arXiv subjects

Ryu Hayakawa

Publications and source records attributed to Ryu Hayakawa.

9 recordsLinked to original sources

arXiv2026

Computational complexity of Berry phase estimation in topological phases of matter

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 $\mathsf{BQP}$-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.

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arXiv2026

Unweighted Gapped Clique Homology is $\mathsf{QMA}_1$-complete

Deciding whether the clique complex of a given graph has nontrivial homology in a given dimension, under vertex-product weighting and an inverse-polynomial spectral gap promise on the combinatorial Hodge Laplacian, is known to be $\mathsf{QMA}_1$-hard and contained in $\mathsf{QMA}$ by King and Kohler (FOCS 2024). The vertex weights are essential in the known proof, where they provide the scale separation needed for the spectral gap analysis. We prove that, when every vertex has weight one, the problem remains $\mathsf{QMA}_1^{g_2}$-hard and is contained in $\mathsf{QMA}_1^{g_2}$, where $\mathsf{QMA}_1^{g_2}$ is $\mathsf{QMA}_1$ with the universal gate set $g_2=\{\mathsf{X},\mathsf{CX},\mathsf{CCX},H\otimes H\}$. The construction replaces weight by expansion: each vertex of the weighted complex is blown up into a clique whose size encodes its weight. The block sizes are chosen so that symmetric averages reproduce the weighted Hodge metric. The symmetric sector therefore carries the weighted Laplacian up to a common scalar factor, while a local averaging argument gives a uniform lower bound on the orthogonal complement to the symmetric sector. Hence the weighted gap analysis transfers to an unweighted clique complex without introducing either additional low-energy states or spurious homology. Containment in $\mathsf{QMA}_1^{g_2}$ follows from Rudolph's exact linear combination of unitaries simulation of sparse integer clique Laplacians. The result shows that the gap promise, rather than vertex weighting, is the source of the complexity of gapped clique homology.

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arXiv2026

Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians

Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. In this paper, we introduce and study the problem of normalized persistence, a practically motivated and easily interpretable version of persistent homology that counts the fraction of holes that persist at different lengthscales. We prove that a variant of normalized persistence is $\mathsf{DQC}_1$-hard and contained in $\mathsf{BQP}$, giving evidence of an exponential quantum speedup for TDA under the standard assumption that $\mathsf{DQC}_1 \not\subseteq \mathsf{BPP}$. These are the first $\mathsf{DQC}_1$-hardness results that are directly applicable to TDA instances. We also find a close connection between normalized persistence and the complexity of estimating spectral quantities in the low-energy subspace of local Hamiltonians. We study a family of such problems, including a low-energy normalized subtrace and spectral density. We show that these are $\mathsf{DQC}_1$-hard for $O(1)$-local Hamiltonians, strengthening previous results that required log-local interactions. We also introduce a variant of $\mathsf{DQC}_1$ with perfect completeness ($\mathsf{SDQC}_1$) to characterize the hardness of problems normalized by an exact kernel. This includes normalized persistence for $O(1)$-local Hamiltonians, which we show is $\mathsf{SDQC}_1$-hard.

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arXiv2026

Provable quantum speedups for computing persistence in topological data analysis

Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We provide an efficient quantum algorithm for a computational problem closely related to a core task in TDA -- determining whether a given hole persists across different length scales. Further, we prove the problem itself is $\mathsf{BQP}_1$-hard, implying that a classical solution is extremely unlikely; this stands in contrast to all previous quantum approaches to TDA, where the problems were also intractable for quantum computers, or where a rigorous proof of classical hardness still remains open. This result implies an {exponential} quantum speedup for this problem under standard complexity-theoretic assumptions. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.

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arXiv2026

Quantum Walks on Simplicial Complexes and Harmonic Homology: Application to Topological Data Analysis with Superpolynomial Speedups

This work investigates whether quantum walks on simplicial complexes exhibit quantum advantages. We introduce a novel quantum walk that encodes the combinatorial Laplacian, a key object reflecting the topology of the simplicial complex. We construct a unitary encoding projecting onto the kernel of the Laplacian, representing the harmonic cycles in the complex's homology. Our efficient construction of quantum walk unitaries for clique complexes paves the way for exploring higher-order interactions within topological structures. Our construction requires $O(n^3\log(1/ε)/λ_k)$ gates, where $n$ is the number of vertices, $λ_k$ is the smallest non-zero eigenvalue of the Laplacian, and $ε$ is the projection error. Our results indicate apparent superpolynomial quantum speedup with quantum walks, without quantum oracles, provided the spectral gap of the Laplacian is inverse-polynomially bounded and efficient simplex sampling is available. Crucially, the walk operates on a state space encompassing both positively and negatively oriented simplices, effectively doubling its size compared to unoriented approaches. Through coherent interference of these paired simplices, we are able to successfully encode the combinatorial Laplacian, which would otherwise be impossible. This is our major technical contribution. We also extend the framework by constructing variant quantum walks that enable us to: (1) estimate normalized persistent Betti numbers throughout a deformation process, (2) verify a specific QMA$_1$-hard problem related to clique complex homology, showcasing potential applications in computational complexity theory, and (3) solve the high-dimensional discrete Dirichlet problem (HDDP), generalizing the classical discrete Dirichlet problem on graphs to simplicial complexes, with an apparent superpolynomial speedup over the best known classical algorithm.

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arXiv2025

Computational complexity of the homology problem with orientable filtration: MA-completeness

We show the existence of an MA-complete homology problem for a certain subclass of simplicial complexes. The problem is defined through a new concept of orientability of simplicial complexes that we call a "uniform orientable filtration", which is related to sign-problem freeness in homology. The containment in MA is achieved through the design of new, higher-order random walks on simplicial complexes associated with the filtration. For the MA-hardness, we design a new gadget with which we can reduce from an MA-hard stoquastic satisfiability problem. Therefore, our result provides the first natural MA-complete problem for higher-order random walks on simplicial complexes, combining the concepts of topology, persistent homology, and quantum computing.

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arXiv2024

Improved Hardness Results for the Guided Local Hamiltonian Problem

Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with $6$-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to $1/2$ with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.

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arXiv2022

Quantum algorithm for persistent Betti numbers and topological data analysis

Topological data analysis (TDA) is an emergent field of data analysis. The critical step of TDA is computing the persistent Betti numbers. Existing classical algorithms for TDA are limited if we want to learn from high-dimensional topological features because the number of high-dimensional simplices grows exponentially in the size of the data. In the context of quantum computation, it has been previously shown that there exists an efficient quantum algorithm for estimating the Betti numbers even in high dimensions. However, the Betti numbers are less general than the persistent Betti numbers, and there have been no quantum algorithms that can estimate the persistent Betti numbers of arbitrary dimensions. This paper shows the first quantum algorithm that can estimate the (normalized) persistent Betti numbers of arbitrary dimensions. Our algorithm is efficient for simplicial complexes such as the Vietoris-Rips complex and demonstrates exponential speedup over the known classical algorithms.

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arXiv2019

Fine-grained quantum supremacy based on Orthogonal Vectors, 3-SUM and All-Pairs Shortest Paths

Fine-grained quantum supremacy is a study of proving (nearly) tight time lower bounds for classical simulations of quantum computing under "fine-grained complexity" assumptions. We show that under conjectures on Orthogonal Vectors (OV), 3-SUM, All-Pairs Shortest Paths (APSP) and their variants, strong and weak classical simulations of quantum computing are impossible in certain exponential time with respect to the number of qubits. Those conjectures are widely used in classical fine-grained complexity theory in which polynomial time hardness is conjectured. All previous results of fine-grained quantum supremacy are based on ETH, SETH, or their variants that are conjectures for SAT in which exponential time hardness is conjectured. We show that there exist quantum circuits which cannot be classically simulated in certain exponential time with respect to the number of qubits first by considering a Quantum Random Access Memory (QRAM) based quantum computing model and next by considering a non-QRAM model quantum computation. In the case of the QRAM model, the size of quantum circuits is linear with respect to the number of qubits and in the case of the non-QRAM model, the size of the quantum circuits is exponential with respect to the number of qubits but the results are still non-trivial.

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