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Quantum Simulation: Solving Schrödinger Equation on a Quantum Computer

Hefeng Wang, Sabre Kais, Alán Aspuru‐Guzik

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Using the MCSCF wave function as the initial guess can improve the success probability dramatically, even just a few CSFs. The method can be generalized to general MRCI scheme and the entire potential energy surface can be explored for ground state and excited states. Instead of starting from a single element as in the HF wave function, MCSCF method starts from a small matrix. This makes the evolution safer and faster. The idea can be generalized to Finite element method and any numerical method.

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Using the MCSCF wave function as the initial guess can improve the success probability dramatically, even just a few CSFs. The method can be generalized to general MRCI scheme and the entire potential energy surface can be explored for ground state and excited states. Instead of starting from a single element as in the HF wave function, MCSCF method starts from a small matrix. This makes the evolution safer and faster. The idea can be generalized to Finite element method and any numerical method.
Simulating a quantum system is more efficient on a quantum computer than on a classical computer. The time required for solving the Schrödinger equation to obtain molecular energies has been demonstrated to scale polynomially with system size on a quantum computer, in contrast to the well-known result of exponential scaling on a classical computer. In this paper, we present a quantum algorithm to obtain the energy spectrum of molecular systems based on the multiconfigurational self-consistent field (MCSCF) wave function. By using a MCSCF wave function as the initial guess, the excited states are accessible. Entire potential energy surfaces of molecules can be studied more efficiently than if the simpler Hartree-Fock guess was employed. We show that a small increase of the MCSCF space can dramatically increase the success probability of the quantum algorithm, even in regions of the potential energy surface that are far from the equilibrium geometry. For the treatment of larger systems, a multi-reference configuration interaction approach is suggested. We demonstrate that such an algorithm can be used to obtain the energy spectrum of the water molecule.
Simulating a quantum system is more efficient on a quantum computer than on a classical computer. The time required for solving the Schro¨ dinger equation to obtain molecular energies has been demonstrated to scale polynomially with system size on a quantum computer, in contrast to the well-known result of exponential scaling on a classical computer. In this paper, we present a quantum algorithm to obtain the energy spectrum of molecular systems based on the multiconfigurational self-consistent field (MCSCF) wave function. By using a MCSCF wave function as the initial guess, the excited states are accessible. Entire potential energy surfaces of molecules can be studied more efficiently than if the simpler Hartree–Fock guess was employed. We show that a small increase of the MCSCF space can dramatically increase the success probability of the quantum algorithm, even in regions of the potential energy surface that are far from the equilibrium geometry. For the treatment of larger systems, a multi-reference configuration interaction approach is suggested. We demonstrate that such an algorithm can be used to obtain the energy spectrum of the water molecule.
As in classical mechanics the number of exactly soluble quantum mechanical problems is very limited and in most cases of practical interest numerical techniques and approximation methods are unavoidable. The availability of fast computers has turned quantum computation into a major industry. This chapter looks at the several approximation techniques. In seeking to solve the Schrodinger equation with the perturbed Hamiltonian, it is convenient to work with series expansions by introducing a dimensionless parameter λ. In many cases the system of interest differs from an exactly soluble system by only a small disturbance, enabling an approximation to be made by expanding in powers of a smallness parameter. The Variational Method provides a very simple method of approximating energy eigenvalues. It is most useful for computing ground state energies and its success depends on our being able to make a reasonable guess for the form of the ground state wave function.
A fast and stable numerical method is formulated to compute the time evolution of a wave function in a magnetic field by solving the time-dependent Schroedinger equation. This computational method is based on the finite element method in real space to improved accuracy without any increase of computational cost. This method is also based on Suzuki's exponential product theory to afford an efficient way to manage the TD-Schroedinger equation with a vector potential. Applying this method to some simple electron dynamics, we have confirmed its efficiency and accuracy.
The full configuration interaction (full-CI) method is capable of providing the numerically best wave functions and energies of atoms and molecules within basis sets being used, although it is intractable for classical computers. Quantum computers can perform full-CI calculations in polynomial time against the system size by adopting a quantum phase estimation algorithm (QPEA). In the QPEA, the preparation of initial guess wave functions having sufficiently large overlap with the exact wave function is recommended. The Hartree-Fock (HF) wave function is a good initial guess only for closed shell singlet molecules and high-spin molecules carrying no spin-β unpaired electrons, around their equilibrium geometry, and thus, the construction of multiconfigurational wave functions without performing post-HF calculations on classical computers is highly desired for applying the method to a wide variety of chemistries and physics. In this work, we propose a method to construct multiconfigurational initial guess wave functions suitable for QPEA-based full-CI calculations on quantum computers, by utilizing diradical characters computed from spin-projected UHF wave functions. The proposed approach drastically improves the wave function overlap, particularly in molecules with intermediate diradical characters.