Considerable effort has been made recently in the development of heuristic quantum algorithms for solving combinatorial optimization problems. Meanwhile, these problems have been studied extensively in classical computing for decades. In this paper, we explore a natural approach to leveraging existing classical techniques to enhance quantum optimization. Specifically, we run a classical algorithm to find an approximate solution and then use a quantum circuit to search its "neighborhood" for higher-quality solutions. We propose the Classically-Boosted Quantum Optimization Algorithm (CBQOA) that is based on this idea and can solve a wide range of combinatorial optimization problems, including all unconstrained problems and many important constrained problems such as Max Bisection, Maximum Independent Set, Minimum Vertex Cover, Portfolio Optimization, Traveling Salesperson and so on. A crucial component of this algorithm is an efficiently-implementable continuous-time quantum walk (CTQW) on a properly-constructed graph that connects the feasible solutions. CBQOA utilizes this CTQW and the output of an efficient classical procedure to create a suitable superposition of the feasible solutions which is then processed in certain way. This algorithm has the merits that it solves constrained problems without modifying their cost functions, confines the evolution of the quantum state to the feasible subspace, and does not rely on efficient indexing of the feasible solutions. We demonstrate the applications of CBQOA to Max 3SAT and Max Bisection, and provide empirical evidence that it outperforms previous approaches on these problems.
Considerable effort has been made recently in the development of heuristic quantum algorithms for solving combinatorial optimization problems. Meanwhile, these problems have been studied extensively in classical computing for decades. In this paper, we explore a natural approach to leveraging existing classical techniques to enhance quantum optimization. Specifically, we run a classical algorithm to find an approximate solution and then use a quantum circuit to search its "neighborhood" for higher-quality solutions. We propose the Classically-Boosted Quantum Optimization Algorithm (CBQOA) that is based on this idea and can solve a wide range of combinatorial optimization problems, including all unconstrained problems and many important constrained problems such as Max Bisection, Maximum Independent Set, Minimum Vertex Cover, Portfolio Optimization, Traveling Salesperson and so on. A crucial component of this algorithm is an efficiently-implementable continuous-time quantum walk (CTQW) on a properly-constructed graph that connects the feasible solutions. CBQOA utilizes this CTQW and the output of an efficient classical procedure to create a suitable superposition of the feasible solutions which is then processed in certain way. This algorithm has the merits that it solves constrained problems without modifying their cost functions, confines the evolution of the quantum state to the feasible subspace, and does not rely on efficient indexing of the feasible solutions. We demonstrate the applications of CBQOA to Max 3SAT and Max Bisection, and provide empirical evidence that it outperforms previous approaches on these problems.
We introduce SamBa-GQW, a novel quantum algorithm for solving binary combinatorial optimization problems of arbitrary degree with no use of any classical optimizer. The algorithm is based on a continuous-time quantum walk on the solution space represented as a graph. The walker explores the solution space to find its way to vertices that minimize the cost function of the optimization problem. The key novelty of our algorithm is an offline classical sampling protocol that gives information about the spectrum of the problem Hamiltonian. Then, the extracted information is used to guide the walker to high quality solutions via a quantum walk with a time-dependent hopping rate. We investigate the performance of SamBa-GQW on several quadratic problems, namely MaxCut, maximum independent set, portfolio optimization, and higher-order polynomial problems such as LABS, MAX- k -SAT and a quartic reformulation of the travelling salesperson problem. We empirically demonstrate that SamBa-GQW finds high quality approximate solutions on problems up to a size of n = 30 qubits by only sampling poly( n ) states among 2 n possible decisions. Furthermore, SamBa-GQW compares in par with classically optimized variational approaches, such as the variational guided quantum walk and QAOA (the latter when run on deep circuits). This places Samba-GQW as a promising heuristic to tackle combinatorial problems beyond the NISQ regime.
