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arXiv:2609.35978

Chern-number estimation: nearly optimal quantum algorithm and provable quantum speedup

Soichiro Imamura, Shintaro Ae, Kazuki Sakamoto, Ryu Hayakawa, Chusei Kiumi

quant-phSubmitted Sep 28, 2026• Announced Sep 30, 2026• 0 citations
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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 $\tilde{O}(L_xL_y/Δ_{\min}^3)$, 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.

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Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

arXiv:2002.12508•Quantum•Dec 14, 2020•189 citations
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 O~(LxLy/Δmin⁡3)\tilde{O}(L_xL_y/Δ_{\min}^3)O~(Lx​Ly​/Δmin3​), where Lx,LyL_x,L_yLx​,Ly​ bound the parameter derivatives of the Hamiltonian and Δmin⁡Δ_{\min}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\mathsf{FBQP}FBQP and is BQP\mathsf{BQP}BQP-hard, and merely deciding whether the Chern number is zero or one is already BQP\mathsf{BQP}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.
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.
Quantum algorithms provide a potential strategy for solving computational problems that are intractable by classical means. Computing the topological invariants of topological matter is one central problem in research on quantum materials, and a variety of numerical approaches for this purpose have been developed. However, the complexity of quantum many-body Hamiltonians makes calculations of topological invariants challenging for interacting systems. Here, we present two quantum circuits for calculating Chern numbers of two-dimensional quantum matter on quantum computers. Both circuits combine a gate-based adiabatic time evolution over the discretized Brillouin zone with particular phase estimation techniques. The first algorithm uses many qubits, and we analyze it using a tensor-network simulator of quantum circuits. The second circuit uses fewer qubits, and we implement it experimentally on a quantum computer based on superconducting qubits. Our results establish a method for computing topological invariants with quantum circuits, taking a step towards characterizing interacting topological quantum matter using quantum computers. Published by the American Physical Society 2024
We investigate the use of variational quantum algorithms to prepare and characterize fractional quantum Hall states on near-term quantum processors. Focusing on the ν=1/3ν=1/3ν=1/3 Laughlin phase described by the V1V_1V1​ Haldane pseudopotential, we formulate the lowest-Landau-level problem in second quantization, and implement particle-number-preserving variational circuits combined with the variational quantum eigensolver (VQE) and variational quantum deflation (VQD). We benchmark the approach in two complementary geometries: Haldane sphere and torus shape. On the Haldane sphere, the target state is a unique zero-energy Laughlin ground state, providing a controlled test of the variational workflow and of excited-state reconstruction. On the torus, the problem retains the genuinely two-dimensional periodic character of the quantum Hall liquid and exhibits the threefold topological ground-state degeneracy expected for the ν=1/3ν=1/3ν=1/3 fractional filling factor. This feature makes the torus a more demanding benchmark than the quasi-one-dimensional cylinder or thin-torus limits commonly exploited in state-preparation quantum protocols. We benchmark the hardware-optimized variational states against exact diagonalization using energy estimates, error-mitigated observables, and subspace-containment diagnostics. Our results show that hybrid quantum algorithms can approximately reconstruct the low-energy structure of small fractional quantum Hall systems, including the topological ground-state manifold on the torus. Beyond serving as a benchmark for quantum hardware, this geometry-resolved approach provides a route toward quantum simulations of fractional Chern insulators and strongly correlated topological phases in realistic two-dimensional materials.
The Berry phase is a fundamental quantity in the classification of topological phases of matter. In this paper, we present a new quantum algorithm and several complexity-theoretical results for the Berry phase estimation (BPE) problems. Our new quantum algorithm achieves BPE in a more general setting than previously known quantum algorithms, with a theoretical guarantee. For the complexity-theoretic results, we consider three cases. First, we prove BQP\mathsf{BQP}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 UQMA∩co\mathsf{UQMA} \cap \mathsf{co}UQMA∩co-UQMA\mathsf{UQMA}UQMA-completeness when we have an a priori\textit{a priori}a priori bound for the ground-state energy. Here, UQMA\mathsf{UQMA}UQMA is the unique witness version of QMA\mathsf{QMA}QMA, and UQMA∩co\mathsf{UQMA} \cap \mathsf{co}UQMA∩co-UQMA\mathsf{UQMA}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 UQMA∩co\mathsf{UQMA} \cap \mathsf{co}UQMA∩co-UQMA\mathsf{UQMA}UQMA. Third, we show PUQMA∩co-UQMA[log]\mathsf{P}^{\mathsf{UQMA} \cap \mathsf{co}\text{-}\mathsf{UQMA}\mathsf{[log]}}PUQMA∩co-UQMA[log]-hardness and containment in PPGQMA[log]\mathsf{P}^{\mathsf{PGQMA[log]}}PPGQMA[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.
Probing multiple properties of ground states is central to understanding quantum matter, yet conventional readout can be prohibitively costly because it consumes many independently prepared copies. Here, in copy-scarce regimes, we establish optimal trade-offs between the number of initial ground-state copies and elapsed Hamiltonian-evolution time required for estimating multiple observables or reconstructing a full classical description of a unique gapped ground state. We derive a unifying inequality that quantifies how well ground states can be distinguished using initial copies and controlled dynamics, even when all copies may be consumed. It yields worst-case lower bounds on elapsed Hamiltonian time that any protocol must incur to perform either task. We construct parallel catalytic readouts that return all initial copies nearly unchanged while attaining these bounds up to poly-logarithmic factors. Our protocols require no coherent preparation circuit or its inverse, and their elapsed Hamiltonian time depends only logarithmically on the inverse of the desired return error. Thus, optimal copy--dynamics trade-offs can be attained together with catalytic return. Numerical benchmarks further show substantial reductions in end-to-end elapsed Hamiltonian time relative to a task-specialized copy-consuming protocol, even under an optimistic preparation-cost model. Our results reveal a copy--dynamics resource landscape connecting copy-scarce learning to copy-only endpoints realized by known protocols for shadow tomography and pure-state tomography.
Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.