Quantum Algorithm for Binary Vector Encoding and Retrieval Utilizing the Permutation Trick

TL;DR

This paper introduces a permutation-based quantum storage method to encode $k$ binary vectors of dimension $m$ into an $m$-qubit state, offering a dense encoding with reduced qubit requirements compared to Ventura and Martinez's QuAM. A key contribution is the permutation trick, which, together with a reduce algorithm, lowers gate counts and enables retrieval via a modified Grover search in $O(\sqrt{k})$ steps, independent of $n=2^m$. The authors contrast this with the VM approach, which requires $O(\sqrt{n})$ Grover rotations, and show that PT can significantly reduce retrieval effort when $k$ is small or highly correlated data permit efficient parallelization. The work addresses fundamental bottlenecks in quantum associative memory and nearest-neighbor retrieval, highlighting practical implications for dense quantum storage and efficient data preparation in quantum systems.

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 algorithm, as Grover algorithm does not function correctly for non uniform distributions. We introduce the permutation trick that enables an exhaustive search by Grover algorithm in square root of k steps for k patterns, independent of n equal two power m. We compare this trick to the Ventura and Martinez trick, which requires square root of n steps for k patterns.

Figures (6)

• Figure 1: Circuit as proposed by generating the distribution $| \hat{\psi} \rangle= \frac{1}{2} \cdot \left( |0011 \rangle + |1001 \rangle + |1111 \rangle +|0110 \rangle \right) \otimes | 000000 \rangle$.
• Figure 2: Proposed circuit generating the distribution $| \psi \rangle= \frac{1}{2} \cdot \left( |0011 \rangle + |1001 \rangle + |1111 \rangle +|0110 \rangle \right)$. We define the ground state $|00\rangle$ by the qubits $2$ and $3$ with uniform distribution of two qubits by two Hadamard gates, qubits $0$ and $1$. Then we address each basis states indicating the flag qubit $4$ by multi controlled not gate, write the required mapping with flag controlled not gates and disentangle the flag qubit with multi controlled not gates gate. Note we do not need to write $|0011 \rangle = |0011 \rangle$ since it already present. We require $2 \cdot 3$ multi controlled multi controlled not gates and $6$ controlled not gates.
• Figure 3: Proposed circuit generating the distribution $| \psi \rangle$ by parallel load as determined by the $reduce$ algorithm.
• Figure 4: Proposed circuit generating the distribution $| \psi \rangle$ by parallel load as determined by the $reduce$ algorithm without unnecessary NOT gates.
• Figure 5: The circuit representing the distribution $\psi$ indicated by the gate $S$ followed by three Grover’s rotations (gate $G$) with twice the cost of marking all four stored patterns, indicating the target pattern $|0110\rangle$.