1,592 research outputs found
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
The approximability of non-Boolean satisfiability problems and restricted integer programming
AbstractIn this paper we present improved approximation algorithms for two classes of maximization problems defined in Barland et al. (J. Comput. System Sci. 57(2) (1998) 144). Our factors of approximation substantially improve the previous known results and are close to the best possible. On the other hand, we show that the approximation results in the framework of Barland et al. hold also in the parallel setting, and thus we have a new common framework for both computational settings. We prove almost tight non-approximability results, thus solving a main open question of Barland et al.We obtain the results through the constraint satisfaction problem over multi-valued domains, for which we develop approximation algorithms and show non-approximability results. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of probabilistically checkable proofs and multi-prover one-round proof systems
The Power of Linear Programming for Valued CSPs
A class of valued constraint satisfaction problems (VCSPs) is characterised
by a valued constraint language, a fixed set of cost functions on a finite
domain. An instance of the problem is specified by a sum of cost functions from
the language with the goal to minimise the sum. This framework includes and
generalises well-studied constraint satisfaction problems (CSPs) and maximum
constraint satisfaction problems (Max-CSPs).
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation. Using this result, we obtain tractability of several novel and
previously widely-open classes of VCSPs, including problems over valued
constraint languages that are: (1) submodular on arbitrary lattices; (2)
bisubmodular (also known as k-submodular) on arbitrary finite domains; (3)
weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo
On parallel versus sequential approximation
In this paper we deal with the class NCX of NP Optimization problems that are approximable within constant ratio in NC. This class is the parallel counterpart of the class APX. Our main motivation here is to reduce the study of sequential and parallel approximability to the same framework. To this aim, we first introduce a new kind of NC-reduction that preserves the relative error of the approximate solutions and show that the class NCX has {em complete} problems under this reducibility.
An important subset of NCX is the class MAXSNP, we show that MAXSNP-complete problems have a threshold on the parallel approximation ratio that is, there are positive constants , such that although the problem can be approximated in P within it cannot be approximated in NC within epsilon_2$, unless P=NC. This result is attained by showing that the problem of approximating the value obtained through a non-oblivious local search algorithm is P-complete, for some values of the approximation ratio. Finally, we show that approximating through non-oblivious local search is in average NC.Postprint (published version
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