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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Crossing The Gap Using Variational Quantum Eigensolver: A Comparative Study

Within the evolving domain of quantum computational chemistry, the Variational Quantum Eigensolver (VQE) has been developed to explore not only the ground state but also the excited states of molecules. In this study, we compare the performance of Variational Quantum Deflation (VQD) and Subspace-Search Variational Quantum Eigensolver (SSVQE) methods in determining the low-lying excited states of $LiH$. Our investigation reveals that while VQD exhibits a slight advantage in accuracy, SSVQE stands out for its efficiency, allowing the determination of all low-lying excited states through a single parameter optimization procedure. We further evaluate the effectiveness of optimizers, including Gradient Descent (GD), Quantum Natural Gradient (QNG), and Adam optimizer, in obtaining $LiH$'s first excited state, with the Adam optimizer demonstrating superior efficiency in requiring the fewest iterations. Moreover, we propose a novel approach combining Folded Spectrum VQE (FS-VQE) with either VQD or SSVQE, enabling the exploration of highly excited states. We test the new approaches for finding all three $H_4$'s excited states. Folded Spectrum SSVQE (FS-SSVQE) can find all three highly excited states near $-1.0$ Ha with only one optimizing procedure, but the procedure converges slowly. In contrast, although Folded spectrum VQD (FS-VQD) gets highly excited states with individual optimizing procedures, the optimizing procedure converges faster.

Chen, I-Chi↗
NASA NTRSDate not supplied

A Look at the Truths and Misconceptions of the Variational Quantum Eigensolver and the Implications of Overparameterization

In this work, we investigate loss landscapes of the variational quantum eigensolver (VQE) by quantifying the number of local minima through empirical analyses. We focus on minimal models in chemistry and physics so that we can do a complete analysis using more computationally expensive tools. We employ Hessian eigenvalue calculations and the nudged elastic band algorithm to characterize these landscapes. Our results expand upon the existing literature by highlighting the optimization challenges faced by VQE. We find that, as the number of parameters in our ansatz increases, the number of basins increases while the corresponding loss function values converge toward the global minimum value. This observation implies that overparameterization may lead to an ``effective convexity'' in VQE loss landscapes, a phenomenon supported by theoretical and numerical work in classical machine learning.

quantum computing↗
DOE OSTI2024

A circuit-generated quantum subspace algorithm for the variational quantum eigensolver

Recent research has shown that wavefunction evolution in real and imaginary time can generate quantum subspaces with significant utility for obtaining accurate ground state energies. Inspired by these methods, we propose combining quantum subspace techniques with the variational quantum eigensolver (VQE). In our approach, the parameterized quantum circuit is divided into a series of smaller subcircuits. The sequential application of these subcircuits to an initial state generates a set of wavefunctions that we use as a quantum subspace to obtain high-accuracy groundstate energies. We call this technique the circuit subspace variational quantum eigensolver (CSVQE) algorithm. By benchmarking CSVQE on a range of quantum chemistry problems, we show that it can achieve significant error reduction in the best case compared to conventional VQE, particularly for poorly optimized circuits, greatly improving convergence rates. Furthermore, we demonstrate that when applied to circuits trapped at local minima, CSVQE can produce energies close to the global minimum of the energy landscape, making it a potentially powerful tool for diagnosing local minima.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Qubit-Efficient Quantum Chemistry with the ADAPT Variational Quantum Eigensolver and Double Unitary Downfolding

Here, in this work, we combine the recently developed double unitary coupled cluster (DUCC) theory with the adaptive, problem-tailored variational quantum eigensolver (ADAPT-VQE) to explore the accuracy of unitary downfolded Hamiltonians for quantum simulation of chemistry. We benchmark the ability of DUCC effective Hamiltonians to recover dynamical correlation energy outside of an active space. We consider the effects of strong correlation, commutator truncation, higher-body terms, and approximate external amplitudes on the accuracy of these effective Hamiltonians. When combining these DUCC Hamiltonians with ADAPT-VQE, we observe similar convergence of the ground state as compared with bare active space Hamiltonians, demonstrating that DUCC Hamiltonians provide increased accuracy without increasing the load on the quantum processor.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Contextual subspace variational quantum eigensolver calculation of the dissociation curve of molecular nitrogen on a superconducting quantum computer

