Improving the Accuracy of Variational Quantum Eigensolvers with Fewer Qubits Using Orbital Optimization

Abstract

Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this work we propose incorporating an orbital optimization scheme into quantum eigensolvers wherein a parametrized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. We find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE when using the same number of qubits and even frequently achieves lower ground state energies than VQE methods using more qubits.

Correlated Reference-Assisted Variational Quantum Eigensolver

“…Several authors have shown that orbital relaxation is important to reduce VQE errors − and the number of qubits . In those studies, the energy of VQE is minimized concerning both cluster amplitudes and orbitals, resulting in a self-consistency that demands higher computational costs than standard VQE.…”

“…Several authors have shown that orbital relaxation is important to reduce VQE errors 39−42 and the number of qubits. 56 In those studies, the energy of VQE is minimized concerning both cluster amplitudes and orbitals, resulting in a self-consistency that demands higher computational costs than standard VQE. It is thus interesting to examine whether correlated orbital reference preoptimized using a lower-level method can improve the accuracy of "single-shot" VQE.…”

Complete Active Space Methods for NISQ Devices: The Importance of Canonical Orbital Optimization for Accuracy and Noise Resilience

“…The idea of a hybrid orbital optimization routine has been explored first by Takeshita et al, and a fully hybrid quantum-classical CASSCF has been reported by Tilly et al where the 1 and 2 body RDMs were sampled independently to mitigate their error and noncanonical orbitals were used. Other quantum multi-configurational SCF implementations have been also reported in the literature. ,− Nevertheless, to our knowledge, no comparison has been made with the results coming from an orbital-optimized wave function in the quantum measurement after VQE. We have implemented a CASSCF routine (Figure ) that evaluates the 1 and 2 body RDMs as auxiliary operators to the Hamiltonian and uses canonical CASSCF orbitals (and thus in particular natural orbitals within the active space) by default.…”

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

“…In this work, we will add a pivotal milestone to the aforementioned adaptive VQE algorithms by combining the adaptive expansion of the quantum circuit and ensuing optimization of its accompanying {θ i } parameter set with simultaneous optimization of the orbital rotation parameters κ in the variational energy minimization as outlined in eq . In contrast to related recent works within the framework of near-term quantum-computing-driven MCSCF-type approaches, − our adaptive, SCF VQE algorithm, denoted in the following as ADAPT-VQE-SCF, greatly benefits from a considerable reduction in the number of two-qubit gates in the final ansatz that are required to reach convergence with respect to chemical precision. Moreover, we not only provide a state-specific energy minimization (eq ) ADAPT-VQE-SCF algorithm but also a SA one that opens up for an optimization of a common set of MOs for a multitude of states i , that is where λ i is a preselected, fixed weighting factor for the i -th state |Φ i ⟩ that satisfy the constraint but are arbitrary otherwise.…”