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Symmetry

Quantum Pattern Classification in a Three-Qubit System

Menna Elmasry, Ahmed Younes, Islam Elkabani, Ashraf Elsayed

Published Apr 8, 2023• 2 citations
Publisher

Abstract

The problem of pattern classification in quantum data has been of great importance over the past few years. This study investigates the effect of deploying Grover’s, the partial diffusion, and the fixed-phase algorithms separately to amplify the amplitudes of a desired pattern in an unstructured dataset. These quantum search operators were applied to symmetric and antisymmetric input superpositions on a three-qubit system for 20 iterations each. After each iteration, different probabilities of classification were calculated in order to determine the accuracy of classification for each of the three quantum search operators. The results indicated that, in the case of applying the three quantum search operators to incomplete superposition input states, the partial diffusion operator outperformed the other operators with a probability of correct classification that reached 100% in certain iterations. It also showed that the classification accuracy of the fixed-phase operator exceeded the accuracy of the other two operators by 40% in most cases when the input state was a uniform superposition, and some of the basis states were phase-inverted.

DOI
10.3390/sym15040883

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The problem of pattern classification in quantum data has been of great importance over the past few years. This study investigates the effect of deploying Grover’s, the partial diffusion, and the fixed-phase algorithms separately to amplify the amplitudes of a desired pattern in an unstructured dataset. These quantum search operators were applied to symmetric and antisymmetric input superpositions on a three-qubit system for 20 iterations each. After each iteration, different probabilities of classification were calculated in order to determine the accuracy of classification for each of the three quantum search operators. The results indicated that, in the case of applying the three quantum search operators to incomplete superposition input states, the partial diffusion operator outperformed the other operators with a probability of correct classification that reached 100% in certain iterations. It also showed that the classification accuracy of the fixed-phase operator exceeded the accuracy of the other two operators by 40% in most cases when the input state was a uniform superposition, and some of the basis states were phase-inverted.
Although the studies on quantum algorithms have been progressing, it is still necessary to broaden the investigation of open quantum systems. In this study, we present the use of an open quantum system to implement a quantum classifier algorithm. Zhang et al. propose a one QuBit system interacting with the environment through a unitary operator that comes from the Hamiltonian. In our proposal, the input data is loaded into the amplitude of the environment instead of being in the unitary operator. This change positively impacts the performance of different databases tested and causes a difference in the system entanglement behavior. For evaluation, the Zhang et al. model and the proposed model were tested in four real-world datasets and seven other toy problems. The results are evaluated according to accuracy and F1-Score. A deeper analysis of the Iris dataset is also done, checking the creation of entanglement and an extensive random search for better parameters on the proposed model. The results show that for most real-world dataset configurations, the proposed model, although having a simpler decision area, performed better than the one inspired by the Zhang et al. model, and that there is no pattern for the system entanglement in the Iris Dataset. Due to an underperform for both models in a linearly separable problem, an exponential kernel was introduced. It resulted in an improvement in the accuracy of both models in most of the evaluated situations.
Grover's algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.
In this paper, we will define a quantum operator that performs the inversion about the mean only on a subspace of the system (partial diffusion operator). This operator is used in a quantum search algorithm that runs in O(N/M) for searching an unstructured list of size N with M matches such that, 1 ⩽ M ⩽ N. We will show that the performance of the algorithm is more reliable than known fixed operator quantum search algorithms especially for multiple matches where we can get a solution after a single iteration with probability over 90% if the number of matches is approximately more than one‐third of the search space. A performance comparison with Grover’s algorithm will be provided.
Grover’s algorithm harnesses the power of quantum computing to swiftly locate specific elements in an unstructured database, outperforming classical computers in tasks like database searching. This algorithm capitalizes on the unique ability of qubits to be in both 0 and 1 states simultaneously (superposition), allowing it to scan the entire search space at once. It then boosts the probability of the target element, making it more prominent. While the foundational concepts of Grover’s algorithm are well-documented, practical implementation using quantum operators, especially for large search spaces, remains less explored beyond basic examples with a small number of qubits. Existing general synthesis techniques often involve numerous operators or are time-consuming. Our proposed methods specifically address the amplitude-amplification component of Grover’s algorithm for any size of search space. These methods detail the required types and quantities of qubits and operators, emphasizing minimal usage and efficient assembly. Developed and evaluated in Python, our methods consistently identified target elements with over 95% accuracy and achieved configurations comparably compact as those in the existing literature but at a faster pace. We anticipate that these methods facilitate practical implementations of Grover’s algorithm across various domains.
Pattern classification represents a challenging problem in machine learning and data science research domains, especially when there is a limited availability of training samples. In recent years, artificial neural network (ANN) algorithms have demonstrated astonishing performance when compared to traditional generative and discriminative classification algorithms. However, due to the complexity of classical ANN architectures, ANNs are sometimes incapable of providing efficient solutions when addressing complex distribution problems. Motivated by the mathematical definition of a quantum bit (qubit), we propose a novel autonomous perceptron model (APM) that can solve the problem of the architecture complexity of traditional ANNs. APM is a nonlinear classification model that has a simple and fixed architecture inspired by the computational superposition power of the qubit. The proposed perceptron is able to construct the activation operators autonomously after a limited number of iterations. Several experiments using various datasets are conducted, where all the empirical results show the superiority of the proposed model as a classifier in terms of accuracy and computational time when it is compared with baseline classification models.
Target item searching in large-scale, unstructured text databases is a major computational task for traditional systems and usually requires algorithms whose time complexity is on the order ofO(N)O(N)O(N). Quantum computing presents a strong competitor by using Grover's algorithm, where theoretically the search complexity comes down toO(N)O(\sqrt{N})O(N​), making faster retrieval possible. This research considers the design and assessment of Grover's algorithm as it is used to find a target word in a paragraph. The process entails encoding a group of unique words into a quantum register and implementing a circuit that incorporates a bespoke oracle and a diffusion operator to amplify the probability amplitude of the target state. Performance was evaluated in two situations: an ideal noise-free simulation with Qiskit's AerSimulator and a real-world run on IBM's quantum hardware. Simulation results demonstrated always-high accuracy in identifying the target word. But tests on real quantum machines indicated that the reliability of the algorithm decreases with an increase in qubits. Though circuits of up to five qubits had tolerable accuracy, larger configurations were plagued by noise, gate errors, and decoherence effects. These results highlight both the potential of quantum search algorithms for text processing and the limitations of NISQ today.