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arXiv:2604.20952

Adiabatic Error Cancellation in Berry Phase Estimation

Chusei Kiumi

quant-phSubmitted Apr 22, 2026• Announced Apr 24, 2026• 0 citations• Revised Jul 21, 2026 (v2)
v1 · Apr 22, 2026v2 · Jul 21, 2026v3 · Sep 16, 2026
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Abstract

The Berry phase encodes the geometry of a closed Hamiltonian path, complementing the dynamical phase determined by the accumulated energy. We uncover an adiabatic error-cancellation principle arising from this geometric character, distinguishing Berry-phase estimation from energy estimation. To extract the Berry phase while canceling the dynamical phase, we combine finite-runtime evolutions generated by $\pm H$ along the loop. This construction also exactly cancels the leading $O(T^{-1})$ and all higher odd-order nonoscillatory phase errors. Richardson extrapolation further reduces the residual error to an oscillatory contribution of order $O(\|\dot H(0)\|^2Δ(0)^{-4}T^{-2})$, whose amplitude is controlled by endpoint data. Beyond this deterministic cancellation, we show that runtime randomization suppresses the remaining oscillatory contribution, reducing the bias after $r$ levels of Richardson extrapolation to $O(T^{-2(r+1)})$ for any fixed $r$. By combining these error-cancellation principles, we obtain a randomized Hadamard-test algorithm for Berry phase estimation over the full range $[0,2π)$. The resulting improvement in runtime scaling turns the geometric origin of the cancellation into an algorithmic advantage, reducing the required coherent evolution time without increasing the asymptotic sample complexity. These features make Berry phase estimation a promising candidate for practical quantum computation in the early fault-tolerant regime.

