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DOE OSTI2022

Quantum Simulation of Molecular Electronic States with a Transcorrelated Hamiltonian: Higher Accuracy with Fewer Qubits

Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron–electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. In conclusion, the transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Improving the Accuracy of Variational Quantum Eigensolvers with Fewer Qubits Using Orbital Optimization

Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this work we propose incorporating an orbital optimization scheme into quantum eigensolvers wherein a parametrized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. Finally, we find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE when using the same number of qubits and even frequently achieves lower ground state energies than VQE methods using more qubits.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

A Stabilizer Framework for the Contextual Subspace Variational Quantum Eigensolver and the Noncontextual Projection Ansatz

Quantum chemistry is a promising application for noisy intermediate-scale quantum (NISQ) devices. However, quantum computers have thus far not succeeded in providing solutions to problems of real scientific significance, with algorithmic advances being necessary to fully utilize even the modest NISQ machines available today. We discuss a method of ground state energy estimation predicated on a partitioning of the molecular Hamiltonian into two parts: one that is noncontextual and can be solved classically, supplemented by a contextual component that yields quantum corrections obtained via a Variational Quantum Eigensolver (VQE) routine. This approach has been termed Contextual Subspace VQE (CS-VQE); however, there are obstacles to overcome before it can be deployed on NISQ devices. The problem we address here is that of the ansatz, a parametrized quantum state over which we optimize during VQE; it is not initially clear how a splitting of the Hamiltonian should be reflected in the CS-VQE ansätze. We propose a “noncontextual projection” approach that is illuminated by a reformulation of CS-VQE in the stabilizer formalism. This defines an ansatz restriction from the full electronic structure problem to the contextual subspace and facilitates an implementation of CS-VQE that may be deployed on NISQ devices. We validate the noncontextual projection ansatz using a quantum simulator and demonstrate chemically precise ground state energy calculations for a suite of small molecules at a significant reduction in the required qubit count and circuit depth.

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

Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method

Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems cost-effectively, setting the stage for more advanced quantum simulations in chemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Improving the accuracy of the variational quantum eigensolver for molecular systems by the explicitly-correlated perturbative [2] R12 -correction

We provide an integration of the universal, perturbative explicitly correlated [2] R12 -correction in the context of the Variational Quantum Eigensolver (VQE). This approach is able to increase the accuracy of the underlying reference method significantly while requiring no additional quantum resources. The proposed approach only requires knowledge of the one- and two-particle reduced density matrices (RDMs) of the reference wavefunction; these can be measured after having reached convergence in the VQE. This computation comes at a cost that scales as the sixth power of the number of electrons. Here, we explore the performance of the VQE + [2] R12 approach using both conventional Gaussian basis sets and our recently proposed directly determined pair-natural orbitals obtained by multiresolution analysis (MRA-PNOs). Both Gaussian orbital and PNOs are investigated as a potential set of complementary basis functions in the computation of [2] R12 . In particular the combination of MRA-PNOs with [2] R12 has turned out to be very promising – persistently throughout our data, this allowed very accurate simulations at a quantum cost of a minimal basis set. Additionally, we found that the deployment of PNOs as complementary basis can greatly reduce the number of complementary basis functions that enter the computation of the correction at a complexity.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Ab initio molecular dynamics on quantum computers

Ab initio molecular dynamics (AIMD) is a valuable technique for studying molecules and materials at finite temperatures where the nuclei evolve on potential energy surfaces obtained from accurate electronic structure calculations. In this work, we present an approach to running AIMD simulations on noisy intermediate-scale quantum (NISQ)-era quantum computers. The electronic energies are calculated on a quantum computer using the variational quantum eigensolver (VQE) method. Algorithms for computation of analytical gradients entirely on a quantum computer require quantum fault-tolerant hardware, which is beyond NISQ-era. Therefore, we compute the energy gradients numerically using finite differences, the Hellmann-Feynman theorem, and a correlated sampling technique. This method only requires additional classical calculations of electron integrals for each degree of freedom without any additional computations on a quantum computer beyond the initial VQE run. As a proof of concept, AIMD simulations are demonstrated for the H-2 molecule on IBM quantum devices. In addition, we demonstrate the validity of the method for larger molecules using full configuration interaction wave functions. As quantum hardware and noise mitigation techniques continue to improve, the method can be utilized for studying larger molecular systems.

