Chern-number estimation: nearly optimal quantum algorithm and provable quantum speedup
Abstract
Topological phases of matter are characterized by global properties of quantum states beyond conventional local order parameters. The Chern number is a central invariant in this characterization, linking the geometry of quantum states to robust physical observables and making its determination key to understanding quantum matter. We construct a nearly optimal quantum algorithm for estimating the Chern number of a smooth Hamiltonian family over a two-dimensional parameter torus with a unique, gapped ground state, including interacting many-body systems. The algorithm transports and reuses two ground-state copies prepared at a single reference point and uses generalized quantum signal processing to coherently accumulate geometric phases. It achieves a Hamiltonian-oracle evolution time of Õ(LxLy/Δmin3), where Lx,Ly bound the parameter derivatives of the Hamiltonian and Δmin 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 FBQP and is BQP-hard, and merely deciding whether the Chern number is zero or one is already 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.
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BibTeX
@misc{imamura2026chernnumber,
title = {{Chern-number estimation: nearly optimal quantum algorithm and provable quantum speedup}},
author = {Soichiro Imamura and Shintaro Ae and Kazuki Sakamoto and Ryu Hayakawa and Chusei Kiumi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35978},
eprint = {2609.35978},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35978},
}
RIS
TY - UNPB
TI - Chern-number estimation: nearly optimal quantum algorithm and provable quantum speedup
AU - Soichiro Imamura
AU - Shintaro Ae
AU - Kazuki Sakamoto
AU - Ryu Hayakawa
AU - Chusei Kiumi
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35978
UR - https://arxiv.org/abs/2609.35978
AB - Topological phases of matter are characterized by global properties of quantum states beyond conventional local order parameters. The Chern number is a central invariant in this characterization, linking the geometry of quantum states to robust physical observables and making its determination key to understanding quantum matter. We construct a nearly optimal quantum algorithm for estimating the Chern number of a smooth Hamiltonian family over a two-dimensional parameter torus with a unique, gapped ground state, including interacting many-body systems. The algorithm transports and reuses two ground-state copies prepared at a single reference point and uses generalized quantum signal processing to coherently accumulate geometric phases. It achieves a Hamiltonian-oracle evolution time of Õ(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 FBQP and is BQP-hard, and merely deciding whether the Chern number is zero or one is already 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.
ER -