Abstract Integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into an unconstrained form through the use of binary variables, which is an indirect and resource-consuming way of solving it. We develop an algorithm that maps and solves an IP problem in its original form to any quantum system possessing a large number of accessible internal degrees of freedom that are controlled with sufficient accuracy. This work leverages the principle of superposition to solve the optimization problem. Using a single Rydberg atom as an example, we associate the integer values to electronic states belonging to different manifolds and implement a selective superposition of different states to solve the full IP problem. The optimal solution is found within a few microseconds for prototypical IP problems with up to eight variables and four constraints. This also includes non-linear IP problems, which are usually harder to solve with classical algorithms when compared to their linear counterparts. Our algorithm for solving IP is benchmarked by a well-known classical algorithm (branch and bound) in terms of the number of steps needed for convergence to the solution. This approach carries the potential to improve the solutions obtained for larger-size problems using hybrid quantum–classical algorithms.
Abstract Integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into an unconstrained form through the use of binary variables, which is an indirect and resource-consuming way of solving it. We develop an algorithm that maps and solves an IP problem in its original form to any quantum system possessing a large number of accessible internal degrees of freedom that are controlled with sufficient accuracy. This work leverages the principle of superposition to solve the optimization problem. Using a single Rydberg atom as an example, we associate the integer values to electronic states belonging to different manifolds and implement a selective superposition of different states to solve the full IP problem. The optimal solution is found within a few microseconds for prototypical IP problems with up to eight variables and four constraints. This also includes non-linear IP problems, which are usually harder to solve with classical algorithms when compared to their linear counterparts. Our algorithm for solving IP is benchmarked by a well-known classical algorithm (branch and bound) in terms of the number of steps needed for convergence to the solution. This approach carries the potential to improve the solutions obtained for larger-size problems using hybrid quantum–classical algorithms.
Integer programming (IP) is an NP-hard combinatorial optimization problem that is widely used to represent a diverse set of real-world problems spanning multiple fields, such as finance, engineering, logistics, and operations research. It is a hard problem to solve using classical algorithms, as its complexity increases with problem size. Most quantum algorithms for solving IP are highly resource-inefficient because they encode integers into qubits. In [1], the issue of resource inefficiency was addressed by mapping integer variables to qudits. However, [1] has limited practical value due to a lack of scalability to multiple qudits to encode larger problems. In this work, by extending upon the ideas of [1], a circuit-based scalable quantum algorithm is presented using multiple interacting qudits for which we show a quantum speed-up for solving the integer programming problem. The quantum algorithm consists of a distillation function that efficiently separates the feasible from the infeasible regions, a phase-amplitude encoding for the cost function, and a quantum phase estimation coupled with a multi-controlled single-qubit rotation for optimization. We prove that our algorithm yields the optimal solution with the maximum measurement probability. The time complexity for the quantum algorithm is shown to be O(rP(mn2logd+dn/2/Nys1/2+nεQPE−1)) for a problem with the number of variables n taking d integer values, satisfying m constraints with a precision of εQPE and rP repetitions required for reaching a success probability P. Compared to the classical time complexity of brute force O(dn) and the classical exact algorithm O((logn)O(n)), it incurs a reduction of ∼dn/2 in the time complexity (for a polynomial number of repetitions and precision requirements) for solving a general polynomial IP problem.
Quantum computing is emerging as a new computing resource that could be superior to conventional computing for certain classes of optimization problems. However, in principle, most existing approaches to quantum optimization are intended to solve unconstrained binary programming problems, while mixed-integer linear programming is of most interest in practice. We attempt to bridge the gap between the capability of quantum computing and real-world applications by developing a new approach for mixed-integer programming. The approach applies Benders decomposition to decompose the mixed-integer programming into binary programming and linear programming sub-problems, which are solved by a noisy intermediate-scale quantum processor and conventional processor, respectively. The algorithm is provably able to reach the optimal solution of the original mixed-integer programming problem. The algorithm is tested on a D-Wave 2000Q quantum processing unit and is shown to be effective for small-scaled test cases. We also test the algorithm on a mixed-integer programming inspired by power system applications. Many insights are drawn from the numerical results for both the capabilities and limitations of the proposed algorithm.
To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing is realized, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.
Quantum computing (QC) is emerging as a new computing resource that could be superior to conventional computing (CC) for certain classes of optimization problems. However, in principle QC can only solve unconstrained binary programming problems, while mixed-integer linear programming (MIP) is of most interest in practice. We attempt to bridge the gap between the capability of QC and real-world applications by developing a new approach for MIP. The idea is decomposing the MIP into binary programming and linear programming (LP) problems, which are respectively solved by QC and conventional computing. We formalize a decomposition approach that ensures that with a sufficient number of back and forth iterations, the algorithm can reach the optimal solution of the original MIP problem. The algorithm is tested on a 2000Q D-Wave quantum processing units (QPU) and is shown to be effective for small-scaled test cases.
Abstract We propose a new approach to utilize quantum computers for binary linear programming (BLP), which can be extended to general integer linear programs (ILP). Quantum optimization algorithms, hybrid or quantum-only, are currently general purpose, standalone solvers for ILP. However, to consider them practically useful, we expect them to overperform the current state of the art classical solvers. That expectation is unfair to quantum algorithms: in classical ILP solvers, after many decades of evolution, many different algorithms work together as a robust machine to get the best result. This is the approach we would like to follow now with our quantum ‘solver’ solutions. In this study we wrap any suitable quantum optimization algorithm into a quantum informed classical constraint generation framework. First we relax our problem by dropping all constraints and encode it into an Ising Hamiltonian for the quantum optimization subroutine. Then, by sampling from the solution state of the subroutine, we obtain information about constraint violations in the initial problem, from which we decide which coupling terms we need to introduce to the Hamiltonian. The coupling terms correspond to the constraints of the initial binary linear program. Then we optimize over the new Hamiltonian again, until we reach a feasible solution, or other stopping conditions hold. Since one can decide how many constraints they add to the Hamiltonian in a single step, our algorithm is at least as efficient as the (hybrid) quantum optimization algorithm it wraps. We support our claim with results on small scale minimum cost exact cover problem instances.
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.