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Quantum simulations of molecular systems with intrinsic atomic orbitals

Stefano Barison, Davide Emilio Galli, Mario Motta

arXiv:2011.08137v3quant-ph

TL;DR

Minimal-basis quantum simulations of molecules use fewer qubits and gates than larger-basis calculations, but have lower accuracy. This paper explores same-size intrinsic atomic orbital bases and quantum algorithms; across the studied species, IAO binding energies agree better with CCSD/aug-cc-pVQZ than the corresponding STO-6G results.

  • Problem

    Minimal-basis molecular quantum simulations use fewer qubits and gates but yield lower accuracy, while increasing basis size typically requires more quantum resources.

    Source: §I · p. 1

  • Method

    The paper uses intrinsic atomic orbitals, which retain minimal-basis size while incorporating molecular-environment polarization, in quantum algorithms for molecular simulations.

    Source: §I · p. 1

  • Results

    For all studied species, FCI/IAO and q-UCCSD/IAO binding energies agree better with CCSD/aug-cc-pVQZ binding energies than their STO-6G counterparts.

    Source: A. Comparison between minimal bases… · p. 5

  • Takeaways & Limitations

    IAOs offer a same-size alternative to minimal bases with more accurate estimates of energy differences and equilibrium geometries.

    Source: §IV · pp. 8–9

    IAOs capture electronic correlation within a valence space, so perturbative or full inclusion of virtual orbitals is needed to cover dynamical correlation.

    Source: §IV · p. 9

Abstract

from arXiv · show

Quantum simulations of molecular systems on quantum computers often employ minimal basis sets of Gaussian orbitals. In comparison with more realistic basis sets, quantum simulations employing minimal basis sets require fewer qubits and quantum gates, but yield results of lower accuracy. A natural strategy to achieve more accurate results is to increase the basis set size, which in turn requires increasing the number of qubits and quantum gates. Here we explore the use of intrinsic atomic orbitals (IAOs) in quantum simulations of molecules, to improve the accuracy of energies and properties at the same computational cost required by a minimal basis. We investigate ground-state energies and one- and two-body density operators in the framework of the variational quantum eigensolver, employing and comparing different Ans\"{a}tze. We also demonstrate the use of this approach in the calculation of ground- and excited-states energies of small molecules by a combination of quantum algorithms, using IBM Quantum computers.

I. INTRODUCTION

The introduction motivates improving molecular quantum simulations without increasing quantum resources, and presents intrinsic atomic orbitals (IAOs) as an alternative to minimal bases. It outlines their construction from polarized atomic orbitals that reproduce occupied molecular orbitals.

I. INTRODUCTION

  • Quantum chemistry simulations commonly use minimal Gaussian bases or classically selected active spaces, but these approaches have not delivered the high accuracy needed for molecular simulation.Increasing accuracy typically requires more qubits and gates, motivating techniques that avoid extra quantum resources and preliminary correlated classical simulations.Source: §I · p. 1
  • IAOs provide atomic core and valence orbitals polarized by the molecular environment and can exactly represent self-consistent-field wavefunctions through a simple algebraic construction.Their construction does not require input from correlated many-body calculations.Source: §I · p. 1

A. Intrinsic Atomic Orbitals

  • The IAO construction combines large-basis molecular orbitals with a valence atomic-orbital basis, then projects occupied orbitals to form polarized atomic orbitals that reproduce them exactly.Unlike molecular orbitals, these orbitals retain atomic association; unlike unpolarized atomic orbitals, they incorporate the molecular environment.Source: A. Intrinsic Atomic Orbitals · pp. 1–2
  • The resulting IAO basis is orthonormalized to preserve canonical anticommutation relations, then Foster-Boys localized to enhance spatial locality.The molecular Hamiltonian is transformed into the orthonormal IAO basis, with a frozen-core approximation used because the employed bases lack core-valence correlation effects.Source: A. Intrinsic Atomic Orbitals · p. 2

B. Ground- and excited-states algorithms

The paper uses VQE and quantum imaginary-time evolution for ground-state studies, with quantum equation-of-motion also considered for excited states. Within VQE, it compares chemically motivated and hardware-efficient ansätze.

