Precision-aware fixed-point emulation of Grover’s algorithm: asymptotic error bounds and design guidelines

We propose a rigorous precision-aware framework for fixed-point emulation of Grover’s quantum search algorithm, systematically analyzing how truncation errors propagate and accumulate through quantum gate operations. Although fixed-point arithmetic significantly reduces resource requirements in quantum computing emulators, limited fractional-bit precision introduces non-negligible numerical errors, adversely impacting algorithm accuracy. To address this, we introduce a simplified two-amplitude quantum-state representation that captures Grover’s algorithmic structure across iterations. Leveraging this model, we analytically derive explicit asymptotic error bounds on the emulated measurement probability distribution, demonstrating that errors scale exponentially as \(O(2^{n - f})\) $O(2^{n-f})$ , where n is the number of qubits and f is fractional-bit precision. Extensive numerical simulations and practical experiments on an actual fixed-point quantum emulator validate our theoretical predictions, confirming the accuracy and robustness of our error analysis. Finally, we present a closed-form expression to determine the minimal fractional-bit precision needed to meet a specified accuracy threshold, providing concrete, actionable guidelines for designing resource-efficient quantum computing emulators.