Quantum Algorithm for the Fixed-Radius Neighbor Search

TL;DR

This work targets the memory-bottleneck of fixed-radius neighbor search (FRANS) by proposing a quantum algorithm (QFRANS) built on fixed-point Grover methods to locate all neighbor pairs within a fixed radius. It encodes the dataset into quantum registers, evaluates pairwise distances with a distance operator, and employs an efficient COMP-based oracle (or a diagonal variant) to mark neighbor pairs, combined with a reflection and a readout phase that uses Bayesian stopping to adaptively terminate. The authors demonstrate that the algorithm achieves a query complexity of $\mathcal{O}(N/\sqrt{M})$ and discuss optimizations such as using Chebyshev distance to reduce distance computation depth to $\mathcal{O}(q_1)$ and structured-data benefits that could yield $\text{poly}(\log N)$ depth. They also address practical considerations, including a Bayes-based stopping criterion validated in 1D simulations, resilience to bit-flip readout noise, and the dominant role of state preparation in circuit depth. Overall, QFRANS offers a conceptual quantum pathway to mitigate memory-hierarchy limitations in FRANS, albeit with current state-preparation challenges that frame it as a proof-of-concept and a stepping stone toward more scalable, domain-structured quantum neighbor search.

Abstract

Neighbor search is a computationally demanding problem, usually both time- and memory-consuming. The main problem of this kind of algorithms is the long execution time due to cache misses. In this work, we propose a quantum algorithm for the Fixed RAdius Neighbor Search problem (FRANS) based on the fixed-point version of Grover's algorithm. We propose an efficient circuit for solving the FRANS with linear query complexity with the number of particles $N$. The quantum circuit returns the list of all the neighbors' pairs within the fixed radius, together with their distance, avoiding the slow down given by cache miss. We analyzed the gate and the query complexity of the circuit. Our FRANS algorithm presents a query complexity of $\mathcal{O}(N/\sqrt{M})$, where $M$ is the number of solutions, reaching the optimal lower bound of the Grover's algorithm. We propose different implementations of the oracle, which must be chosen depending on the precise structure of the database. Among these, we present an implementation using the Chebyshev distance with depth $\mathcal{O}(q_1)$, where $2^{q_1}$ is the number of grid points used to discretize a spatial dimension. State-of-the-art algorithms for state preparation allow for a trade-off between depth and width of the circuit, with a volume (depth$\times$ width) of $\mathcal{O}(N\log(N))$. This unfavorable scaling can be brought down to $\mathcal{O}(\text{poly}(\log N))$ in case of structured datasets. We proposed a stopping criterion based on Bayes interference and tested its validity on $1D$ simulations. Finally, we accounted for the readout complexity and assessed the resilience of the model to the readout error, suggesting an error correction-free strategy to check the accuracy of the results.

Figures (10)

• Figure 1: Toy model illustrating the cache miss problem in neighbor search algorithms. An evenly spaced grid contains eight particles distributed across four memory blocks (indicated by red boundaries).Each particle's data is stored in memory locations denoted by matching colors. Note that spatially adjacent particles might be stored in different memory locations.
• Figure 2: Circuit used for the amplitude amplification process. Based on Mizel search algorithm. The parameter $\alpha$ is adjusted accordingly to the iteration. The measurement on the ancilla qubit serves as control: if $\ket{0}$ is the outcome, it means that the quantum register is ready for measurement; otherwise repeat the process, varying the angle $\alpha$.
• Figure 3: In (a), the average number of oracle calls before success (using $M=1$ and changing the database size $N$) for the constant critical value of $\alpha_C=\alpha_i$ and for the variable decreasing sequence, defined in \ref{['eq:varying_alpha']} and referenced as $\alpha_V$. This is compared with the references $\sqrt{N}$ and $N$ using a logarithmic scale on both the horizontal and vertical axes. The fixed point search algorithm provides a quadratic advantage with respect to classical search, for the considered angle's schedules. In (b), a comparison between the cumulative probabilities for the two choices with respect to a classical algorithm. The vertical lines represent the average number of queries to the oracle for the respective cases. In (c) and (d) the time evolution of the coefficients in Eq. \ref{['eq:sincos_FPS']}, the instantaneous and cumulative probabilities $p_i, p_{\text{cum}}$ for the case $M=1$, $N=1000$ with $\alpha_C$ and $\alpha_V$ respectively.
• Figure 4: The circuit for the QFRANS algorithm, which has to be repeated until the state measured in the register $a_0$ is $|0\rangle$.
• Figure 5: A possible implementation of the $\text{PREP}$ operator, where $\hat{L}$ creates a balanced superposition of the labels of the particles and $\hat{E}$ assigns to each particle its position.