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arXiv:2507.22460 · Advanced Quantum Technologies

Toward Practical Quantum Phase Estimation: A Modular, Scalable, and Adaptive Approach

Alok Shukla, Prakash Vedula

Published Mar 1, 2026• 1 citation
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Abstract

ABSTRACT Quantum Phase Estimation (QPE) is a cornerstone algorithm in quantum computing, with applications ranging from integer factorization to quantum chemistry simulations. However, the resource demands of standard QPE, which require a large number of coherent qubits and deep circuits, pose significant challenges for current Noisy Intermediate Scale Quantum (NISQ) devices. In this work, we introduce the Adaptive Windowed Quantum Phase Estimation (AWQPE) algorithm, a novel method designed to address the limitations of standard QPE. AWQPE utilizes small, independent blocks of control qubits to estimate multiple phase bits simultaneously within a “window,” thereby significantly reducing the number of coherent qubits required to achieve a desired precision. These independent blocks are amenable to parallelization and, when combined with a robust least‐significant‐bit (LSB) to most‐significant‐bit (MSB) ambiguity resolution mechanism, enhance the algorithm's accuracy while mitigating the risk of error propagation. Our numerical simulations demonstrate AWQPE's accuracy and improved robustness relative to standard QPE under representative noise models, showcasing a favorable balance between resource efficiency and computational speed. This makes AWQPE particularly well‐suited for near‐term quantum platforms.

DOI
10.1002/qute.202500683

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ABSTRACT Quantum Phase Estimation (QPE) is a cornerstone algorithm in quantum computing, with applications ranging from integer factorization to quantum chemistry simulations. However, the resource demands of standard QPE, which require a large number of coherent qubits and deep circuits, pose significant challenges for current Noisy Intermediate Scale Quantum (NISQ) devices. In this work, we introduce the Adaptive Windowed Quantum Phase Estimation (AWQPE) algorithm, a novel method designed to address the limitations of standard QPE. AWQPE utilizes small, independent blocks of control qubits to estimate multiple phase bits simultaneously within a “window,” thereby significantly reducing the number of coherent qubits required to achieve a desired precision. These independent blocks are amenable to parallelization and, when combined with a robust least‐significant‐bit (LSB) to most‐significant‐bit (MSB) ambiguity resolution mechanism, enhance the algorithm's accuracy while mitigating the risk of error propagation. Our numerical simulations demonstrate AWQPE's accuracy and improved robustness relative to standard QPE under representative noise models, showcasing a favorable balance between resource efficiency and computational speed. This makes AWQPE particularly well‐suited for near‐term quantum platforms.
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 phase estimation (QPE) is one of the core algorithms for quantum computing. It has been extensively studied and applied in a variety of quantum applications such as the Shor's factoring algorithm, quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. The QPE algorithm has been combined with Kitaev's algorithm and the inverse quantum Fourier transform (IQFT) which are utilized as a fundamental component of such quantum algorithms. In this paper, we explore the computational challenges of implementing QPE algorithms on noisy intermediate-scale quantum (NISQ) machines using the IBM Q Experience (e.g., the IBMQX4, 5-qubit quantum computing hardware platform). Our experimental results indicate that the accuracy of finding the phase using these QPE algorithms is severely constrained by the NISQ computer's physical characteristics such as coherence time and error rates. To mitigate these physical limitations, we propose implementing a modified solution by reducing the number of controlled rotation gates and phase shift operations, thereby increasing the accuracy of the finding phase in near-term quantum computers.
Many researchers have been heavily investigated on quantum phase estimation (QPE) algorithms to find the unknown phase, since QPE is the core building block of the most quantum algorithms such as the Shor's factoring algorithm, quantum sampling algorithms, and finding the eigenvalues of unitary matrices. Kitaev's algorithm and QPE algorithms using inverse Quantum Fourier transform were proposed and widely used by researchers as a key component for their quantum algorithms. In this paper, we explore the experimental challenges of QPE algorithms on Noisy Intermediate-Scale Quantum (NISQ) computers by implementing various QPE algorithms on the state-of-the-art IBM quantum computer. Our experimental results demonstrate that the accuracy of finding the phase using these algorithms are severely constrained by NISQ's physical characteristics such as coherence time and error rates. To mitigate such physical limitations, we propose modified solutions of these algorithms by reducing the number of control gates and phase shift operations. Our experimental results showed that our solutions can significantly increase the accuracy of the finding phase in near-term quantum computers.
Quantum Phase Estimation (QPE) routines are known to fail probabilistically even with perfect gates and input states. This effect stems from an incompatibility of finite-sized quantum registers to capture a phase within QPE with phase angles of infinite precision, and the effect extend even beyond what would be reasonably expected from rounding. This effect can be partially mitigated by biasing the phase register with a window, or taper state, from classical signal processing. This paper focuses on how windowing a coherent QPE used as a subroutine can improve the accuracy of the overall algorithm. Specifically we study the quantum task of estimating observables where window-assisted coherent QPE is used as a subroutine to implement a reflection about an eigenstate. Quantum resource estimates show over 2-orders-of-magnitude reduction in Toffoli counts over the previous costed techniques -- also assisted by the use of improved block encoding techniques -- demonstrating an encouraging decrease in resources for quantum computation of molecular observables. Since QPE, as one of only a few quantum building blocks, appears as a subroutine in many algorithms; this analysis also provides a model for understanding how window functions propagate to an improved error in composite algorithms.
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.
Quantum phase estimation~(QPE) is central to numerous quantum algorithms, yet its standard implementation demands an $\calO(m^{2})$-gate quantum Fourier transform~(QFT) on mmm control qubits-a prohibitive overhead on near-term noisy intermediate-scale quantum (NISQ) devices. We introduce the \emph{Phase-Fidelity-Aware Truncated QFT} (PFA-TQFT), a family of approximate QFT circuits parameterised by a truncation depth~ddd that omits controlled-phase rotations below a hardware-calibrated fidelity threshold~$\eps$. Our central result establishes $\TV(P_{\varphi},P_{\varphi}^{d})\leqπ(m{-}d)/2^{d}$, showing that for $d=\calO(\log m)$ circuit size collapses from $\calO(m^{2})$ to $\calO(m\log m)$ while estimation error grows by at most $\calO(2^{-d})$. We characterise $\dstar=\Floor{\log_{2}(2π/\eps_{2q})}$ directly from native gate fidelities, demonstrating 31.3 -43.7\% at m = 30, gate-count reduction on IBM Eagle/Heron and IonQ~Aria with negligible accuracy loss. Numerical experiments on the transverse-field Ising model confirm all theoretical predictions and reveal a \emph{noise-truncation synergy}: PFA-TQFT outperforms full QFT under NISQ noise $\eps_{2q}\gtrsim 2\times10^{-3}$.