B. Ground- and excited-states algorithms

1. Variational Quantum Eigensolver

  • VQE prepares a parameterized ansatz state and uses classical optimization to minimize its energy expectation as an approximation to the ground state.The circuit form determines the heuristic method's accuracy and computational cost.Source: §1 · p. 2
  • Within VQE, the comparison includes q-UCCSD with single and double excitations, the hardware-efficient Ry ansatz with linear connectivity, and the hardware-efficient SO(4) ansatz.Source: §1 · p. 2

2. Quantum imaginary-time evolution

Quantum imaginary-time evolution provides an ansatz-independent approach to ground-state search. The paper also describes evaluating one- and two-body density matrices from an optimized quantum state.

2. Quantum imaginary-time evolution

  • Quantum imaginary-time evolution complements heuristic ground-state methods by applying imaginary-time evolution to a trial wavefunction without relying on a chosen ansatz.The reported QITE simulations use a two-orbital space.Source: §2 · p. 3

C. Evaluation of density matrices

  • The paper considers one- and two-body density matrices as expectation values of qubit operators evaluated after finding the optimal state.These matrices are used in applications including correlation functions, electron entanglement, molecular bonding, and orbital relaxation.Source: C. Evaluation of density matrices · p. 3
  • The fermionic operators for density-matrix evaluation can be mapped to qubit operators using standard techniques, including the Jordan-Wigner representation.The IAO basis size determines n in the mapping, while σ_μ denotes a Pauli operator.Source: C. Evaluation of density matrices · p. 3

D. Variational quantum subspace expansion

The paper uses a simplified virtual quantum subspace expansion to incorporate contributions from orbitals outside an active space. Classical preprocessing and quantum simulations supply the inputs for the reported molecular calculations.

D. Variational quantum subspace expansion

  • VQSE partly addresses missing dynamical correlation by adding contributions from non-valence virtual orbitals outside a chosen active space.Its reference state is constructed from active orbitals that are linear combinations of IAOs; expansion amplitudes are obtained from a generalized eigenvalue equation.Source: D. Variational quantum subspace… · pp. 3–4
  • VQSE matrix elements are evaluated using quantum-device data, then the resulting matrices are classically diagonalized to obtain the lowest eigenvalue.The calculations focus on two-electron problems, where the matrices are defined by active-space one- and two-body density matrices; the authors note Davidson's algorithm could improve scaling with basis size.Source: D. Variational quantum subspace… · p. 4

E. Software for classical and quantum simulations

  • Classical preprocessing generates mean-field states, IAO-basis Hamiltonian elements, and the NH3 → NH2 + H reaction path; quantum simulations use Qiskit and IBM Quantum hardware.The hardware calculations used 5-qubit IBM devices, while tapering was used where possible to reduce qubit requirements.Source: E. Software for classical and quantum… · p. 4

F. Error mitigation techniques

IAO-based simulations improve agreement with larger-basis reference results, while hardware experiments illustrate both the value of adding outside-orbital correlation and practical limits of error mitigation. Across the reported two-qubit experiments, measurement calibration was sufficient for good-quality results.

F. Error mitigation techniques

  • Four calibration circuits were sufficient for good-quality results in experiments using two qubits and no more than two CNOTs, where measurement error was dominant.As qubit counts increase, calibration-matrix computation quickly becomes unfeasible; deeper circuits also make gate errors significant.Source: F. Error mitigation techniques · p. 4

A. Comparison between minimal bases and IAO

  • Across studied species, FCI/IAO and q-UCCSD/IAO binding energies agree better with CCSD/aug-cc-pVQZ than their STO-6G counterparts.The table compares dissociation energies and equilibrium bondlengths across molecules, VQE Ansätze, and basis levels.Source: A. Comparison between minimal bases… · p. 5, Table I · p. 5
  • VQE/q-UCCSD gives FCI-like accuracy for NH3 dissociation, while Ry and SO(4) are qualitatively correct but lower quality at lower computational cost.The largest deviation between hardware-efficient Ansätze and q-UCCSD occurs near R ≃ 1.75 Å, where the wavefunction is maximally multireference.Source: A. Comparison between minimal bases… · p. 5, Figure 1 · p. 6

