We propose an approach for quantum amplitude estimation (QAE) designed to enhance computational efficiency while minimizing the reliance on quantum resources. Our method leverages quantum computers to generate a sequence of signals, from which the quantum amplitude is inferred through classical post-processing techniques. Unlike traditional methods that use quantum phase estimation (QPE), which requires numerous controlled unitary operations and the quantum Fourier transform, our method avoids these complex and resource-demanding steps. By integrating quantum computing with classical post-processing techniques, our method significantly reduces the need for quantum gates and qubits, thus optimizing the utilization of quantum hardware. We present numerical simulations to validate the effectiveness of our method and provide a comprehensive analysis of its computational complexity and error. This hybrid strategy not only improves the practicality of QAE but also broadens its applicability in quantum computing.
We propose an approach for quantum amplitude estimation (QAE) designed to enhance computational efficiency while minimizing the reliance on quantum resources. Our method leverages quantum computers to generate a sequence of signals, from which the quantum amplitude is inferred through classical post-processing techniques. Unlike traditional methods that use quantum phase estimation (QPE), which requires numerous controlled unitary operations and the quantum Fourier transform, our method avoids these complex and resource-demanding steps. By integrating quantum computing with classical post-processing techniques, our method significantly reduces the need for quantum gates and qubits, thus optimizing the utilization of quantum hardware. We present numerical simulations to validate the effectiveness of our method and provide a comprehensive analysis of its computational complexity and error. This hybrid strategy not only improves the practicality of QAE but also broadens its applicability in quantum computing.
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.
In this work, we provide the first QFT-free algorithm for Quantum Amplitude Estimation (QAE) that is asymptotically optimal while maintaining the leading numerical performance. QAE algorithms appear as a subroutine in many applications for quantum computers. The optimal query complexity achievable by a quantum algorithm for QAE is log queries, providing a speedup of a factor of 1/ε over any other classical algorithm for the same problem. The original algorithm for QAE utilizes the quantum Fourier transform (QFT) which is expected to be a challenge for near-term quantum hardware. To solve this problem, there has been interest in designing a QAE algorithm that avoids using QFT. Recently, the iterative QAE algorithm (IQAE) [1] was introduced with a near-optimal query complexity and small constant factors. In this work, we combine ideas from the preceding line of work to introduce a QFT-free QAE algorithm that maintains the asymptotically optimal query complexity while retaining small constant factors. We supplement our analysis with numerical experiments comparing our performance with IQAE where we find that our modifications retain the high performance, and in some cases even improve the numerical results.
Quantum Amplitude Estimation (QAE) is a key primitive in quantum computing, but its standard implementation using Quantum Phase Estimation is resource-intensive, requiring a large number of coherent qubits in a single circuit block to achieve high precision. This presents a significant challenge for near-term Noisy Intermediate-Scale Quantum (NISQ) devices. To address this, we introduce the Adaptive Windowed Quantum Amplitude Estimation (AWQAE) framework, a modular, scalable and adaptive approach that decouples estimation precision from the number of physical qubits required in a single circuit. AWQAE operates by iteratively estimating the phase bits in small, fixed-size chunks, using a number of smaller, independent quantum circuits, which are amenable to parallel processing. A key technical contribution of this work is introduction of a phase resolution circuit and an ancilla-guided mechanism that enables accurate chunk assignment and eigenphase reconstruction in the presence of multiple eigenstates. This design is inherently NISQ-friendly, by lowering circuit depth and qubit count per block to reduce decoherence and noise effects. A key component of our approach is a robust classical post-processing algorithm that resolves measurement ambiguities that arise during the iterative process. This post-processing routine uses a least-significant-bit (LSB)-to-most-significant-bit (MSB) correction to reconstruct the full, high-precision phase estimate, ensuring accuracy. By combining a modular quantum-classical loop with an ambiguity-aware reconstruction method, AWQAE offers a powerful and flexible solution for performing high-precision QAE on resource-constrained quantum hardware. Our approach demonstrates enhanced scalability, and adaptability, making it a promising candidate for practical applications of QAE in the NISQ era.
Quantum Amplitude Estimation (QAE) -- a technique by which the amplitude of a given quantum state can be estimated with quadratically fewer queries than by standard sampling -- is a key sub-routine in several important quantum algorithms, including Grover search and Quantum Monte-Carlo methods. An obstacle to implementing QAE in near-term noisy intermediate-scale quantum (NISQ) devices has been the need to perform Quantum Phase Estimation (QPE) -- a costly procedure -- as a sub-routine. This impediment was lifted with various QPE-free methods of QAE, wherein Grover queries of varying depths / powers (often according to a "schedule") are followed immediately by measurements and classical post-processing techniques like maximum likelihood estimation (MLE). Existing analyses as to the optimality of various query schedules in these QPE-free QAE schemes have hitherto assumed noise-free systems. In this work, we analyse QPE-free QAE under common noise models that may afflict NISQ devices and report on the optimality of various query schedules in the noisy regime. We demonstrate that, given an accurate noise characterization of one's system, one must choose a schedule that balances the trade-off between the greater ideal performance achieved by higher-depth circuits, and the correspondingly greater accumulation of noise-induced error.
Quantum Amplitude Estimation (QAE) is a fundamental component of many quantum computing algorithms and applications. The practical implementation of QAE workflows on current Noisy Intermediate-Scale Quantum (NISQ) platforms requires accurate probability amplitude representation and reliable measurement recovery. Understanding the behavior of individual computational stages is therefore essential for improving the overall performance of practical quantum estimation procedures. This study presents a component-wise validation of the amplitude encoding process within a controlled quantum simulation environment. The experimental implementation was developed using the Qiskit framework, where classical probability values were encoded into single-qubit quantum states through parameterized rotation operations and subsequently recovered from measurement statistics. A systematic evaluation was performed across the probability interval from 0.1 to 0.9 using 8192 measurement shots for each experiment. The experimental results demonstrate that the recovered measurement probabilities closely match the corresponding target values throughout the investigated domain. Statistical evaluation produced a Mean Absolute Error (MAE) of 0.003521, a Mean Squared Error (MSE) of 1.67 × 10⁻⁵, and a Root Mean Squared Error (RMSE) of 0.004087, indicating high numerical consistency under ideal simulation conditions. The findings suggest that the amplitude encoding stage accurately represents classical probability values and provides a reliable baseline for studying more advanced quantum estimation workflows. The component-wise validation methodology adopted in this work offers a practical framework for investigating individual stages of quantum algorithms and supports future research on hybrid quantum-classical processing strategies for NISQ-era quantum computing.
We introduce a variant of Quantum Amplitude Estimation (QAE), called Iterative QAE (IQAE), which does not rely on Quantum Phase Estimation (QPE) but is only based on Grover’s Algorithm, which reduces the required number of qubits and gates. We provide a rigorous analysis of IQAE and prove that it achieves a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo simulation with provably small constant overhead. Furthermore, we show with an empirical study that our algorithm outperforms other known QAE variants without QPE, some even by orders of magnitude, i.e., our algorithm requires significantly fewer samples to achieve the same estimation accuracy and confidence level.