Quantum Optimization Problems

Krentel [J. Comput. System. Sci., 36, pp.490--509] presented a framework for an NP optimization problem that searches an optimal value among exponentially-many outcomes of polynomial-time computations. This paper expands his framework to a quantum optimization problem using polynomial-time quantum computations and introduces the notion of an ``universal'' quantum optimization problem similar to a classical ``complete'' optimization problem. We exhibit a canonical quantum optimization problem that is universal for the class of polynomial-time quantum optimization problems. We show in a certain relativized world that all quantum optimization problems cannot be approximated closely by quantum polynomial-time computations. We also study the complexity of quantum optimization problems in connection to well-known complexity classes.Comment: date change

An Integrated Programming and Development Environment for Adiabatic Quantum Optimization

Adiabatic quantum computing is a promising route to the computational power afforded by quantum information processing. The recent availability of adiabatic hardware has raised challenging questions about how to evaluate adiabatic quantum optimization programs. Processor behavior depends on multiple steps to synthesize an adiabatic quantum program, which are each highly tunable. We present an integrated programming and development environment for adiabatic quantum optimization called JADE that provides control over all the steps taken during program synthesis. JADE captures the workflow needed to rigorously specify the adiabatic quantum optimization algorithm while allowing a variety of problem types, programming techniques, and processor configurations. We have also integrated JADE with a quantum simulation engine that enables program profiling using numerical calculation. The computational engine supports plug-ins for simulation methodologies tailored to various metrics and computing resources. We present the design, integration, and deployment of JADE and discuss its potential use for benchmarking adiabatic quantum optimization programs by the quantum computer science community.Comment: 28 pages, 17 figures, feedback welcomed, even if it's criticism; v2 manuscript updated based on reviewer feedback; v3 manuscript updated based on reviewer feedback, title modifie

Efficiency of quantum versus classical annealing in non-convex learning problems

Quantum annealers aim at solving non-convex optimization problems by exploiting cooperative tunneling effects to escape local minima. The underlying idea consists in designing a classical energy function whose ground states are the sought optimal solutions of the original optimization problem and add a controllable quantum transverse field to generate tunneling processes. A key challenge is to identify classes of non-convex optimization problems for which quantum annealing remains efficient while thermal annealing fails. We show that this happens for a wide class of problems which are central to machine learning. Their energy landscapes is dominated by local minima that cause exponential slow down of classical thermal annealers while simulated quantum annealing converges efficiently to rare dense regions of optimal solutions.Comment: 31 pages, 10 figure

Quantum Annealing: from Viewpoints of Statistical Physics, Condensed Matter Physics, and Computational Physics

In this paper, we review some features of quantum annealing and related topics from viewpoints of statistical physics, condensed matter physics, and computational physics. We can obtain a better solution of optimization problems in many cases by using the quantum annealing. Actually the efficiency of the quantum annealing has been demonstrated for problems based on statistical physics. Then the quantum annealing has been expected to be an efficient and generic solver of optimization problems. Since many implementation methods of the quantum annealing have been developed and will be proposed in the future, theoretical frameworks of wide area of science and experimental technologies will be evolved through studies of the quantum annealing.Comment: 57pages, 15figures, to appear in "Lectures on Quantum Computing, Thermodynamics and Statistical Physics," Kinki University Series on Quantum Computing (World Scientific, 2012

Achieving robust and high-fidelity quantum control via spectral phase optimization

Achieving high-fidelity control of quantum systems is of fundamental importance in physics, chemistry and quantum information sciences. However, the successful implementation of a high-fidelity quantum control scheme also requires robustness against control field fluctuations. Here, we demonstrate a robust optimization method for control of quantum systems by optimizing the spectral phase of an ultrafast laser pulse, which is accomplished in the framework of frequency domain quantum optimal control theory. By incorporating a filtering function of frequency into the optimization algorithm, our numerical simulations in an abstract two-level quantum system as well as in a three-level atomic rubidium show that the optimization procedure can be enforced to search optimal solutions while achieving remarkable robustness against the control field fluctuations, providing an efficient approach to optimize the spectral phase of the ultrafast laser pulse to achieve a desired final quantum state of the system.Comment: 17 pages, 8 figure