ABSTRACT We propose a variational quantum regression (VQR) algorithm using a hardware‐efficient ansatz (HEA) structure. This approach enables the quantum state to directly encode classical tabular data with variational parameters corresponding to real‐valued regression coefficients, ensuring high interpretability and efficient optimization without sacrificing expressiveness. By combining a variational quantum circuit with a classical optimizer, our method predicts the ground‐state energy of a hydrogen molecule using full configuration interaction (FCI) data. We quantify the expressibility of the VQR HEA circuit via the Kullback–Leibler (KL) divergence D KL and show the advantages of our R y − R x gate sequence in balancing expressibility. Performed for pennylane using an idealized quantum simulator, our 4‐qubit, 5‐layer HEA‐based VQR model achieves an accuracy of ∼0.99, mean squared error (MSE) < 10 −6 Hartree 2 , and mean absolute error (MAE) around 0.1 × 10 −2 Hartree compared to FCI benchmarks. We further demonstrate how qubit number and layer depth influence model accuracy, providing insights for task‐specific quantum circuit design. Our results advance practical applications of quantum computing for electronic structure problems by introducing an expressive, interpretable ansatz tailored for high‐precision simulations.
ABSTRACT We propose a variational quantum regression (VQR) algorithm using a hardware‐efficient ansatz (HEA) structure. This approach enables the quantum state to directly encode classical tabular data with variational parameters corresponding to real‐valued regression coefficients, ensuring high interpretability and efficient optimization without sacrificing expressiveness. By combining a variational quantum circuit with a classical optimizer, our method predicts the ground‐state energy of a hydrogen molecule using full configuration interaction (FCI) data. We quantify the expressibility of the VQR HEA circuit via the Kullback–Leibler (KL) divergence D KL and show the advantages of our R y − R x gate sequence in balancing expressibility. Performed for pennylane using an idealized quantum simulator, our 4‐qubit, 5‐layer HEA‐based VQR model achieves an accuracy of ∼0.99, mean squared error (MSE) < 10 −6 Hartree 2 , and mean absolute error (MAE) around 0.1 × 10 −2 Hartree compared to FCI benchmarks. We further demonstrate how qubit number and layer depth influence model accuracy, providing insights for task‐specific quantum circuit design. Our results advance practical applications of quantum computing for electronic structure problems by introducing an expressive, interpretable ansatz tailored for high‐precision simulations.
Hybrid variational quantum algorithms are promising for solving practical problems, such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. However, variational quantum algorithms (derived from randomized hardware-efficient ansatz or adaptive ansatz) become a black box, not trustworthy for model interpretation, and not to mention for application deployment in informing critical decisions. In this paper, we construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also exploit the encoded data structure to reduce the gate complexity of computing the regression map. To reduce circuit depth in nonlinear regression, our algorithm can be extended by directly constructing nonlinear features via classical preprocessing, such as independent encoded column vectors. By design, the model performance is determined by the cost function measurement results C synchronous to the mean squared errors (MSE) for the regression models. We derived the read-out errors induced by one-hot encoding and compact encoding; the required physical qubit resources are exponentially compressed for the compact encoding to be favorable for noisy quantum devices. We also derive the cost function dependent sample complexity $ \in \mathcal{O}\left(σ^{2}(\mathcal{C}) \ln (1/α)/ε^{2}\right)$ under the error budget ε and confidence tolerance α.
Hybrid variational quantum algorithms (VQAs) are promising for solving practical problems such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. However, with typical random ansatz or quantum alternating operator ansatz, derived variational quantum algorithms become a black box that cannot be trusted for model interpretation, not to mention deploying as applications in informing critical decisions: the results of these variational parameters are just rotational angles for the quantum gates and have nothing to do with interpretable values that a model can provide directly. In this paper, we construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also take advantage of the encoded data structure to reduce the time complexity of computing the regression map. To shorten the circuit depth for nonlinear regression, our algorithm can be extended by building nonlinear features by classical preprocessing as the independent encoded column vectors. Even though the realization of compressed encoding in superconducting qubits has been achieved by the less noisy compressed encoding recently by the authors, we envision potential quantum utilities with multi-qubit gates implemented in neutral cold atoms and ions.
High Resolution Image Download MS PowerPoint Slide The variational quantum eigensolver (VQE) algorithm has positioned quantum chemistry calculation methods as promising applications for noisy intermediate-scale quantum (NISQ) devices. In NISQ devices, where the quantum noise is considerable, it is desirable to have an ansatz for the quantum circuit that is practical to implement and capable of achieving high chemical accuracy. Hardware-efficient Ansätze (HEA), such as the RyRz linear ansatz (RLA) and its modified form that preserves physical quantities─the symmetry preserving ansatz (SPA)─are constructed using quantum gates that can be implemented easily on a quantum computer. However, the extent to which HEA can accurately yield the electronic state energies of actual molecules and the number of quantum gates required to achieve chemical accuracy─which, in practical cases, should be within 1 kcal/mol of the exact energy value─are not well understood. In this study, we aimed to gain a detailed understanding of these aspects by performing noiseless simulations to obtain the ground and low-lying excited state using high-depth HEA quantum circuits for several molecules, including LiH, H 2 O, BeH 2, CH 4, and N 2 . From the results, we demonstrated that symmetry preserving HEA, such as SPA, can achieve accurate computational results that maintain CCSD-level chemical accuracy by increasing the number of layers. We quantitatively analyze how the expressibility and entangling capability of RLA and SPA quantum circuits evolve with circuit depth. Our results reveal that while increased layers generally expand accessible Hilbert space and entanglement, differences in their limitations highlight the importance of considering physically allowed entanglement for performance of a parametric quantum circuit in VQE. We also studied the potential energy surface of the dissociation of these molecules and found that quantum circuits using SPA can capture static electron correlation, which is challenging to address with classical single-reference quantum chemistry methods such as CCSD. These results from SPA demonstrate its potential to represent electronic states, as it achieves highly accurate results with fewer gate operations compared to physically inspired quantum circuits such as the Unitary Coupled Cluster (UCC) method. This study establishes a possible path for the application of quantum devices in solving quantum chemistry problems.