Abstract We present an experimental demonstration of the Contextual Subspace Variational Quantum Eigensolver on superconducting hardware. Calculating the potential energy curve of molecular nitrogen proves challenging for many conventional quantum chemistry techniques, since static correlation dominates in the dissociation limit. Our quantum simulations retain good agreement with the Full Configuration Interaction energy, outperforming all benchmarked single-reference wavefunction techniques in capturing the bond-breaking appropriately. Moreover, our methodology is competitive with multiconfigurational approaches but at a saving of quantum resource, meaning larger active spaces can be treated for a fixed qubit allowance. To achieve this result, we deploy an error mitigation/suppression strategy comprised of Dynamical Decoupling, Measurement-Error Mitigation and Zero-Noise Extrapolation. Circuit parallelization also provides passive noise-averaging and improves the effective shot yield to reduce the measurement overhead. Furthermore, we introduce a modified adaptive ansatz construction algorithm that incorporates hardware awareness into our variational circuits, minimizing the transpilation cost for the target qubit topology.

Physics↗
DOE OSTI2025

Classical Benchmarks for Variational Quantum Eigensolver Simulations of the Hubbard Model

Simulating the Hubbard model is of great interest to a wide range of applications within condensed matter physics, however its solution on classical computers remains challenging in dimensions larger than one. The relative simplicity of this model, embodied by the sparseness of the Hamiltonian matrix, allows for its efficient implementation on quantum computers, and for its approximate solution using variational algorithms such as the variational quantum eigensolver. While these algorithms have been shown to reproduce the qualitative features of the Hubbard model, their quantitative accuracy in terms of producing true ground state energies and other properties, and the dependence of this accuracy on the system size and interaction strength, the choice of variational ansatz, and the degree of spatial inhomogeneity in the model, remains unknown. Here we present a rigorous classical benchmarking study, demonstrating the potential impact of these factors on the accuracy of the variational solution of the Hubbard model on quantum hardware, for systems with up to 32 qubits. We find that even when using the most accurate wavefunction ansätze for the Hubbard model, the error in its ground state energy and wavefunction plateaus for larger lattices, while stronger electronic correlations magnify this issue. Concurrently, spatially inhomogeneous parameters and the presence of off-site Coulomb interactions only have a small effect on the accuracy of the computed ground state energies. Our study highlights the capabilities and limitations of current approaches for solving the Hubbard model on quantum hardware, and we discuss potential future avenues of research.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Error mitigation in variational quantum eigensolvers using tailored probabilistic machine learning

Quantum computing technology has the potential to revolutionize the simulation of materials and molecules in the near future. A primary challenge in achieving near-term quantum advantage is effectively mitigating the noise effects inherent in current quantum processing units (QPUs). This challenge is also decisive in the context of quantum-classical hybrid schemes employing variational quantum eigensolvers (VQEs) that have attracted significant interest in recent years. In this paper, we present a method that employs parametric Gaussian process regression (GPR) within an active learning framework to mitigate noise in quantum computations, focusing on VQEs. Our approach, grounded in probabilistic machine learning, exploits a custom prior based on the VQE ansatz to capture the underlying correlations between VQE outputs for different variational parameters, thereby enhancing both accuracy and efficiency. We demonstrate the effectiveness of our method on a two-site Anderson impurity model and a eight-site Heisenberg model, using the IBM open-source quantum computing framework, Qiskit, showcasing substantial improvements in the accuracy of VQE outputs while reducing the number of direct QPU energy evaluations. This paper contributes to the ongoing efforts in quantum-error mitigation and optimization, bringing us a step closer to realizing the potential of quantum computing in quantum matter simulations. Published by the American Physical Society 2024

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Nonunitary Variational Quantum Eigensolver with the Localized Active Space Method and Cost Mitigation

