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We investigate the use of amplitude amplification on the gate-based model of quantum computing as a means for solving combinatorial optimization problems. This study focuses primarily on quadratic unconstrained binary optimization (QUBO) problems, which are well-suited for qubit superposition states. Specifically, we demonstrate circuit designs which encode QUBOs as ‘cost oracle’ operations UC, which distribute phases across the basis states proportional to a cost function. We then show that when UC is combined with the standard Grover diffusion operator Us, one can achieve high probabilities of measurement for states corresponding to optimal and near optimal solutions while still only requiring O(π42N/M) iterations. In order to achieve these probabilities, a single scalar parameter ps is required, which we show can be found through a variational quantum–classical hybrid approach and can be used for heuristic solutions.

We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding the minimum or maximum solutions on a weighted directed graph. We focus on the geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by Gaussian distributions. We then demonstrate how an oracle that encodes these distributions can be used to solve for the optimal path via amplitude amplification. And finally, we explore the degree to which this algorithm is capable of solving cases that are generated using randomized weights, as well as a theoretical application for solving the Traveling Salesman problem.

Quantum computing has the potential to revolutionize the way hard computational problems are solved in terms of speed and accuracy. Quantum hardware is an active area of research and different hardware platforms are being developed. Quantum algorithms target each hardware implementation and bring advantages to specific applications. The focus of this paper is to compare how well quantum annealing techniques and the QAOA models constrained optimization problems. As a use case, a constrained optimization problem called mission covering optimization is used. Quantum annealing is implemented in adiabatic hardware such as D-Wave, and QAOA is implemented in gate-based hardware such as IBM. This effort provides results in terms of cost, timing, constraints held, and qubits used.

As superconducting qubits continue to advance technologically, the realization of quantum algorithms from theoretical abstraction to physical implementation requires knowledge of both quantum circuit construction as well as hardware limitations. In this study, we present results from experiments run on IBM’s 20-qubit “Poughkeepsie” architecture, with the goal of demonstrating various qubit qualities and challenges that arise in designing quantum algorithms. These include experimentally measuring T1 and T2 coherence times, gate fidelities, sequential CNOT gates, techniques for handling ancilla qubits, and finally CCNOT and QFT† circuits implemented on several different qubit geometries. Our results demonstrate various techniques for improving quantum circuits, which must compensate for limited connectivity, either through the use of SWAP gates or additional ancilla qubits.