9,394 research outputs found
Quantum Optimization From a Computer Science Perspective
Optimization problems are ubiquitous in but not limited to the sciences, engineering, and applied mathematics. Examples range from the fastest way USPS can route packages through a delivery network to the best way an autonomous vehicle can navigate through a given traffic environment. Classical optimization algorithms dominate the way we solve these problems. However, with the rapid advance of quantum computers, we are looking at novel, quantum-inspired ways of solving old problems to achieve some speedup over classical algorithms. Specifically, we are looking at the Quantum Approximate Optimization Algorithm (QAOA). We show that QAOA provides a tunable, optimization algorithm whose quantum circuit grows linearly with the number of constraints for MAXSAT, an NP-complete problem
Quantum Gate-Model Approaches to Exact and Approximate Optimization
Many of the most challenging computational problems arising in practical applications are tackled by heuristic algorithms which have not been rigorously proven to outperform other approaches but rather have been empirically demonstrated to be effective. While quantum heuristics have been proposed since the early days of quantum computing, true empirical evaluation of the real-world performance of these algorithms is only becoming possible now as increasingly powerful quantum gate-model devices continue to come online.In this talk, I will give an overview of the NASA QuAIL team's ongoing investigation into quantum gate-model heuristic algorithms for exact and approximate optimization. In particular, we consider the performance of the Quantum Approximate Optimization Algorithm on NP-hard optimization problems, and describe algorithm parameter setting strategies for real-world quantum hardware. We then show a generalization of QAOA circuits, the Quantum Alternating Operator Ansatz, especially suitable for low-resource implementations of QAOA for problems with hard (feasibility) constraints. The talk will conclude with a discussion of research challenges, particularly for optimization and sampling applications of QAOA, and the potential of more general quantum heuristics to give advantages over classical computers
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
Quantum-Assisted Solution Paths for the Capacitated Vehicle Routing Problem
Many relevant problems in industrial settings result in NP-hard optimization
problems, such as the Capacitated Vehicle Routing Problem (CVRP) or its reduced
variant, the Travelling Salesperson Problem (TSP). Even with today's most
powerful classical algorithms, the CVRP is challenging to solve classically.
Quantum computing may offer a way to improve the time to solution, although the
question remains open as to whether Noisy Intermediate-Scale Quantum (NISQ)
devices can achieve a practical advantage compared to classical heuristics. The
most prominent algorithms proposed to solve combinatorial optimization problems
in the NISQ era are the Quantum Approximate Optimization Algorithm (QAOA) and
the more general Variational Quantum Eigensolver (VQE). However, implementing
them in a way that reliably provides high-quality solutions is challenging,
even for toy examples. In this work, we discuss decomposition and formulation
aspects of the CVRP and propose an application-driven way to measure solution
quality. Considering current hardware constraints, we reduce the CVRP to a
clustering phase and a set of TSPs. For the TSP, we extensively test both QAOA
and VQE and investigate the influence of various hyperparameters, such as the
classical optimizer choice and strength of constraint penalization. Results of
QAOA are generally of limited quality because the algorithm does not reach the
energy threshold for feasible TSP solutions, even when considering various
extensions such as recursive, warm-start and constraint-preserving mixer QAOA.
On the other hand, the VQE reaches the energy threshold and shows a better
performance. Our work outlines the obstacles to quantum-assisted solutions for
real-world optimization problems and proposes perspectives on how to overcome
them.Comment: Submitted to the IEEE for possible publicatio
Ising formulations of many NP problems
We provide Ising formulations for many NP-complete and NP-hard problems,
including all of Karp's 21 NP-complete problems. This collects and extends
mappings to the Ising model from partitioning, covering and satisfiability. In
each case, the required number of spins is at most cubic in the size of the
problem. This work may be useful in designing adiabatic quantum optimization
algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more
references; v3: substantial revision and extension, to-be-published versio
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