Variational quantum computing for quantum simulation: principles, implementations, and challenges

TL;DR

This paper surveys variational quantum computing as a practical framework for quantum simulation on noisy intermediate-scale devices, arguing that quantum data and hybrid quantum–classical workflows can tackle problems beyond classical capabilities under appropriate trainability and noise conditions. It unifies VQAs and QML into a common toolkit, detailing ansatz design (PMA vs. HHA), cost-function crafting, and gradient estimation via the parameter-shift rule, while candidly addressing barren plateaus and measurement-noise limitations. The implementations section inventories ground- and excited-state methods (VQE, VQD, qUCC), dynamical simulations (closed and open systems with McLachlan’s principle and Lindblad dynamics), thermal-state preparation (VQT, QITE), and quantum-learning approaches, illustrating how these techniques enable practical quantum-simulation tasks such as molecular energies, time evolution, and Gibbs-state preparation. Collectively, the work highlights opportunities for hardware-aware algorithm design, co-design of hardware and software, and targeted application domains where variational methods may yield meaningful quantum advantage in the near term, while outlining persistent challenges that guide future research.

Abstract

This work presents a comprehensive overview of variational quantum computing and their key role in advancing quantum simulation. This work explores the simulation of quantum systems and sets itself apart from approaches centered on classical data processing, by focusing on the critical role of quantum data in Variational Quantum Algorithms (VQA) and Quantum Machine Learning (QML). We systematically delineate the foundational principles of variational quantum computing, establish their motivational and challenges context within the noisy intermediate-scale quantum (NISQ) era, and critically examine their application across a range of prototypical quantum simulation problems. Operating within a hybrid quantum-classical framework, these algorithms represent a promising yet problem-dependent pathway whose practicality remains contingent on trainability and scalability under noise and barren-plateau constraints.This review serves to complement and extend existing literature by synthesizing the most recent advancements in the field and providing a focused perspective on the persistent challenges and emerging opportunities that define the current landscape of variational quantum computing for quantum simulation.

Figures (6)

• Figure 1: Schematic of digital and analog quantum simulations. First, an initial state $\ket{\psi}$ is prepared, and then the desired evolution $U$ is mapped onto an effective operator $U'$ implementable in the simulator. After applying $U'$, measurements of the system are performed to extract the physical quantities of interest associated with the simulated state $\ket{\upsilon(t)}$.
• Figure 2: General representation of a quantum circuit. The circuit is read from left to right and represents a controlled operation: the top line is the control qubit and the bottom is the target qubit. When the control qubit is in the $\ket{1}$ state, the unitary operation $U$ is applied to the target qubit. Finally, the target qubit is measured in an appropriate basis (represented by the semicircular gauge).
• Figure 3: General representation of a variational quantum algorithm. Inputs: initial parameters ${\theta}_0$, cost function $C({\theta})$, parametrized ansatz $U({\theta})$, and the input state $\lvert\psi\rangle$. The ansatz is applied on an input state $\lvert\psi\rangle$ or training instances $\rho_k$, followed by an optional basis–change or post–processing block $W$ and measurements of observables $\{O_k\}$. The measured expectations define the cost function $C({\theta})=\sum_k f_k({\theta},\rho_k)$, which a classical optimizer minimizes by updating the initial parameters ${\theta}\!\to\!{\theta}^\ast$. The procedure outputs the optimized parameters ${\theta}^\ast$, minimized cost $C({\theta}^\ast)$, the optimized ansatz $U({\theta}^\ast)$, and the prepared state $\lvert\Psi({\theta}^\ast)\rangle$. The inset cost landscape (bottom right) illustrates the “hypersurface”, where variational training corresponds to searching for the global minimum under NISQ constraints.
• Figure 4: The cost function landscapes with different behaviors. On the left, the mostly flat cost function represents the vanishing of the gradient function, which makes the minimum inaccessible. On the right, the smooth loss function allows for the identification of the minimum point.
• Figure 5: Opportunities in variational quantum computing for quantum simulation. Variational quantum computing offers a versatile framework for quantum simulation, with multiple algorithms demonstrating diverse applications across quantum system characterization. The VQE enables ground state determination peruzzo2014variational, while its extension VQD targets excited states Higgott2019variationalquantum. For dynamical simulations, VQS handles both conservative li2017efficient and open quantum systems PhysRevLett.125.010501doi:10.1021/acs.jpclett.4c00576, complemented by VFF for long-time evolution cirstoiu2020variational. Thermal state preparation is addressed through both QITE motta2020imaginary and VQT approaches verdon2019quantumhamiltonianbasedmodelsvariationalSelisko_2024. Quantum machine learning further expands these capabilities, with QNNs applied to electronic structure xia2018quantum, state preparation kan2026machinewang2021variational, dynamics gibbs2024dynamicalgardas2018quantumlong2024quantum, and high-energy physics guan2021quantum. Additional QML architectures include QGANs for many-body system dynamics kim2024hamiltonian, QSVMs for classification tasks in high-energy physics wu2021application, and QCNNs for phase transition detection baul2025quantum. This diverse algorithmic landscape, employing cost functions ranging from energy expectation values to fidelity-based metrics, demonstrates the rich potential of variational methods for advancing quantum simulation across multiple scientific domains.