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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 Sep 16, 2026 (v3)
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 that arises from this geometric character and has no counterpart for energy-based quantities. To extract the Berry phase while canceling the dynamical phase, we combine finite-runtime evolutions generated by $\pm H$ along the loop. This construction exactly cancels the leading $O(T^{-1})$ and all higher odd-order nonoscillatory phase errors. Richardson extrapolation further reduces the leading residual error to an oscillatory contribution whose amplitude is set by the endpoints of the loop alone. Beyond this deterministic cancellation, runtime randomization suppresses the remaining oscillatory contribution, reducing the bias after $r$ Richardson levels to $O(T^{-(2r+2)})$ for fixed $r$. The same average also suppresses the bias from an imperfectly prepared guiding state, so that the admissible preparation infidelity no longer depends on the target accuracy \(\varepsilon_B\). Combining these principles, we obtain a randomized Hadamard-test algorithm for Berry phase estimation that reduces the coherent evolution time to \(O(\varepsilon_B^{-1/(2r+2)})\) while retaining the standard \(O(\varepsilon_B^{-2})\) sample complexity. This turns the geometric origin of the cancellation into an algorithmic advantage and makes Berry phase estimation a promising task for early fault-tolerant quantum computers.

Comments
28 pages, 3 figures

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This walkthrough derives, from substrate primitives, three closely related quantities of standard quantum mechanics: the Berry connection Aₙ(R) = i⟨n(R) | ∇ᴿ n(R)⟩ on parameter space; the Berry phase γₙ[C] = ∮₍C₎ Aₙ(R) · dR accumulated under adiabatic cyclic evolution around a closed loop; and the Berry curvature Ωₙ(R) = ∇ᴿ × Aₙ(R), with γₙ[C] = ∫₍S₎ Ωₙ · dS for any surface S bounded by C. Each is derived, not posited. Berry’s 1984 derivation observed that adiabatic cyclic evolution leaves an instantaneous eigenstate |n(R)⟩ unchanged in identity but multiplied by a phase factor e^{iγₙ} above and beyond the standard dynamical phase. The construction proceeds: write |ψ(t)⟩ = e^{−i∫₀ᵗ Eₙ(t′) dt′ / ℏ} e^{iγₙ(t)} |n(R(t))⟩, substitute into Schrödinger, project onto ⟨n|, and read off γ̇ₙ = i⟨n | ∇ᴿ n⟩ · Ṙ. This is mathematically correct. It is also mechanistically opaque. Standard QM does not say what makes the parameter‑dependent eigenstate |n(R)⟩ a coherent object across R‑values that are never simultaneously instantiated, why parallel transport produces a connection rather than a flat structure, what makes the geometric phase rate‑independent, or what physical object the Berry curvature describes. The construction is presented as a derived consequence of QM postulates, but the postulates themselves are unexplained at any deeper ontological level. The Berry connection and curvature are objects in parameter space, not physical space — their meaning at the level of underlying ontology is left unstated. This walkthrough shows how the Event Density framework forces all three quantities from substrate ontology when participation rules are parameter‑dependent. The connection one‑form is the substrate‑level statement that adiabatic identity‑preservation under parameter change requires a path‑dependent compensation. The geometric‑phase rate‑independence is the substrate‑level statement that the connection lives on parameter space rather than time. The curvature is the substrate‑level rule‑type curvature in parameter space — directly analogous to the rule‑type curvature of a gauge field in physical space (T17), but lifted to the space of rule parameters. The substrate‑level mechanism, in one sentence: the Berry phase is the holonomy of the rule‑type connection on the bundle of parameter‑dependent participation rules. The derivation chain runs from five substrate primitives (parameter‑dependent participation rule, identity alignment with pre‑individuation amplitudes, channels and adiabatic evolution as slow parameter change, global ED‑geometry in parameter space, closed loops as closed families of rules) through parallel‑transport‑forced connection and Stokes‑theorem‑related curvature, with the geometric‑vs‑dynamical phase separation made explicit.

