Quantum Phase Estimation (QPE) is a foundational algorithm for molecular ground-state energy estimation, but its deep circuit requirements make direct hardware execution impractical on Noisy Intermediate-Scale Quantum (NISQ) devices. We present an analytically grounded variational surrogate framework in which a shallow Variational Quantum Circuit (VQC) is trained to reproduce the QPE measurement distribution without any quantum circuit simulation. The training target is computed entirely classically via the Dirichlet kernel, evaluated directly from the Full Configuration Interaction (FCI) ground-state energy, the ancilla qubit count, and the time evolution parameter, eliminating the exponentially scaling simulation bottleneck of prior surrogate approaches. We apply this framework to the hydrogen molecule (H
2) with a symmetry-tapered Hamiltonian, conducting a four-stage experimental investigation on IBM Quantum hardware. Stage 1 compares linear and full entangler topologies for the
RY-
RZ-
CZ ansatz, with and without XpXm Dynamical Decoupling (DD), across four distributional metrics (Hellinger distance, fidelity error, total variation distance, Jensen-Shannon divergence), identifying the linear entangler as optimal. Stage 2 varies VQC layers (
p=1 to
5) for the linear-entangler ansatz, identifying single-layer depth as optimal under hardware noise. Stage 3 applies this configuration to the reduced
RY-
CZ ansatz, comparing ideal and noisy simulator-trained parameters. A supplementary noise analysis at
p∈{8,64} characterizes the depth-dependent interplay between circuit depth and DD effectiveness. The framework enables faithful QPE mimicry using a linearly scaling VQC, recovering the ground-state energy within the chemical accuracy threshold (1 kcal/mol), constituting a scalable, hardware-efficient paradigm for QPE-based molecular energy estimation on NISQ devices.
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.
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.
Abstract Quantum phase estimation (QPE) is a fundamental tool in quantum computing, facilitating efficient simulations of complex problems in quantum chemistry and materials science. While most phase estimation algorithms are deterministic, recent advancements indicate that incorporating randomness can enhance performance. This study introduces a framework for randomized QPE that merges the benefits of randomized compilation with phase estimation algorithms based on quantum signal processing. Our proposed algorithms effectively reduce circuit depths by eliminating the need for precise Hamiltonian time evolution, making them advantageous for digital quantum computers estimating the eigenvalue and eigenvector properties of Hamiltonians. Notably, our findings show that the quantum stochastic drift protocol (qDRIFT)-based randomized algorithm surpasses the original phase estimation with qDRIFT, especially in scaling inverse failure probabilities. We also establish that a circuit depth of O ( log ( M ) ) suffices for estimating M distinct observables. The protocol is executed through multiple iterations of the randomized algorithms combined with classical shadow techniques. Overall, our framework retains many advantages of the randomized compilation technique, making it a compelling solution for challenges in quantum chemistry.
Background Quantum phase estimation (QPE) is a foundational subroutine in quantum algorithms that ranges from Shor’s factoring scheme to variational quantum eigensolvers and quantum metrology protocols. Classical post-processing replacements for the inverse quantum Fourier transform (iQFT) —specifically Kitaev-style iterative estimation and Bayesian update schemes—substantially reduce circuit depth, making them practical for noisy intermediate-scale quantum (NISQ) hardware. Despite this promise, the statistical performance of these methods under the conditions that actually arise in practice—non-eigenstate inputs characterised by eigenstate overlap p ≡ O = | ψ | ϕ | 2 < 1 and mixed-state purity γ ≡ P = Tr ( ρ 2 ) < 1 —remains uncharacterised in the literature. Methods We develop a rigorous information-theoretic framework for Bayesian QPE (BQPE) beyond the eigenstate assumption. Starting from the binary measurement likelihood attenuated by the joint factor O P , we derive exact classical Fisher information expressions and apply the Cramér–Rao inequality to obtain tight variance lower bounds. We then construct an adaptive measurement protocol based on von Mises posterior updates, prove its asymptotic efficiency in the frequentist sense, and establish matching upper and lower bounds on sample complexity. All theoretical results are validated against 2,000 -trial Monte Carlo simulations and cross-checked against published nitrogen-vacancy (NV) centre and photonic experimental data. Results We prove that the fundamental variance floor for any unbiased frequentist estimator is ( 1 − p γ ) / ( N p γ n ̄ 2 ) 1 / ( N ( p γ ) 2 n ̄ 2 ) , where N
Quantum Phase Estimation(QPE) is a fundamental quantum algorithm that is used for the estimation of eigenphases of unitary operators. Its main goal is to determine the phase associated with each eigenstate. Usually, it take steps such as prepare quantum states, apply controlled unitaries, inverse quantum fourier transformation, and measurement. This study uses the OpenAI Gym framework to build a customized QPE environment. Here, the phase of a randomly generated target unitary operator is estimated using a quantum circuit. Through interaction with this environment, the DQN agent learns the best course of action to increase phase estimation accuracy. It exhibits more flexibility in noisy environments and reduces estimating mistakes. With its insights and approaches for further study in this area, this effort represents a significant advancement in the use of Deep Reinforcement Learning in quantum computing. A Comparative analysis between IBM Quantum(ibm kyiv) and the Aer Simulator on the OpenAI Gym environment using RL agents has been done.