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26 pages, 4 figures

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The Berry phase encodes the geometry of a closed Hamiltonian path, complementing the dynamical phase determined by the accumulated energy. We uncover an adiabatic error-cancellation principle arising from this geometric character, distinguishing Berry-phase estimation from energy estimation. To extract the Berry phase while canceling the dynamical phase, we combine finite-runtime evolutions generated by ±H\pm H±H along the loop. This construction also exactly cancels the leading O(T−1)O(T^{-1})O(T−1) and all higher odd-order nonoscillatory phase errors. Richardson extrapolation further reduces the residual error to an oscillatory contribution of order O(∥H˙(0)∥2Δ(0)−4T−2)O(\|\dot H(0)\|^2Δ(0)^{-4}T^{-2})O(∥H˙(0)∥2Δ(0)−4T−2), whose amplitude is controlled by endpoint data. Beyond this deterministic cancellation, we show that runtime randomization suppresses the remaining oscillatory contribution, reducing the bias after rrr levels of Richardson extrapolation to O(T−2(r+1))O(T^{-2(r+1)})O(T−2(r+1)) for any fixed rrr. By combining these error-cancellation principles, we obtain a randomized Hadamard-test algorithm for Berry phase estimation over the full range [0,2π)[0,2π)[0,2π). The resulting improvement in runtime scaling turns the geometric origin of the cancellation into an algorithmic advantage, reducing the required coherent evolution time without increasing the asymptotic sample complexity. These features make Berry phase estimation a promising candidate for practical quantum computation in the early fault-tolerant regime.
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The adiabatic theorem in quantum mechanics implies that if a system is in a discrete eigenstate of a Hamiltonian and the Hamiltonian evolves in time arbitrarily slowly, the system will remain in the corresponding eigenstate of the evolved Hamiltonian. Understanding corrections to the adiabatic result that arise when the evolution of the Hamiltonian is slow, but not arbitrarily slow, has become increasingly important, especially since adiabatic evolution has been proposed as a method of state preparation in quantum computing. This paper identifies two regimes, an adiabatic regime in which corrections are generically small and can depend on details of the evolution throughout the path and a hyperadiabatic regime in which the error is given by a form similar to an asymptotic expansion in the inverse of the evolution time with the coefficients depending principally on the behavior at the endpoints. However, the error in this hyperadiabatic regime is neither given by a true asymptotic series nor solely dependent on the endpoints: The coefficients combine the contributions from both endpoints, with relative phase factors that depend on the average spectral gaps along the trajectory, multiplied by the evolution time. The central result of this paper is to identify a quantity, referred to as the typical error, which is obtained by appropriately averaging the error over evolution times that are small compared to the evolution time itself. This typical error is characterized by an asymptotic series and depends solely on the endpoints of the evolution, remaining independent of the details of the intermediate evolution.
The Berry phase is a fundamental geometric quantity for characterizing the geometry and topology of quantum many-body systems. Previous work established a super-polynomial quantum advantage in Berry phase estimation at inverse-polynomial precision, given an ansatz state approximating the initial ground state. However, whether this hardness persists for Berry phases quantized by symmetry, which can be distinguished at constant precision, remained open. In this work, we characterize the computational complexity of quantized Berry phase estimation. First, we prove that deciding whether the Berry phase is exactly 000 or πππ is BQP\mathsf{BQP}BQP-complete when the spectral gap is inverse-polynomially small. This result implies that distinguishing the quantized Berry phase still has a super-polynomial quantum advantage, assuming BPP≠BQP\mathsf{BPP}\neq \mathsf{BQP}BPP=BQP. To prove this result, we introduce an encoding that maps the output of any quantum computation to a Berry phase that is exactly 000 or πππ. Second, we extend this hardness to physically motivated Hamiltonians on a 2D square lattice, including Heisenberg and XY interactions. Third, we provide a classical polynomial-time algorithm for geometrically local Hamiltonians on fixed-dimensional lattices with constant spectral gaps. Therefore we identify that the spectral gap is a key resource for the classical tractability, rather than precision. These results establish a super-polynomial quantum advantage for computing a quantized topological invariant that is unchanged under symmetry- and gap-preserving deformations, and provide new steps towards practical quantum advantage in quantum physics.
We present an adaptive variational quantum algorithm to estimate the Berry phase accumulated by a nondegenerate ground state under cyclic, adiabatic evolution of a time-dependent Hamiltonian. Our method leverages cyclic adiabatic evolution of the Hamiltonian and employs adaptive variational quantum algorithms for state preparation and evolution, optimizing circuit efficiency while maintaining high accuracy. We benchmark our approach on dimerized Fermi–Hubbard chains with four sites, demonstrating precise Berry phase simulations in both noninteracting and interacting regimes. Our results show that circuit depths reach up to 106 layers for noninteracting systems and increase to 279 layers for interacting systems due to added complexity. In addition, we demonstrate the robustness of our scheme across a wide range of parameters governing adiabatic evolution and variational algorithms. These findings highlight the potential of adaptive variational quantum algorithms for advancing quantum simulations of topological materials and computing geometric phases in strongly correlated systems.
We formulate a continuous-variable quantum computing (CVQC) algorithm to study Berry's phase on photonic quantum computers. We demonstrate that CVQC allows the simulation of charged particles with orbital angular momentum under the influence of an adiabatically changing B⃗\vec{B}B field. Although formulated entirely in the CVQC setting, our construction uses only passive linear-optical operations (beam splitters and phase shifts), which act identically in single-photon photonic architectures. This enables experimental realisation on the Quandella Ascella platform, where we observe the Berry's phase phenomenon with interferometric measurement. We also generalise the framework to more rapid non-adiabatic evolution. By concatenating Aharonov-Anandan cycles for opposing magnetic fields we demonstrate that one can engineer a circuit in which dynamical phases and leading non-geometric errors cancel by symmetry, leaving the intrinsically robust geometric phase contribution.
Recently, it has been proposed to perform quantum computation through the Berry phase (adiabatic cyclic geometric phase) shift with NMR. This geometric quantum gate will hopefully be fault tolerant to certain types of error because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through the Aharonov-Anandan phase (non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. Moreover, in the scheme we have chosen the zero dynamic phase time evolution loop in obtaining the geometric phase shift; we need not perform any extra operation to cancel the dynamic phase.