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arXiv:2404.07512 · Journal of Chemical Theory and Computation

Numerical Investigation of the Quantum Inverse Algorithm on Small Molecules

Mauro Cainelli, Reo Baba, Yuki Kurashige

Published Sep 11, 2024• 0 citations
arXivHTMLPDF

Abstract

We evaluate the accuracy of the quantum inverse (Q-Inv) algorithm, in which the multiplication of Ĥ – k to the reference wave function is replaced by the Fourier transformed multiplication of e –iλ Ĥ, as a function of the integration parameters and the iteration power k for various systems, including H 2, LiH, BeH 2 and the notorious H 4 molecule at square geometry. We further consider the possibility of employing the Gaussian-quadrature rule as an alternate integration method and compared it to the results employing trapezoidal integration. The Q-Inv algorithm is compared to the inverse iteration method using the Ĥ –1 inverse (I-Iter) and the exact inverse by lower-upper decomposition. Energy values are evaluated as the expectation values of the Hamiltonian. Results suggest that the Q-Inv method provides lower energy results than the I-Iter method up to a certain k, after which the energy increases due to errors in the numerical integration that are dependent on the integration interval. A combined Gaussian-quadrature and trapezoidal integration method proved to be more effective at reaching convergence while decreasing the number of operations. For systems like H 4, in which the Q-Inv cannot reach the expected error threshold, we propose a combination of Q-Inv and I-Iter methods to further decrease the error with k at lower computational cost. Finally, we summarize the recommended procedure when treating unknown systems.

