Quantum self-consistent equation-of-motion method for computing molecular excitation energies, ionization potentials, and electron affinities on a quantum computer
Ayush Asthana, Ashutosh Kumar, Vibin Abraham, Harper R. Grimsley, Yu Zhang, Łukasz Cincio, Sergei Tretiak, Pavel A. Dub, Sophia E. Economou, Edwin Barnes, Nicholas J. Mayhall
O, and LiH molecules to test our quantum self-consistent equation-of-motion (q-sc-EOM) method and compare it to other current state-of-the-art methods. q-sc-EOM makes use of self-consistent operators to satisfy the vacuum annihilation condition, a critical property for accurate calculations. It provides real and size-intensive energy differences corresponding to vertical excitation energies, ionization potentials and electron affinities. We also find that q-sc-EOM is more suitable for implementation on NISQ devices as it is expected to be more resilient to noise compared with the currently available methods.
O, and LiH molecules to test our quantum self-consistent equation-of-motion (q-sc-EOM) method and compare it to other current state-of-the-art methods. q-sc-EOM makes use of self-consistent operators to satisfy the vacuum annihilation condition, a critical property for accurate calculations. It provides real and size-intensive energy differences corresponding to vertical excitation energies, ionization potentials and electron affinities. We also find that q-sc-EOM is more suitable for implementation on NISQ devices as it is expected to be more resilient to noise compared with the currently available methods.
Near-term quantum devices promise to revolutionize quantum chemistry, but simulations using the current noisy intermediate-scale quantum (NISQ) devices are not practical due to their high susceptibility to errors. This motivated the design of NISQ algorithms leveraging classical and quantum resources. While several developments have shown promising results for ground-state simulations, extending the algorithms to excited states remains challenging. This paper presents two cost-efficient excited-state algorithms inspired by the classical Davidson algorithm. We implemented the Davidson method into the quantum self-consistent equation-of-motion unitary coupled-cluster (q-sc-EOM-UCC) excited-state method adapted for quantum hardware. The circuit strategies for generating desired excited states are discussed, implemented, and tested. We demonstrate the performance and accuracy of the proposed algorithms (q-sc-EOM-UCC/Davidson and its variational variant) by simulations of H 2, H 4, LiH, and H 2 O molecules. Similar to the classical Davidson scheme, q-sc-EOM-UCC/Davidson algorithms are capable of targeting a small number of excited states of the desired character.
Solving challenging problems in quantum chemistry is one of the most promising applications of quantum computers. Within the quantum algorithms proposed for problems in excited-state quantum chemistry, subspace-based quantum algorithms, including quantum subspace expansion (QSE), quantum equation of motion (qEOM), and quantum self-consistent equation-of-motion (q-sc-EOM), are promising for pre-fault-tolerant quantum devices. The working equation of QSE and qEOM requires solving a generalized eigenvalue equation with associated matrix elements measured on a quantum computer. Our careful analytical and numerical analysis of the standard and generalized eigenvalue problems, especially in the context of excited-state methods, shows that the errors in eigenvalues magnify drastically with an increase in the condition number of the overlap matrix when a generalized eigenvalue equation is solved in the presence of statistical sampling errors. This makes methods such as QSE unstable for errors that are unavoidable when using quantum computers. Further, at very high condition numbers of the overlap matrix, the QSE's working equation could not be solved without any additional steps in the presence of sampling errors, as it becomes ill-conditioned. It was possible to use the thresholding technique in this case to solve the equation, but the solutions achieved had missing excited states, which may be a problem for future chemical studies. We also show that excited-state methods that have an eigenvalue equation as the working equation, such as q-sc-EOM, do not have the problems associated with the condition number and could be generally more stable to errors and therefore more suitable candidates for excited-state quantum chemistry calculations using quantum computers.
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) to O(N5), 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.
Solving challenging problems in quantum chemistry is one of the most promising applications of quantum computers. Within the quantum algorithms proposed for problems in excited state quantum chemistry, subspace-based quantum algorithms, including quantum subspace expansion (QSE), quantum equation of motion (qEOM) and quantum self-consistent equation-of-motion (q-sc-EOM), are promising for pre-fault-tolerant quantum devices. The working equation of QSE and qEOM requires solving a generalized eigenvalue equation with associated matrix elements measured on a quantum computer. Our careful analytical and numerical analysis of the standard and generalized eigenvalue problems, especially in the context of excited-state methods, shows that the errors in eigenvalues magnify drastically with an increase in the condition number of the overlap matrix when a generalized eigenvalue equation is solved in the presence of statistical sampling errors. This makes methods such as QSE unstable to errors that are unavoidable when using quantum computers. Further, at very high condition numbers of the overlap matrix, the QSE's working equation could not be solved without any additional steps in the presence of sampling errors, as it becomes ill-conditioned. It was possible to use the thresholding technique in this case to solve the equation, but the solutions achieved had missing excited states, which may be a problem for future chemical studies. We also show that excited-state methods that have an eigenvalue equation as the working equation, such as q-sc-EOM, do not have the problems associated with the condition number and could be generally more stable to errors, and therefore, more suitable candidates for excited-state quantum chemistry calculations using quantum computers.
Band structure is a cornerstone to understand the electronic properties of materials. Accurate band structure calculations using a high-level quantum chemistry theory can be computationally very expensive. It is promising to speed up such calculations with a quantum computer. In this study, we present a quantum algorithm for band structure calculations based on the equation-of-motion (EOM) theory. First, we introduce a new variational quantum eigensolver algorithm named ADAPT-C for ground-state quantum simulation of solids, where the wave function is built adaptively from a complete set of anti-Hermitian operators. Then, on top of the ADAPT-C ground state, quasiparticle energies and the band structure can be calculated using the EOM theory in a quantum-subspace-expansion style, where the projected excitation operators guarantee that the killer condition is satisfied. As a proof of principle, such an EOM-ADAPT-C protocol is used to calculate the band structures of silicon and diamond using a quantum computer simulator.
Abstract Quantum computing in the noisy intermediate-scale quantum (NISQ) era has foregrounded the importance of Variational Quantum algorithms (VQAs). These algorithms are crucial for addressing complex quantum mechanical problems that challenge classical computers. One such problem is the electron-phonon (e-ph) interaction, which is essential for determining the zero-point renormalization (ZPR) of electronic structure properties. The calculation of ZPR of fundamental gap relies on the accurate computation of ionization potential (IP) and electron affinity (EA) energy levels in molecular systems, where the VQAs offer the promising solutions. Despite the critical importance of IP, EA energies and ZPR in quantum chemistry calculations, research into the application of quantum algorithms for these calculations remains limited. To address these challenges, we propose two quantum algorithms for ZPR of fundamental gap calculation using Variational Quantum Deflation (VQD) and Quantum equation of Motion (QEOM) algorithm for several molecular systems. This work opens up new possibilities for the accurate and efficient study of e-ph interaction in electronic structure calculations, even with NISQ-era hardware.