Quantum computers could significantly outperform their classical counterparts for solving hard optimization problems. One prominent classical technique for such problems is known as branch-and-bound. This work describes a quantum algorithm for accelerating general branch-and-bound methods, and gives an application to spin glasses.
Due to the expected disparity in quantum vs. classical clock speeds, quantum advantage for branch and bound algorithms is more likely achievable in settings involving large search trees and low operator evaluation costs. Therefore, in this paper, we describe and experimentally validate an exact classical branch and bound solver for quadratic unconstrained binary optimization (QUBO) problems that matches these criteria. Our solver leverages cheap-to-implement bounds from the literature previously proposed for Ising models, including that of Hartwig, Daske, and Kobe from 1984. We detail a variety of techniques from high-performance computing and operations research used to boost solver performance, including a global variable reordering heuristic, a primal heuristic based on simulated annealing, and a truncated computation of the recursive bound. We also outline a number of simple and inexpensive bound extrapolation techniques. Finally, we conduct an extensive empirical analysis of our solver, comparing its performance to state-of-the-art QUBO and MaxCut solvers, and discuss the challenges of a speedup via quantum branch and bound beyond those faced by any quadratic quantum speedup.
Mixed Integer Programs (MIPs) model many optimization problems of interest in Computer Science, Operations Research, and Financial Engineering. Solving MIPs is NP-Hard in general, but several solvers have found success in obtaining near-optimal solutions for problems of intermediate size. Branch-and-Cut algorithms, which combine Branch-and-Bound logic with cutting-plane routines, are at the core of modern MIP solvers. Montanaro proposed a quantum algorithm with a near-quadratic speedup compared to classical Branch-and-Bound algorithms in the worst case, when every optimal solution is desired. In practice, however, a near-optimal solution is satisfactory, and by leveraging tree-search heuristics to search only a portion of the solution tree, classical algorithms can perform much better than the worst-case guarantee. In this paper, we propose a quantum algorithm, Incremental-Quantum-Branch-and-Bound, with universal near-quadratic speedup over classical Branch-and-Bound algorithms for every input, i.e., if classical Branch-and-Bound has complexity Q on an instance that leads to solution depth d, Incremental-Quantum-Branch-and-Bound offers the same guarantees with a complexity of O~(Qd). Our results are valid for a wide variety of search heuristics, including depth-based, cost-based, and A∗ heuristics. Universal speedups are also obtained for Branch-and-Cut as well as heuristic tree search. Our algorithms are directly comparable to commercial MIP solvers, and guarantee near quadratic speedup whenever Q≫d. We use numerical simulation to verify that Q≫d for typical instances of the Sherrington-Kirkpatrick model, Maximum Independent Set, and Portfolio Optimization; as well as to extrapolate the dependence of Q on input size parameters. This allows us to project the typical performance of our quantum algorithms for these important problems.
In the era of quantum technology, benchmarking classical algorithms is necessary for certifying the results given by the quantum device to the optimization problem that is wanted to solve. Here we analyse a new algorithm that gives an upper and a lower bound to the ground state solution of Ising type optimization problems. We test its performance in solution planted problems and compare it with simulated annealing, a common algorithm to solve optimization problems, getting in general more reliable results in the same amount of time. Additionally, we verify the tunable hardness of the planting schemes used to generate the optimization problems.
We propose a complete quantum-classical hybrid branch-and-bound algorithm (QCBB) to solve binary linear programs with equality constraints. That includes bound calculation, convergence metrics and optimality guarantee to the quantum optimization based algorithm, which makes our method directly comparable to classical methods. Key aspects of the proposed algorithm are (i) encapsulation of the quantum optimization method, (ii) utilization of noisy samples for problem reduction, (iii) classical approximation based bound calculation, (iv) branch and bound traits like gap-based stopping criterion and monotonic increase in solution quality, (v) integrated composition of many different solutions that can be improved individually. We show numerical results on set partitioning problem instances and provide many details about the characteristics of the different steps of the algorithm.
Mathematical optimization is a field where the potential of quantum algorithms is actively being explored. The Quantum Approximate Optimization Algorithm (QAOA) serves as a notable quantum-classical hybrid approach aimed at solving combinatorial optimization challenges on Noisy Intermediate-Scale Quantum (NISQ) devices. In QAOA, classical variables find representation in qubits via a one-to-one mapping. A newer entrant, Quantum Random Access Optimization (QRAO), encodes problems using fewer qubits than QAOA by leveraging the Quantum Random Access Code (QRAC) to structure a problem into its Hamiltonian form. Notably, the QRAO Hamiltonian, dubbed the Quantum Relaxation Hamiltonian, boasts of a ground state energy inferior to that of the QAOA Hamiltonian. Nonetheless, the QRAO's ground state exhibits entanglement, making it starkly different from QAOA's ground state. As a workaround, preceding studies formulated solutions to the original problem using a rounding algorithm. Our research introduces the Quantum Relaxation based Branch-and-Bound (QR-BnB) algorithm. This novel method marries Quantum Relaxation and the classical exact solution strategy, Branch-and-Bound. Within the QR-BnB framework, a classical device shoulders the responsibility of the branch-and-bound operation, while its quantum counterpart dedicates itself to determining each ground state energy for the subproblem's Quantum Relaxation Hamiltonian. This fusion eliminates the prerequisite of the rounding algorithm by crafting the optimal solution through branching operations. Preliminary simulations of the QR-BnB approach attest to its capability in accurately solving a MaxCut problem. As future work, exploring its applicability on diverse combinatorial optimization problems holds promise. Our algorithm signals a step forward in harnessing NISQ devices for tackling expansive optimization challenges.
In this thesis, we have considered two important problems, Boolean satisfiability (SAT) and derivative free optimization in the context of large scale quantum computers. In the first part, we survey well known classical techniques for solving satisfiability. We compute the approximate time it would take to solve SAT instances using quantum techniques and compare it with state-of-the heart classical heuristics employed annually in SAT competitions. In the second part of the thesis, we consider a few classically well known algorithms for derivative free optimization which are \nubiquitously employed in engineering problems. We propose a quantum speedup to this classical algorithm by using techniques of the quantum minimum finding algorithm. In the third part of the thesis, we consider practical applications in the fields of bio-informatics, petroleum refineries and civil engineering which involve solving either satisfiability or derivative free optimization. We investigate if using known quantum techniques to speedup these algorithms directly translate to \nthe benefit of industries which invest in technology to solve these problems. In the last section, we propose a few open problems which we feel are immediate hurdles, either from an algorithmic or architecture perspective to getting a convincing speedup for the practical problems considered.