B. Hardware experiments

  • IAO/aug-cc-pVTZ VQE results for H2 are statistically compatible with IAO-basis FCI, but omit virtual orbitals needed to recover full aug-cc-pVTZ results.VQSE energies are statistically compatible with full aug-cc-pVTZ FCI energies.Source: B. Hardware experiments · p. 5, Figure 2 · p. 6
  • For NH3, VQE in the active space improves significantly over RHF and yields results in qualitative agreement with FCI; its circuit uses an SO(4) Ansatz in the HONO/LUNO subspace.Source: B. Hardware experiments · p. 6, Figure 3 · p. 7
  • Within the NH3 HONO/LUNO active space, VQE reproduces FCI quantities, but gives a slightly shorter equilibrium bondlength and less accurate binding energy than the full IAO basis.The active-space approximation affects the electronic structure to varying extents along the dissociation profile.Source: B. Hardware experiments · p. 7, Table II · p. 7

1. Assessment of accuracy

For NH3 dissociation, the SO(4) VQE improves substantially over RHF in the active space and agrees qualitatively with FCI. Spin and purity diagnostics show increasing hardware-state deviations as dissociation proceeds.

1. Assessment of accuracy

  • For R ≥1.5 Å, deviations of the hardware VQE wavefunction from S² = 0 become more intense as the lowest-energy singlet and triplet states approach degeneracy.The wavefunction remains essentially in the singlet manifold, but is not an eigenfunction of S².Source: §1 · p. 7, Figure 5 · p. 8
  • Figure 4 shows the NH3 ground-state potential energy surface along the NH3 → NH2 + H reaction path, comparing RHF, FCI, and depth-1 SO(4) VQE using an HONO/LUNO active space.The active space is based on an MP2 calculation at IAO/aug-cc-pVQZ level; the bottom panel shows deviations of RHF and VQE energies from FCI.Source: Figure 4 · p. 7
  • For R ≥1.5 Å, the VQE density-operator purity is approximately 0.955, and its decrease signals decoherent interaction with the environment that limits simulation accuracy.Figure 5 compares hardware calculations with a classical simulator using an ibmq manila noise model.Source: §1 · p. 7, Figure 5 · p. 8
  • The authors attribute the observed accuracy loss to decoherence, measurement, and gate errors, with gates identified as the main error source.The noise model captures decoherence only partially, so its data are treated as indicative.Source: §1 · pp. 7–8

2. Density matrices

One- and two-body density matrices track the changing electronic structure along NH3 dissociation, while QITE experiments examine readout mitigation. The discussion frames IAOs as a minimal-size alternative with residual correlation limitations.

2. Density matrices

  • One- and two-body density matrices provide an alternative to quantum state tomography and can be obtained by measuring up to O(N_q^5) qubit operators.The passage notes that QST is expensive as the number of qubits grows.Source: §2 · p. 8
  • The one-body density-matrix eigenvalues evolve from (1, 0) to (1/2, 1/2) as R increases, indicating increasing electron entanglement as H separates from NH2.For small R, the spin-resolved two-body matrix is peaked at 0000; at larger R, its entries indicate a linear combination of two closed-shell singlet wavefunctions.Source: §2 · p. 8, Figure 6 · p. 9, Figure 7 · p. 9

C. QITE hardware experiments

  • QITE readout-error mitigation affects the binding energy by approximately 0.015 Eh, but does not change the equilibrium bondlength within statistical uncertainties.Its effect is more pronounced for R ≥1.75, where the wavefunction begins acquiring multireference character.Source: C. QITE hardware experiments · p. 8, Table III · p. 8, Figure 8 · p. 10

IV. DISCUSSION

  • IAOs have minimal-basis size and offer more accurate energy differences and equilibrium geometries, but their valence-space correlation treatment leaves dynamical correlation to perturbative or full virtual-orbital inclusion.The discussion presents IAOs as an alternative until larger basis sets become routine in quantum hardware and classical simulators.Source: §IV · pp. 8–9

Appendix A: Comparison of IAO against other bases

IAOs perform similarly to high-occupancy MP2 natural orbitals and better than low-energy Hartree-Fock and CASSCF orbitals for binding energies against the cited CCSD reference. The qEOM appendix applies the method to excited states and identifies degeneracy as a source of ill-conditioning.

