arXiv:1909.09415 · Physical Review A

Berry phase estimation in gate-based adiabatic quantum simulation

Bruno Murta, Gonçalo Catarina, Joaquín Fernández-Rossier

Published Feb 11, 2020• 17 citations

Abstract

Gate-based quantum computers can in principle simulate the adiabatic dynamics of a large class of Hamiltonians. Here, we consider the cyclic adiabatic evolution of a parameter in the Hamiltonian. We propose a quantum algorithm to estimate the Berry phase and use it to classify the topological order of both single-particle and interacting models, highlighting the differences between the two. This algorithm is extensible to an interacting topological system. Our results evidence the potential of near-term quantum hardware for the topological classification of quantum matter.

DOI: 10.1103/physreva.101.020302

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arXiv:1909.09415 · Physical Review A

Berry phase estimation in gate-based adiabatic quantum simulation

Bruno Murta, Gonçalo Catarina, Joaquín Fernández-Rossier

Published Feb 11, 2020• 17 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.101.020302

Related papers

Efficient Berry phase calculation via adaptive variational quantum computing approach

Martin Mootz, Yongxin Yao

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 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.

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We develop an adiabatic quantum simulation framework on gate-based quantum computers to probe topological signatures of the one-dimensional fermionic Su--Schrieffer--Heeger--Hubbard (SSHH) model. We present explicit quantum-circuit constructions for initial-state preparation and time evolution, together with a practical measurement protocol and classical post-processing procedure for extracting the many-body Berry phase and the spatial profile of the sublattice polarization. Using classical simulations of the proposed circuits, we demonstrate -- for the first time within a genuine many-body framework -- that the topological characteristics of the SSH model remain robust against weak Hubbard interactions but eventually break down as the chiral-symmetry-breaking component of the interaction exceeds a threshold. The required qubit number, gate complexity, measurement shots, and classical pre- and post-processing costs all scale polynomially with system size. Our results provide a proof-of-concept framework for probing topological properties of interacting many-body systems via adiabatic quantum simulation on future large-scale quantum computers.We develop an adiabatic quantum simulation framework on gate-based quantum computers to probe topological signatures of the one-dimensional fermionic Su--Schrieffer--Heeger--Hubbard (SSHH) model. We present explicit quantum-circuit constructions for initial-state preparation and time evolution, together with a practical measurement protocol and classical post-processing procedure for extracting the many-body Berry phase and the spatial profile of the sublattice polarization. Using classical simulations of the proposed circuits, we demonstrate -- for the first time within a genuine many-body framework -- that the topological characteristics of the SSH model remain robust against weak Hubbard interactions but eventually break down as the chiral-symmetry-breaking component of the interaction exceeds a threshold. The required qubit number, gate complexity, measurement shots, and classical pre- and post-processing costs all scale polynomially with system size. Our results provide a proof-of-concept framework for probing topological properties of interacting many-body systems via adiabatic quantum simulation on future large-scale quantum computers.

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Adiabatic Error Cancellation in Berry Phase Estimation

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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 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.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.

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quant-pharXiv:2609.35978Sep 30, 20260 citations

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\mathsf{BQP}

-completeness when we are given a guiding state that has a large overlap with the ground state. This result establishes an exponential quantum speedup for estimating the Berry phase. Second, we prove

\mathsf{UQMA} \cap \mathsf{co}

\mathsf{UQMA}

-completeness when we have an

\textit{a priori}

bound for the ground-state energy. Here,

\mathsf{UQMA}

is the unique witness version of

\mathsf{QMA}

, and

\mathsf{UQMA} \cap \mathsf{co}

\mathsf{UQMA}

precisely captures the complexity of BPE without the known guiding state. Remarkably, this problem is, to our knowledge, the first natural problem complete for

\mathsf{UQMA} \cap \mathsf{co}

\mathsf{UQMA}

. Third, we show

\mathsf{P}^{\mathsf{UQMA} \cap \mathsf{co}\text{-}\mathsf{UQMA}\mathsf{[log]}}

-hardness and containment in

\mathsf{P}^{\mathsf{PGQMA[log]}}

when we have no additional assumption. These results advance the role of quantum computing in the study of topological phases of matter and provide a pathway for clarifying the connection between topological phases of matter and computational complexity.

\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.

\tilde{O}(L_xL_y/Δ_{\min}^3)

, where

L_x,L_y

bound the parameter derivatives of the Hamiltonian and

Δ_{\min}

is a known spectral-gap lower bound. We prove a matching worst-case oracle lower bound, even when the Chern number is promised to be zero or one, establishing optimality up to polylogarithmic factors. For local Hamiltonian families with an inverse-polynomial gap and a supplied guiding state, exact Chern-number computation is in

\mathsf{FBQP}

and is

\mathsf{BQP}

-hard, and merely deciding whether the Chern number is zero or one is already

\mathsf{BQP}

-complete. These results establish how efficiently quantum computers can estimate a quantized topological invariant and provide evidence for quantum advantage in characterizing topological phases.