification-inspired QAOA variants are introduced and implemented, (ii) the variants are experimentally compared with a standard QAOA implementation, and (iii) the impact on the success probability and ansatz complexity is analyzed. The experiment results show that an improvement in the success probability can be achieved with only a moderate increase in circuit complexity.
The quantum approximate optimization algorithm (QAOA) is a promising method for solving certain classical combinatorial optimization problems on near-term quantum devices. When employing the QAOA to 3-SAT and Max-3-SAT problems, the quantum cost exhibits an easy-hard-easy or easy-hard pattern, respectively, as the clause density is changed. The quantum resources needed in the hard-region problems are out of reach for current noisy intermediate-scale quantum (NISQ) devices. We show by numerical simulations with up to 14 variables and analytical arguments that the adaptive-bias QAOA (ab-QAOA) greatly improves performance in the hard region of the 3-SAT problems and hard region of the Max-3-SAT problems. For similar accuracy, on average, ab-QAOA needs 3 levels for 10-variable 3-SAT problems as compared to 22 for QAOA. For 10-variable Max-3-SAT problems, the numbers are 7 levels and 62 levels. The improvement comes from a more targeted and more limited generation of entanglement during the evolution. We demonstrate that classical optimization is not strictly necessary in the ab-QAOA since local fields are used to guide the evolution. This leads us to propose an optimization-free ab-QAOA that can solve the hard-region 3-SAT and Max-3-SAT problems effectively with significantly fewer quantum gates as compared to the original ab-QAOA. Our work paves the way for realizing quantum advantages for optimization problems on NISQ devices.
One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k -SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n , which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024
The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational approach for solving Boolean satisfiability (SAT) problems. Existing SAT-to-QAOA workflows first translate Boolean formulas into penalty Hamiltonians before applying generic quantum compilation, thereby discarding SAT-specific information such as clause structure, logical operators, and variable dependencies. Consequently, subsequent compiler optimizations cannot exploit the underlying Boolean formulation. This paper presents SAQC, a SAT-aware compilation framework that introduces a SAT Intermediate Representation (SIR) to preserve Boolean structure during compilation. The proposed framework supports both CNF and XOR-extended CNF (eCNF) formulations and performs SAT-aware optimizations including clause rewriting, dependency analysis, and clause scheduling prior to quantum lowering. Building on the optimized SIR, SAQC provides a unified framework for both penalty Hamiltonian generation and direct clause-to-ansatz synthesis, together with ansatz optimizations based on relative-phase decomposition, ancilla-assisted synthesis, and dynamic uncomputation. Experimental results demonstrate significant reductions in circuit depth, two-qubit gate count, and compilation time compared with conventional Hamiltonian-based workflows while remaining compatible with existing quantum compilation frameworks.
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.
Abstract At present, the quantum approximate optimization algorithm (QAOA) faces scalability challenges in high-dimensional combinatorial optimization problems due to exponentially growing computational costs and reachability deficits for noisy intermediate-scale quantum (NISQ) devices. This study focuses on the multiscale quantum approximate optimization algorithm (MQAOA), which integrates renormalization group (RG) transformations with QAOA to address these limitations. Based on the connections between the variables in the problem to be solved, the weighted maximal matching method is employed to generate a variable partitioning strategy guiding the RG transformation. This approach not only extends the applicability of MQAOA to satisfiability (SAT) problems — including those with three-body and higher-order interactions in the problem Hamiltonian — but also eliminates the algorithm’s sensitivity to problem density. Validations conducted on quantum simulators show that, after running two-round MQAOA, its capability is enhanced to identify optimal solutions with approximately 97% success probability as defined by the ground-state overlap for Max-2-SAT problems (78% success probability for Max-3-SAT problems). The results confirm the feasibility of MQAOA and establish it as a resource-efficient framework for complex combinatorial optimization problems, providing a pathway for NISQ-era deployment.
The main object of study of this work is the Quantum Approximate Optimization Algorithm (QAOA) —the best-known heuristic quantum algorithm for addressing combinatorial optimization problems. Despite its popularity, QAOA poses major challenges. Namely, its performance remains poorly understood, especially in the low depth regime most relevant to near-term quantum computing. Besides, the algorithm is of variational nature, relying on several hyperparameters to be classically trained. In this work, we contribute to addressing these issues by developing techniques to understand the performance of QAOA at low depth, and deducing from these pretrained hyperparameters, thereby eliminating or considerably reducing the need for a classical optimization loop. We first consider applying QAOA to constraint satisfaction problems with the goal of maximizing the algorithm’s approximation ratio (fraction of satisfied constraints). As a particular example, we consider QAOA applied to MAXCUT on non-regular graphs, showing how analytic techniques provide good guess hyperparameters in this setting. Next, we analyze the performance of QAOA on a well-studied constraint satisfaction problem: random k-SAT. In this case, the performance is measured by the probability of finding an assignment satisfying all constraints (“success probability”). To estimate this quantity, we develop new techniques, mapping quantum expectation values of the QAOA circuits to classical spin models. We further show how these techniques can be generalized to a larger family of random constraint satisfaction problems. We conclude the work by two feasibility studies inspired by real-world use cases encountered in collaborations with industrial partners: peptide sampling, and solving flow problems. These problems are naturally phrased by encoding combinatorial variables in qudits, a relatively unexplored setting in the existing literature. We present partial analytic and numerical results for optimal QAOA hyperparameters and the scaling of its success probability in this framework, underlining potential specific difficulties when using QAOA with qudits.