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

Quantum subspace expansion in the presence of hardware noise

Finding ground state energies on current quantum processing units (QPUs) using algorithms such as the variational quantum eigensolver (VQE) continues to pose challenges. Hardware noise severely affects both the expressivity and trainability of parameterized quantum circuits, limiting them to shallow depths in practice. Here, we demonstrate that both issues can be addressed by synergistically integrating VQE with a quantum subspace expansion, allowing for an optimal balance between quantum and classical computing capabilities and costs. We perform a systematic benchmark analysis of the iterative quantum-assisted eigensolver in the presence of hardware noise. We determine ground state energies of 1D and 2D mixed-field Ising spin models on noisy simulators and the IBM QPUs ibmq_quito (5 qubits) and ibmq_guadalupe (16 qubits). To maximize accuracy, we propose a suitable criterion to select the subspace basis vectors according to the trace of the noisy overlap matrix. Finally, we show how to systematically approach the exact solution by performing controlled quantum error mitigation based on probabilistic error reduction on the noisy backend fake_guadalupe.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Artificial-intelligence-driven shot reduction in quantum measurement

Variational Quantum Eigensolver (VQE) provides a powerful solution for approximating molecular ground state energies by combining quantum circuits and classical computers. However, estimating probabilistic outcomes on quantum hardware requires repeated measurements (shots), incurring significant costs as accuracy increases. Optimizing shot allocation is thus critical for improving the efficiency of VQE. Current strategies rely heavily on hand-crafted heuristics requiring extensive expert knowledge. This paper proposes a reinforcement learning (RL)-based approach that automatically learns shot assignment policies to minimize total measurement shots while achieving convergence to the minimum of the energy expectation in VQE. The RL agent assigns measurement shots across VQE optimization iterations based on the progress of the optimization. This approach reduces VQE's dependence on static heuristics and human expertise. When the RL-enabled VQE is applied to a small molecule, a shot reduction policy is learned. The policy demonstrates transferability across systems and compatibility with other wavefunction Ansätze. In addition to these specific findings, this work highlights the potential of RL for automatically discovering efficient and scalable quantum optimization strategies.

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

Chemical applications of variational quantum eigenvalue-based quantum algorithms: Perspective and survey

Exploring many-body chemical systems on classical computers often involves solving the Schrödinger equation. However, this approach is frequently limited by the exponential increase in the dimensionality of the Hamiltonian as the number of degrees of freedom increases. In contrast, quantum computing, specifically through the variational quantum eigensolver (VQE) framework, shows promise in overcoming this exponential cost. VQE can utilize the collective properties of quantum states to model the wavefunction in polynomial time. Despite the current limitations of quantum hardware, significant advances have been made in the development of VQE-based algorithms. Here, in this review, we provide an overview of emerging protocols, focusing on their applications in simulating the ground state, excited state, and vibrational properties of chemical systems. By examining notable algorithmic advancements and applications, this review aims to shed light on the challenges and potential of VQE-based algorithms in addressing relevant chemical problems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Qubit coupled cluster singles and doubles variational quantum eigensolver ansatz for electronic structure calculations

Variational quantum eigensolver (VQE) for electronic structure calculations is believed to be one major potential application of near term quantum computing. Among all proposed VQE algorithms, the unitary coupled cluster singles and doubles excitations (UCCSD) VQE ansatz has achieved high accuracy and received a lot of research interest. However, the UCCSD VQE based on fermionic excitations needs extra terms for the parity when using Jordan–Wigner transformation. Here we introduce a new VQE ansatz based on the particle preserving exchange gate to achieve qubit excitations. The proposed VQE ansatz has gate complexity up-bounded to O ( n 4 ) for all-to-all connectivity where n is the number of qubits of the Hamiltonian. Numerical results of simple molecular systems such as BeH 2 , H 2 O, N 2 , H 4 and H 6 using the proposed VQE ansatz gives very accurate results within errors about 10 -3 Hartree.

Physics↗
DOE OSTI2024

Quantum Davidson algorithm for excited states

Abstract Excited state properties play a pivotal role in various chemical and physical phenomena, such as charge separation and light emission. However, the primary focus of most existing quantum algorithms has been the ground state, as seen in quantum phase estimation and the variational quantum eigensolver (VQE). Although VQE-type methods have been extended to explore excited states, these methods grapple with optimization challenges. In contrast, the quantum Krylov subspace (QKS) method has been introduced to address both ground and excited states, positioning itself as a cost-effective alternative to quantum phase estimation. However, conventional QKS methodologies depend on a pre-generated subspace through real or imaginary-time evolutions. This subspace is inherently expansive and can be plagued with issues like slow convergence or numerical instabilities, often leading to relatively deep circuits. Our research presents an economic QKS algorithm, which we term the quantum Davidson (QDavidson) algorithm. This innovation hinges on the iterative expansion of the Krylov subspace and the incorporation of a pre-conditioner within the Davidson framework. By using the residues of eigenstates to expand the Krylov subspace, we manage to formulate a compact subspace that aligns closely with the exact solutions. This iterative subspace expansion paves the way for a more rapid convergence in comparison to other QKS techniques, such as the quantum Lanczos. Using quantum simulators, we employ the novel QDavidson algorithm to delve into the excited state properties of various systems, spanning from the Heisenberg spin model to real molecules. Compared to the existing QKS methods, the QDavidson algorithm not only converges swiftly but also demands a significantly shallower circuit. This efficiency establishes the QDavidson method as a pragmatic tool for elucidating both ground and excited state properties on quantum computing platforms.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2021

