Abstract By exploiting the invariance of the molecular Hamiltonian by a unitary transformation of the orbitals it is possible to significantly shorter the depth of the variational circuit in the variational quantum eigensolver (VQE) algorithm by using the wavefunction adapted Hamiltonian through orbital rotation (WAHTOR) algorithm. This work introduces a non-adiabatic version of the WAHTOR algorithm and compares its efficiency with three implementations by estimating quantum processing unit (QPU) resources in prototypical benchmarking systems. Calculating first and second-order derivatives of the Hamiltonian at fixed VQE parameters does not introduce a significant QPU overload, leading to results on small molecules that indicate the non-adiabatic Newton–Raphson method as the more convenient choice. On the contrary, we find out that in the case of Hubbard model systems the trust region non-adiabatic optimization is more efficient. The preset work therefore clearly indicates the best optimization strategies for empirical variational ansatzes, facilitating the optimization of larger variational wavefunctions for quantum computing.
Abstract By exploiting the invariance of the molecular Hamiltonian by a unitary transformation of the orbitals it is possible to significantly shorter the depth of the variational circuit in the variational quantum eigensolver (VQE) algorithm by using the wavefunction adapted Hamiltonian through orbital rotation (WAHTOR) algorithm. This work introduces a non-adiabatic version of the WAHTOR algorithm and compares its efficiency with three implementations by estimating quantum processing unit (QPU) resources in prototypical benchmarking systems. Calculating first and second-order derivatives of the Hamiltonian at fixed VQE parameters does not introduce a significant QPU overload, leading to results on small molecules that indicate the non-adiabatic Newton–Raphson method as the more convenient choice. On the contrary, we find out that in the case of Hubbard model systems the trust region non-adiabatic optimization is more efficient. The preset work therefore clearly indicates the best optimization strategies for empirical variational ansatzes, facilitating the optimization of larger variational wavefunctions for quantum computing.
The use of the variational quantum eigensolver (VQE) for quantum chemistry is one of the most promising applications for noisy intermediate-scale quantum (NISQ) devices. A major limitation is represented by the need to build compact and shallow circuit ansatzes having the variational flexibility to catch the complexity of the electronic structure problem. To alleviate this drawback, we introduce a modified VQE scheme in which the form of the molecular Hamiltonian is adapted to the circuit ansatz through an optimization procedure. Exploiting the invariance of the Hamiltonian by molecular orbital rotations, we can optimize it using gradients that can be calculated without significant computational overload. The proposed method, named Wavefunction Adapted Hamiltonian Through Orbital Rotation (WAHTOR), has been applied to small molecules in numerical state vector simulations. The results demonstrate that, at variance with standard VQE, the method is less dependent on circuit topology and less prone to be trapped into high-energy local minima. It is able to recover a significant amount of electron correlation even with only empirical ansatzes with shallow circuit depth. Noisy calculations demonstrate the robustness and feasibility of the proposed methodology and indicate the hardware requirements to effectively apply the procedure using forthcoming NISQ devices.
In the evolving landscape of quantum computing, variational algorithms have emerged as powerful tools for studying molecular systems. Accurately representing these systems' wavefunctions is challenging due to the limitations of current noisy quantum computers, which can only handle short-depth circuits. This thesis addresses this challenge by providing improved descriptions of wavefunctions using heuristic Ansätze with shallow circuit depth. Two primary approaches are explored: one focused on enhancing wave functions via gate-based circuits, and the other involving pulse-level control of hardware. The non-unitary VQE (nu-VQE) algorithm is introduced to improve trial wave functions using a non-unitary quantum operator, aiming to work with compact circuit Ansätze. Another approach, the Wavefunction-Adapted Hamiltonian Through Orbital Rotation (WAHTOR), exploits the invariance of the molecular Hamiltonian through orbital rotations to adapt it to the empirical circuit Ansatz. Both adiabatic and non-adiabatic optimization strategies are explored for the WAHTOR algorithm, demonstrating its resilience to Ansatz topology variations and noise effect mitigation capability. Furthermore, the convergence of the WAHTOR algorithm, with a computational cost not significantly higher than the VQE, leads to optimized orbitals coinciding with the natural ones. The Pulse-VQE strategy directly encodes variational parameters as pulse amplitudes and durations, significantly reducing the pulse schedule and circuit duration. Both gate-based and pulse-based strategies improve energy results while enhancing circuit compactness, thereby reducing the impact of quantum noise. Finally, ongoing research includes studying wave functions that preserve spin-inversion symmetry and developing entanglement pulses for devices with fixed-frequency transmons coupled by tunable buses.
Abstract The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
Quantum computing promises to efficiently and accurately solve many important problems in quantum chemistry which elude classical solvers, such as the electronic structure problem of highly correlated materials. Two leading methods in solving the ground state problem are the Variational Quantum Eigensolver (VQE) and Adiabatic Quantum Computing (AQC) algorithms. VQE often struggles with convergence due to the energy landscape being highly non-convex and the existence of barren plateaux, and implementing AQC is beyond the capabilities of current quantum devices as it requires deep circuits. Adiabatically-inspired algorithms aim to fill this gap. In this paper, we first present a unifying framework for these algorithms and then benchmark the following methods: the Adiabatically Assisted VQE (AAVQE) (Garcia-Saez and Latorre (2018)), the Variational Adiabatic Quantum Computing (VAQC) (Harwood et al (2022)), and the Adiabatic Quantum Computing with Parametrized Quantum Circuits (AQC-PQC) (Kolotouros et al (2025)) algorithms. Second, we introduce a novel hybrid approach termed G-AQC-PQC, which generalizes the AQC-PQC method, and combines adiabatic-inspired initialization with the low-memory BFGS optimizer, reducing the quantum computational cost of the method. Third, we compare the accuracy of the methods for chemistry applications using the beryllium hydride molecule (BeH2). We compare the approaches across a number of different choices (ansätze types, depth, discretization steps, initial Hamiltonian, adiabatic schedules and method used). Our results show that the G-AQC-PQC outperforms conventional VQE. We further discuss limitations such as the zero-gradient problem and identify regimes where adiabatically-inspired methods offer a tangible advantage for near-term quantum chemistry applications.
Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that, instead of fixing an ansatz upfront, grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware.
Abstract The use of the variational quantum eigensolver (VQE) on near-term quantum computers for solving Kohn–Sham Hamiltonian is still underexplored. In this study, we apply the weighted subspace search variational quantum eigensolver (weighted SSVQE) to obtain the occupied KS orbitals, their corresponding energies, and the total energy, exploring the effects of varying entanglement unitary blocks, circuit depths, and optimization strategies. While the choice of optimization algorithm has minimal impact on the total energy, it significantly influences convergence behavior, stability, and computational time. Our results further reveal the existence of an optimal weight configuration that achieves both faster convergence and improved energy accuracy. Furthermore, our findings highlight the critical role of ansatz depth and entanglement unitary blocks on total energy calculations across molecular geometries. The most effective design begins with a low-expressibility, low-entangling ansatz, such as linear entanglement unitary blocks, and repeats it until a sufficient number of layers is achieved. Using homoatomic and heteroatomic molecule as test sets, the approach remains valid and effective across systems exhibiting different types of electron correlation. This approach proves highly efficient, enabling shallow circuits that avoid convergence issues and closely match standard density functional calculations. This work thus offers a design strategy for shallow, convergence-friendly VQE circuits that match standard density functional energies.