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This review presents a comprehensive overview of the Unitary Coupled Cluster (UCC) ansatz and related ansätze which are used to solve the electronic structure problem on quantum computers.

We present a basis-set-free approach to the variational quantum eigensolver using an adaptive representation of the spatial part of molecular wave functions. Our approach directly determines system-specific representations of qubit Hamiltonians while fully omitting globally defined basis sets. In this work, we use directly determined pair-natural orbitals on the level of second-order perturbation theory. This results in compact qubit Hamiltonians with high numerical accuracy. We demonstrate initial applications with compact Hamiltonians on up to 22 qubits where conventional representation would for the same systems require 40−100 or more qubits. We further demonstrate reductions in the quantum circuits through the structure of the pair-natural orbitals.

We develop computationally affordable and encoding independent gradient evaluation procedures for unitary coupled-cluster type operators, applicable on quantum computers.

MADNESS (multiresolution adaptive numerical environment for scientific simulation) is a high-level software environment for solving integral and differential equations in many dimensions that uses adaptive and fast harmonic analysis methods with guaranteed precision based on multiresolution analysis and separated representations. Underpinning the numerical capabilities is a powerful petascale parallel programming environment that aims to increase both programmer productivity and code scalability. This paper describes the features and capabilities of MADNESS and briefly discusses some current applications in chemistry and several areas of physics.

Variational quantum algorithms are currently the most promising class of algorithms for deployment on near-term quantum computers. In contrast to classical algorithms, there are almost no standardized methods in quantum algorithmic development yet, and the field continues to evolve rapidly. As in classical computing, heuristics play a crucial role in the development of new quantum algorithms, resulting in a high demand for flexible and reliable ways to implement, test, and share new ideas. Inspired by this demand, we introduce tequila, a development package for quantum algorithms in python, designed for fast and flexible implementation, prototyping and deployment of novel quantum algorithms in electronic structure and other fields. tequila operates with abstract expectation values which can be combined, transformed, differentiated, and optimized. On evaluation, the abstract data structures are compiled to run on state of the art quantum simulators or interfaces.

An efficient representation of molecular correlated wave functions is proposed, which features regularization of the Coulomb electron–electron singularities via the F12-style explicit correlation and a pair-natural orbital factorization of the correlation components of the wave function expressed in the real space. The pair-natural orbitals are expressed in an adaptive multiresolution basis and computed directly by iterative variational optimization. The approach is demonstrated by computing the second-order Moller–Plesset energies of small- and medium-sized molecules. The resulting MRA-PNO-MP2-F12 method allows for the first time to compute correlated wave functions in a real-space representation for systems with dozens of atoms (as demonstrated here by computations on alkanes as large as C10H22), with precision exceeding what is achievable with the conventional explicitly correlated MP2 approaches based on the atomic orbital representations.

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