Self-Consistent Field Approach for the Variational Quantum Eigensolver: Orbital Optimization Goes Adaptive

Abstract

We present a self-consistent field (SCF) approach within the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression that is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition-metal complex ferrocene [Fe (C 5 H 5 ) 2 ], indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to the complete active space SCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on nearterm quantum computers. Hence, ADAPT-VQE-SCF paves the way toward a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets.

Which Options Exist for NISQ-Friendly Linear Response Formulations?

“…The orbital-optimized unitary coupled cluster (oo-UCC) method is a recently developed, chemistry-inspired ansatz for the wave function on a quantum computer. − Its classic counterpart is the wave function obtained via the multiconfigurational self-consistent field (MCSCF) method . Just like MCSCF, oo-UCC splits the full space into smaller subspaces.…”

“…The energy of the ground state can now be found by variational minimization in the parameters θ and κ , E normalg normals = min bold-italicθ , bold-italicκ ⟨ normalU normalC normalC false( bold-italicθ false) false| Ĥ ( κ ) false| normalU normalC normalC false( bold-italicθ false) ⟩ Here, false| U C C ⟩ = exp ( − t̂ false( bold-italicθ false) ) false| C S F ⟩ . Performing this minimization process on quantum architecture is known as the orbital-optimized variational quantum eigensolver (oo-VQE) algorithm. − Its major advantage is splitting the problem into the active space simulations on the quantum computer and performing all operator applications outside the active space, i.e., the orbital rotations, classically.…”

“…The energy of the ground state can now be found by variational minimization in the parameters θ and κ , Here, . Performing this minimization process on quantum architecture is known as the orbital-optimized variational quantum eigensolver (oo-VQE) algorithm. − Its major advantage is splitting the problem into the active space simulations on the quantum computer and performing all operator applications outside the active space, i.e., the orbital rotations, classically.…”

Electric Field Gradient Calculations for Ice VIII and IX Using Polarizable Embedding: A Comparative Study on Classical Computers and Quantum Simulators

“…In the following, we introduce the PE-VQE-SCF formalism, developed by Kjellgren et al, which builds upon the ADAPT-VQE-SCF method originally developed by Fitzpatrick et al Further information can also be found in ref . We begin by expressing the spin-free, nonrelativistic, electronic Hamiltonian in second quantization as Ĥ = prefix∑ p q h p q Ê p q + 1 2 ∑ p q r s g q p r s ( Ê p q Ê r s − δ q r Ê p s ) where Ê pq is again a singlet one-electron excitation operator and h pq and g pqrs are, respectively, the one- and two-electron integrals over the molecular orbitals ϕ p ( r ): h p q = prefix∫ ϕ p * ( r ) ĥ ϕ q ( r ) .25em normald boldr g p q r s = prefix∫ ϕ p * ( r 1 ) ϕ r * …”

“…In this paper, we will use the implementation of the PE model for quantum computers in combination with the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver self-consistent field approach (ADAPT-VQE-SCF) and test its performance on calculating the EFGs of ice VIII and ice IX. We aim to reproduce the experimental results with conventional CASSCF and the ADAPT-VQE-SCF approach on a quantum simulator for ice VIII and ice IX.…”