Establishing quantum advantage for variational quantum algorithms is an important direction in quantum computing. In this work, we apply the Quantum Approximate Optimisation Algorithm (QAOA) -- a popular variational quantum algorithm for general combinatorial optimisation problems -- to a variant of the satisfiability problem (SAT): Not-All-Equal SAT (NAE-SAT). We focus on regimes where the problems are known to have solutions with low probability and introduce a novel classical solver that outperforms existing solvers. Extensively benchmarking QAOA against this, we show that while the runtime of both solvers scales exponentially with the problem size, the scaling exponent for QAOA is smaller for large enough circuit depths. This implies a polynomial quantum speedup for solving NAE-SAT.
Establishing quantum advantage for variational quantum algorithms is an important direction in quantum computing. In this work, we apply the Quantum Approximate Optimisation Algorithm (QAOA) -- a popular variational quantum algorithm for general combinatorial optimisation problems -- to a variant of the satisfiability problem (SAT): Not-All-Equal SAT (NAE-SAT). We focus on regimes where the problems are known to have solutions with low probability and introduce a novel classical solver that outperforms existing solvers. Extensively benchmarking QAOA against this, we show that while the runtime of both solvers scales exponentially with the problem size, the scaling exponent for QAOA is smaller for large enough circuit depths. This implies a polynomial quantum speedup for solving NAE-SAT.
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 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.
We present new advances towards achieving exponential quantum speedups for solving optimization problems by low-depth quantum algorithms. Specifically, we focus on families of combinatorial optimization problems that exhibit symmetry and contain planted solutions. We rigorously prove that the 1-step Quantum Approximate Optimization Algorithm (QAOA) can achieve a success probability of Ω(1/n), and sometimes Ω(1), for finding the exact solution in many cases. This allows us to prove a separation of O(1) quantum queries and Ω(n/logn) classical queries required to find the planted solution in the latter setting. Furthermore, we construct near-symmetric optimization problems by randomly sampling the individual clauses of symmetric problems, and prove that the QAOA maintains a strong success probability in this setting even when the symmetry is broken. Finally, we construct various families of near-symmetric Max-SAT problems and benchmark state-of-the-art classical solvers, discovering instances where all known general-purpose classical algorithms require exponential time. Therefore, our results indicate that low-depth QAOA may achieve an exponential quantum speedup for optimization problems.
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.
Quantum Approximate Optimization algorithm (QAOA) is one of the candidates to achieve a near-term quantum advantage. As QAOA seems only capable of solving optimization problems, there is a folklore that QAOA cannot see the difference between easy problems such as 2-SAT and hard problems such as 3-SAT -- although 2-SAT is in the polynomial-time (P) class, its optimization version is also nondeterministic polynomial-time (NP)-hard. In this paper, we show that the folklore is not true. We find a computational phase transition in QAOA when solving a variant of 3-SAT -- the amplitude of gradient and the success probability achieve their minimum at the well-known SAT-UNSAT phase transition. On the contrary, for 2-SAT, such a phenomenon is absent at SAT-UNSAT phase transition and the success probability is unity for a reasonable circuit depth. We connect the gradient transition to the dynamical Lie algebra of the QAOA circuit. In solving the NP-hard optimization versions of SAT, we identify quantum advantages over a classical approximate algorithm at quite a shallow depth of p=4 for the problem size of n=10.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.