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Register a new userGlobally optimizing QAOA circuit depth for constrained optimization problems
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We develop a global variable substitution method that reduces $n$-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to $3$-SAT and analyze the optimal quantum circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark $3$-SAT problems, we find that the upper bound of the circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.
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Present-day, noisy, small or intermediate-scale quantum processors---although far from fault-tolerant---support the execution of heuristic quantum algorithms, which might enable a quantum advantage, for example, when applied to combinatorial optimization problems. On small-scale quantum processors, validations of such algorithms serve as important technology demonstrators. We implement the quantum approximate optimization algorithm (QAOA) on our hardware platform, consisting of two superconducting transmon qubits and one parametrically modulated coupler. We solve small instances of the NP-complete exact-cover problem, with 96.6% success probability, by iterating the algorithm up to level two.
In recent years, there is a growing interest in using quantum computers for solving combinatorial optimization problems. In this work, we developed a generic, machine learning-based framework for mapping continuous-space inverse design problems into surrogate quadratic unconstrained binary optimization (QUBO) problems by employing a binary variational autoencoder and a factorization machine. The factorization machine is trained as a low-dimensional, binary surrogate model for the continuous design space and sampled using various QUBO samplers. Using the D-Wave Advantage hybrid sampler and simulated annealing, we demonstrate that by repeated resampling and retraining of the factorization machine, our framework finds designs that exhibit figures of merit exceeding those of its training set. We showcase the frameworks performance on two inverse design problems by optimizing (i) thermal emitter topologies for thermophotovoltaic applications and (ii) diffractive meta-gratings for highly efficient beam steering. This technique can be further scaled to leverage future developments in quantum optimization to solve advanced inverse design problems for science and engineering applications.
One of the major application areas of interest for both near-term and fault-tolerant quantum computers is the optimization of classical objective functions. In this work, we develop intuitive constructions for a large class of these algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. This is connected to effects from wavefunction confinement, phase randomization, and shadow defects lurking in the objective far away from the ideal solution. As an example, we explore the surprising deficiency of many quantum methods in solving uncoupled spin problems and how this is both predictive of performance on some more complex systems while immediately suggesting simple resolutions. Further examination of canonical problems like the Hamming ramp or bush of implications show that entanglement can be strictly detrimental to performance results from the underlying mechanism of solution in approaches like QAOA. Kinetic energy and graph Laplacian perspectives provide new insights to common initialization and optimal solutions in QAOA as well as new methods for more effective layerwise training. Connections to classical methods of continuous extensions, homotopy methods, and iterated rounding suggest new directions for research in quantum optimization. Throughout, we unveil many pitfalls and mechanisms in quantum optimization using a physical perspective, which aim to spur the development of novel quantum optimization algorithms and refinements.
In this paper, we extend a bio-inspired algorithm called the porcellio scaber algorithm (PSA) to solve constrained optimization problems, including a constrained mixed discrete-continuous nonlinear optimization problem. Our extensive experiment results based on benchmark optimization problems show that the PSA has a better performance than many existing methods or algorithms. The results indicate that the PSA is a promising algorithm for constrained optimization.
This paper considers a general convex constrained problem setting where functions are not assumed to be differentiable nor Lipschitz continuous. Our motivation is in finding a simple first-order method for solving a wide range of convex optimization problems with minimal requirements. We study the method of weighted dual averages (Nesterov, 2009) in this setting and prove that it is an optimal method.
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