Variational Quantum Eigensolver (VQE) algorithm is one of few approaches where the hope for near-term quantum advantage concentrates. However, they face challenges connected with measurement stochastic noise, barren plateaus, and optimization difficulties in periodic parameter spaces. While most of the efforts concentrates on optimizing the quantum part of the procedure, here we aim to enhance the classical optimization by utilizing a modified version of Harmonic Oscillator-based Particle Swarm Optimization (HOPSO). By adapting its dynamics to respect the periodicity of quantum parameters and enhance noise resilience, we show its strengths on hydrogen (H2) and lithium hydride (LiH) molecules modeled as 4- and 8-qubit Hamiltonians. HOPSO achieves competitive ground-state energy approximations and demonstrates improved robustness compared to COBYLA, Differential Evolution (DE), and standard Particle Swarm Optimization (PSO) methods in all situations and outperforms other methods under realistic noise conditions. These results suggest that a properly tailored classical part of VQE algorithms can tackle with current problems and gives hope for its scalability for larger systems.
Variational Quantum Eigensolver (VQE) algorithm is one of few approaches where the hope for near-term quantum advantage concentrates. However, they face challenges connected with measurement stochastic noise, barren plateaus, and optimization difficulties in periodic parameter spaces. While most of the efforts concentrates on optimizing the quantum part of the procedure, here we aim to enhance the classical optimization by utilizing a modified version of Harmonic Oscillator-based Particle Swarm Optimization (HOPSO). By adapting its dynamics to respect the periodicity of quantum parameters and enhance noise resilience, we show its strengths on hydrogen (H2) and lithium hydride (LiH) molecules modeled as 4- and 8-qubit Hamiltonians. HOPSO achieves competitive ground-state energy approximations and demonstrates improved robustness compared to COBYLA, Differential Evolution (DE), and standard Particle Swarm Optimization (PSO) methods in all situations and outperforms other methods under realistic noise conditions. These results suggest that a properly tailored classical part of VQE algorithms can tackle with current problems and gives hope for its scalability for larger systems.
In the field of finding ground and excited states, where quantum computation holds significant promise, using a variational quantum eigensolver (VQE) is a typical approach. However, the success of this approach is vulnerable to two factors: classical optimization for the ansätz parameters and noise from quantum devices. To address these challenges, we adopted particle swarm optimization (PSO) based on swarm intelligence for VQE and presented its performance. Furthermore, a modified PSO, gradient-based adaptive quantum-behaved particle swarm optimization (GAQPSO), is proposed. This algorithm adaptively upgrades parameters based on gradients or shared information within the swarm, enhancing optimization capability and noise resistance. We tested this algorithm using VQE simulations on several molecular systems with different geometries and found that, when using random initial values, GAQPSO achieves accurate results even in the presence of noise, whereas traditional PSO, QPSO, COBYLA, and gradient-based algorithms (GD and L-BFGS-B) fail. The GAQPSO algorithm presents a novel approach to optimizing VQE, demonstrating superior performance in parameter optimization and noise resilience compared to traditional optimization algorithms. This effectively advances the process of achieving a quantum advantage for quantum computing in quantum chemistry problems.
While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum-classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy, standard L2 squared-norm regularization, can systematically stabilize hybrid quantum-classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty proportional to the squared norm of the parameters, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians, H2, LiH, and the Random Field Ising Model (RFIM), we observe improved performance over a broad window of the regularization strength. Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.
Abstract While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum--classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy---standard L2 squared-norm regularization---can systematically stabilize hybrid quantum--classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty (R(θ)=λ∥θ∥2), without modifying the quantum circuit or measurement process. Across all tested Hamiltonians---H(2), LiH, and the Random Field Ising Model (RFIM)---we observe improved performance over a broad window of the regularization strength (λ). Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.
The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024
The variational quantum eigensolver (or VQE), first developed by Peruzzo et al. (2014), has received significant attention from the research community in recent years. It uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits facing exact modeling of the exponentially growing electronic wavefunction for these many-electron systems. The VQE may be used to model these complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. One important advantage is that variational algorithms have been shown to present some degree of resilience to the noise in the quantum hardware. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm, but without clear descriptions of how the diverse parts fit together. The potential practical advantages of the algorithm are also widely discussed in the literature, but with varying conclusions. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims at disentangling the relevant literature to provide a comprehensive overview of the progress that has been made on the different parts of the algorithm, and to discuss future areas of research that are fundamental for the VQE to deliver on its promises. All the different components of the algorithm are reviewed in detail. These include the representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process to find ground state energies, the post processing mitigation of quantum errors, and suggested best practices. We identify four main areas of future research: (1) optimal measurement schemes for reduction of circuit repetitions required; (2) large scale parallelization across many quantum computers; (3) ways to overcome the potential appearance of vanishing gradients in the optimization process for large systems, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated in a tractable manner. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.
Recent practical approaches for the use of current generation noisy quantum devices in the simulation of quantum many-body problems have been dominated by the use of a variational quantum eigensolver (VQE). These coupled quantum-classical algorithms leverage the ability to perform many repeated measurements to avoid the currently prohibitive gate depths often required for exact quantum algorithms, with the restriction of a parametrized circuit to describe the states of interest. In this work, we show how the calculation of zero-temperature dynamic correlation functions defining the linear response characteristics of quantum systems can also be recast into a modified VQE algorithm, which can be incorporated into the current variational quantum infrastructure. This allows for these important physical expectation values describing the dynamics of the system to be directly converged on the frequency axis, and they approach exactness over all frequencies as the flexibility of the parametrization increases. The frequency resolution hence does not explicitly scale with gate depth, which is approximately twice as deep as a ground-state VQE. We apply the method to compute the single-particle Green's function of ab initio dihydrogen and lithium hydride molecules, and demonstrate the use of a practical active space embedding approach to extend to larger systems. While currently limited by the fidelity of two-qubit gates, whose number is increased compared to the ground-state algorithm on current devices, we believe the approach shows potential for the extraction of frequency dynamics of correlated systems on near-term quantum processors.