DOI
10.1021/acs.jctc.4c00483

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We evaluate the accuracy of the quantum inverse (Q-Inv) algorithm, in which the multiplication of Ĥ – k to the reference wave function is replaced by the Fourier transformed multiplication of e –iλ Ĥ, as a function of the integration parameters and the iteration power k for various systems, including H 2, LiH, BeH 2 and the notorious H 4 molecule at square geometry. We further consider the possibility of employing the Gaussian-quadrature rule as an alternate integration method and compared it to the results employing trapezoidal integration. The Q-Inv algorithm is compared to the inverse iteration method using the Ĥ –1 inverse (I-Iter) and the exact inverse by lower-upper decomposition. Energy values are evaluated as the expectation values of the Hamiltonian. Results suggest that the Q-Inv method provides lower energy results than the I-Iter method up to a certain k, after which the energy increases due to errors in the numerical integration that are dependent on the integration interval. A combined Gaussian-quadrature and trapezoidal integration method proved to be more effective at reaching convergence while decreasing the number of operations. For systems like H 4, in which the Q-Inv cannot reach the expected error threshold, we propose a combination of Q-Inv and I-Iter methods to further decrease the error with k at lower computational cost. Finally, we summarize the recommended procedure when treating unknown systems.
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 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\"atze. We also demonstrate the use of this approach in the calculation of ground- and excited-state energies of small molecules by a combination of quantum algorithms using IBM quantum computers.
For a complex quantum system, the dimensionality of the Hamiltonian grows exponentially with the size of the system, making it difficult to perform fast large-scale simulations and calculations on a classical computer. The first Variational Quantum Eigenvalue solver (VQE) proposed by Peruzzo et al. This gives the possibility of highly accurate molecular simulations and calculations in the field of quantum chemical computing. VQE is roughly based on the principle of using a quantum computer and set up a trial wave function |ψ(θ)⟩=U(θ)|ψ0⟩ to measure the energy expectation of the wave function |ψ(θ)⟩ at the corresponding initialization parameter, and then using a classical optimization algorithm to iterate θ0→θn over the energy to minimum until convergence. The significance of this paper is to verify the accuracy as well as the feasibility of the VQE algorithm for the calculation of LiH molecules in STO-3g basis, and to perform an error analysis and to verify the exponential acceleration of the classical algorithm implementation. These results shed light on guiding further exploration focusing on quantum computingt.
An alternative approach to conventional quantum chemistry techniques for molecular studies is described.This approach uses the quantum Monte Carlo method, which was developed and used primarily in nuclear and condensed-matter physics.In this approach the many-body Schrodinger equation is re-interpreted as a diffusion equation.Simulation of the appropriate random walk process allows one to calculate expectation values of molecular properties.In principle these expectation values can be calculated exactly, subject only to statistical errors (which may be made arbitrarily small).In preliminary work on small molecules (H 2 , LiH, Li 2 , and H 2 0), we use a simple but accurate approximation to ease the treatment of Fermi statistics.In that approximation, the calculated total energy remains an upper bound to the true energy.How good the bound is, as well as the magnitude of the statistical error, depends on the quality of an "importance function" which guides the diffusion into more probable regions of phase space.With relatively simple importance functions, we obtain from 75-100% of the correlation energy of the above-mentioned molecules.
In this work, we introduce a quantum inverse power iteration (QIPI) algorithm based on the quantum singular value transformation (QSVT) to target arbitrary excited states. Given an energy shift ωωω, QIPI prepares the target excited state by iteratively applying an approximation of the shifted inverse Hamiltonian (H−ωI)−1(H-ωI)^{-1}(H−ωI)−1 to a trial state. Prior quantum inverse power approaches typically relied on Fourier decompositions of the inverse Hamiltonian, with numerical quadrature used to reconstruct the transformation, but such methods are highly sensitive to hyperparameter choices and have been observed to be numerically unstable, effectively restricting their use to ground-state preparation. To enable robust excited-state targeting, we investigate two alternative transformation techniques: a Chebyshev decomposition of the inverse (Cheb-inv) and an eigenstate filtering (EF) approach based on QSVT. We find that EF-based QIPI is substantially more robust than Cheb-inv and other decomposition-based approaches due to the symmetry of the applied filtering polynomial, avoiding divergence with respect to the choice of ωωω and efficiently suppressing off-target eigenstates even in closely spaced spectra. Numerical simulations for molecular Hamiltonians of H2_22​, LiH, and BeH2_22​ show improved convergence and enhanced access to higher excited states relative to other quantum power methods. Assuming standard oracle access to the Hamiltonian, we further provide logical resource estimates in fault-tolerant settings in terms of T gate counts, and conclude that QIPI can achieve high target state amplification with modest polynomial degrees, thereby making it a promising candidate for scalable excited-state preparation in fault-tolerant quantum chemistry applications.
An iterative version of the qubit coupled cluster (QCC) method [I. G. Ryabinkin et al., J. Chem. Theory Comput. 2019, 14, 6317] is proposed. The new method seeks to find ground electronic energies of molecules on noisy intermediate-scale quantum devices. Each iteration involves a canonical transformation of the Hamiltonian and employs constant-size quantum circuits at the expense of increasing the Hamiltonian size. We numerically studied the convergence of the method on ground-state calculations for LiH, H 2 O, and N 2 molecules and found that the exact ground-state energies can be systematically approached only if the generators of the QCC ansatz are sampled from a specific set of operators. We report an algorithm for constructing this set that scales linearly with the size of the Hamiltonian.
High Resolution Image Download MS PowerPoint Slide The recent developments of quantum computing present novel potential pathways for quantum chemistry as the scaling of the computational power of quantum computers could be harnessed to naturally encode and solve electronic structure problems. Theoretically exact quantum algorithms for chemistry have been proposed (e.g., quantum phase estimation), but the limited capabilities of current noisy intermediate-scale quantum devices motivated the development of less demanding hybrid algorithms. In this context, the variational quantum eigensolver (VQE) algorithm was successfully introduced as an effective method to compute the ground-state energies of small molecules. This study investigates the folded spectrum (FS) method as an extension of the VQE algorithm for the computation of molecular excited states. It provides the possibility of directly computing excited states around a selected target energy using the same or quantum circuit as for the ground-state calculation. Inspired by the variance-based methods from the quantum Monte Carlo literature, the FS method minimizes the energy variance, thus, in principle, requiring a computationally expensive squared Hamiltonian to be applied. We alleviate this potentially poor scaling by employing a Pauli grouping procedure to identify sets of commuting Pauli strings that can be evaluated simultaneously. This allows for a significant reduction in the computational cost. We applied the FS-VQE method to small molecules (H 2, LiH), obtaining all electronic excited states with chemical accuracy on ideal quantum simulators. Furthermore, we explore the application of quantum error mitigation techniques, demonstrating improved energy accuracy on noisy simulators compared with simulations without mitigation.