Quantum Sketches, Hashing, and Approximate Nearest Neighbors
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917286905905152 |
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| author | Hashemian, Sajjad |
| author_facet | Hashemian, Sajjad |
| contents | Motivated by Johnson--Lindenstrauss dimension reduction, amplitude encoding, and the view of measurements as hash-like primitives, one might hope to compress an $n$-point approximate nearest neighbor (ANN) data structure into $O(\log n)$ qubits. We rule out this possibility in a broad quantum sketch model, the dataset $P$ is encoded as an $m$-qubit state $ρ_P$, and each query is answered by an arbitrary query-dependent measurement on a fresh copy of $ρ_P$. For every approximation factor $c\ge 1$ and constant success probability $p>1/2$, we exhibit $n$-point instances in Hamming space $\{0,1\}^d$ with $d=Θ(\log n)$ for which any such sketch requires $m=Ω(n)$ qubits, via a reduction to quantum random access codes and Nayak's lower bound. These memory lower bounds coexist with potential quantum query-time gains and in candidate-scanning abstractions of hashing-based ANN, amplitude amplification yields a quadratic reduction in candidate checks, which is essentially optimal by Grover/BBBV-type bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_19259 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Sketches, Hashing, and Approximate Nearest Neighbors Hashemian, Sajjad Quantum Physics Data Structures and Algorithms Motivated by Johnson--Lindenstrauss dimension reduction, amplitude encoding, and the view of measurements as hash-like primitives, one might hope to compress an $n$-point approximate nearest neighbor (ANN) data structure into $O(\log n)$ qubits. We rule out this possibility in a broad quantum sketch model, the dataset $P$ is encoded as an $m$-qubit state $ρ_P$, and each query is answered by an arbitrary query-dependent measurement on a fresh copy of $ρ_P$. For every approximation factor $c\ge 1$ and constant success probability $p>1/2$, we exhibit $n$-point instances in Hamming space $\{0,1\}^d$ with $d=Θ(\log n)$ for which any such sketch requires $m=Ω(n)$ qubits, via a reduction to quantum random access codes and Nayak's lower bound. These memory lower bounds coexist with potential quantum query-time gains and in candidate-scanning abstractions of hashing-based ANN, amplitude amplification yields a quadratic reduction in candidate checks, which is essentially optimal by Grover/BBBV-type bounds. |
| title | Quantum Sketches, Hashing, and Approximate Nearest Neighbors |
| topic | Quantum Physics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2602.19259 |