The Quantum Approximate Optimization Algorithm (QAOA) represents a significant opportunity for practical quantum computing applications, particularly in the era before error correction is fully realized. This algorithm is especially relevant for addressing constraint satisfaction problems (CSPs), which are critical in various fields such as supply chain management, energy distribution, and financial modeling. In our study, we conduct a numerical comparison of three different strategies for incorporating linear constraints into QAOA: transforming them into an unconstrained format, introducing penalty dephasing, and utilizing the quantum Zeno effect. We assess the efficiency and effectiveness of these methods using the knapsack problem as a case study. Our findings provide insights into the potential applicability of different encoding methods for various use cases.
The Quantum Approximate Optimization Algorithm (QAOA) represents a significant opportunity for practical quantum computing applications, particularly in the era before error correction is fully realized. This algorithm is especially relevant for addressing constraint satisfaction problems (CSPs), which are critical in various fields such as supply chain management, energy distribution, and financial modeling. In our study, we conduct a numerical comparison of three different strategies for incorporating linear constraints into QAOA: transforming them into an unconstrained format, introducing penalty dephasing, and utilizing the quantum Zeno effect. We assess the efficiency and effectiveness of these methods using the knapsack problem as a case study. Our findings provide insights into the potential applicability of different encoding methods for various use cases.
When tackling binary optimization problems using quantum algorithms, the conventional Ising representation and Quantum Approximate Optimization Algorithm (QAOA) encounter difficulties in efficiently handling errors for large-scale problems involving multiple constraints. To address these challenges, this paper presents a hybrid framework that combines the use of standard Ising Hamiltonians to solve a subset of the constraints, while employing non-Ising formulations to represent and address the remaining constraints. The resolution of these non-Ising constraints is achieved through either penalty dephasing or the quantum Zeno effect. This innovative approach leads to a collection of quantum circuits with adaptable structures, depending on the chosen representation for each constraint. Furthermore, this paper introduces a novel technique that utilizes the quantum Zeno effect by frequently measuring the constraint flag, enabling the resolution of any optimization constraint. Theoretical properties of these algorithms are discussed, and their performance in addressing practical aircraft loading problems is highly promising, showcasing significant potential for a wide range of industrial applications.
Traditional methods for handling (inequality) constraints in the quantum approximate optimization (QAOA) typically rely on penalty terms and slack variables, which increase problem complexity and expand the search space. More sophisticated mixer-based QAOA variants restrict the search within the feasible assignments but often suffer from prohibitive circuit complexity. This paper presents a penalty-free formalism for incorporating inequality constraints into the cost function of QAOA using an oracle-based subroutine that evaluates constraint satisfaction in an additional register, subsequently called indicator function QAOA (IF-QAOA). Applied to the knapsack problem, we demonstrate in numerical simulations the superior performance of IF-QAOA over conventional penalty-based approaches. Using advanced QAOA simulation techniques, we find that IF-QAOA achieves significantly higher solution quality and a faster time to solution in 82% of our benchmark cases, even though circuit depth is approximately three times larger. Analysis of the scaling behavior shows favorable scaling of IF-QAOA compared to penalty-based methods. Also, benchmarks against the recently developed quantum tree generator QAOA for knapsack problems (P. Christiansen , ) demonstrated higher solution quality for circuits of similar depth. Additionally, this paper introduces a method for approximating the indicator function when the number of ancillary qubits is limited or the constraint function is noninteger. With a specialized simulation algorithm based on projective measurements, we empirically demonstrate that this formalism can encode general inequality constraints using a fixed number of ancillary qubits.
Combinatorial Optimization (CO) is one of the most important areas in the field of optimization, with practical applications found in every industry, including both the private and public sectors. In recent years, it was discovered that a mathematical formulation known as QUBO (Quadratic Unconstrained Binary Optimization) problem can embrace an exceptional variety of important CO problems found in the industry. In this work, we explore the Quantum Approximate Optimization Algorithm (QAOA) as a quantum computing approach to solve the Knapsack Problem efficiently by formulating the problem as a QUBO model. Here we implemented QAOA on a quantum emulator, leveraging quantum superposition and entanglement to explore multiple solutions simultaneously. The method offers a promising route for solving large, complex scheduling and resource allocation problems that are tough for classical algorithms, with potential applications in logistics, HPC resource scheduling, and energy optimization.
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.
The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical variational algorithm that offers the potential to handle combinatorial optimization problems. Introducing constraints in such combinatorial optimization problems poses a significant challenge in the extensions of QAOA to support relevant larger-scale problems. This paper introduces a quantum machine learning approach to learn the mixer Hamiltonian required to hard constrain the search subspace. We show that this method can be used for encoding any general form of constraints. One can directly plug the learnt unitary into the QAOA framework using an adaptable ansatz. This procedure gives the flexibility to control the depth of the circuit at the cost of the accuracy of enforcing the constraint, thus having immediate application in the Noisy Intermediate Scale Quantum (NISQ) era. We also develop an intuitive metric that uses Wasserstein distance to assess the performance of general approximate optimization algorithms with/without constraints. Finally, using this metric, we evaluate the performance of the proposed algorithm.
Combinatorial optimization is a promising area for achieving quantum speedup. The quantum approximate optimization algorithm (QAOA) is designed to search for low-energy states of the Ising model, which correspond to near-optimal solutions of combinatorial optimization problems (COPs). However, effectively dealing with constraints of COPs remains a significant challenge. Existing methods, such as tailoring mixing operators, are typically limited to specific constraint types, like one-hot constraints. To address these limitations, we introduce a method for engineering a compressed space that represents the feasible solution space with fewer qubits than the original. Our approach includes a scalable technique for determining the unitary transformation between the compressed and original spaces on gate-based quantum computers. We then propose compressed space QAOA, which seeks near-optimal solutions within this reduced space, while utilizing the Ising model formulated in the original Hilbert space. Experimental results on a quantum simulator demonstrate the effectiveness of our method in solving various constrained COPs.