Preserving Symmetries for Variational Quantum Eigensolvers in the Presence of Noise

One of the most promising applications of noisy intermediate-scale quantum computers is the simulation of molecular Hamiltonians using the variational quantum eigensolver (VQE). Here, we show that encoding symmetries of the simulated Hamiltonian in the VQE ansatz reduces both classical and quantum resources compared to other widely available ansatze. Through simulations of the H 2 molecule and of a Heisenberg model on a two-dimensional lattice, we verify that these improvements persist in the presence of noise. This is done using both real IBM devices and classical simulations. We also demonstrate how these techniques can be used to find molecular excited states of various symmetries using a noisy processor. We use error-mitigation techniques to further improve the quality of our results.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Lipkin model on a quantum computer

Atomic nuclei are important laboratories for exploring and testing new insights into the universe, such as experiments to directly detect dark matter or explore properties of neutrinos. The targets of interest are often heavy, complex nuclei that challenge our ability to reliably model them (as well as quantify the uncertainty of those models) with classical computers. Hence there is great interest in applying quantum computation to nuclear structure for these applications. As an early step in this direction, especially with regards to the uncertainties in the relevant quantum calculations, we develop circuits to implement variational quantum eigensolver (VQE) algorithms for the Lipkin-Meshkov-Glick model, which is often used in the nuclear physics community as a testbed for many-body methods. Here, we present quantum circuits for VQE for two and three particles and discuss the construction of circuits for more particles. Implementing the VQE for a two-particle system on the IBM Quantum Experience, we identify initialization and two-qubit gates as the largest sources of error. We find that error mitigation procedures reduce the errors in the results significantly, but additional quantum hardware improvements are needed for quantum calculations to be sufficiently accurate to be competitive with the best current classical methods.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Quantum computing for heavy quarkonium spectroscopy

Here, we report a first demonstration for the application of quantum computing to heavy quarkonium spectroscopy study. Based on a Cornell-potential model for the heavy quark and antiquark system, we show how this Hamiltonian problem can be formulated and solved with the variational quantum eigensolver (VQE) approach on the IBM cloud quantum computing platform. Errors due to a global depolarizing noise channel are corrected with a zero-noise extrapolation method, resulting in good agreement with the expected value. We also generalize the VQE method for solving excited states by orthogonalization with respect to the ground state. This new approach is demonstrated to be successful for the quarkonium system on a noiseless quantum simulator and can easily be adapted for solving similar excited state problems in many other physical systems.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Noise robust detection of quantum phase transitions

Quantum computing allows for the manipulation of highly correlated states whose properties quickly go beyond the capacity of any classical method to calculate. Thus one natural problem which could lend itself to quantum advantage is the study of ground-states of condensed matter models, and the transitions between them. However, current levels of hardware noise can require extensive application of error-mitigation techniques to achieve reliable computations. In this work, we use several IBM devices to explore a finite-size spin model with multiple “phaselike” regions characterized by distinct ground-state configurations. Using preoptimized Variational Quantum Eigensolver (VQE) solutions, we demonstrate that in contrast to calculating the energy, where zero-noise extrapolation is required in order to obtain qualitatively accurate yet still unreliable results, calculations of the energy derivative, two-site spin correlation functions, and the fidelity susceptibility yield accurate behavior across multiple regions, even with minimal or no application of error-mitigation approaches. Taken together, these sets of observables could be used to identify level crossings in a simple, noise-robust manner which is agnostic to the method of ground state preparation. This work shows promising potential for near-term application to identifying quantum phase transitions, including avoided crossings and nonadiabatic conical intersections in electronic structure calculations. Published by the American Physical Society 2024

Lively, Kevin (ORCID:0000000320981494)↗
DOE OSTI2021

Numerical Simulations of Noisy Variational Quantum Eigensolver Ansatz Circuits

This is a case study of the variational quantum eigensolver (VQE) method using numerical simulations to test the influence of noise on the accuracy of the underlying circuit ansatz. We investigate a computational chemistry application of VQE to calculate the electronic ground state and its energy for Sodium Hydride (NaH), a prototypical two-electron problem. Using a one-parameter ansatz derived from unitary coupled cluster (UCC) theory, we simulate the effects of noise on the energy expectation value and variance with respect to the ansatz parameter. These numerical simulations provide insights into the accuracy of the prepared quantum state and the efficiency of the classical optimizer that iteratively refines the ansatz. We conduct a comparative study between analytical results derived for the UCC ansatz in the absence of noise and the noisy numerical simulation results obtained using an isotropic depolarizing noise model for each gate. We also compare the relative increase in noise on logically equivalent UCC ansatz circuits generated by randomized compiling. Notably, we observe that the intrinsic variance in the energy due to the simplicity of the ansatz itself compares with the noise induced by the bare circuit.

Gowrishankar, Meenambika↗