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arXiv:2509.07716 · Physics Letters A

Measuring the non-abelian quantum phase with the algorithm of quantum phase estimation

Seng Ghee Tan, Son-Hsien Chen, Yaocheng Yang, Yen‐Fu Chen, Yen‐Lin Chen, Jia-Hsiu Hsieh

Published Jul 18, 2026• 0 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2026.132019

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Quantum Phase Estimation: A Novel Adaptive Optimization Algorithm for Quantum State Estimation

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Quantum phase estimation (QPE) is a fundamental technique for determining the phase of a quantum state, crucial for various quantum algorithms and quantum measurements. This paper introduces a novel adaptive optimization algorithm, dubbed the "Adaptive Quantum Phase Estimation" (AQPE) algorithm, designed to enhance the efficiency of QPE by dynamically adjusting parameters during the state estimation process. We demonstrate the effectiveness of AQPE through rigorous theoretical analysis and extensive numerical simulations, achieving significantly improved convergence rates compared to traditional optimization techniques. The algorithm leverages a feedback mechanism based on the observed phase error during the estimation, allowing it to iteratively refine the parameter set and ultimately locate the optimal phase estimation. We present preliminary results demonstrating the algorithm's capability to mitigate the challenges associated with conventional QPE, particularly in the presence of noisy data. The implications of this research extend to broader advancements in quantum algorithm design and computational power.

•Aug 31, 2026•0 citations
We consider the time evolution of a particle on a ring with a long solenoid through and show that due to the Aharonov-Bohm effect this system naturally makes up a physical implementation of the quantum phase estimation algorithm for a U(1)\text{U}(1)U(1) unitary operator. The implementation of the full quantum phase estimation algorithm with a U(N)\text{U}(N)U(N) unitary operator is realized through the non-Abelian Aharonov-Bohm effect. The implementation allows for a more physically intuitive understanding of the algorithm. As an example we use the path integral formulation of the implemented quantum phase estimation algorithm to analyze the classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$.
We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues and eigenvectors. This proposal combines ideas of frequent measurement, measured quantum Fourier transform, and quantum state tomography. It provides a generalization of the conventional phase estimation algorithm, which is limited to Hermitian or unitary matrices.
Quantum algorithms for quantum chemistry and other many-body fermionic systems work by expressing the Hamiltonian in a basis of qubits and fragmenting the Hamiltonian into a sum of products of Pauli operators whose exponentials are easily encoded on a quantum device. Applying the product of exponentials, known as Trotterization, leads to an error associated with the non-commutativity of operators. This error can lead to breaking the symmetries of the Hamiltonian because the fragments are not symmetry conserving in general. Nonetheless, many algorithms for time evolution rely on Trotterization, including time evolution and quantum phase estimation. We show that we can express the Hamiltonian in terms of Hermitian excitation operators which map to sums of commuting Pauli strings for any encoding and conserve symmetries corresponding to Abelian groups of symmetry operators. Symmetries corresponding to non-Abelian groups, on the other hand, are not fully conserved by Trotterized Hermitian excitation operators, so we developed ``operator kirigami'' to cut the sum of non-commuting operators by orthogonal projection and to fold terms together using unitary rotations. We tested pools of operators for small molecules and basis sets, and found that electron number and spin symmetry conserving pools led to greater errors that decreased for larger molecules and were negated with second-order Trotterization. Our work shows the potential for testing quantum computing algorithms on classical computers by adapting tools used in electronic structure theory with conserved symmetries.
Quantum phase estimation (QPE) is a fundamental technique for determining the phase of a quantum state, crucial for various quantum algorithms and quantum measurements. This paper introduces a novel adaptive optimization algorithm, dubbed the "Adaptive Quantum Phase Estimation" (AQPE) algorithm, designed to enhance the efficiency of QPE by dynamically adjusting parameters during the state estimation process. We demonstrate the effectiveness of AQPE through rigorous theoretical analysis and extensive numerical simulations, achieving significantly improved convergence rates compared to traditional optimization techniques. The algorithm leverages a feedback mechanism based on the observed phase error during the estimation, allowing it to iteratively refine the parameter set and ultimately locate the optimal phase estimation. We present preliminary results demonstrating the algorithm's capability to mitigate the challenges associated with conventional QPE, particularly in the presence of noisy data. The implications of this research extend to broader advancements in quantum algorithm design and computational power.