Accurately describing strongly correlated systems with affordable quantum resources remains a central challenge for quantum chemistry applications on near and intermediate term quantum computers. The localized active space self-consistent field (LASSCF) approximates the complete active space self-consistent field (CASSCF) by generating active space-based wave functions within specific fragments while treating interfragment correlation with mean-field approach, hence is computationally less expensive. Hardware-efficient ansatzes (HEA) offer affordable and shallower circuits, yet they often fail to capture the necessary correlation. Previously, Jastrow-factor-inspired nonunitary qubit operators were proposed to use with HEA for variational quantum eigensolver (VQE) calculations (so-called nuVQE), as they do not increase circuit depths and recover correlation beyond the mean-field level for Hartree–Fock initial states. Here, in this study, we explore running nuVQE with LASSCF as the initial state. The method, named LAS-nuVQE, is shown to recover interfragment correlations, reach chemical accuracy with a small number of gates (<70) in both H 4 and square cyclobutadiene (C 4 H 4 ), and produces more accurate energetics than its HEA counterparts at all circuit depths. To further address the inherent symmetry-breaking in HEA, we implemented spin-constrained LAS-nuVQE to extend the capabilities of HEA further and show spin-pure results for square cyclobutadiene. We also mitigate the increased measurement overhead of nuVQE via Pauli grouping and shot-frugal sampling, reducing measurement costs by up to 2 orders of magnitude compared to ungrouped operator, and show that one can achieve better accuracy with a small number of shots (10 3–4 ) per one expectation value calculation compared to noiseless simulations with one or two orders of magnitude more shots. Finally, wall clock time estimates show that, with our measurement mitigation protocols, nuVQE becomes a cheaper and more accurate alternative than vanilla VQE with HEA. Taken together, these developments illustrate a practical pathway toward performing multireference chemical simulations with accuracy and affordable resources on today’s quantum hardware, achieving both accuracy and affordability in challenging correlated systems.

Wang, Qiaohong [Univ. of Chicago, IL (United State↗
DOE OSTI2025

Surrogate-constructed scalable-circuits adaptive variational quantum eigensolver in the Schwinger model

Inspired by recent advancements in simulating periodic systems on quantum computers, we develop an approach to further advance the simulation of these systems, named (SC) 2 -ADAPT-VQE. Our approach extends the scalable-circuits ADAPT-VQE framework, which builds an ansatz from a pool of coordinate-invariant operators defined for arbitrarily large, though not arbitrarily small, volumes. Our method uses a classically tractable “surrogate constructed” method to remove irrelevant operators from the pool, reducing the minimum size for which the scalable circuits are defined. Bringing together the scalable circuits and the surrogate constructed approaches forms the core of the (SC) 2 methodology. Our approach allows for a wider set of classical computations on small volumes, which can be used for a more robust extrapolation protocol. While developed in the context of lattice models, the surrogate construction portion is applicable to a wide variety of problems where information about the relative importance of operators in the pool is available. As an example, we use it to compute the properties of the Schwinger model—quantum electrodynamics for a single, massive fermion in 1 +1 dimensions—and show that our method can be used to accurately extrapolate to the continuum limit.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Compressing Hamiltonians with ab initio downfolding for simulating strongly-correlated materials on quantum computers

The accurate first-principles description of strongly correlated materials is an important and challenging problem in condensed matter physics. Ab initio downfolding has emerged as a way of deriving compressed many-body Hamiltonians that maintain the essential physics of strongly correlated materials. The solution of these material-specific models is still exponentially difficult to generate on classical computers, but quantum algorithms allow for a significant speed-up in obtaining the ground states of these compressed Hamiltonians. Here, we demonstrate that using quantum algorithms to obtain the properties of downfolded Hamiltonians can indeed yield high-fidelity solutions. By combining ab initio downfolding and variational quantum eigensolvers, we correctly predict the antiferromagnetic state of one-dimensional cuprate Ca 2 Cu O 3 , the excitonic ground state of monolayer W Te 2 , and the charge-ordered state of correlated metal Sr VO 3 . Numerical simulations using a classical tensor network implementation of variational quantum eigensolvers allow us to simulate large models with up to 54 qubits and encompassing up to four bands in the correlated subspace, which is indicative of the complexity that our framework can address. Through these methods we demonstrate the potential of classical preoptimization and downfolding techniques for enabling efficient materials simulation using quantum algorithms.

Alvertis, Antonios M. [NASA, Ames; LBNL, Berkeley]↗
NASA NTRSDate not supplied

Quantum Hardware-Enabled Molecular Dynamics via Transfer Learning

The ability to perform ab initio molecular dynamics simulations using potential energy surfaces provided by quantum computers would open the door to virtually exact dynamics for a variety of chemical and biochemical systems, with impacts on catalysis and biophysics. Nonetheless, performing molecular dynamics on surfaces produced by quantum hardware has been hampered by the noisy energies typically produced by quantum computers and challenges associated with computing gradients and scaling to large systems interest. A recent set of advances in machine learning, known as transfer learning, provides a new path forward for molecular dynamics simulations on quantum hardware. Transfer learning offers a workaround, where one first trains models on larger, less accurate classical datasets and then refines them on smaller, more accurate quantum datasets. We explore this approach by training machine learning models to predict a molecule's potential energy based on its geometric structure using Behler-Parrinello neural networks. When successfully trained, the model enables energy gradient predictions necessary for dynamic simulations. To reduce the quantum resources needed, the model is initially trained with data derived from classical density functional theory and subsequently refined with a smaller dataset obtained from a variational quantum eigensolver optimization of the unitary coupled cluster ansatz. We show that this approach significantly reduces the size of the needed quantum training dataset while capturing the high accuracies needed within quantum chemistry simulations. The success of this two-step training method opens more opportunities to apply machine learning models on quantum data, a significant stride towards efficient quantum-classical hybrid computational models.

