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.
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.
The ability of the quantum approximate optimization algorithm (QAOA) to deliver a quantum advantage on combinatorial optimization problems is still unclear. Recently, a scaling advantage over a classical solver was postulated to exist for random 8-SAT at the satisfiability threshold. At the same time, the viability of quantum error mitigation for deep circuits on near-term devices has been put into doubt. Here we analyze the QAOA's performance on random Max- k XOR as a function of k and the clause-to-variable ratio. As a classical benchmark, we use the mean-field approximate optimization algorithm and find that it performs better than or equal to the QAOA on average. Still, for large k and numbers of layers p , there may remain a window of opportunity for the QAOA. However, by extrapolating our numerical results, we find that reaching high levels of satisfaction would require extremely large p , which must be considered rather difficult both in the variational context and on near-term 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
In the search for quantum advantage in real--world problems, one promising avenue is to use a quantum algorithm to improve on the solution found using an efficient classical algorithm. The quantum approximate optimization algorithm (QAOA) is particularly well adapted for such a "warm start" approach, and can be combined with the powerful classical Goemans-Williamson (GW) algorithms based on semi-definite programming. Nonetheless, the best way to leverage the power of the QAOA remains an open question. Here we propose a general model that describes a class of QAOA variants, and use it to explore routes to quantum advantages in a canonical optimization problem, MaxCut. For these algorithms we derive analytic expectation values of the cost Hamiltonian for the MaxCut problem in the level-1 case. Using these analytic results we obtain reliable averages over many instances for fairly large numbers of qubits. We find that the warm start adaptive-bias QAOA (WS-ab-QAOA) initialized by the GW algorithm outperforms previously proposed warm start variants on problems with 40 to 180 qubits. To assess whether a quantum advantage exists with this algorithm, we did numerical simulations with up to 1000 qubits to see whether the level-1 WS-ab-QAOA can improve the GW solution for 3-regular graphs. In fact the improvement in the 1000-qubit case even in level 1 can only be matched by the GW algorithm after about 105.5 random projections performed after the semi-definite program stage. This work gives evidence that the final stage of optimization after an efficient classical algorithm has produced an approximate solution may be a place where quantum advantages can be realized.
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.
Molecular docking, a critical task in drug discovery, represents a highly promising application domain for quantum computing. While the Quantum Approximate Optimization Algorithm (QAOA) shows potential, its performance on near-term quantum devices is limited by noise susceptibility and optimization challenges such as barren plateaus. To address these limitations, we apply Adaptive Bias Fields QAOA (ABF-QAOA) to solve molecular docking problems mapped to Ising Hamiltonians. Our results demonstrate that ABF-QAOA achieves accelerated convergence and higher solution accuracy compared to standard QAOA. In addition, under simulated quantum noise conditions, ABF-QAOA maintains higher solution quality, showcasing significantly enhanced noise resilience. This work establishes ABF-QAOA as a robust quantum approach for molecular docking on noisy intermediate-scale quantum (NISQ) hardware.
Abstract Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA’s performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified.