13 research outputs found
Quantum Optimization of Fully-Connected Spin Glasses
The Sherrington-Kirkpatrick model with random couplings is programmed
on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type
graph. The performance of the optimizer compares and correlates to simulated
annealing. When considering the effect of the static noise, which degrades the
performance of the annealer, one can estimate an improvement on the comparative
scaling of the two methods in favor of the D-Wave machine. The optimal choice
of parameters of the embedding on the Chimera graph is shown to be associated
to the emergence of the spin-glass critical temperature of the embedded
problem.Comment: includes supplemental materia
Hadwiger number of graphs with small chordality
The Hadwiger number of a graph G is the largest integer h such that G has the
complete graph K_h as a minor. We show that the problem of determining the
Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved
in polynomial time on cographs and on bipartite permutation graphs. We also
consider a natural generalization of this problem that asks for the largest
integer h such that G has a minor with h vertices and diameter at most . We
show that this problem can be solved in polynomial time on AT-free graphs when
s>=2, but is NP-hard on chordal graphs for every fixed s>=2
Contracting to a Longest Path in H-Free Graphs
The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P? and polynomial-time solvable if H = P?. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction
Contracting to a longest path in H-free graphs
The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction
Contracting edges to destroy a pattern: A complexity study
Given a graph G and an integer k, the objective of the -Contraction
problem is to check whether there exists at most k edges in G such that
contracting them in G results in a graph satisfying the property . We
investigate the problem where is `H-free' (without any induced copies of
H). It is trivial that H-free Contraction is polynomial-time solvable if H is a
complete graph of at most two vertices. We prove that, in all other cases, the
problem is NP-complete. We then investigate the fixed-parameter tractability of
these problems. We prove that whenever H is a tree, except for seven trees,
H-free Contraction is W[2]-hard. This result along with the known results
leaves behind three unknown cases among trees.Comment: 30 pages, 10 figures, a short version is accepted to FCT 202
Quantum-classical generative models for machine learning
The combination of quantum and classical computational resources towards more effective algorithms is one of the most promising research directions in computer science. In such a hybrid framework, existing quantum computers can be used to their fullest extent and for practical applications. Generative modeling is one of the applications that could benefit the most, either by speeding up the underlying sampling methods or by unlocking more general models. In this work, we design a number of hybrid generative models and validate them on real hardware and datasets. The quantum-assisted Boltzmann machine is trained to generate realistic artificial images on quantum annealers. Several challenges in state-of-the-art annealers shall be overcome before one can assess their actual performance. We attack some of the most pressing challenges such as the sparse qubit-to-qubit connectivity, the unknown effective-temperature, and the noise on the control parameters. In order to handle datasets of realistic size and complexity, we include latent variables and obtain a more general model called the quantum-assisted Helmholtz machine. In the context of gate-based computers, the quantum circuit Born machine is trained to encode a target probability distribution in the wavefunction of a set of qubits. We implement this model on a trapped ion computer using low-depth circuits and native gates. We use the generative modeling performance on the canonical Bars-and-Stripes dataset to design a benchmark for hybrid systems. It is reasonable to expect that quantum data, i.e., datasets of wavefunctions, will become available in the future. We derive a quantum generative adversarial network that works with quantum data. Here, two circuits are optimized in tandem: one tries to generate suitable quantum states, the other tries to distinguish between target and generated states