We propose and implement a family of quantum-informed recursive optimization (QIRO) algorithms for combinatorial optimization problems. Our approach leverages quantum resources to obtain information that is used in problem-specific classical reduction steps that recursively simplify the problem. These reduction steps address the limitations of the quantum component (e.g., locality) and ensure solution feasibility in constrained optimization problems. Additionally, we use backtracking techniques to further improve the performance of the algorithm without increasing the requirements on the quantum hardware. We showcase the capabilities of our approach by informing QIRO with correlations from classical simulations of shallow circuits of the quantum approximate optimization algorithm, solving instances of maximum independent set and maximum satisfiability problems with hundreds of variables. We also demonstrate how QIRO can be deployed on a neutral atom quantum processor to find large independent sets of graphs. In summary, our scheme achieves results comparable to classical heuristics even with relatively weak quantum resources. Furthermore, enhancing the quality of these quantum resources improves the performance of the algorithms. Notably, the modular nature of QIRO offers various avenues for modifications, positioning our work as a template for a broader class of hybrid quantum-classical algorithms for combinatorial optimization. Published by the American Physical Society 2024
The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm that finds approximate solutions to problems in combinatorial optimization, especially those that can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. In prior work, researchers have considered various ways of "warm-starting" QAOA by constructing an initial quantum state using classically-obtained solutions or information; these warm-starts typically cause QAOA to yield better approximation ratios at much lower circuit depths. For the Max-Cut problem, one warm-start approaches constructs the initial state using the high-dimensional vectors that are output from an SDP relaxation of the corresponding Max-Cut problem. This work leverages these semidefinite warmstarts for a broader class of problem instances by using a standard reduction that transforms any QUBO instance into a Max-Cut instance. We empirically compare this approach to a "QUBO-relaxation" approach that relaxes the QUBO directly. Our results consider a variety of QUBO instances ranging from randomly generated QUBOs to QUBOs corresponding to specific problems such as the traveling salesman problem, maximum independent set, and portfolio optimization. We find that the best choice of warmstart approach is strongly dependent on the problem type.
Combinatorial optimization problems are at the heart of modern computational challenges; however, their NP-hard nature often makes exact classical solutions intractable. The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising candidate for NISQ-era optimization. However, a significant performance gap exists between unconstrained and constrained formulations. The former refers to problems in which each solution is valid, either bad or good. The latter, to tasks in which there exist illegal solutions. While native constrained QAOA mixers offer high solution quality, they often suffer from prohibitive runtimes. In this paper, we introduce LE-QAOA (L-reduction Enhanced QAOA), a framework that leverages classical complexity theory to bridge this gap. By utilizing L-reductions to map diverse problems into a canonical Max2SAT form, we are able to solve constrained problems using efficient unconstrained subroutines followed by a classical repairing tournament. Our results on the Maximum Independent Set (MIS) problem show that LE-QAOA achieves solution quality that matches and frequently exceeds that of native constrained solvers, while requiring as little as 10% of the execution time. This approach effectively combines the scalability of unconstrained quantum evolution with the precision of constrained optimization. Moreover, this approach makes it possible to solve combinatorial optimization by reduction to a single chosen problem whose QAOA encoding is optimized, rather than explicitly building specific Hamiltonians.
This paper investigates the potential of quantum computing to accelerate combinatorial optimization algorithms. Traditional combinatorial optimization problems, such as the Traveling Salesperson Problem (TSP) and the Integer Linear Programming (ILP), often require exponential time to solve with classical algorithms. We propose a framework leveraging quantum algorithms, specifically the Variational Quantum Eigensolver (VQE), to tackle these problems. The core idea involves mapping the optimization problem into a quantum circuit, where the VQE algorithm iteratively refines a quantum state to find the optimal solution. We explore the theoretical underpinnings of this approach, focusing on the challenges and opportunities associated with quantum state preparation, measurement, and error mitigation. The paper provides a conceptual outline of the methodology and discusses potential improvements and future research directions. We demonstrate the feasibility of applying quantum computing to combinatorial optimization and highlight the potential for significant speedups compared to classical methods, especially as quantum hardware matures.