Quantum Phase Estimation (QPE) is a foundational algorithm for molecular ground-state energy estimation, but its deep circuit requirements make direct hardware execution impractical on Noisy Intermediate-Scale Quantum (NISQ) devices. We present an analytically grounded variational surrogate framework in which a shallow Variational Quantum Circuit (VQC) is trained to reproduce the QPE measurement distribution without any quantum circuit simulation. The training target is computed entirely classically via the Dirichlet kernel, evaluated directly from the Full Configuration Interaction (FCI) ground-state energy, the ancilla qubit count, and the time evolution parameter, eliminating the exponentially scaling simulation bottleneck of prior surrogate approaches. We apply this framework to the hydrogen molecule (H2) with a symmetry-tapered Hamiltonian, conducting a four-stage experimental investigation on IBM Quantum hardware. Stage 1 compares linear and full entangler topologies for the RY-RZ-CZ ansatz, with and without XpXm Dynamical Decoupling (DD), across four distributional metrics (Hellinger distance, fidelity error, total variation distance, Jensen-Shannon divergence), identifying the linear entangler as optimal. Stage 2 varies VQC layers (p=1 to 5) for the linear-entangler ansatz, identifying single-layer depth as optimal under hardware noise. Stage 3 applies this configuration to the reduced RY-CZ ansatz, comparing ideal and noisy simulator-trained parameters. A supplementary noise analysis at p∈{8,64} characterizes the depth-dependent interplay between circuit depth and DD effectiveness. The framework enables faithful QPE mimicry using a linearly scaling VQC, recovering the ground-state energy within the chemical accuracy threshold (1 kcal/mol), constituting a scalable, hardware-efficient paradigm for QPE-based molecular energy estimation on NISQ devices.
Abstract The variational quantum Eigensolver (VQE) provides a scalable route to molecular ground-state energies on near-term quantum hardware, but its accuracy degrades rapidly as system size grows because hardware-efficient ansätze cannot span the full correlation subspace. We introduce a physics-informed ridge regression correction that is trained on the difference between VQE and Hartree–Fock (HF) energies and applied as a postprocessing step requiring no additional quantum resources. The feature set─the VQE–HF energy gap ΔE, its square ΔE2, the active-space qubit count, the number of active electrons, and two cross-term interactions─captures the dominant drivers of ansatz error in a low-dimensional, interpretable form. The model is benchmarked on H2, H2O, NH3, CH4, C2H6, and CH3OH using the STO-3G basis set and a state vector VQE simulator using a two-layer Ry-CNOT hardware-efficient ansatz. To assess cross-molecule generalizability, the corrected framework is further tested on CO2, N2, and HCN under both untrained and trained conditions. Absolute energy errors fall from a range of 0.02–0.79 Ha (raw VQE) to 0.0007–0.094 Ha after correction, corresponding to improvement factors of 3× to 640× relative to CASCI reference values. Zero-shot generalization tests yield factors of 9–58×, and including molecules in training improves results by a further order of magnitude in several cases. The three extended molecules exhibit improvement factors of 4–6× in untrained settings and 5–6× after training, providing preliminary evidence of transferability across chemically distinct systems. These findings suggest that the VQE–HF energy gap encodes physically transferable information about correlation-energy saturation, and that even the smallest training set can produce practically useful corrections for near-term quantum chemistry.
Variational quantum algorithms (VQAs) are promising methods to demonstrate quantum advantage on near-term devices as the required resources are divided between a quantum simulator and a classical optimizer. As such, designing a VQA which is resource-efficient and robust against noise is a key factor to achieve a potential advantage with the existing noisy quantum simulators. It turns out that a fixed VQA circuit design, such as the widely used hardware-efficient ansatz, is not necessarily robust against imperfections. In this work, we propose a genome-length-adjustable evolutionary algorithm to design a robust VQA circuit that is optimized over variations of both circuit ansatz and gate parameters, without any prior assumptions on circuit structure or depth. Remarkably, our method not only generates a noise-effect-minimized circuit with shallow depth, but also accelerates the classical optimization by substantially reducing the number of parameters. In this regard, the optimized circuit is far more resource-efficient with respect to both quantum and classical resources. As applications, based on two typical error models in VQA, we apply our method to calculate the ground energy of the hydrogen and the water molecules as well as the Heisenberg model. Simulations suggest that, compared with conventional hardware-efficient ansatz, our circuit-structure-tunable method can generate circuits apparently more robust against both coherent and incoherent noise and hence is more likely to be implemented on near-term devices.