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 \(1\%\) 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.
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%) 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.
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.
Given an array a[1..n], the Range Minimum Query (RMQ) problem is to maintain a data structure that supports RMQ queries: given a range [l,r], find the index of the minimum element among a[l..r], i.e., argmini∈[l,r]a[i]. In this paper, we propose a quantum data structure that supports RMQ queries and range updates, with an optimal time complexity Θ(nq) for performing q=O(n) operations without preprocessing, compared to the classical Θ(n+q). As an application, we obtain a time-efficient quantum algorithm for k-minimum finding without the use of quantum random access memory.
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
The rapidly increasing number of heterogeneous data ecosystems, such as structured databases, semi-structured data lakes, real-time streams, distributed knowledge graphs, has further complicated the issue of query planning and optimization in modern information systems extensively. The optimization of queries is essentially an NP-hard combinatorial problem especially when based on multi-join queries, distributed execution plan, dynamic cost model, and uncertainty in the data access patterns. Even decades of concept and engineering advances, classical database optimizers are rapidly becoming unable to offer a globally optimal execution plan when faced with high-dimensional data, high data volatility, with a very tight latency constraint. Good news Recent breakthroughs in quantum computing propose promising computational paradigms that can be used to solve combinatorial optimization problems via quantum parallelism, superposition, and entanglement. Nevertheless, even the near-term quantum hardware is limited in the number of available qubits, noise, and decoherence, making it impractical to achieve fully quantum database systems in any perceived future. This is also the reason why quantum-classical hybrid algorithms are being developed: designed to add quantum optimization subroutines to classical database architectures is guaranteed without breaking their compatibility with existing execution engines. The following paper presents a framework of scalable quantum-classical hybrid computing of complex query planning in heterogeneous data ecosystems. The framework breaks down query optimization into classical and quantum-amenable optimization, which allows NP-hard subproblems, such as join ordering, operator placement, and cost minimization, to be transformed into quantum optimization models such as Quadratic Unconstrained Binary Optimization (QUBO) and Ising Hamiltonians. Classical elements handle query parsing, semantic analysis, constraint verification, and orchestration, whereas quantum processors are called selectively to solve combinatorial decision-making tasks. The suggested architecture will consist of a layered execution, hybrid execution, cost-model encoding approach, and adaptive fallback mechanisms to make them robust to quantum hardware variability. This has been experimentally tested with simulated quantum annealing and variational quantum algorithms that exhibit a quantifiable improvement in the quality and scalability of optimization and heuristics based on entirely classical algorithms, especially in high-dimensional query plans. The findings confirm the practicability of quantum query optimization and put down a feasible route to implement quantum algorithms in the serving machine of the next-generation data management systems.
Quantum Random Access Memory (qRAM) is a critical component for achieving quantum advantage in algorithms ranging from database search to quantum machine learning. In a recently introduced model [arXiv:2508.02855], we proposed a resource-efficient qRAM architecture based on discrete-time quantum walkers. This article serves as a comprehensive technical follow-up, providing the full mathematical derivations, detailed protocol specifications, and in-depth resource analysis. Moreover, we extend the original proposal with novel techniques for the purpose of making the qRAM implementation more realistic. Our model resolves the primary drawbacks of leading qRAM proposals: it avoids the exponential number of active nodes O(2n) required by the "Bucket Brigade" architecture by employing a number of quantum walkers that scales linearly with the address and message sizes, n and m, respectively. Simultaneously, it overcomes the spatial bottlenecks of previous quantum-walker schemes by eliminating the need for multiple parallel trees. We propose two algorithmic paradigms: the long- and the short-range approaches, which differ by the length of interaction of the main routing gates employed in the qRAM. We formalize the routing, message-copy, and walker retrieval phases for both variants and we show how, while the long-range scheme employs controlled gates having a multitude of target systems, the short-range approach decomposes these interactions into sequences of 2- and 3-body local gates, which improves the architecture's feasibility for near-term experimental implementation. Finally, our comprehensive resource analysis confirms that the short-range approach achieves the optimal O(n+m) circuit depth. Within these two paradigms, we explore the potential of different types of quantum walkers, namely bosons, dual-rail qubits and four-level qudits.
Abstract Quantum search algorithms offer a remarkable advantage of quadratic reduction in query complexity using quantum superposition principle. However, how an actual architecture may access and handle the database in a quantum superposed state has been largely unexplored so far; the quantum state of data was simply assumed to be prepared and accessed by a black-box operation—so-called oracle, even though this process, if not appropriately designed, may adversely diminish the quantum query advantage. Here, we introduce an efficient quantum data-access process, dubbed as quantum data-access machine (QDAM), and present a general architecture for quantum search algorithm. We analyze the runtime of our algorithm in view of the fault-tolerant quantum computation (FTQC) consisting of logical qubits within an effective quantum error correction code. Specifically, we introduce a measure involving two computational complexities, i.e. quantum query and T-depth complexities, which can be critical to assess performance since the logical non-Clifford gates, such as the T (i.e. π / 8 rotation) gate, are known to be costliest to implement in FTQC. Our analysis shows that for N searching data, a QDAM model exhibiting a logarithmic, i.e. O ( log N ) , growth of the T-depth complexity can be constructed. Further analysis reveals that our QDAM-embedded quantum search requires O ( N × log N ) runtime cost. Our study thus demonstrates that the quantum data search algorithm can truly speed up over classical approaches with the logarithmic T-depth QDAM as a key component.