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1. Noise-tolerant quantum speedups in quantum annealing without fine tuning

   https://doi.org/10.1088/2058-9565/abd59a  

   Kapit, Eliot; Oganesyan, Vadim (February 2021, Quantum Science and Technology)

   Abstract Quantum annealing is a powerful alternative model of quantum computing, which can succeed in the presence of environmental noise even without error correction. However, despite great effort, no conclusive demonstration of a quantum speedup (relative to state of the art classical algorithms) has been shown for these systems, and rigorous theoretical proofs of a quantum advantage (such as the adiabatic formulation of Grover’s search problem) generally rely on exponential precision in at least some aspects of the system, an unphysical resource guaranteed to be scrambled by experimental uncertainties and random noise. In this work, we propose a new variant of quantum annealing, called RFQA, which can maintain a scalable quantum speedup in the face of noise and modest control precision. Specifically, we consider a modification of flux qubit-based quantum annealing which includes low-frequency oscillations in the directions of the transverse field terms as the system evolves. We show that this method produces a quantum speedup for finding ground states in the Grover problem and quantum random energy model, and thus should be widely applicable to other hard optimization problems which can be formulated as quantum spin glasses. Further, we explore three realistic noise channels and show that the speedup from RFQA is resilient to 1/f-like local potential fluctuations and local heating from interaction with a sufficiently low temperature bath. Another noise channel, bath-assisted quantum cooling transitions, actually accelerates the algorithm and may outweigh the negative effects of the others. We also detail how RFQA may be implemented experimentally with current technology. 

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2. Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions

   Harrow, Aram W.; Wei, AY (January 2020, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms)

   Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference. 

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3. A Diffusion Approximation Theory of Momentum Stochastic Gradient Descent in Nonconvex Optimization

   https://doi.org/10.1287/stsy.2021.0083  

   Liu, Tianyi; Chen, Zhehui; Zhou, Enlu; Zhao, Tuo (December 2021, Stochastic Systems)

   Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks. 

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4. Quantum Social Computing Approaches for Influence Maximization

   https://doi.org/10.1109/GLOBECOM48099.2022.10000698  

   Dinh, Thang; Nguyen, An; Nguyen, Uyen; Nguyen, Giang (January 2023, GLOBECOM 2022 - 2022 IEEE Global Communications Conference)

   Influence Maximization (IM), which seeks a small set of important nodes that spread the influence widely into the network, is a fundamental problem in social networks. It finds applications in viral marketing, epidemic control, and assessing cascading failures within complex systems. Despite the huge amount of effort, finding near-optimal solutions for IM is difficult due to its NP-completeness. In this paper, we propose the first social quantum computing approaches for IM, aiming to retrieve near-optimal solutions. We propose a two-phase algorithm that 1) converts IM into a Max-Cover instance and 2) provides efficient quadratic unconstrained binary optimization formulations to solve the Max-Cover instance on quantum annealers. Our experiments on the state-of-the-art D-Wave annealer indicate better solution quality compared to classical simulated annealing, suggesting the potential of applying quantum annealing to find high-quality solutions for IM. 

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5. Optimization Applications as Quantum Performance Benchmarks

   https://doi.org/10.1145/3678184  

   Lubinski, Thomas; Coffrin, Carleton; McGeoch, Catherine; Sathe, Pratik; Apanavicius, Joshua; Bernal_Neira, David; Development, Quantum Economic (September 2024, ACM Transactions on Quantum Computing)

   Combinatorial optimization is anticipated to be one of the primary use cases for quantum computation in the coming years. The Quantum Approximate Optimization Algorithm and Quantum Annealing can potentially demonstrate significant run-time performance benefits over current state-of-the-art solutions. Inspired by existing methods to characterize classical optimization algorithms, we analyze the solution quality obtained by solving Max-cut problems using gate-model quantum devices and a quantum annealing device. This is used to guide the development of an advanced benchmarking framework for quantum computers designed to evaluate the trade-off between run-time execution performance and the solution quality for iterative hybrid quantum-classical applications. The framework generates performance profiles through compelling visualizations that show performance progression as a function of time for various problem sizes and illustrates algorithm limitations uncovered by the benchmarking approach. As an illustration, we explore the factors that influence quantum computing system throughput, using results obtained through execution on various quantum simulators and quantum hardware systems. 

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