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arXiv:2510.07354 · Quantum Machine Intelligence

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

Andreas Wichert

Published Oct 12, 2026• 0 citations
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Abstract

Abstract We present a novel quantum storage algorithm for $k$ binary vectors of dimension $m$ into a superposition of a $m$-qubit quantum state based on a permutation technique. We compare this algorithm to the storage algorithm proposed by Ventura and Martinez. The permutation technique is simpler and can lead to an additional reduction through the reduce algorithm. To retrieve a binary vector from the superposition of $k$ vectors represented by a $m$-qubit quantum state, we must use a modified version of Grover’s algorithm, as Grover’s algorithm does not function correctly for non-uniform distributions. We introduce the permutation trick that enables an exhaustive search by Grover’s algorithm in $O(\sqrt{k})$ steps for $k$ patterns, independent of $n=2^m$. We compare this trick to the Ventura and Martinez trick, which requires $O(\sqrt{n})$ steps for $k$ patterns

DOI
10.1007/s42484-026-00454-w

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Quantum encoding is a process to transform classical information into the quantum states. It plays a crucial role in using quantum algorithm. Basis encoding is the most intuitive way to encode the classical binary vectors into a quantum state. It encodes n -dimensional binary vector to an n -qubit quantum state represented computational basis state. A set of binary vectors is represented by the basis encoding. A uniform superposition of binary vectors can be easily generated by the Hadamard gates. The method of entanglement of binary pattern is easier to understand and implement an allows us to address the patterns through the index.
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IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
This paper describes a method for using Grovers algorithm to create a quantum vector database, the database stores embeddings based on Controlled-S gates, which represent a binary numerical value. This value represents the embeddings value. The process of creating meaningful embeddings is handled by a classical computer and the search process is handled by the quantum computer. This search approach might be beneficial for a large enough database, or it could be seen as a very qubit-efficient (super dense) way for storing data on a quantum computer, since the proposed circuit stores many embeddings inside one quantum register simultaneously.