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International Scientific Journal of Engineering and Management

Hierarchical Quantum-Classical Multidimensional Tree (HQCMT): A Novel Data Structure for Enhanced Multidimensional Query Processing

Shaikh Javed

Published Jul 12, 2025• 1 citation
Publisher

Abstract

This paper introduces the Hierarchical Quantum-Classical Multidimensional Tree (HQCMT), a novel hybrid data structure that combines classical hierarchical organization with quantum computational advantages for multidimensional data processing. The HQCMT integrates skip octree structures, quantum random access memory (QRAM), and quantum B+ tree methodologies to achieve unprecedented performance in range queries and multidimensional searches. Our theoretical analysis demonstrates up to 251× speedup in memory access operations compared to classical approaches, with time complexity improvements from 𝑂(log 𝑁 + 𝑘) to 𝑂(logB 𝑁) for range queries independent of output size. The structure supports native multidimensional clustering, quantum superposition-based parallel processing, and maintains fault tolerance through hybrid error correction mechanisms. Experimental validation shows significant performance gains across various workloads, positioning HQCMT as a breakthrough in quantum-enhanced data structures for next-generation computing applications. Keywords: Quantum Data Structures, Hierarchical Indexing, Multidimensional Queries, Hybrid Computing, QRAM, Quantum B+ Trees

