Quantum algorithm for binary vector encoding and retrieval utilizing the permutation trick

Andreas Wichert

doi:10.1007/s42484-026-00454-w

Quantum algorithm for binary vector encoding and retrieval utilizing the permutation trick

2026-10-12

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Andreas Wichert *

DOI：10.1007/s42484-026-00454-w

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Storing multiple binary vectors in a quantum superposition and subsequently retrieving a specific one based on its content is a fundamental capability of quantum associative memory and quantum nearest neighbor search. This paper presents a novel encoding scheme, referred to as the permutation technique, for loading k binary vectors of dimension m into a superposition over an m-qubit register. Compared to the storage algorithm of Ventura and Martinez, the proposed method exhibits a structurally simpler structure and, when the stored vectors exhibit strong correlations, can be further compressed using the reduce algorithm. A significant challenge in retrieval lies in the fact that the resulting superposition is generally non-uniform, and Grover’s algorithm assumes a uniform distribution to achieve quadratic speedup. Applying it directly to a non uniform superposition of stored patterns consequently fails to amplify the correct target. To address this, the paper introduces a modified, permutation-based variant of Grover’s algorithm, known as the permutation trick, which restores correct behavior over the non-uniform superposition and completes retrieval in $$O(\sqrt{k})$$ oracle calls for k stored patterns, irrespective of the ambient dimension $$n = 2^m$$ . This is contrasted with the Ventura–Martinez retrieval approach, which scales as $$O(\sqrt{n})$$ and is therefore less favorable when $$k \ll n$$ . The storage and retrieval principles are demonstrated through two applications: quantum associative memory and quantum nearest neighbor search, thereby showcasing that the permutation trick provides a simpler and more scalable route to pattern storage and content-based retrieval on near-term quantum hardware.

Keywords:

Basis Encoding

Non-Uniform Distribution

Quantum Nearest Neighbor

Quantum Associative Memory

Grover’s Algorithm

Quantum algorithm for binary vector encoding and retrieval utilizing the permutation trick

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