Reducing Circuit Resources in Grover's Algorithm via Constraint-Aware Initialization

TL;DR

This work proposes a framework for constraint-aware initialization of Grover's algorithm to reduce the effective search space in problems with linear constraints. By classically preprocessing to find disjoint constraint sets and encoding constraints with Dicke states (cardinality) or GHZ-type states (parity) in the initial state, the approach narrows the search effort prior to oracle diffusion steps. Although constraint encoding adds circuit-level cost, a conservative resource analysis shows potential improvements in total gate count and circuit depth, particularly when multiple disjoint constraint sets are used. Numerical experiments on exact-cover problems corroborate the theoretical gains and demonstrate robustness to noise, positioning constraint-aware initialization as a practical baseline for more resource-efficient Grover implementations. The framework also points to future directions, including general Dicke-state constructions, inequality constraints, and integration with optimization-era Grover variants.

Abstract

Grover's search algorithm provides a quadratic speedup over classical brute-force search in terms of query complexity and is widely used as a versatile subroutine in numerous quantum algorithms, including those for combinatorial problems with large search spaces. For such problems, it is natural to reduce the effective search space by incorporating problem constraints at the initialization step, which in Grover's algorithm can be achieved by preparing structured initial states that encode constraint information. In this work, we present a systematic framework with a simple preprocessing procedure for constraint-aware initialization in Grover's algorithm, focusing on problems with linear constraints. While such structured initial states can reduce the number of oracle queries required to obtain a solution, their preparation incurs additional circuit-level costs. We therefore offer a conservative circuit-level resource analysis, showing that the resulting constraint-aware initialization can improve resource efficiency in terms of gate counts and circuit depth. The validity of the framework is further demonstrated numerically using the exact-cover problem. Overall, our results indicate that this approach serves as a practical baseline for achieving more resource-efficient implementations of Grover's algorithm compared to the standard uniform initialization.

Figures (6)

• Figure 1: Schematic of Grover's algorithm with constraint-aware initialization. The initial state $\ket{\psi_{F}^{(0)}}=V_{F}\ket{0}^{\otimes n}$ is a uniform superposition over the search subspace $F$, which may encode some or all of the problem's constraints. Each query consists of applying the oracle operator $O_{f}$ followed by the diffusion operator $R_{F}=V_{F}(2\ket{0}\bra{0}^{\otimes n} - \mathbf{I})V_{F}^{\dagger}$. After $\kappa_{F}^{opt}$ queries, a solution can be obtained with high probability upon measurement.
• Figure 2: (a) Quantum circuit for preparing $\ket{D_{1}^{\mu}}$ on $\mu$ qubits. (b) Quantum circuit for preparing $\ket{GHZ_{\mu,\nu}^{(X)}}$ on $\mu$ qubits. From the circuits, the total gate count, the two-qubit gate count, and the circuit depth are $2\mu-1$, $2\mu-2$, and $2\mu-1$ for (a), and $2\mu+1$, $\mu-1$, and $\mu+1$ for (b), respectively.
• Figure 3: Quantum circuit for preparing $\ket{D_{1,1}^{\mu}}$ on $\mu$ qubits.
• Figure 4: Comparison of the solution counts for the exact-cover instance in Eq. (\ref{['ex:exact-cover']}) under different initialization strategies. The standard uniform initialization is compared with two constraint-aware initialization (CA init.) schemes incorporating $(C'_{1}, C'_{2})$ and $(C'_{1}, C'_{2}, R'_{3})$, respectively. Solid (dashed) curves correspond to the ideal (noisy) model, where depolarizing noise with error rates of $10^{-5}$ and $10^{-4}$ is applied to one- and two-qubit gates. Each data point is obtained by averaging over 20 independent circuit executions, each consisting of 1000 measurement shots. Bidirectional arrows indicate the difference in the solution counts between the ideal and noisy models, evaluated at the optimal number of queries for each initialization scheme.
• Figure 5: Difference in solution counts between the ideal and noisy models, evaluated at the respective optimal number of queries for different initialization strategies, where the bars show the solution counts in the noisy model and $\Delta$ denotes the corresponding difference. All simulation settings are the same as in Figure \ref{['Figure04']}. The strategies include the standard uniform initialization and several constraint-aware initializations incorporating single constraint sets $(C'_{2})$ and $(C'_{1})$, as well as multiple constraint sets obtained through preprocessing, namely $(C'_{1}, C'_{2})$ and $(C'_{1}, C'_{2}, R'_{3})$.

Theorems & Definitions (5)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3