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Quantum Science and Technology

Resource-efficient quantum phase estimation by randomization

Youle Wang

Published May 30, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1088/2058-9565/addee9

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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.
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.
Randomized protocols are procedures that incorporate probabilistic choices during their execution and they play a central role in quantum algorithms, spanning Hamiltonian simulation, noise mitigation, and measurement tasks. In practical implementations, the dominant cost of such protocols typically arises from circuit execution and measurement, and depends on hardware-specific resources such as gate counts, circuit depth, runtime, or dissipated energy. We introduce a general framework for applying classical importance sampling to randomized quantum protocols. Given a cost function for running quantum circuits, the proposed approach minimizes a net-cost figure of merit that jointly captures the computational expense per circuit and the estimator variance. We further extend the framework to scenarios where the quantum computation is subject to errors arising either from algorithmic approximations or from physical noise, proving that importance sampling preserves estimator bias despite altering the sampling distribution, and to settings with error-detection schemes, where we characterize the resulting changes in the optimal sampling strategy and achievable net-cost reduction. Representative applications include the Qdrift protocol, dephasing channels, mixed-states simulation, composite observables estimation, classical shadows, and probabilistic error cancellation. Overall, our results establish a principled approach for reducing the computational resources required by randomized quantum protocols through classical sampling optimization.
Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision εεε in Heisenberg-limited time T=Θ(1/ε)T=Θ(1/ε)T=Θ(1/ε). Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling T=O(1εlog⁡(δ−1))T = O\left( \frac{1}ε \log\left(δ^{-1}\right)\right)T=O(ε1​log(δ−1)) in both the precision εεε and failure probability δδδ. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.
The quantum stochastic drift protocol, also known as qDRIFT, has become a popular algorithm for implementing time-evolution of quantum systems using randomised compiling. In this work we develop qFLO, a higher order randomised algorithm for time-evolution. To estimate an observable expectation value at time TTT to precision εεε, we show it is sufficient to use circuit depths of O(T2log⁡(1/ε))O(T^2\log(1/ε))O(T2log(1/ε)) -- an exponential improvement over standard qDRIFT requirements with respect to εεε. The protocol achieves this using O(1/ε2)O(1/ε^2)O(1/ε2) repeated runs of the standard qDRIFT protocol combined with classical post-processing in the form of Richardson extrapolation. Notably, it requires no ancillary qubits or additional control gates making it especially promising for near-term quantum devices. Furthermore, it is well-conditioned and inherits many desirable properties of randomly compiled simulation methods, including circuit depths that do not explicitly depend on the number of terms in the Hamiltonian.
Quantum phase estimation is a central primitive in quantum algorithms and sensing, where performance is governed by the sensitivity of measurement signals to the target parameter. While existing methods have developed increasingly sophisticated inference and adaptive design strategies, the signal family used for phase learning is often largely pre-specified. Here we propose a programmable signal design framework for quantum phase estimation based on quantum signal processing, which enables the measurement signal to be tailored to the current uncertainty region. We cast phase estimation as a max-min optimization problem over admissible signals and introduce a sensitivity efficiency parameter that quantifies information gain per query depth. The resulting iterative algorithm combines optimized quantum signal transformations with structured classical inference, retaining Heisenberg-limited scaling while improving sensitivity efficiency and practical resource prefactors. Numerical results show reduced estimation variance compared with standard protocols such as robust phase estimation. Our framework also extends to Hamiltonian eigenvalue estimation in higher dimensions and establishes a quantum-classical co-design paradigm through programmable signal shaping.