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_B N). 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 times) to 64.70 times. 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.

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