Quantum control aims to manipulate quantum systems toward specific quantum states or desired operations. Designing highly accurate and effective control steps is vitally important to various quantum applications, including energy minimization and circuit compilation. We consider discrete binary quantum control problems and propose optimization techniques with provable error bounds. We apply an alternating direction method of multipliers (ADMM) algorithm to solve the continuous relaxation of the penalized model, and round the resulting solution. We show how to amortize the computationally expensive factorization of the Hamiltonians by exploiting the discrete nature of the controls. Our algorithms can obtain high-quality control results, as demonstrated by numerical studies on diverse quantum control examples.

With the advancement of quantum hardware and theory, quantum algorithms have demonstrated their effectiveness in finding high-quality solutions for Quadratic Unconstrained Binary Optimization (QUBO) problems. Moreover, there exists a myriad of novel methods aimed at finding reasonable solutions to QUBO problems efficiently, named Ising solvers, given the equivalence of QUBO to the transverse field Ising problem in statistical physics. However, optimization problems of interest to process system engineering are more general than QUBOs, usually involving discrete non-binary and continuous variables and constraints. Although it is possible to map such problems onto QUBOs, the size of solvable QUBOs is inherently limited by the number of qubits in available quantum computers. Moreover, the global optimality of the solution cannot be guaranteed due to the thermal noise and the nature of the algorithms implementable for this problem class in quantum devices. Guaranteeing global optimality has been recently addressed using hybrid quantum-classical algorithms.
This work presents a hybrid quantum branch-and-bound method to address these challenges. This method utilizes the computational strengths of quantum hardware alongside the global search capabilities of the branch-and-bound method. The strategy of breaking the problem into smaller subproblems aligns well with the capabilities of quantum hardware. Contrary to previous work on characterizing the hardness of a branch-and-bound enhanced with quantum algorithms for exploring the nodes, this work provides a practical implementation of the method relying on a classical branch-and-bound enhanced with Ising solvers, potentially leveraging quantum mechanics, in an open-access repository (https://github.com/SECQUOIA/QuantumBranchAndBound). The Ising solver serves as a heuristic method, and we further explore the strategy of when and where to apply it within the branch and bound framework. Particularly, limitations in quantum annealers related to the embedding problem set such limitations.
Additionally, a customized branching rule and an optimized embedding of QUBO problems are applied to accelerate the convergence of the algorithm, where even without enhancing the commercial solver Gurobi 11 with the solutions coming from Ising solvers, we can achieve a speedup of 20\% with respect to the default branching rule in Gurobi. The performance of the proposed method is evaluated on hundreds of QUBO instances from QUBOLib.jl (https://github.com/SECQUOIA/QUBOLib.jl), using Gurobi as the branch-and-bound solver, several physics-inspired classical heuristics and the D-Wave quantum annealer as the quantum solver. Detailed computational results are presented, and the proposed method, Gurobi, and the hybrid B\&B method are compared with classical heuristic approaches. Improvements in the solution time go up to 14\%, and the reduction of the number of nodes explored to prove optimality reaches 16\% using Ising solvers with respect to default Gurobi when solving QUBO problems to 0.01\% of optimality, highlighting the value of this hybrid approach.

In the last two decades, there has been much effort in finding secure protocols for two-party cryptographic tasks. It has since been discovered that even with quantum mechanics, many such protocols are limited in their security promises. In this work, we use stochastic selection, an idea from stochastic programming, to circumvent such limitations. For example, we find a way to switch between bit commitment, weak coin flipping, and oblivious transfer protocols to improve their security. We also use stochastic selection to turn trash into treasure yielding the first quantum protocol for Rabin oblivious transfer.