•May 12, 2026•0 citations
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 that arises from this geometric character and has no counterpart for energy-based quantities. 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 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 leading residual error to an oscillatory contribution whose amplitude is set by the endpoints of the loop alone. Beyond this deterministic cancellation, runtime randomization suppresses the remaining oscillatory contribution, reducing the bias after rrr Richardson levels to O(T−(2r+2))O(T^{-(2r+2)})O(T−(2r+2)) for fixed rrr. The same average also suppresses the bias from an imperfectly prepared guiding state, so that the admissible preparation infidelity no longer depends on the target accuracy (\varepsilon_B). Combining these principles, we obtain a randomized Hadamard-test algorithm for Berry phase estimation that reduces the coherent evolution time to (O(\varepsilon_B^{-1/(2r+2)})) while retaining the standard (O(\varepsilon_B^{-2})) sample complexity. This turns the geometric origin of the cancellation into an algorithmic advantage and makes Berry phase estimation a promising task for early fault-tolerant quantum computers.
In the quantum-computation scenario, the geometric phase gates are becoming increasingly attractive for their intrinsic fault tolerance to disturbance. With an adiabatic cyclic evolution, Berry phase appears to realize a geometric transformation. Performing the quantum gates as many as possible within the timescale of coherence, however, remains an inconvenient bottleneck due to the systematic errors. Here we propose an accelerated adiabatic quantum gate based on the Berry phase, the transitionless driving, and the dynamical decoupling. It reconciles a high fidelity with a high speed in the presence of control noise or imperfection. We optimize the dynamical-decoupling sequence in the time domain under a popular Gaussian noise spectrum following the inversely quadratic power law.
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.
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.
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.
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.
This walkthrough derives, from substrate primitives, three closely related quantities of standard quantum mechanics: the Berry connection Aₙ(R) = i⟨n(R) | ∇ᴿ n(R)⟩ on parameter space; the Berry phase γₙ[C] = ∮₍C₎ Aₙ(R) · dR accumulated under adiabatic cyclic evolution around a closed loop; and the Berry curvature Ωₙ(R) = ∇ᴿ × Aₙ(R), with γₙ[C] = ∫₍S₎ Ωₙ · dS for any surface S bounded by C. Each is derived, not posited. Berry’s 1984 derivation observed that adiabatic cyclic evolution leaves an instantaneous eigenstate |n(R)⟩ unchanged in identity but multiplied by a phase factor e^{iγₙ} above and beyond the standard dynamical phase. The construction proceeds: write |ψ(t)⟩ = e^{−i∫₀ᵗ Eₙ(t′) dt′ / ℏ} e^{iγₙ(t)} |n(R(t))⟩, substitute into Schrödinger, project onto ⟨n|, and read off γ̇ₙ = i⟨n | ∇ᴿ n⟩ · Ṙ. This is mathematically correct. It is also mechanistically opaque. Standard QM does not say what makes the parameter‑dependent eigenstate |n(R)⟩ a coherent object across R‑values that are never simultaneously instantiated, why parallel transport produces a connection rather than a flat structure, what makes the geometric phase rate‑independent, or what physical object the Berry curvature describes. The construction is presented as a derived consequence of QM postulates, but the postulates themselves are unexplained at any deeper ontological level. The Berry connection and curvature are objects in parameter space, not physical space — their meaning at the level of underlying ontology is left unstated. This walkthrough shows how the Event Density framework forces all three quantities from substrate ontology when participation rules are parameter‑dependent. The connection one‑form is the substrate‑level statement that adiabatic identity‑preservation under parameter change requires a path‑dependent compensation. The geometric‑phase rate‑independence is the substrate‑level statement that the connection lives on parameter space rather than time. The curvature is the substrate‑level rule‑type curvature in parameter space — directly analogous to the rule‑type curvature of a gauge field in physical space (T17), but lifted to the space of rule parameters. The substrate‑level mechanism, in one sentence: the Berry phase is the holonomy of the rule‑type connection on the bundle of parameter‑dependent participation rules. The derivation chain runs from five substrate primitives (parameter‑dependent participation rule, identity alignment with pre‑individuation amplitudes, channels and adiabatic evolution as slow parameter change, global ED‑geometry in parameter space, closed loops as closed families of rules) through parallel‑transport‑forced connection and Stokes‑theorem‑related curvature, with the geometric‑vs‑dynamical phase separation made explicit.