264 research outputs found
Nonlinear Quantum Optimization Algorithms via Efficient Ising Model Encodings
Despite extensive research efforts, few quantum algorithms for classical
optimization demonstrate realizable advantage. The utility of many quantum
algorithms is limited by high requisite circuit depth and nonconvex
optimization landscapes. We tackle these challenges to quantum advantage with
two new variational quantum algorithms, which utilize multi-basis graph
encodings and nonlinear activation functions to outperform existing methods
with remarkably shallow quantum circuits. Both algorithms provide a polynomial
reduction in measurement complexity and either a factor of two speedup
\textit{or} a factor of two reduction in quantum resources. Typically, the
classical simulation of such algorithms with many qubits is impossible due to
the exponential scaling of traditional quantum formalism and the limitations of
tensor networks. Nonetheless, the shallow circuits and moderate entanglement of
our algorithms, combined with efficient tensor method-based simulation, enable
us to successfully optimize the MaxCut of high-connectivity global graphs with
up to nodes (qubits) on a single GPU.Comment: 10 pages, 4 figure
Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints
Semidefinite programs are optimization methods with a wide array of
applications, such as approximating difficult combinatorial problems. One such
semidefinite program is the Goemans-Williamson algorithm, a popular integer
relaxation technique. We introduce a variational quantum algorithm for the
Goemans-Williamson algorithm that uses only qubits, a constant number
of circuit preparations, and expectation values in order to
approximately solve semidefinite programs with up to variables and constraints. Efficient optimization is achieved by encoding the
objective matrix as a properly parameterized unitary conditioned on an auxilary
qubit, a technique known as the Hadamard Test. The Hadamard Test enables us to
optimize the objective function by estimating only a single expectation value
of the ancilla qubit, rather than separately estimating exponentially many
expectation values. Similarly, we illustrate that the semidefinite programming
constraints can be effectively enforced by implementing a second Hadamard Test,
as well as imposing a polynomial number of Pauli string amplitude constraints.
We demonstrate the effectiveness of our protocol by devising an efficient
quantum implementation of the Goemans-Williamson algorithm for various NP-hard
problems, including MaxCut. Our method exceeds the performance of analogous
classical methods on a diverse subset of well-studied MaxCut problems from the
GSet library.Comment: 21 pages, 6 figures. Updated files to the version of manuscript
accepted by Quantu
Multi-objective gene-pool optimal mixing evolutionary algorithms
The recently introduced Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA), with a lean, but sufficient, linkage model and an efficient variation operator, has been shown to be a robust and efficient methodology for solving single objective (SO) optimization problems with superior performance compared to classic genetic algorithms (GAs) and estimation-of-distribution algorithms (EDAs). In this paper, we bring the strengths of GOMEAs to the multi-objective (MO) optimization realm. To this end, we modify the linkage learning procedure and the variation operator of GOMEAs to better suit the need of finding the whole Pareto-optimal front rather than a single best solution. Based on state-of-the-art studies on MOEAs, we further pinpoint and incorporate two other essential components for a scalable MO optimizer. First, the use of an elitist archive is beneficial for keeping track of non-dominated solutions when the main population size is limited. Second, clustering can be crucial if different parts of the Pareto-optimal front need to be handled differently. By combining these elements, we construct a multi-objective GOMEA (MO-GOMEA). Experimental results on various MO optimization problems confirm the capability and scalability of our MO-GOMEA that compare favorably with those of the well-known GA NSGA-II and the more recently introduced EDA mohBOA
Quantum and Classical Multilevel Algorithms for (Hyper)Graphs
Combinatorial optimization problems on (hyper)graphs are ubiquitous in science and industry. Because many of these problems are NP-hard, development of sophisticated heuristics is of utmost importance for practical problems. In recent years, the emergence of Noisy Intermediate-Scale Quantum (NISQ) computers has opened up the opportunity to dramaticaly speedup combinatorial optimization. However, the adoption of NISQ devices is impeded by their severe limitations, both in terms of the number of qubits, as well as in their quality. NISQ devices are widely expected to have no more than hundreds to thousands of qubits with very limited error-correction, imposing a strict limit on the size and the structure of the problems that can be tackled directly. A natural solution to this issue is hybrid quantum-classical algorithms that combine a NISQ device with a classical machine with the goal of capturing “the best of both worlds”.
Being motivated by lack of high quality optimization solvers for hypergraph partitioning, in this thesis, we begin by discussing classical multilevel approaches for this problem. We present a novel relaxation-based vertex similarity measure termed algebraic distance for hypergraphs and the coarsening schemes based on it. Extending the multilevel method to include quantum optimization routines, we present Quantum Local Search (QLS) – a hybrid iterative improvement approach that is inspired by the classical local search approaches. Next, we introduce the Multilevel Quantum Local Search (ML-QLS) that incorporates the quantum-enhanced iterative improvement scheme introduced in QLS within the multilevel framework, as well as several techniques to further understand and improve the effectiveness of Quantum Approximate Optimization Algorithm used throughout our work
QuOp_MPI: a framework for parallel simulation of quantum variational algorithms
QuOp_MPI is a Python package designed for parallel simulation of quantum
variational algorithms. It presents an object-orientated approach to quantum
variational algorithm design and utilises MPI-parallelised sparse-matrix
exponentiation, the fast Fourier transform and parallel gradient evaluation to
achieve the highly efficient simulation of the fundamental unitary dynamics on
massively parallel systems. In this article, we introduce QuOp_MPI and explore
its application to the simulation of quantum algorithms designed to solve
combinatorial optimisation algorithms including the Quantum Approximation
Optimisation Algorithm, the Quantum Alternating Operator Ansatz, and the
Quantum Walk-assisted Optimisation Algorithm.Comment: Software available at: https://github.com/Edric-Matwiejew/QuOp_MP
Multi-objective Gene-pool Optimal Mixing Evolutionary Algorithm with the interleaved multi-start scheme
The Multi-objective Gene-pool Optimal Mixing Evolutionary Algorithm (MO-GOMEA) has been shown to be a promising solver for multi-objective combinatorial optimization problems, obtaining an excellent scalability on both standard benchmarks and real-world applications. To attain optimal performance, MO-GOMEA requires its two parameters, namely the population size and the number of clusters, to be set properly with respect to the problem instance at hand, which is a non-trivial task for any EA practitioner. In this article, we present a new version of MO-GOMEA in combination with the so-called Interleaved Multi-start Scheme (IMS) for the multi-objective domain that eliminates the manual setting of these two parameters. The new MO-GOMEA is then evaluated on multiple benchmark problems in comparison with two well-known multi-objective evolutionary algorithms (MOEAs): Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D). Experiments suggest that MO-GOMEA with the IMS is an easy-to-use MOEA that retains the excellent performance of the original MO-GOMEA
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