We present a novel approach to accelerate the Goemans-Williamson (GW) randomized rounding procedure for quadratic unconstrained binary optimization (QUBO) problems. Instead of solving the conventional semi-definite programming (SDP) relaxation, which is computationally expensive, we employ a difference-of-convex (DC) optimization framework to efficiently approximate the SDP solution. The DC optimization produces candidate vectors that are then used within the GW randomized rounding scheme to generate high-quality binary solutions. Furthermore, we perform direct expectation minimization over manifolds of matrices with limited rank to further enhance the solution quality. Our method is benchmarked on real-world QUBO instances, including inverse kinematics problems, and compared against state-of-the-art solvers, such as quantum-inspired algorithms, demonstrating competitive approximation guarantees alongside substantial computational gains.
Quadratic Unconstrained Binary Optimization (QUBO) problems are prevalent in various applications and are known to be NP-hard. The seminal work of Goemans and Williamson introduced a semidefinite programming (SDP) relaxation for such problems, solvable in polynomial time that upper bounds the optimal value. Their approach also enables randomized rounding techniques to obtain feasible solutions with provable performance guarantees. In this work, we identify instances of QUBO problems where matrix multiplicative weight methods lead to quantum and quantum-inspired algorithms that approximate the Goemans-Williamson SDP exponentially faster than existing methods, achieving polylogarithmic time complexity relative to the problem dimension. This speedup is attainable under the assumption that the QUBO cost matrix is sparse when expressed as a linear combination of Pauli strings satisfying certain algebraic constraints, and leverages efficient quantum and classical simulation results for quantum Gibbs states. We demonstrate how to verify these conditions efficiently given the decomposition. Additionally, we explore heuristic methods for randomized rounding procedures and extract the energy of a feasible point of the QUBO in polylogarithmic time. While the practical relevance of instances where our methods excel remains to be fully established, we propose heuristic algorithms with broader applicability and identify Kronecker graphs as a promising class for applying our techniques. We conduct numerical experiments to benchmark our methods. Notably, by utilizing tensor network methods, we solve an SDP with
D=250 variables and extract a feasible point which is certifiably within
0.15% of the optimum of the QUBO through our approach on a desktop, reaching dimensions millions of times larger than those handled by existing SDP or QUBO solvers, whether heuristic or rigorous.
The Maximum Cut (MaxCut) problem is NP-Complete, and obtaining its optimal solution is NP-hard in the worst case. As a result, heuristic-based algorithms are commonly used, though their design often requires significant domain expertise. More recently, learning-based methods trained on large (un)labeled datasets have been proposed; however, these approaches often struggle with generalizability and scalability. A well-known approximation algorithm for MaxCut is the Goemans-Williamson (GW) algorithm, which relaxes the Quadratic Unconstrained Binary Optimization (QUBO) formulation into a semidefinite program (SDP). The GW algorithm then applies hyperplane rounding by uniformly sampling a random hyperplane to convert the SDP solution into binary node assignments. In this paper, we propose a training-data-free approach based on a non-episodic reinforcement learning formulation, in which an agent learns to select improved rounding hyperplanes that yield better cuts than those produced by the GW algorithm. By optimizing over a Markov Decision Process (MDP), our method consistently achieves better cuts across large-scale graphs with varying densities and degree distributions.
We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erdös-Rényi graphs. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into approximations of the original quadratic optimization problem.
This repository contains the source code, computational experiments, analysis, and benchmarks accompanying the paper A Direct SDP-Based Framework for Certified Bounds for Constrained Quadratic Binary Optimization Problems. The repository implements a direct semidefinite programming (SDP) approach for constrained quadratic binary optimization, designed to obtain strong bounds for exact optimization. The SDP relaxation is strengthened using quadratized linear constraints and triangle (hypermetric) inequalities. To address the numerical infeasibility of SDP solutions obtained from numerical solvers such as MOSEK, the framework includes a correction procedure that converts approximate SDP solutions into mathematically certified bounds suitable for use in Branch-and-Bound algorithms. The repository also implements an adaptation of the Goemans–Williamson randomized hyperplane rounding scheme to constrained binary optimization, together with problem-specific feasibility restoration. Computational experiments are provided for the Sparsest k-Subgraph and Maximum Independent Set problems. The work presented in this repository was funded and developed under the QBIQ project (project no. 644), funded by the Slovenian Research and Innovation Agency (ARIS) through the Applied Projects programme. The code and accompanying data are intended to facilitate reproducibility of the computational results and provide a foundation for further research on exact SDP-based methods for constrained quadratic binary optimization.
This thesis is devoted to four modern quantum optimisation algorithms and their applications. In the first chapter, we provide a complete Quantum Approximate Optimisation Algorithm (QAOA) pipeline for the discrete global minimum-variance portfolio (DGMVP) problem. We introduce an ansatz with efficient initial-state preparation, new hard mixing operators satisfying the constraints of the DGMVP problem, and scalable cost operators utilising binary encoding of the qubits. Benchmarking the method with the most effective optimisation strategies---dual annealing and unfrozen layerwise optimisation---indicates a scaling advantage over classical methods in finding the global optimum of the DGMVP problem. In the second chapter, we show an end-to-end quantum/quantum-inspired Goemans-Williamson semidefinite programming (SDP) method for relaxing Quadratic Unconstrained Binary Optimisation (QUBO) problems and extracting their rounding energy. Our method identifies a class of QUBO instances that is effectively solvable in polylogarithmic time relative to the problem dimension, under the assumptions of Pauli sparsification of the QUBO cost matrix, reducibility of the diagonal constraints, and efficient evaluation of the corresponding Gibbs-state expectation values. Further, we identify the commuting 1D Hamiltonian and the Kronecker graph as practically important instances. Our numerical experiments show that our method solves the SDP and QUBO with up to
250 variables for 1D Hamiltonian instances and achieves results competitive with the best classical solver for Kronecker graphs, giving strong evidence of the power of our algorithm. In the third chapter, we improve the current QAOA method with an iterative warm-start initial-state preparation method. We implement the
t-th order statistic state as the warm-start initial state generated from each QAOA optimisation loop. We achieve an improved approximation ratio when solving the 3-regular MaxCut problem, and find that the worst-case approximation ratio converges to that of the best classical algorithm for
p=1 QAOA. Additionally, our method achieves more favourable scaling than standalone QAOA, single-string warm-started QAOA, and a classical constrained-sampling approach when solving the DGMVP problem. In the fourth chapter, we provide a hard-constrained Grover adaptive search (GAS) method for fixed-sum quadratic binary optimisation problems, and an extension of the alternating direction method of multipliers (ADMM) framework for fixed-sum quartic binary optimisation. Our method guarantees at least a quadratic speed-up over the classical method, and exponentially reduces the gate count for solving fixed-sum binary optimisation when compared with the soft-constrained GAS in the fault-tolerant quantum computing regime. In summary, our contributions include efficient algorithms and novel frameworks that substantially improve performance and broaden the applicability of these quantum optimisation methods. The proposed methods exhibit favourable scaling behaviour and indicate potential quantum advantages in solving challenging optimisation problems, thereby motivating further research at the interface of quantum computing and optimisation.
There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.