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

S. Choi, Kyeongwon Lee, Jong‐In Choi, Chaebin Jung, Woojoo Lee

Published Jun 12, 2026• 1 citation
Publisher

Abstract

Abstract 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(2n−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.

DOI
10.1007/s11128-026-05235-9

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Abstract 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(2n−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.
With rapid advancement in quantum computing technology, continuous efforts are being directed to simulation and emulation of quantum algorithms on classical platforms. A well-known limitation to classical emulation of quantum circuits is scalability. Existing hardware emulators implement gate-based circuit models of quantum circuits that result in heavy resource utilization and degrade the scalability of the system. Also, current quantum emulation hardware use fixedpoint arithmetic, which has an adverse effect on accuracy when the system is scaled up. In this work, we employ a complexmultiply-and-accumulate (CMAC) and lookup-based emulation approach that greatly reduces resource utilization and improves system scalability in terms of number of emulated qubits. We demonstrate emulation of up to 16 fully-entangled qubits which is highest among existing work. We design fully-pipelined, highthroughput hardware architectures that use floating-point precision for higher accuracy. Experimental evaluation and analysis of the architectures in terms of speed and area is also provided. The emulator is prototyped on a high-performance reconfigurable computing (HPRC) system and our results demonstrate quantitative improvement over existing Field-Programmable-Gate-Array (FPGA)-based hardware emulators.
Quantum computing holds immense promise for solving complex problems beyond the reach of classical computers. However, the realization of practical quantum computers faces significant challenges, including its inherent noise and error rates. To address these challenges and explore the capabilities of quantum computers, accurate and efficient quantum computer simulations are essential. This paper presents a novel quantum computer simulation framework designed to accommodate a wide range of custom reduced precision floating-point formats. Leveraging our proposed quantum computation algorithm, our framework enables the quantum computer simulations using exceptionally low precision floating-point numbers, while maintaining high accuracy levels. Through a systematic analysis, we explore the impact of floating-point precision on the accuracy of the quantum computer simulation, providing optimized bit-widths tailored for floating-point numbers. Particularly, using our proposed quantum computation algorithm, our framework achieves quantum state vector outcomes comparable to full-precision quantum simulations, even with extremely low precision floating-point formats ranging from 4 to 6 bits. Overall, these findings highlight the viability of precision reduction techniques in enhancing the efficiency of quantum computer simulations.
Quantum hardware emulators using field programmable gate arrays (FPGAs) have been demonstrated to efficiently exploit the inherent parallelism of quantum algorithms and perform better than sequential software simulators. However, one key problem in mapping quantum circuits on FPGAs is the exponential increase in resource requirements with the increase in the number of quantum bits (qubits). As a result, the precision of hardware emulators is often reduced to save resources and accommodate larger quantum circuits which results in lower accuracy and lower throughput. In this paper, we present a full precision FPGA quantum emulator that uses 32-bit floating-point arithmetic and has a fully pipelined architecture. Our emulator is capable of emulating an entangled 4-qubit quantum system performing Quantum Fourier Transform, Grover's Search algorithms on a single FPGA node. Our experimental work was conducted on a highly scalable multi-node (multi-FPGA) state-of-the-art high-performance reconfigurable computer (HPRC). Experimental results show that our emulator is functionally accurate and has a higher operating frequency relative to other emulators. For emulating larger-scale circuits we propose using a multi-node architecture scheme with low latency, high bandwidth inter-node communication.
Fixed-point quantum search algorithms succeed at finding one of MMM target items among NNN total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property---the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.
Grover’s algorithm represents a cornerstone quantum attack against symmetric cryptography, offering a quadratic speedup for exhaustive key search. While its theoretical properties are well established, the practical robustness of Grover’s algorithm under non-ideal oracle execution remains insufficiently quantified, particularly in security-relevant contexts. This work presents a comprehensive quantitative analysis of Grover’s robustness against intermittent oracle failures, modeling realistic execution imperfections that may arise in practical quantum attack scenarios. Using a reduced two-dimensional representation of amplitude amplification, we define and measure a practical reliability threshold beyond which Grover’s success probability collapses abruptly. Our results demonstrate that this threshold decreases exponentially with problem size, scaling as p∗(n)∝2−n/2 p^{\ast}(n) \propto 2^{-n/2} p∗(n)∝2−n/2, implying that cryptographically relevant problem sizes require unrealistically high oracle reliability to preserve quantum advantage. For n=40 n=40 n=40, the oracle must fail less than once every ∼800,000 \sim 800{,}000 ∼800,000 iterations to preserve practical advantage, a requirement that becomes exponentially more stringent for larger key sizes. These findings do not contradict Grover’s theoretical correctness but highlight critical robustness limitations that must be explicitly considered when assessing quantum threats to symmetric cryptography. We provide actionable insights for post-quantum security assessment by translating abstract quantum assumptions into concrete reliability requirements, demonstrating that the practical feasibility of Grover-based attacks may be significantly lower than commonly assumed under realistic hardware constraints.
Grover's quantum search algorithm provides a fundamental quadratic speedup for unstructured datasets, reducing query complexity from O(N)\mathcal{O}(N)O(N) to O(N)\mathcal{O}(\sqrt{N})O(N​). However, the algorithm's reliance on precise iteration counts leads to the ``soufflé problem,'' where over-rotation results in a sharp decline in success probability. This limitation is particularly restrictive when the number of solution states, MMM, is unknown. In this work, we present an ancilla-mediated fixed-point quantum search algorithm that achieves robust convergence by mapping the solution amplitude to a dedicated ancilla qubit. Unlike existing phase-matching fixed-point methods, our approach utilizes Grover's real-plane reflections, thereby maintaining the intuitive geometric architecture of the original algorithm. We demonstrate that this method achieves a success probability of at least 92.6%92.6\%92.6% with a query complexity of approximately O(N/M)\mathcal{O}(\sqrt{N/M})O(N/M​), effectively bridging the gap between standard amplitude amplification and robust fixed-point convergence.