This dissertation addresses fundamental challenges in solving constrained combinatorial optimization problems using hybrid quantum-classical computing approaches. Quantum computing shows promise for tackling computationally intractable problems, current approaches face significant limitations. The path forward in computing involves combining quantum and classical computing approaches. Quantum processing units (QPUs) will play a key role in accelerating algorithms in optimization, simulation, and machine learning applications beyond what classical computers can achieve on their own, similar to how graphical processing units (GPUs) have become integral to general-purpose computing. The current generation of quantum devices is limited by noise and circuit depth constraints, making it challenging to gain practical utility. Hybrid quantum-classical approaches, which combine quantum computing resources with classical computing resources, are essential for addressing these limitations on near-term devices. In the area of combinatorial optimization, current practical state-of-the-art hybrid approaches are variational in nature, relying on parameterized quantum circuits to encode solutions and classical optimization techniques to find optimal parameters. While the ultimate goal of solving computationally intractable problems is to find provably optimal solutions, practical constraints of real-world scenarios often necessitate focusing on efficiently obtaining high-quality, near-optimal solutions. The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical approach for tackling these challenging problems that are encoded using quadratic and higher-order unconstrained binary optimization problems (QUBO and HUBO). QAOA alternates between two operators: a problem-specific cost Hamiltonian that encodes the optimization objective, and a mixer Hamiltonian that explores the solution space. The algorithm's parameters are iteratively optimized using classical methods to maximize the probability of sampling high-quality solutions. However, these methods often struggle with solution feasibility, especially for problems where the solutions must satisfy specific constraints. Penalty-based encodings require careful parameter tuning and risk producing infeasible solutions, while feasibility-preserving methods demand deeper quantum circuits that are challenging to implement on near-term quantum devices. In this dissertation, we present novel strategies that reduce or omit dependency on penalty parameters while maintaining solution quality, effectively capturing both optimal and near-optimal solutions, and balancing the trade-offs between circuit depth and solution feasibility. We present SCOOP, a novel QAOA-based framework for solving constrained optimization problems. SCOOP transforms a constrained problem into an unconstrained counterpart, forming SCOOP problem twins. QAOA operates on the unconstrained twin to identify potential optimal and near-optimal solutions. Effective classical post-processing reduces the solution set to the constrained problem space. Our SCOOP approach is solution-enhanced, objective-function-compatible, and scalable. We demonstrate the effectiveness of the SCOOP framework on selected NP-hard problems, some of which can be encoded quadratically (such as Minimum Vertex Cover and Maximum Independent Set), and some that can be encoded using higher-order terms (such as Minimum Dominating Set, Minimum Maximal Matching, and Minimum Set Cover). We also apply our framework to constrained combinatorial optimization problems that can be solved in polynomial time, such as Minimum Edge Cover and Maximum Matching. Experimental results on full statevector simulators and tensor network simulators show that SCOOP significantly improves the probability of finding high-quality feasible solutions across various problem classes, on problems that can be encoded quadratically as well as problems that can be encoded using higher-order terms. We also demonstrate the effectiveness of SCOOP on real quantum hardware, specifically IBM Quantum Heron R2 processors with 156 qubits, showing that SCOOP can be applied to practical problems with current quantum hardware limitations. This work makes significant contributions toward achieving quantum utility by enabling systematic investigation of constrained optimization problems on quantum devices. While current quantum computing research primarily focuses on unconstrained problems like MaxCut, real-world applications typically involve constraints. SCOOP provides a scalable framework for encoding constrained problems, extending quantum computing's practical utility beyond simple unconstrained cases. By eliminating dependency on penalty parameters and providing natural problem formulations, SCOOP makes it feasible to explore quantum solutions for constrained problems where classical exact methods become intractable. This capability is crucial for identifying promising avenues for demonstrating quantum utility across diverse constrained optimization problems, advancing the practical applicability of quantum computing approaches.