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arXiv:2307.07478 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Modal Analysis on Quantum Computers via Qubitization

Yasunori Lee, Keita Kanno

Published Aug 30, 2025• 2 citations
arXivHTMLPDF

Abstract

Natural frequencies and normal modes are basic properties of a structure which play important roles in analyses of its vibrational characteristics. As their computation reduces to solving eigenvalue problems, it is a natural arena for application of quantum phase estimation algorithms, in particular for large systems. In this paper, we take up some simple examples of (classical) coupled oscillators and show how the algorithm works by using qubitization methods based on a sparse structure of the matrix. We explicitly construct block-encoding oracles along the way, showing an exponential advantage over any known classical methods. We also propose a way to prepare initial states, and briefly touch on a more generic oracle construction for systems with repetitive structure. As a demonstration, we also give some estimates of the necessary number of physical qubits and actual runtime it takes when carried out on a fault-tolerant quantum computer.

DOI
10.1109/qce65121.2025.00237

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Natural frequencies and normal modes are basic properties of a structure which play important roles in analyses of its vibrational characteristics. As their computation reduces to solving eigenvalue problems, it is a natural arena for application of quantum phase estimation algorithms, in particular for large systems. In this paper, we take up some simple examples of (classical) coupled oscillators and show how the algorithm works by using qubitization methods based on a sparse structure of the matrix. We explicitly construct block-encoding oracles along the way, showing an exponential advantage over any known classical methods. We also propose a way to prepare initial states, and briefly touch on a more generic oracle construction for systems with repetitive structure. As a demonstration, we also give some estimates of the necessary number of physical qubits and actual runtime it takes when carried out on a fault-tolerant quantum computer.
We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. The functional form of these response functions can be mapped to a corresponding eigenproblem of a Hermitian matrix 𝐻, thus suggesting the use of quantum phase estimation. Our proposed quantum algorithm operates in the standard 𝑠-sparse, oracle-based query access model. For a network of 𝑁 oscillators with maximum norm ∥𝐻∥max, and when the eigenvalue tolerance ɛ is much smaller than the minimum eigenvalue gap, we use 𝒪(log(𝑁⁢𝑠⁢∥𝐻∥max⁢/ɛ) algorithmic qubits and obtain a rigorous worst-case query complexity upper bound 𝒪⁢(𝑠⁢∥𝐻∥max/(𝛿2⁢ɛ)) up to logarithmic factors, where 𝛿 denotes the desired precision on the coefficients appearing in the response functions. Crucially, our proposal does not suffer from the infamous state preparation bottleneck and can as such potentially achieve large quantum speedups compared to relevant classical methods. As a proof-of-principle of exponential quantum speedup, we show that a simple adaptation of our algorithm solves the random glued-trees problem in polynomial time. We discuss practical limitations as well as potential improvements for quantifying finite-size, end-to-end complexities for application to relevant instances.
We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. The functional form of these response functions can be mapped to a corresponding eigenproblem of a Hermitian matrix H , thus suggesting the use of quantum phase estimation. Our proposed quantum algorithm operates in the standard s -sparse, oracle-based query access model. For a network of N oscillators with maximum norm ∥ H ∥ max , and when the eigenvalue tolerance ɛ is much smaller than the minimum eigenvalue gap, we use O ( log ( N s ∥ H ∥ max / ɛ ) algorithmic qubits and obtain a rigorous worst-case query complexity upper bound O ( s ∥ H ∥ max / ( δ 2 ɛ ) ) up to logarithmic factors, where δ denotes the desired precision on the coefficients appearing in the response functions. Crucially, our proposal does not suffer from the infamous state preparation bottleneck and can as such potentially achieve large quantum speedups compared to relevant classical methods. As a proof-of-principle of exponential quantum speedup, we show that a simple adaptation of our algorithm solves the random glued-trees problem in polynomial time. We discuss practical limitations as well as potential improvements for quantifying finite-size, end-to-end complexities for application to relevant instances.
