The weighted MAX $$k$$ k -CUT problem consists of finding a k-partition of a given weighted undirected graph G(V, E), such that the sum of the weights of the crossing edges is maximized. The problem is of particular interest as it has a multitude of practical applications. We present a formulation of the weighted MAX $$k$$ k -CUT suitable for running the quantum approximate optimization algorithm (QAOA) on noisy intermediate scale quantum (NISQ) devices to get approximate solutions. The new formulation uses a binary encoding that requires only $$|V|\log _2k$$ | V | log 2 k qubits. The contributions of this paper are as follows: (i) a novel decomposition of the phase-separation operator based on the binary encoding into basis gates is provided for the MAX $$k$$ k -CUT problem for $$k>2$$ k > 2 . (ii) Numerical simulations on a suite of test cases comparing different encodings are performed. (iii) An analysis of the resources (number of qubits, CX gates) of the different encodings is presented. (iv) Formulations and simulations are extended to the case of weighted graphs. For small k and with further improvements when k is not a power of two, our algorithm is a possible candidate to show quantum advantage on NISQ devices.

The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find approximate solutions of combinatorial optimization problems. Most of the literature is limited to quadratic problems without constraints. However, many practically relevant optimization problems do have (hard) constraints that need to be fulfilled. In this article, we present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space given by these constraints. We generalize the “XY”-mixer designed to preserve the subspace of “one-hot” states to the general case of subspaces given by a number of computational basis states. We expose the underlying mathematical structure which reveals more of how mixers work and how one can minimize their cost in terms of the number of CX gates, particularly when Trotterization is taken into account. Our analysis also leads to valid Trotterizations for an “XY”-mixer with fewer CX gates than is known to date. In view of practical implementations, we also describe algorithms for efficient decomposition into basis gates. Several examples of more general cases are presented and analyzed.

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In the past years, we have witnessed an increasing interest in smart buildings, in particular for optimal energy management, renewable energy sources, and smart appliances. In this paper, we investigate the problem of scheduling smart appliance operation in a given time horizon with a set of energy sources and accumulators. Appliance operation is modeled in terms of uninterruptible sequential phases with a given power demand, with the goal of minimizing the energy bill fulfilling duration, energy, and user preference constraints. A mixed-integer linear programming (MIP) model and greedy heuristic algorithm are given, which are used in a synergic way. We show how a general purpose (off-the-shelf) MIP-refining procedure can be effectively used for improving, in short computing time, the quality of the solutions provided by the initial greedy heuristic. Computational results confirm the viability of the overall approach, in terms of both solution quality and speed. PrologueMany successful matheuristic schemes use a black-box mixed integer linear programming (MIP) solver to generate high-quality heuristic solutions for difficult optimization problems. The hallmark of this approach is the availability of an (possibly incomplete) MIP model, and an external metascheme that iteratively solves sub-MIPs obtained by introducing invalid constraints (e.g., variable fixings) defining "interesting" neighborhoods of certain solutions. The goal of the approach is to iteratively refine the incumbent solution, producing a sequence of improved feasible solutions in short (or, at least, acceptable) computing times. The above solution-refinement approach is completely general, that is, it can in principle be applied to the original MIP without the need of ad hoc adaptations.