We introduce the qumode subspace variational quantum eigensolver (QSS-VQE), a hybrid quantum-classical algorithm for computing molecular excited states using the Fock basis of bosonic qumodes in circuit quantum electrodynamics (cQED) devices. This approach harnesses the native universal gate sets of qubit-qumode architectures to construct highly expressive variational ansatze, offering potential advantages over conventional qubit-based methods. In QSS-VQE, the electronic structure Hamiltonian is first mapped to a qubit representation and subsequently embedded into the Fock space of bosonic qumodes, enabling efficient state preparation and reduced quantum resource requirements. We demonstrate the performance of QSS-VQE through simulations of molecular excited states, including dihydrogen and a conical intersection in cytosine. Additionally, we explore a bosonic model Hamiltonian to assess the expressivity of qumode gates, identifying regimes where qumode-based implementations outperform purely qubit-based approaches. These results highlight the promise of leveraging bosonic degrees of freedom for enhanced quantum simulation of complex molecular systems.
We introduce the qumode subspace variational quantum eigensolver (QSS-VQE), a hybrid quantum-classical algorithm for computing molecular excited states using the Fock basis of bosonic qumodes in circuit quantum electrodynamics (cQED) devices. This approach harnesses the native universal gate sets of qubit-qumode architectures to construct highly expressive variational ansatze, offering potential advantages over conventional qubit-based methods. In QSS-VQE, the electronic structure Hamiltonian is first mapped to a qubit representation and subsequently embedded into the Fock space of bosonic qumodes, enabling efficient state preparation and reduced quantum resource requirements. We demonstrate the performance of QSS-VQE through simulations of molecular excited states, including dihydrogen and a conical intersection in cytosine. Additionally, we explore a bosonic model Hamiltonian to assess the expressivity of qumode gates, identifying regimes where qumode-based implementations outperform purely qubit-based approaches. These results highlight the promise of leveraging bosonic degrees of freedom for enhanced quantum simulation of complex molecular systems.
Quantum harmonic oscillators, or qumodes, provide a promising and versatile framework for quantum computing. Unlike qubits, which are limited to two discrete levels, qumodes have an infinite-dimensional Hilbert space, making them well-suited for a wide range of quantum simulations. In this work, we focus on the molecular electronic structure problem. We propose an approach to map the electronic Hamiltonian into a qumode bosonic problem that can be solved on bosonic quantum devices using the variational quantum eigensolver (VQE). Our approach is demonstrated through the computation of ground potential energy surfaces for benchmark model systems, including H 2 and the linear H 4 molecule. The preparation of trial qumode states and the computation of expectation values leverage universal ansatzes based on the echoed conditional displacement (ECD), or the selective number-dependent arbitrary phase (SNAP) operations. These techniques are compatible with circuit quantum electrodynamics (cQED) platforms, where microwave resonators coupled to superconducting transmon qubits can offer an efficient hardware realization. This work establishes a new pathway for simulating many-fermion systems, highlighting the potential of hybrid qubit-qumode quantum devices in advancing quantum computational chemistry.
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 and linear H4, 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 and H2S to spectroscopic accuracy with per-fragment circuits that scale as O(N) in single-qumode gates and O(N2) in beam-splitter gates for N qumodes. For the case of CO2, 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.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.
Harnessing the full power of nascent quantum processors requires the efficient management of a limited number of quantum bits with finite coherent lifetimes.Hybrid algorithms, such as the variational quantum eigensolver (VQE), leverage classical resources to reduce the required number of quantum gates.Experimental demonstrations of VQE have resulted in calculation of Hamiltonian ground states, and a new theoretical approach based on a quantum subspace expansion (QSE) has outlined a procedure for determining excited states that are central to dynamical processes.We use a superconducting-qubit-based processor to apply the QSE approach to the H 2 molecule, extracting both ground and excited states without the need for auxiliary qubits or additional minimization.Further, we show that this extended protocol can mitigate the effects of incoherent errors, potentially enabling larger-scale quantum simulations without the need for complex error-correction techniques.
Abstract Molecular simulations with the variational quantum eigensolver (VQE) are a promising application for emerging noisy intermediate-scale quantum computers. Constructing accurate molecular ansatze that are easy to optimize and implemented by shallow quantum circuits is crucial for the successful implementation of such simulations. Ansatze are, generally, constructed as series of fermionic-excitation evolutions. Instead, we demonstrate the usefulness of constructing ansatze with ``qubit-excitation evolutions', which, contrary to fermionic excitation evolutions, obey ``qubit commutation relations'. We show that qubit excitation evolutions, despite the lack of some of the physical features of fermionic excitation evolutions, accurately construct ansatze, while requiring asymptotically fewer gates. Utilizing qubit excitation evolutions, we introduce the iterative qubit excitation based VQE (IQEB-VQE) algorithm. The IQEB-VQE performs molecular simulations using a problem-tailored ansatz, grown iteratively by appending evolutions of single and double qubit excitation operators. By performing numerical simulations for small molecules, we benchmark the IQEB-VQE, and compare it against other competitive VQE algorithms. In terms of circuit efficiency and time complexity, we find that the IQEB-VQE systematically outperforms the previously most circuit-efficient, practically scalable VQE algorithms.
Abstract Variational quantum‐classical hybrid algorithms are emerging as important tools for simulating quantum chemistry with quantum devices. These algorithms can be applied to evaluate various molecular properties, including potential energy surfaces. Here in, recent progresses on the development of the so‐called variational quantum eigensolver (VQE) are surveyed. The eigensolver aims at reducing the consumption of quantum resources as much as possible. The key feature of VQE is that variation quantum states are optimized by a feedback process, where the measurement of the Hamiltonian is implemented term by term. This approach avoids the need of encoding all of the information about the molecular Hamiltonian in a quantum circuit. The VQE method is also compatible with classical methods in quantum chemistry, such as unitary coupled‐cluster ansatz. Furthermore, basic elements of VQE are covered, such as qubit encoding, mapping rules of the fermionic operators, ansatz preparation, together with several techniques for improving the performance, including constraining, and error mitigation.