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Proposed hybrid algorithms encode a combinatorial cost function into a problem Hamiltonian and optimize its energy by varying over a set of states with low circuit complexity. Classical processing is typically only used for the choice of variational parameters following gradient descent. As a consequence, these approaches are limited by the descriptive power of the associated states. We argue that for certain combinatorial optimization problems, such algorithms can be hybridized further, thus harnessing the power of efficient non-local classical processing. Specifically, we consider combining a quantum variational ansatz with a greedy classical post-processing procedure for the MaxCut-problem on $3$-regular graphs. We show that the average cut-size produced by this method can be quantified in terms of the energy of a modified problem Hamiltonian. This motivates the consideration of an improved algorithm which variationally optimizes the energy of the modified Hamiltonian. We call this a twisted hybrid algorithm since the additional classical processing step is combined with a different choice of variational parameters. We exemplify the viability of this method using the quantum approximate optimization algorithm (QAOA), giving analytic lower bounds on the expected approximation ratios achieved by twisted QAOA. These show that the necessary non-locality of the quantum ansatz can be reduced compared to the original QAOA: We find that for levels $p=2,\ldots, 6$, the level $p$ can be reduced by one while roughly maintaining the expected approximation ratio. This reduces the circuit depth by $4$ and the number of variational parameters by $2$., Comment: 29 pages, 12 figures

Molecular docking is an important step of the drug discovery process which aims at calculating the preferred position and shape of one molecule to a second when they are bound to each other. During such analysis, 3D representations of molecules are manipulated according to their degree of freedoms: rigid roto-translation and fragment rotations along the rotatable bonds. In our work, we focussed on one specific phase of the molecular docking procedure i.e. molecular unfolding (MU), which is used to remove the initial bias of a molecule by expanding it to an unfolded shape simpler to manipulate within the target cavity. The objective of the MU problem is to find the configuration that maximizes the molecular area, or equivalently, that maximizes the internal distances between atoms inside the molecule. We propose a quantum annealing approach to MU by formulating it as a high-order unconstrained binary optimization which was possible to solve on the latest D-wave annealing hardware (2000Q and advantage). Results and performances obtained with quantum annealers are compared with state of art classical solvers.

In this paper, we introduce two powerful graph reductions for the maximum weighted stable set (mwss) in general graphs. We show that these reductions allow to reduce the mwss in claw-free graphs to the mwss in a class of quasi-line graphs, that we call bipolar-free. For this latter class, we provide a new algorithmic decomposition theorem running in polynomial time. We then exploit this decomposition result and our reduction tools again to transform the problem to either a single matching problem or a longest path computation in an acyclic auxiliary graph (in this latter part we use some results of Pulleyblank and Shepherd [10]). Putting all the pieces together, the main contribution of this paper is a new polynomial time algorithm for the mwss in claw-free graphs. A rough analysis of the complexity of this algorithm gives a time bound of O(n6), where n is the number of vertices in the graph, and which we hope can be improved by a finer analysis. Incidentally, we prove that the mwss problem can be solved efficiently for any class of graphs that admits a "suitable" decomposition into pieces where the mwss is easy.