Quantum computing for molecular systems

The VQE algorithm uses a parameterized quantum circuit (PQC) |ψ(θ)⟩ to construct a quantum state that approximates the ground state of the system. The parameters of the quantum circuit θ are optimized to its optimal value θ∗ using a classical optimization algorithm, such as gradient descent or Newton’s method, to minimize the energy of the quantum state E(θ)=⟨ψ(θ)|H^|ψ(θ)⟩. According to the Rayleigh-Ritz variational principle, E(θ∗)≥Eground and the equity is reached when |ψ(θ∗)⟩ is the ground state wave function. Thus, given an expressive PQC, |ψ(θ∗)⟩ is a good estimation of the ground state wave function.

For molecular systems, the ab initio Hamiltonian is written as

where hpq and hpqrs=[ps|qr] are one-electron and two-electron integrals, and a^p†,a^p are fermionic creation and annihilation operators, respectively. For chemical systems, the VQE algorithm is composed of several steps. The first step is to calculate the integrals in the Hamiltonian under the molecular orbital basis. Then, the second-quantized fermion Hamiltonian is mapped to a spin Hamiltonian using fermion-qubit mapping, since quantum computers are built based on the spin model. In this work, we employ the parity transformation for saving two qubits

Here c^ is the qubit annihilation operator 12(X^+iY^), and X^, Y^ and Z^ are Pauli operators. The transformation ensures the preservation of the commutation and anti-commutation properties of fermion operators. After the fermion-qubit mapping, the Hamiltonian in Eq. (1) is transformed to a summation of the products of Pauli operators. More formally, the Hamiltonian can be written as H^=∑jMαjP^j where αj is the coefficient and P^j is the product of Pauli operators. M is the total number of terms. Each Pj can be measured on a quantum computer and subsequently, the overall energy is obtained by taking the weighted summation.

The active-space approximation is employed to reduce computational cost and enhance accuracy. The approximation adopts the Hartree-Fock state as the baseline state and chooses an “active space” that is treated with a high-accuracy computational method. In classical computation, the high-accuracy method is usually full configuration interaction (FCI) or density matrix renormalization group (DMRG)⁶⁵. In our case, quantum computers are employed to solve the problem with the VQE algorithm. The active space is usually constructed in the molecular orbital space. Most commonly, orbitals that have the closest energy with the HOMO and LUMO orbitals will be included in the active space. Meanwhile, the inner shell orbitals are treated at the mean-field level. Thus, this approximation is sometimes also called the frozen core approximation. Denote the set of frozen occupied spin-orbitals by Λ. The frozen core provides an effective repulsion potential Veff to the remaining electrons

The frozen core also bears the mean-field core energy

The ab initio Hamiltonian with the active space approximation is rewritten as

where the indices p, q, r and s refer to spin-orbitals in the active space.