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arXiv:2503.23983 · Theoretical Chemistry Accounts

Quantum computing within a bosonic context: assessing finite basis effects on prototypical vibrational Hamiltonian spectra

Joachim Knapik, Bruno Senjean, Benjamin Lasorne, Yohann Scribano

Published Dec 20, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Quantum computing has recently been emerging in theoretical chemistry as a realistic avenue meant to offer computational speedup to challenging eigenproblems in the context of strongly-correlated molecular systems or extended materials. Most studies so far have been devoted to the quantum treatment of electronic structure, for which the transformation of fermionic operators into the qubit space is quite transparent. In contrast, only a few were directed to the quantum treatment of vibrational structure, which at the moment remains not devoid of unknowns. When simulating an anharmonic vibrational mode under bosonic second quantization, the biggest problem, which we analyze in detail in the present work, is the disruption of the resolution of the identity when truncating the infinite harmonic-oscillator basis set of primitive modals and how this alters the canonical commutation relation. This may lead to serious incorrectness in the evaluation of the Hamiltonian matrix elements when assembling them from finite matrix products, which eventually entail a nonvariational behavior with respect to the finite size of the computational basis. As we show here, a simple cure occurs to be obtained upon using Wick's normal order, for reasons that are not standardly evoked. We also provide a detailed comparison between the boson-to-qubit unary and binary mappings under two different representations: one for the bosonic ladder operators within the harmonic-oscillator primitive basis, and one for the so-called n-mode representation within any type of computational basis. In addition, we discuss the impact of choosing an adequate primitive basis set in terms of quantum computing with respect to its variational convergence efficiency (number of basis functions, hence of qubits) and as regards the magnitude of the 1-norm of the encoded Hamiltonian (a measure of the computational complexity of the quantum algorithm). Such fundamental aspects are illustrated numerically on a one-dimensional anharmonic Hamiltonian model corresponding to a symmetric double-well potential, of interest both for vibrational spectroscopy and chemical reactivity, and which is a challenging situation for numerical convergence due to fine tunneling splitting.

DOI
10.1007/s00214-025-03269-3

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Bosonic quantum devices, which utilize harmonic oscillator modes to encode information, are emerging as a promising alternative to conventional qubit-based quantum devices, especially for the simulation of vibrational dynamics and spectroscopy. We present a framework for digital quantum simulation of vibrational dynamics under anharmonic potentials on these bosonic devices. In our approach, the vibrational Hamiltonian is decomposed into solvable fragments that can be used for Hamiltonian simulation on currently available bosonic hardware. Specifically, we have extended the Cartan subalgebra approach [Yen, T.C.; Izmaylov, A. F. PRX Quantum 2, 2021; 040320]− a method for decomposing quantum Hamiltonians into solvable parts− to bosonic operators, enabling us to construct anharmonic Hamiltonian fragments that can be efficiently diagonalized using Bogoliubov transforms. The approach is tested using a simulation of tunneling dynamics in a model two-dimensional double-well potential and calculations of vibrational eigenenergies for small molecules. Our fragmentation scheme provides a new approach for digital quantum simulations on bosonic quantum hardware for multimode anharmonic vibrational dynamics.
Simulating vibrational dynamics is essential for understanding molecular structure, unlocking useful applications such as vibrational spectroscopy for high-fidelity chemical detection. Quantum algorithms for vibrational dynamics are emerging as a promising alternative to resource-demanding classical approaches, but this domain is largely underdeveloped compared with quantum simulations of electronic structure. In this work, we describe in detail three distinct forms of the vibrational Hamiltonian: canonical bosonic quantization, real space representation, and the Christiansen second-quantized form. Leveraging the Lie algebraic properties of each, we develop efficient fragmentation schemes to enable the use of Trotter product formulas for simulating time evolution. We introduce circuits required to implement time evolution in each form and highlight factors that contribute to the simulation cost, including the choice of vibrational coordinates. Using a perturbative approach for the Trotter error, we obtain tight estimates of the T gate cost for the simulation of time evolution in each form, enabling their quantitative comparison. Combining tight Trotter error estimates and efficient fragmentation schemes, we find that for the CH 4 molecule with 9 vibrational modes, time evolution for approximately 1.8 ps may be simulated using as few as 36 qubits and approximately 3 × 10 8 T gates─an order-of-magnitude speedup over state-of-the-art algorithms. Finally, we present calculations of vibrational spectra using each form to demonstrate the fidelity of our algorithms. This work presents a unified and highly optimized framework that makes simulating vibrational dynamics an attractive use case for quantum computers.
Variational quantum algorithms on bosonic quantum processors are an emerging paradigm for quantum chemistry calculations, exploiting the natural alignment between molecular structure and harmonic oscillator-based hardware. We introduce the qumode-based variational quantum deflation framework (QumVQD) for finding both electronic and vibrational excited state energies on qumode-based architectures. We validate the approach through electronic structure calculations on H2_{\text{2}}2​ and linear H4_{\text{4}}4​, where we introduce Hamming-weight filtering of the Fock basis to enforce particle number conservation and eliminate spurious eigenstates by reducing the required Hilbert space, which reduces the required number of qumodes in turn. We achieve agreement with full configuration interaction (FCI) using the STO-3G basis set within the chemical accuracy threshold at most points along the potential energy surfaces. Extending to the vibrational structure, we combine QumVQD with an existing Hamiltonian fragmentation approach based on Cartan subalgebra, allowing us to compute the vibrational eigenenergies of CO2_{\text{2}}2​ and H2_{\text{2}}2​S to spectroscopic accuracy with per-fragment circuits that scale as O(N)O(N)O(N) in single-qumode gates and O(N2)O(N^2)O(N2) in beam-splitter gates for NNN qumodes. For the case of CO2_{\text{2}}2​, we get total gate counts more than an order of magnitude smaller than those reported for qubit-based vibrational algorithms at this system size. These results demonstrate that bosonic quantum devices are a viable platform for excited-state quantum chemistry, particularly for vibrational problems where qubit-based methods incur substantial boson-to-qubit mapping overhead.
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.
Bosonic quantum devices offer a novel approach to realize quantum computations, where the quantum two-level system ( qubit ) is replaced with the quantum (an)harmonic oscillator ( qumode ) as the fundamental building block of the quantum simulator. The simulation of chemical structure and dynamics can then be achieved by representing or mapping the system Hamiltonians in terms of bosonic operators. In this Perspective, we review recent progress and future potential of using bosonic quantum devices for addressing a wide range of challenging chemical problems, including the calculation of molecular vibronic spectra, the simulation of gas-phase and solution-phase adiabatic and nonadiabatic chemical dynamics, the efficient solution of molecular graph theory problems, and the calculations of electronic structure.
One of the primary challenges prohibiting demonstrations of practical quantum advantages for near-term devices amounts to excessive measurement overheads for estimating relevant physical quantities such as ground state energies. However, with major differences between the electronic and vibrational structures of molecules, the question of how the resource requirements of computing anharmonic, vibrational states can be reduced remains relatively unexplored compared to its electronic counterpart. Importantly, bosonic commutation relations, distinguishable Hilbert spaces and vibrational coordinates allow manipulations of the vibrational system that can be exploited to minimize resource requirements. In this work, we investigate the impact of different coordinate systems and measurement schemes on the number of measurements needed to estimate anharmonic, vibrational states for a variety of three-mode (six-mode) molecules. We demonstrate an average of 3-fold (1.5-fold), with up to 7-fold (2.5-fold), reduction in the number of measurements required by employing appropriate coordinate transformations, based on an automized construction of qubit Hamiltonians from a conventional vibrational structure program.