We investigate the quantum algorithm of Babbush et al. (arXiv:2303.13012v3) for simulating coupled harmonic oscillators, which promises exponential speedups over classical methods. Focusing on linearly connected oscillator chains, we bridge the gap between theory and implementation by developing and comparing three concrete realizations of the algorithm. First, we implement a sparse initial state preparation combined with product-formula (Suzuki-Trotter) Hamiltonian simulation. Second, we implement a fully quantum, oracle-based framework in which classical data are accessed via oracles, the Hamiltonian is block-encoded, and time evolution is performed using QSVT-based Hamiltonian simulation. Third, we propose an efficient alternative that combines the sparse state-preparation routine of the first approach with the oracle and block-encoding-based simulation pipeline of the second. We provide these implementations on Classiq, a high-level quantum design platform and provide appropriate resource benchmarks. Our simulation results show that the complex initial state preparation proposed by Babbush et al. can be circumvented at least in the linear-chain case. Finally, we illustrate two physical applications-extracting normal modes and simulating coarse-grained energy propagation-demonstrating how the algorithm connects to measurable observables. Our results clarify the resource requirements of the algorithm and provide concrete pathways toward practical quantum advantage.
Quantum computation of vibrational properties of molecules is a promising platform to obtain computational advantages for computational chemistry. However, fault-tolerant quantum computations of vibrational properties remain a relatively unexplored field in quantum computing. In this work, we present different algorithms for efficient encodings of vibrational Hamiltonians using qubitization. Specifically, we investigate different encoding representations, high-order tensor decomposition to obtain low rank approximations for the vibrational Hamiltonian, rectilinear, and polyspherical coordinate systems, parallelization, and grouping algorithms. To investigate the performance of the different methods, we perform benchmark computations for both small and large molecules with more than one hundred vibrational modes with two- and three-mode coupled Hamiltonian. We present the first resource estimates for qubitization using multimode vibrational Hamiltonians with 1 cm –1 spectral resolution for molecules with up to one hundred vibrational modes. Additionally, we show that tensor decomposition and operator parallelization effectively reduce the T gate depth of the quantum algorithm.
Simulation of molecular vibrations is emerging as a promising application of quantum computing in quantum chemistry. In this work, we develop new strategies for block-encoding the second-quantized nnn-mode representation of the vibrational Hamiltonian, whose algebraic structure closely parallels that of the second-quantized electronic Hamiltonian. We derive linear-combination-of-unitaries (LCU) representations of this Hamiltonian based on Pauli decomposition and tensor hypercontraction (THC) and construct efficient circuits for the corresponding block encodings. We also extend the Block-Invariant Symmetry Shift (BLISS) technique to vibrational Hamiltonians, yielding a lower LCU 1-norm and, consequently, a lower overall circuit cost. For the molecules considered here, BLISS-THC reduces the Toffoli cost of quantum phase estimation by more than two orders of magnitude relative to existing block-encoding approaches. In comparison with recent Trotter-based methods, it shows up to a ninefold reduction in the T-gate count, with the relative savings tending to increase with system size. Asymptotically, and up to polylogarithmic factors, both the Pauli and THC LCUs trade additional ancillas for a quadratic improvement in scaling with the number of modes, while THC provides an additional quadratic improvement in scaling with the number of modals. These results highlight the importance of Hamiltonian factorizations and Toffoli-qubit tradeoffs for reducing the computational cost of quantum simulations of molecular vibrations.
We study the problem of simulating the time evolution of a system of 2nclassical coupled oscillators (e.g., 2nballs connected by springs) on a quantum computer. We map Newton’s equation for harmonic potentials to Schrödinger’s equation, such that the amplitudes of an O(n)\mathcal{O}(n)O(n)-qubit quantum state encode the momenta and displacements of the 2nclassical oscillators. Given oracle access to the masses and spring constants, we describe a quantum algorithm with query and time complexity poly (n) that solves this problem when certain parameters are polynomially bounded and the initial state is easy to prepare. As an example application, we apply our quantum algorithm to efficiently estimate the normalized kinetic energy of an oscillator at any time. We then show that any classical algorithm solving the same problem must make 2Ω(n)2^{\Omega(n)}2Ω(n) queries to the oracle and we also show that when the oracles are instantiated by poly (n)-size circuits, the problem is BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers.