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We present an algorithm for quantum-assisted cluster analysis that makes use of the topological properties of a D-Wave 2000Q quantum processing unit. Clustering is a form of unsupervised machine learning, where instances are organized into groups whose members share similarities. The assignments are, in contrast to classification, not known a prior...
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... feature maps (SOFMs) are used to project high- dimensional data onto a low-dimensional map while trying preserve the neighboring structure of data. This means that data close in distance in an n-dimensional space should also stay close in distance in the low-dimensional map-the neighboring structure is kept. SOFMs inventor, Teuvo Kohonen, was inspired by the sensory and motor parts of the human brain [33]. 1-Self organizing feature map-the scheme of a SOFM shows that every component of the input vector x is represented by an input neuron and is connected with the low dimensional layer (in this case) above it. During a learning phase, the weight vectors of a SOFM are adapted by self-organization [34]. As other artificial neural networks (ANNs), the SOFM consists of neurons (n 1 , . . . , n n ), each having a weight vector w i and a distance to a neighbor neuron. The distance between the neurons n i and n j is n ij ,. As Figure 1-shows, each neuron is allocated a position in the low-dimensional map space. As in all other ANNs, initially the neuron weights are randomized. During learning, the similarity of each input vector to the weights of all neurons on the map is calculated, meaning that each member of the set of all weight vectors D is compared with the input vector d ∈ D. The SOFMs learning algorithm therefore belongs to the group of unsupervised learning algorithms. The neuron showing the highest similarity, having the smallest distance d small to d ∈ D is then selected as the winning neuron n win (Equation 2) ...
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... the example depicted in Figure 1, the SOFM is a two- dimensional lattice of nodes, and depending on a presented instance, different nodes will fire with different strengths. The ones firing with the greatest amplitude give the cluster assignment. The QACA works similar in the sense that the two-dimensional topological properties of the D-Wave are exploited for cluster assignments. Assuming we embed two- dimensional clusters on the chip (higher-dimensional structures can be mapped as well-see the explanations in chapter 3), an assignment of cluster points to qubits may look as described in Figure 2: Figure 2 shows schematically that qubits 1-8, and 17, 18, 21 would "fire, " thus take the value 1 in the result-vector, and qubits 9-16 and 19, 20, 22-24 would not fire, thus take the value 0. We need to set the couplings accordingly, so that when a candidate instance is fed into the cluster-form (see QUBO- form and embedding, Figure 3) and embedded onto the QPU, the result allows us identify "areas" of activity or groups of qubits set to 1 for similar instances. Figure 3 shows how an instance is fed into the cluster-form. − → X = (x 1 , . . . , x n ) represents the input ...
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... Many problems have already been solved using quantum annealing, giving reasonable solutions in real time [24] or giving optimal or very good solutions faster than classical alternatives [25]. Applications of quantum annealing are diverse and include traffic optimisation [24], finance [26], cyber security problems [27] and machine learning [13,25,28]. In quantum annealing, qubits are brought in an initial superposition state, after which a problem-specific Hamiltonian is applied to the qubits. ...
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... Quantum machine learning refers to quantum algorithms that are used to solve machine learning tasks, for instance, the outcome of the measurement of a qubit can represent the result of a binary classification task. Here we discuss different machine learning problems that have been addressed by making use of QA: classification [127] and reinforcement learning problems [135], cluster analysis [104,134], and matrix factorization [67,142,145]. For an introduction on the general topic, see, for example, [124,132]. ...
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... Several methods inspired by classical clustering have been proposed and developed recently. For example, quantum-assisted clustering based on similarities 28 , K-means [29][30][31] and K-Medoids 32 clustering on a quantum annealer, as well as K-means clustering on a gate-model quantum computer 38 , have been proposed. K-means-like clustering on a digital Ising machine has also been investigated 36 . ...
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... Reinforcement learning: [121], [38]. Cluster analysis: [120], [93]. Matrix factorization: [128], [132], [59]. ...
... For instance, the outcome of the measurement of a qubit can represent the result of a binary classification task. Here we discuss different machine learning problems that have been addressed by making use of QA: classification [114] and reinforcement learning problems [121], cluster analysis [93,120], and matrix factorization [59,128,132]. For an introduction on the general topic, see, for example, Refs. ...
... A quantum-assisted clustering algorithm (QACA) was introduced by Neukart et al. [120]. The algorithm is described as showing similarities with the self-organizing feature map, in the sense that the topological properties of the D-Wave QPU are exploited for cluster assignments. ...
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... The quantum computers that realize quantum annealing can efficiently find a solution to the QUBO problem 1 . Therefore, researchers are investigating potential applications of QUBO (such as machine learning [13], traffic flow optimization [12], and automated guided vehicle control in a factory [14]). ...
In this paper, we study the computational complexity of the quadratic unconstrained binary optimization (QUBO) problem under the functional problem FP^NP categorization. We focus on three sub-classes: (1) When all coefficients are integers QUBO is FP^NP-complete. When every coefficient is an integer lower bounded by a constant k, QUBO is FP^NP[log]-complete. (3) When coefficients can only be in the set {1, 0, -1}, QUBO is again FP^NP[log]-complete. With recent results in quantum annealing able to solve QUBO problems efficiently, our results provide a clear connection between quantum annealing algorithms and the FP^NP complexity class categorization. We also study the computational complexity of the decision version of the QUBO problem with integer coefficients. We prove that this problem is DP-complete.
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... Other works have also shown how to reformulate parts of classical clustering algorithms as quantum subroutines that can be executed on error-corrected gate-model QPUs (Alexander et al. 2018;Aimeur et al. 2013;Wiebe et al. 2015;Horn and Gottlieb 2001). In quantum annealing, a similar approach has been shown in which the objective function of the clustering task (minimizing distance metrics between high-dimensional vectors) has been directly translated to a QUBO, with each vector's possible assignment represented via one-hot encoding to physical qubits (Kumar et al. 2018;Neukart et al. 2018). ...
In this paper we develop methods to solve two problems related to time series (TS) analysis using quantum computing: reconstruction and classification. We formulate the task of reconstructing a given TS from a training set of data as an unconstrained binary optimization (QUBO) problem, which can be solved by both quantum annealers and gate-model quantum processors. We accomplish this by discretizing the TS and converting the reconstruction to a set cover problem, allowing us to perform a one-versus-all method of reconstruction. Using the solution to the reconstruction problem, we show how to extend this method to perform semi-supervised classification of TS data. We present results indicating our method is competitive with current semi- and unsupervised classification techniques, but using less data than classical techniques.
... Other works have also shown how to reformulate parts of classical clustering algorithms as quantum subroutines that can be executed on error-corrected gate-model QPUs [6,4,57,25]. In quantum annealing, a similar approach has been shown in which the objective function of the clustering task (minimizing distance metrics between high-dimensional vectors) has been directly translated to a QUBO, with each vector's possible assignment represented via one-hot encoding to physical qubits [31,40]. ...
In this paper we develop methods to solve two problems related to time series (TS) analysis using quantum computing: reconstruction and classification. We formulate the task of reconstructing a given TS from a training set of data as an unconstrained binary optimization (QUBO) problem, which can be solved by both quantum annealers and gate-model quantum processors. We accomplish this by discretizing the TS and converting the reconstruction to a set cover problem, allowing us to perform a one-versus-all method of reconstruction. Using the solution to the reconstruction problem, we show how to extend this method to perform semi-supervised classification of TS data. We present results indicating our method is competitive with current semi- and unsupervised classification techniques, but using less data than classical techniques.
