45 research outputs found
Reducing Binary Quadratic Forms for More Scalable Quantum Annealing
Recent advances in the development of commercial quantum annealers such as the D-Wave 2X allow solving NP-hard optimization problems that can be expressed as quadratic unconstrained binary programs. However, the relatively small number of available qubits (around 1000 for the D-Wave 2X quantum annealer) poses a severe limitation to the range of problems that can be solved. This paper explores the suitability of preprocessing methods for reducing the sizes of the input programs and thereby the number of qubits required for their solution on quantum computers. Such methods allow us to determine the value of certain variables that hold in either any optimal solution (called strong persistencies) or in at least one optimal solution (weak persistencies). We investigate preprocessing methods for two important NP-hard graph problems, the computation of a maximum clique and a maximum cut in a graph. We show that the identification of strong and weak persistencies for those two optimization problems is very instance-specific, but can lead to substantial reductions in the number of variables
Posiform planting: generating QUBO instances for benchmarking
We are interested in benchmarking both quantum annealing and classical algorithms for minimizing quadratic unconstrained binary optimization (QUBO) problems. Such problems are NP-hard in general, implying that the exact minima of randomly generated instances are hard to find and thus typically unknown. While brute forcing smaller instances is possible, such instances are typically not interesting due to being too easy for both quantum and classical algorithms. In this contribution, we propose a novel method, called posiform planting, for generating random QUBO instances of arbitrary size with known optimal solutions, and use those instances to benchmark the sampling quality of four D-Wave quantum annealers utilizing different interconnection structures (Chimera, Pegasus, and Zephyr hardware graphs) and the simulated annealing algorithm. Posiform planting differs from many existing methods in two key ways. It ensures the uniqueness of the planted optimal solution, thus avoiding groundstate degeneracy, and it enables the generation of QUBOs that are tailored to a given hardware connectivity structure, provided that the connectivity is not too sparse. Posiform planted QUBOs are a type of 2-SAT boolean satisfiability combinatorial optimization problems. Our experiments demonstrate the capability of the D-Wave quantum annealers to sample the optimal planted solution of combinatorial optimization problems with up to 5, 627 qubits
Parallel Quantum Annealing
Quantum annealers of D-Wave Systems, Inc., offer an efficient way to compute
high quality solutions of NP-hard problems. This is done by mapping a problem
onto the physical qubits of the quantum chip, from which a solution is obtained
after quantum annealing. However, since the connectivity of the physical qubits
on the chip is limited, a minor embedding of the problem structure onto the
chip is required. In this process, and especially for smaller problems, many
qubits will stay unused. We propose a novel method, called parallel quantum
annealing, to make better use of available qubits, wherein either the same or
several independent problems are solved in the same annealing cycle of a
quantum annealer, assuming enough physical qubits are available to embed more
than one problem. Although the individual solution quality may be slightly
decreased when solving several problems in parallel (as opposed to solving each
problem separately), we demonstrate that our method may give dramatic speed-ups
in terms of Time-to-Solution (TTS) for solving instances of the Maximum Clique
problem when compared to solving each problem sequentially on the quantum
annealer. Additionally, we show that solving a single Maximum Clique problem
using parallel quantum annealing reduces the TTS significantly.Comment: 13 pages. v4: format improvement
Posiform Planting: Generating QUBO Instances for Benchmarking
We are interested in benchmarking both quantum annealing and classical
algorithms for minimizing Quadratic Unconstrained Binary Optimization (QUBO)
problems. Such problems are NP-hard in general, implying that the exact minima
of randomly generated instances are hard to find and thus typically unknown.
While brute forcing smaller instances is possible, such instances are typically
not interesting due to being too easy for both quantum and classical
algorithms. In this contribution, we propose a novel method, called posiform
planting, for generating random QUBO instances of arbitrary size with known
optimal solutions, and use those instances to benchmark the sampling quality of
four D-Wave quantum annealers utilizing different interconnection structures
(Chimera, Pegasus, and Zephyr hardware graphs) as well as the simulated
annealing algorithm. Posiform planting differs from many existing methods in
two key ways. It ensures the uniqueness of the planted optimal solution, thus
avoiding groundstate degeneracy, and it enables the generation of QUBOs that
are tailored to a given hardware connectivity structure, provided that the
connectivity is not too sparse. Posiform planted QUBOs are a type of 2-SAT
boolean satisfiability combinatorial optimization problems. Our experiments
demonstrate the capability of the D-Wave quantum annealers to sample the
optimal planted solution of combinatorial optimization problems with up to
qubits
On-Line and Dynamic Shortest Paths Through Graph Decompositions
We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only time, where is the number of vertices of the digraph. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem. Our results can be extended to hold for digraphs of genus
Optimizing embedding-related quantum annealing parameters for reducing hardware bias
Quantum annealers have been designed to propose near-optimal solutions to
NP-hard optimization problems. However, the accuracy of current annealers such
as the ones of D-Wave Systems, Inc., is limited by environmental noise and
hardware biases. One way to deal with these imperfections and to improve the
quality of the annealing results is to apply a variety of pre-processing
techniques such as spin reversal (SR), anneal offsets (AO), or chain weights
(CW). Maximizing the effectiveness of these techniques involves performing
optimizations over a large number of parameters, which would be too costly if
needed to be done for each new problem instance. In this work, we show that the
aforementioned parameter optimization can be done for an entire class of
problems, given each instance uses a previously chosen fixed embedding.
Specifically, in the training phase, we fix an embedding E of a complete graph
onto the hardware of the annealer, and then run an optimization algorithm to
tune the following set of parameter values: the set of bits to be flipped for
SR, the specific qubit offsets for AO, and the distribution of chain weights,
optimized over a set of training graphs randomly chosen from that class, where
the graphs are embedded onto the hardware using E. In the testing phase, we
estimate how well the parameters computed during the training phase work on a
random selection of other graphs from that class. We investigate graph
instances of varying densities for the Maximum Clique, Maximum Cut, and Graph
Partitioning problems. Our results indicate that, compared to their default
behavior, substantial improvements of the annealing results can be achieved by
using the optimized parameters for SR, AO, and CW
Graph Partitioning using Quantum Annealing on the D-Wave System
In this work, we explore graph partitioning (GP) using quantum annealing on
the D-Wave 2X machine. Motivated by a recently proposed graph-based electronic
structure theory applied to quantum molecular dynamics (QMD) simulations, graph
partitioning is used for reducing the calculation of the density matrix into
smaller subsystems rendering the calculation more computationally efficient.
Unconstrained graph partitioning as community clustering based on the
modularity metric can be naturally mapped into the Hamiltonian of the quantum
annealer. On the other hand, when constraints are imposed for partitioning into
equal parts and minimizing the number of cut edges between parts, a quadratic
unconstrained binary optimization (QUBO) reformulation is required. This
reformulation may employ the graph complement to fit the problem in the Chimera
graph of the quantum annealer. Partitioning into 2 parts, 2^N parts
recursively, and k parts concurrently are demonstrated with benchmark graphs,
random graphs, and small material system density matrix based graphs. Results
for graph partitioning using quantum and hybrid classical-quantum approaches
are shown to equal or out-perform current "state of the art" methods