quantum computing↗
DOE OSTI2025

Advancing quantum simulations of the nuclear shell model with Gray-code–based resource-efficient protocols

Background: Some of the computational limitations in solving the nuclear many-body problem could be overcome by utilizing quantum computers. The nuclear shell-model calculations providing deeper insights into the properties of atomic nuclei are one such case with high demand for resources, as the size of the Hilbert space grows exponentially with the number of particles involved. Quantum algorithms are being developed to overcome these challenges and advance such calculations. Purpose: To develop quantum circuits for the nuclear shell-model, leveraging the capabilities of noisy intermediate-scale quantum (NISQ) devices. Here, we aim to minimize resource requirements (specifically in terms of qubits and gates) and strive to reduce the impact of noise by employing relevant mitigation techniques. Methods: We achieve noise resilience by designing an optimized Ansatz for the variational quantum eigensolver (VQE) based on Givens rotations and incorporating qubit-ADAPT-VQE in combination with variational quantum deflation (VQD) to compute ground and excited states, incorporating the zero-noise extrapolation mitigation technique. Furthermore, the qubit requirements are significantly reduced by mapping the basis states to qubits using Gray-code encoding and generalizing transformations of fermionic operators to efficiently represent many-body states. Results: By employing the resource-efficient protocols, we achieve the ground and excited state energy levels of 38 Ar and 6 Li with better accuracy. These energy levels are presented for noiseless simulations, noisy conditions, and after applying noise mitigation techniques. Results are compared for Jordan-Wigner and Gray-code encoding using VQE, qubit-ADAPT-VQE, and VQD. Conclusions: Our work highlights the potential of resource-efficient protocols to leverage the full potential of NISQ devices in scaling the nuclear shell model calculations, offering a pathway toward more complex quantum simulations in nuclear physics. This approach establishes a framework for studying other nuclear systems with improved quantum resource efficiency, marking a significant advancement in applying quantum computing to realistic nuclear physics applications.

Physics - Nuclear physics and radiation physics↗
NASA NTRSDate not supplied

Surrogate Optimization for Quantum Circuits

Variational quantum Eigensolvers are touted as a near-term algorithm capable of impacting many applications. However, the potential has yet to be realized with few claims of quantum advantage and high resource estimates mainly due to the need for optimization in the presence of noise. Finding algorithms and methods to improve the convergence is essential to accelerate the capabilities of near-term hardware for VQE or more broad applications of hybrid methods in which optimization is required. To this goal we look to use modern approaches recently developed in circuit simulations and stochastic classical optimization that can be combined in a surrogate optimization approach to classical circuits. Using an approximate state vector simulator, we efficiently calculate an approximate Hessian, fed as an input for a detailed quantum circuit simulator. We demonstrate the capabilities of such an approach with and without sampling noise. We also show that this method outperforms Powell in the presence of quantum circuit shot noise by a factor of 2-4

quantum computing↗
DOE OSTI2026

Quantum solver for single-impurity Anderson models with particle-hole symmetry

Quantum embedding methods, such as dynamical mean-field theory (DMFT), provide a powerful framework for investigating strongly correlated materials. A central computational bottleneck in DMFT is in solving the Anderson impurity model (AIM), whose exact solution is classically intractable for large bath sizes. In this work, we benchmark a quantum-classical hybrid solver tailored for particle-hole symmetric AIMs, using the variational quantum eigensolver to prepare the ground state of the model with shallow quantum circuits. The solver uses shallow quantum ansätze and one set of variational parameters to prepare the ground state and its particle and hole excitations, enabling the construction of the impurity Green’s function through a continued-fraction expansion. We evaluate the performance of this approach across a few bath sizes and interaction strengths under noisy, shot-limited conditions. We compare three optimization routines (COBYLA, Adam, and L-BFGS-B) in terms of convergence and fidelity, assess the benefits of estimating a quantum-computed moment correction to the variational energies, and benchmark the approach by comparing the density of states computed from the impurity Green’s function against that obtained using a classical pipeline. Our results demonstrate the feasibility of Green’s function construction on near-term devices and establish practical benchmarks for quantum impurity solvers embedded within self-consistent DMFT loops.