DOI
10.55041/isjem04770

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This paper introduces the Hierarchical Quantum-Classical Multidimensional Tree (HQCMT), a novel hybrid data structure that combines classical hierarchical organization with quantum computational advantages for multidimensional data processing. The HQCMT integrates skip octree structures, quantum random access memory (QRAM), and quantum B+ tree methodologies to achieve unprecedented performance in range queries and multidimensional searches. Our theoretical analysis demonstrates up to 251× speedup in memory access operations compared to classical approaches, with time complexity improvements from 𝑂(log 𝑁 + 𝑘) to 𝑂(logB 𝑁) for range queries independent of output size. The structure supports native multidimensional clustering, quantum superposition-based parallel processing, and maintains fault tolerance through hybrid error correction mechanisms. Experimental validation shows significant performance gains across various workloads, positioning HQCMT as a breakthrough in quantum-enhanced data structures for next-generation computing applications. Keywords: Quantum Data Structures, Hierarchical Indexing, Multidimensional Queries, Hybrid Computing, QRAM, Quantum B+ Trees
Quantum computing is a popular topic in computer science, which has recently attracted many studies in various areas such as machine learning and network. However, the topic of quantum data structures seems neglected. There is an open problem in the database area: Can we improve existing data structures by quantum techniques? Consider a dataset of key-record pairs. Given an interval as a query range, a classical B+ tree can report all the records with keys within this interval, which is called a range query, in O(log N + k) time, where N is the total number of records and k is the output size. It is asymptotically optimal in a classical computer but not efficient enough in a quantum computer, because it is expected that the execution time and the output size are linear in a quantum computer. In this paper, we propose the quantum range query problem. Different from the classical range queries, a quantum range query returns the results in quantum bits, which has broad potential applications due to the foreseeable advance of quantum computers and quantum algorithms. To the best of our knowledge, we design the first tree-like quantum data structure called the quantum B+ tree. Based on this data structure, we propose a hybrid quantum-classical algorithm to do the range search. It answers a static quantum range query in O(log_B N) time, which is asymptotically optimal in quantum computers. Since the execution time does not depend on the output size (i.e., k, which could be as large as O(N)), it is significantly faster than the classical data structure. Moreover, we extend our quantum B+ tree to answer the dynamic and d-dimensional quantum range queries efficiently in O(log^2_B N) and O(log^d_B N) time, respectively. Our experimental results show that our proposed quantum data structures achieve up to 1000x improvement in the number of memory accesses compared to their classical competitors.
Quantum Random Access Memory (QRAM) is a crucial architectural component for querying classical or quantum data in superposition, enabling algorithms with wide-ranging applications in quantum arithmetic, quantum chemistry, machine learning, and quantum cryptography. In this work, we introduce Fat-Tree QRAM, a novel query architecture capable of pipelining multiple quantum queries simultaneously while maintaining desirable scalings in query speed and fidelity. Specifically, Fat-Tree QRAM performs O(log⁡(N))O(\log (N))O(log(N)) independent queries in O(log⁡(N))O(\log (N))O(log(N)) time using O(N)O(N)O(N) qubits, offering immense parallelism benefits over traditional QRAM architectures. To demonstrate its experimental feasibility, we propose modular and on-chip implementations of Fat-Tree QRAM based on superconducting circuits and analyze their performance and fidelity under realistic parameters. Furthermore, a query scheduling protocol is presented to maximize hardware utilization and access the underlying data at an optimal rate. These results suggest that Fat-Tree QRAM is an attractive architecture in a shared memory system for practical quantum computing.
Quantum computing is an emerging field of study and gains importance due to the fact that with the introduction of quantum computers, many challenges and changes are presented for the existing algorithms. The main reason for this is the exponential speed of such computers. This study analyzes some of the benefits and implications of quantum computing on geometrical problems such as the multidimensional search for window queries with R-Trees. A review of the window query on R-Trees in classical computing is done to consider its adaptability to quantum computers by applying the Grover's quantum search algorithm from a theoretical point of view. Thereby, the query time complexity in worst-case scenarios could be improved to quadratic search time.
Range query is a fundamental task in geospatial data search and many other downstream applications. Classic range queries often rely on tree-based spatial indexes, of which the query speed depends on the number of indexed points k within the queried range. For instance, a classical B+ tree answers a range query in O(log N + k). For a long time, this speed has long been considered asymptotically optimal in classic database systems, until the recent emergence of quantum computing, where a quantum B+ tree may requires only O(log BN). This paper presents Quantum Range Query (QRQ) via a hybrid quantum-classic algorithm to return the range query results in quantum superpositions. In this context, QRQ is designed to accelerate classic range query on spatial-temporal-semantic trajectory geodata using quantum algorithms. Specifically, QRQ develops quantum variants of R-tree, TB-tree and KD-tree, where the physical slots of a node, including unused padding slots, are treated as an array that a quantum random-access memory (QRAM) can read in superpositions. Evaluations on three common trajectory datasets, namely GeoLife, T-Drive, and GDP Drifter, and 10, 000 queries per setting, the QRQ speedup at 1%1\%1% target selectivity ranges from 2.03 × to 64.70 ×. More importantly, QRQ shows a great potential in optimizing existing trajectory range query, so it becomes the shared primitive on which human mobility analysis, and later searches specified by large language models (LLMs) and geo-foundation models (GeoFMs), can rest.
Operating on the principles of quantum mechanics, quantum algorithms hold the promise for solving problems that are beyond the reach of the best-available classical algorithms. An integral part of realizing such speedup is the implementation of quantum queries, which read data into forms that quantum computers can process. Quantum random access memory (QRAM) is a promising architecture for realizing quantum queries. However, implementing QRAM in practice poses significant challenges, including query latency, memory capacity and fault-tolerance.
The rapid advancement of quantum computing presents unprecedented opportunities for transforming information retrieval systems, specifically through the development of quantum algorithms and data structures. This chapter explores the innovative applications of quantum algorithms, such as Grover's search and quantum hashing, which offer substantial improvements in processing speed and efficiency over classical counterparts. Additionally, the integration of quantum data structures, including quantum trees and databases, enhances data management capabilities by leveraging quantum phenomena like superposition and entanglement. The chapter also addresses critical challenges, including scalability, error rates, and security concerns, which must be overcome to fully harness the potential of quantum technologies. By examining case studies across various industries, this work highlights practical implementations and the future trends shaping the evolution of information retrieval in the quantum era. The insights presented herein underscore the transformative impact of quantum computing on data processing, ultimately paving the way for next-generation information retrieval systems.