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arXiv:2409.11210 · Quantum Science and Technology

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

Harper R. Grimsley, Francesco A. Evangelista

Published Dec 16, 2024• 16 citations
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Abstract

Abstract 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.

DOI
10.1088/2058-9565/ad9fa2

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Abstract 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.
Problems in quantum chemical simulations, especially achieving accurate excited-state potential energy surfaces, are among the primary applications to achieve quantum utility. On near-term quantum hardware, variants of the variational quantum eigensolver (VQE) algorithms are the primary choice for chemistry simulation. In this study, a combination of leading ground and excited state quantum algorithms for general excited states, namely, ADAPT-VQE/LUCJ and q-sc-EOM, are utilized to calculate accurate excited state potential energy surfaces in challenging bond-breaking scenarios and compared with the classical scalable EOM-CCSD method. This work investigates avenues toward quantum utility in excited-state quantum chemistry using the q-sc-EOM approach. We assess its accuracy while mitigating major scaling bottlenecks through the Davidson algorithm and basis rotation grouping, reducing the measurement scaling from O(N12^{12}12) to O(N5^{5}5), and implementing the method on quantum hardware with various error mitigation strategies to reduce gate and measurement errors in excited states. The hardware implementation of the q-sc-EOM algorithm, augmented by mitigation of M3 readout error and symmetry projection, produces reasonably accurate excited-state energies with gate noise identified as the predominant source of error. This paves the way for accurate and scalable, generally applicable quantum excited-state methods with potential for quantum utility while identifying critical problems that require advancements.
Calculations of molecular spectral properties, like photodissociation rates and absorption bands, rely on knowledge of the excited state energies of the molecule of interest. Protocols based on the variational quantum eigensolver (VQE) are promising candidates to calculate such energies on emerging noisy intermediate-scale quantum (NISQ) computers. The successful implementation of these protocols on NISQ computers, relies on ans\"atze that can accurately approximate the molecular states and that can be implemented by shallow quantum circuits. We introduce the excited qubit-excitation-based adaptive (e-QEB-ADAPT)-VQE protocol to calculate molecular-excited-state energies. The e-QEB-ADAPT-VQE constructs efficient problem-tailored ans\"atze by iteratively appending evolutions of qubit excitation operators. The e-QEB-ADAPT-VQE also improves on previous ADAPT-VQE protocol in that it is designed to be independent on the choice of initial reference state. We perform classical numerical simulations for LiH and BeH2{\mathrm{BeH}}_{2}BeH2​ to benchmark the performance of the e-QEB-ADAPT-VQE. We demonstrate that the e-QEB-ADAPT-VQE can construct highly accurate ans\"atze that require at least an order of magnitude fewer cnot gates than standard fixed unitary coupled-cluster ans\"atze, such as the UCCSD and the GUCCSD. We also show that the e-QEB-ADAPT-VQE is more successful in constructing ans\"atze for excited molecular states than other ADAPT-VQE protocols.
Calculations of molecular spectral properties, like photodissociation rates and absorption bands, rely on knowledge of the excited state energies of the molecule of interest. Protocols based on the variational quantum eigensolver (VQE) are promising candidates to calculate such energies on emerging noisy intermediate scale quantum (NISQ) computers. The successful implementation of these protocols on NISQ computers, relies on ansätze that can accurately approximate the molecular states and that can be implemented by shallow quantum circuits. In this paper, we introduce the excited qubit-excitation-based adaptive (e-QEB-ADAPT)-VQE protocol to calculate molecular excited state energies. The e-QEB-ADAPT-VQE constructs efficient problem-tailored ansätze by iteratively appending evolutions of qubit excitation operators. The e-QEB-ADAPT-VQE is an adaptation of the QEB-ADAPT-VQE protocol, which is designed to be independent on the choice of an initial reference state. We perform classical numerical simulations for LiH and BeH2_22​ to benchmark the performance of the e-QEB-ADAPT-VQE. We demonstrate that the e-QEB-ADAPT-VQE can construct highly accurate ansätze that require at least an order of magnitude fewer CNOTCNOTCNOTs than standard fixed UCC ansätze, such as the UCCSD and the GUCCSD.
A quantum computing algorithm is proposed to obtain low-lying excited states in many-body interacting systems. The approximate eigenstates are obtained by using a quantum space diagonalization method in a subspace of states selected from the convergence path of the ADAPT-VQE (adaptive derivative-assembled pseudo-Trotter Ansatz variational quantum eigensolver) towards the ground state of the many-body problem. This method is shown to be accurate with only a small overhead in terms of quantum resources required to get the ground state. We also show that the quantum algorithm might be used to facilitate the convergence of the ADAPT-VQE method itself. Successful applications of the technique are made to like-particle pairing as well as neutron-proton pairing. Finally, the H 4 molecule’s dissociation also illustrates the technique, demonstrating its accuracy and versatility.
The simulation of molecular excited states is a key challenge in quantum chemistry and a promising application for quantum computing. In this work, we investigate the efficacy of the truncated eigenvalue parametrized initial density adaptive variational algorithm (TEPID-ADAPT-VQE) for computing low-lying excited states and potential energy surfaces. TEPID-ADAPT variationally diagonalizes a truncated low-temperature Gibbs state, enabling the simultaneous preparation of multiple excited states within a single optimization. We apply the method to H2_22​, LiH, and linear H4_44​ across bond lengths spanning weakly and strongly correlated regimes. The adaptive derivative-assembled problem-tailored (ADAPT) ansatz construction yields compact circuits suitable for near-term hardware. We also implement a modified version of the MORE-ADAPT-VQE algorithm for comparison with TEPID-ADAPT. We find that both algorithms accurately reproduce excited-state spectra and potential energy curves within chemical accuracy for all the molecules and geometries studied. However, TEPID-ADAPT has the advantage of utilizing only a single, physically motivated hyperparameter (temperature) that controls the energy scale at which excited states are targeted, while MORE-ADAPT utilizes multiple hyperparameters whose optimal values depend sensitively on the target problem. These results demonstrate that combining adaptive ansatz construction with density-matrix-based formulations provides an efficient framework for excited-state quantum chemistry on near-term devices.
Variational Quantum Eigensolver (VQE) provides a lucrative platform to determine molecular energetics in near-term quantum devices. While the VQE is traditionally tailored to determine the ground state wavefunction with the underlying Rayleigh-Ritz principle, the access to specific symmetry-adapted excited states remains elusive. This often requires high depth circuit or additional ancilla qubits along with prior knowledge of the ground state wavefunction. We propose a unified VQE framework that treats the ground and excited states in the same footings. With the knowledge of the irreducible representations of the spinorbitals, we construct a multi-determinantal reference that is adapted to a given spatial symmetry where additionally, the determinants are entangled through appropriate Clebsch-Gordan coefficients to ensure the desired spin-multiplicity. We introduce the notion of totally symmetric, spin-scalar unitary which maintains the purity of the reference at each step of the optimization. The state-selectivity safeguards the method against any variational collapse while leading to any targeted low-lying eigenroot of arbitrary symmetry. The direct access to the excited states shields our approach from the cumulative error that plagues excited state calculations in a quantum computer and with few parameter count, it is expected to be realized in near-term quantum devices.