Karabin, Mariia [ORNL]↗
DOE OSTI2024

Reducing measurement costs by recycling the Hessian in adaptive variational quantum algorithms

Abstract Adaptive protocols enable the construction of more efficient state preparation circuits in variational quantum algorithms (VQAs) by utilizing data obtained from the quantum processor during the execution of the algorithm. This idea originated with Adaptive Derivative-Assembled Problem-Tailored variational quantum eigensolver (ADAPT-VQE), an algorithm that iteratively grows the state preparation circuit operator by operator, with each new operator accompanied by a new variational parameter, and where all parameters acquired thus far are optimized in each iteration. In ADAPT-VQE and other adaptive VQAs that followed it, it has been shown that initializing parameters to their optimal values from the previous iteration speeds up convergence and avoids shallow local traps in the parameter landscape. However, no other data from the optimization performed at one iteration is carried over to the next. In this work, we propose an improved quasi-Newton optimization protocol specifically tailored to adaptive VQAs. The distinctive feature in our proposal is that approximate second derivatives of the cost function are recycled across iterations in addition to optimal parameter values. We implement a quasi-Newton optimizer where an approximation to the inverse Hessian matrix is continuously built and grown across the iterations of an adaptive VQA. The resulting algorithm has the flavor of a continuous optimization where the dimension of the search space is augmented when the gradient norm falls below a given threshold. We show that this inter-optimization exchange of second-order information leads the approximate Hessian in the state of the optimizer to be consistently closer to the exact Hessian. As a result, our method achieves a superlinear convergence rate even in situations where the typical implementation of a quasi-Newton optimizer converges only linearly. Our protocol decreases the measurement costs in implementing adaptive VQAs on quantum hardware as well as the runtime of their classical simulation.

Ramôa, Mafalda (ORCID:0000000302187801)↗
DOE OSTI2025

Challenging excited states from adaptive quantum eigensolvers: subspace expansions vs. state-averaged strategies

The prediction of electronic structure for strongly correlated molecules represents a promising application for near-term quantum computers. Significant attention has been paid to ground state wavefunctions, but excited states of molecules are relatively unexplored. In this work, we consider the adaptive, problem-tailored (ADAPT)-variational quantum eigensolver (VQE) algorithm, a single-reference approach for obtaining ground states, and its state-averaged generalization for computing multiple states at once. We demonstrate for both rectangular and linear H4, as well as for BeH2, that this approach, which we call multistate-objective, Ritz-eigenspectral (MORE)-ADAPT-VQE, can make better use of small excitation manifolds than an analogous method based on a single-reference ADAPT-VQE calculation, q-sc-EOM. In particular, MORE-ADAPT-VQE is able to accurately describe both avoided crossings and crossings between states of different symmetries. In addition to more accurate excited state energies, MORE-ADAPT-VQE can recover accurate transition dipole moments in situations where traditional ADAPT-VQE and q-sc-EOM struggle. These improvements suggest a promising direction toward the use of quantum computers for difficult excited state problems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Distributionally Robust Variational Quantum Algorithms With Shifted Noise

Given their potential to demonstrate near-term quantum advantage, variational quantum algorithms (VQAs) have been extensively studied. Although numerous techniques have been developed for VQA parameter optimization, it remains a significant challenge. A practical issue is the high sensitivity of quantum noise to environmental changes, and its propensity to shift in real time. This presents a critical problem as an optimized VQA ansatz may not perform effectively under a different noise environment. For the first time, we explore how to optimize VQA parameters to be robust against unknown shifted noise. We model the noise level as a random variable with an unknown probability density function (PDF), and we assume that the PDF may shift within an uncertainty set. This assumption guides us to formulate a distributionally robust optimization problem, with the goal of finding parameters that maintain effectiveness under shifted noise. We utilize a distributionally robust Bayesian optimization solver for our proposed formulation. This provides numerical evidence in both the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE) with hardware-efficient ansatz, indicating that we can identify parameters that perform more robustly under shifted noise. We regard this work as the first step towards improving the reliability of VQAs influenced by real-time noise.

97 MATHEMATICS AND COMPUTING↗