Exact bounds on quantum partial search algorithm and improving the parallel search

TL;DR

This work provides strong evidence for the strict optimality of the GRK operator sequence among all admissible compositions of global and local Grover operators, and derives an asymptotically tight upper bound on the maximal success probability for partial search and a matching lower bound on the minimal expected number of oracle queries.

Abstract

Grover's algorithm provides a quadratic speedup over classical algorithms for searching unstructured databases and is known to be strictly optimal in oracle query complexity, with tight bounds on its success probability. Although the standard Grover search cannot be further accelerated in the full-search setting, a trade-off between accuracy and query complexity gives rise to the partial search problem. The Grover-Radhakrishnan-Korepin (GRK) algorithm is widely regarded as the optimal protocol for this task. In this work, we provide strong evidence for the strict optimality of the GRK operator sequence among all admissible compositions of global and local Grover operators. By exhaustively examining all operator sequences with a fixed number of oracle queries, we show that the GRK structure universally maximizes the success probability. Building on this result, we derive an asymptotically tight upper bound on the maximal success probability for partial search and establish a matching lower bound on the minimal expected number of oracle queries. Furthermore, we investigate parallel quantum search within the partial-search framework. While a direct GRK-based parallelization does not outperform established parallel Grover schemes, we demonstrate that a hybrid strategy combining partial and full search protocols achieves a strictly improved parallel efficiency. Our results clarify the fundamental limits of quantum partial search and its role in optimizing parallel quantum search algorithms.

Figures (5)

• Figure 1: Success probabilities of the quantum partial search algorithm as a function of the total number of oracle queries $k_\text{tot}$, for a database of size $N=2^8$. Values marked with red circles represent $\mathrm{Pr}^\text{max}_{x=t_1}(k_{\text{tot}})$, as defined in Eq. (\ref{['eq:pr_max']}), obtained by optimizing over all possible operator sequences. Blue triangles correspond to the case where the partial search is performed using only the global Grover operator. The analytical expression for this probability is given in Eq. (\ref{['eq:pr_grover']}). The optimal sequences associated with $\mathrm{Pr}^\text{max}_{x=t_1}(k_{\text{tot}})$ are listed in Table \ref{['tab:n=8,pr']}.
• Figure 2: expected iteration numbers for the quantum partial search algorithm as a function of the total oracle queries $k_\text{tot}$, for a database of size $N=2^8$. The red circles show the values obtained from the optimized success probability $\mathrm{Pr}^\text{max}_{x=t_1}(k_{\text{tot}})$ defined in Eq. (\ref{['eq:pr_max']}). The blue triangles correspond to using only the global Grover operator, with the success probability $\mathrm{Pr}_{x=t_1}(k_{\text{tot}})$ given in Eq. (\ref{['eq:pr_grover']}). The corresponding numerical values are detailed in Table \ref{['tab:n=8,k']}.
• Figure 3: Maximal success probability of the quantum partial search algorithm with (a) $m=10$ and (b) $m=20$, for a database of size $N=2^{30}$. The analytical curves are plotted from Eq. \ref{['eq:pr_GRK']}. Numerical results are obtained by maximizing the success probability over $G_nG_m^{k_2}G_n^{k_1}$, under the constraint $k_\text{tot} = 1+k_1+k_2$. The value of $k_2$ used analytically is given by $k_2 = \pi\sqrt b/6$.
• Figure 4: (a) Minimal expected iteration number of the quantum partial search algorithm, and the corresponding iteration numbers (b) $k_\text{tot}$ and (c) $k_2$. The red dashed lines show the analytical results established in Theorem \ref{['theorem:k_opt_GRK_partial_search']}. The green dash-dotted line represents the GRK algorithm with success probability close to unity.
• Figure 5: Minimal expected iteration number as a function of the parallelism $l$ for the four parallel schemes: the inner, outer, GRK‑based, and hybrid parallel search algorithms, defined respectively in Defs. \ref{['def:inner']}, \ref{['def:outer']}, \ref{['def:GRK']}, and \ref{['def:hybrid']}. The database size is fixed at $N=2^5$.

Theorems & Definitions (20)

• Theorem 1
• proof
• Corollary 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Definition 5
• Definition 6