In this paper we are dealing with the frequent itemset mining. We concentrate on the special case that we only want to identify the most frequent itemset of length N. To do that, we present a pattern on how to consider this search as an optimization problem. First, we extract the frequency of all possible 2-item-sets. Then the optimization problem is to find the N objects, for which the minimal frequency of all containing 2-item-sets is maximal. This combinatorial optimization problem can be solved by any optimization algorithm. We will solve them with Quantum Annealing and QUBO with QbSolv by D-Wave. The advantages of MFIO in comparison to the state-of-the-art-approach are the enormous reduction of time need, reduction of memory need and the omission of a threshold. The disadvantage is that there is no guaranty for accuracy of the result. The evaluation indicates good results.
In this paper we are dealing with the frequent itemset mining. We concentrate on the special case that we only want to identify the most frequent itemset of length N. To do that, we present a pattern on how to consider this search as an optimization problem. First, we extract the frequency of all possible 2-item-sets. Then the optimization problem is to find the N objects, for which the minimal frequency of all containing 2-item-sets is maximal. This combinatorial optimization problem can be solved by any optimization algorithm. We will solve them with Quantum Annealing and QUBO with QbSolv by D-Wave. The advantages of MFIO in comparison to the state-of-the-art-approach are the enormous reduction of time need, reduction of memory need and the omission of a threshold. The disadvantage is that there is no guaranty for accuracy of the result. The evaluation indicates good results.
Association rules mining (ARM) is one of the most important problems in knowledge discovery and data mining. Given a transaction database that has a large number of transactions and items, the task of ARM is to acquire consumption habits of customers by discovering the relationships between itemsets (sets of items). In this paper, we address ARM in the quantum settings and propose a quantum algorithm for the key part of ARM, finding frequent itemsets from the candidate itemsets and acquiring their supports. Specifically, for the case in which there are Mf(k) frequent k-itemsets in the Mc(k) candidate k-itemsets (${M}_{f}^{(k)}\ensuremath{\le}{M}_{c}^{(k)}$), our algorithm can efficiently mine these frequent k-itemsets and estimate their supports by using parallel amplitude estimation and amplitude amplification with complexity $O(\frac{k\sqrt{{M}_{c}^{(k)}{M}_{f}^{(k)}}}{\ensuremath{\epsilon}})$, where $\ensuremath{\epsilon}$ is the error for estimating the supports. Compared with the classical counterpart, i.e., the classical sampling-based algorithm, whose complexity is $O(\frac{k{M}_{c}^{(k)}}{{\ensuremath{\epsilon}}^{2}})$, our quantum algorithm quadratically improves the dependence on both $\ensuremath{\epsilon}$ and Mc(k) in the best case when ${M}_{f}^{(k)}\ensuremath{\ll}{M}_{c}^{(k)}$ and on $\ensuremath{\epsilon}$ alone in the worst case when ${M}_{f}^{(k)}\ensuremath{\approx}{M}_{c}^{(k)}$.
Maximum frequent pattern generation from a large database of transactions and items for association rule mining is an important research topic in data mining. Association rule mining aims to discover interesting correlations, frequent patterns, associations, or causal structures between items hidden in a large database. By exploiting quantum computing, we propose an efficient quantum search algorithm design to discover the maximum frequent patterns. We modified Grover’s search algorithm so that a subspace of arbitrary symmetric states is used instead of the whole search space. We presented a novel quantum oracle design that employs a quantum counter to count the maximum frequent items and a quantum comparator to check with a minimum support threshold. The proposed derived algorithm increases the rate of the correct solutions since the search is only in a subspace. Furthermore, our algorithm significantly scales and optimizes the required number of qubits in design, which directly reflected positively on the performance. Our proposed design can accommodate more transactions and items and still have a good performance with a small number of qubits.
Introduction. Quantum computers provide several times faster solutions to several NP-hard combinatorial optimization problems in comparison with computing clusters. The trend of doubling the number of qubits of quantum computers every year suggests the existence of an analog of Moore's law for quantum computers, which means that soon they will also be able to get a significant acceleration of solving many applied large-scale problems. The purpose of the article is to review methods for creating algorithms of quantum computer mathematics for combinatorial optimization problems and to analyze the influence of the qubit-to-qubit coupling and connections strength on the performance of quantum data processing. Results. The article offers approaches to the classification of algorithms for solving these problems from the perspective of quantum computer mathematics. It is shown that the number and strength of connections between qubits affect the dimensionality of problems solved by algorithms of quantum computer mathematics. It is proposed to consider two approaches to calculating combinatorial optimization problems on quantum computers: universal, using quantum gates, and specialized, based on a parameterization of physical processes. Examples of constructing a half-adder for two qubits of an IBM quantum processor and an example of solving the problem of finding the maximum independent set for the IBM and D-wave quantum computers are given. Conclusions. Today, quantum computers are available online through cloud services for research and commercial use. At present, quantum processors do not have enough qubits to replace semiconductor computers in universal computing. The search for a solution to a combinatorial optimization problem is performed by achieving the minimum energy of the system of coupled qubits, on which the task is mapped, and the data are the initial conditions. Approaches to solving combinatorial optimization problems on quantum computers are considered and the results of solving the problem of finding the maximum independent set on the IBM and D-wave quantum computers are given. Keywords: quantum computer, quantum computer mathematics, qubit, maximal independent set for a graph.
We consider two combinatorial problems. The first we call ``search with wildcards'': given an unknown n-bit string x, and the ability to check whether any subset of the bits of x is equal to a provided query string, the goal is to output x. We give a nearly optimal O(nlogn) quantum query algorithm for search with wildcards, beating the classical lower bound of Ω(n) queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most k special items out of a set of n items, given the ability to make queries of the form ``does the set S contain any special items?''\ for any subset S of the n items. We give a simple quantum algorithm which uses O(k) queries to solve this problem, as compared with the classical lower bound of Ω(klog(n/k)) queries.
This paper investigates the application of quantum annealing for solving complex combinatorial optimization problems. The core claim is that quantum annealing leverages its global search capabilities to efficiently address such problems. The proposed mechanism involves formulating the combinatorial optimization problem as a quantum annealing problem and utilizing the simulated annealing process within quantum annealing to conduct a search for the optimal solution. This research explores the potential of quantum annealing in practical applications, acknowledging that its application in combinatorial optimization is still in its exploratory phase. The algorithm is designed to minimize the energy of a system described by a Hamiltonian, which evolves over time according to the time-dependent Schrödinger equation. The core of the algorithm lies in mapping the problem's constraints and objective function into this Hamiltonian. The search process is governed by quantum fluctuations, represented by the operator H^, which encourages transitions to lower energy states, effectively navigating the solution space. The effectiveness of the algorithm is evaluated based on the quality of the solution found and the computational resources required. This work provides a foundational exploration of quantum annealing's potential within the field of combinatorial optimization.