Appendix A: Comparison of IAO against other bases

  • IAOs perform similarly to high-occupancy MP2 natural orbitals, and both give binding energies in better agreement with CCSD/aug-cc-pVQZ than low-energy Hartree-Fock and CASSCF orbitals.Low-energy Hartree-Fock orbitals also give lower-accuracy total and correlation energies; the authors attribute this to including Rydberg rather than antibonding orbitals.Source: Appendix A · p. 13, Figure 9 · p. 13, Figure 10 · p. 14

Appendix B: Quantum Equation-of-Motion

  • qEOM approximates excited states by applying suitable excitation operators to the ground state, with low-rank operators used to obtain accurate approximations for selected states.The expansion coefficients are determined by solving a generalized eigenvalue equation.Source: Appendix B · p. 13

1. qEOM hardware experiments

  • The mean deviations between exact and computed first-, second-, and third-excited-state energies are 0.019784, 0.027757, and 0.029781, respectively.The two-orbital active space detects only a subset of excited states, and energies near equilibrium are significantly biased.Source: §1 · p. 14, Figure 11 · p. 14

2. Details of qEOM simulations

  • For R ≥2.5, det(G) approaches zero as singlet and triplet states become degenerate, making the qEOM eigenvalue equation ill-conditioned.The authors report lower excited-state accuracy in the long-R regime than at short R.Source: §1 · p. 14, §2 · p. 15, Figure 12 · p. 15

Appendix C: Details of VQE simulations

The SO(4) VQE optimization uses analytical energy gradients with gradient descent and a manual line search. Fidelity between the VQE density operator and the RHF projector decreases alongside other indicators of multireference character.

1. Optimization on quantum hardware

  • The SO(4) VQE energy gradient can be evaluated analytically with 12 n_gates energy measurements, and parameters are optimized by gradient descent with a manually performed line search.The illustrated optimization initializes parameters at zero and continues until energy convergence within statistical uncertainties.Source: §1 · p. 15, Figure 13 · p. 15

2. Fidelity between VQE and Hartree-Fock states

  • From R ≥1.5 Å, declining fidelity between the VQE density operator and the RHF projector coincides with deviations from S² = 0 and decreasing purity, signaling multireference character.Figure 14 plots the fidelity against reaction coordinate and marks the ideal statevector result.Source: §2 · p. 16, Figure 14 · p. 16

Appendix D: Details of QITE simulations

The QITE simulations use imaginary-time evolution under a two-qubit Hamiltonian, with a KAK decomposition keeping circuit resources constant as projection time increases. The technique is designed for two-qubit systems, and broader generalization is under study.

Appendix D: Details of QITE simulations

  • QITE used ∆τ = 0.5Eh and total projection time β = 7.0Eh.Source: Appendix D · p. 16
  • The evolution uses a two-qubit Hamiltonian without Trotter-Suzuki or similar approximations, but the technique is specifically designed for two-qubit systems.Source: Appendix D · p. 16
  • A KAK decomposition reduces the QITE unitary for n time steps to a circuit with 2 CNOT gates, keeping depth and gate count constant as β increases.Source: Appendix D · p. 16

Appendix E: Variational quantum subspace expansion

The appendix describes a simplified VQSE implementation that builds a reference in active orbitals from a large basis and evaluates its matrices using quantum-hardware data. Its simplifications apply to two-electron active spaces; larger spaces require higher-body density matrices.

Appendix E: Variational quantum subspace expansion

  • The appendix shows that VQSE matrices can be evaluated using data from quantum hardware.Source: Appendix E · p. 16
  • VQSE starts from a reference function in active orbitals from a large basis, with active-space orbitals formed as linear combinations of IAOs.Source: Appendix E · p. 16
  • The expansion operators excite electrons from active to generic orbitals, while contractions over orbitals outside the active space can be evaluated analytically using Wick’s theorem.Source: Appendix E · pp. 16–17
  • Real amplitudes are determined by solving the generalized eigenvalue equation Hv = ESv, with H and S defined by bilinear forms.Source: Appendix E · p. 16
  • The stated simplifications hold for two-electron active spaces; otherwise VQSE requires three- and four-body active-space density matrices, with computational cost stated as O(N^8).Source: Appendix E · p. 17
  • Full-basis one- and two-body density matrices can be computed from their active-space counterparts, which are embedded into full-basis counterparts.Source: Appendix E · p. 17
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