Quantum number preserving ansätze and error mitigation studies for the variational quantum eigensolver

Computational chemistry has advanced rapidly in the last decade on the back of the progress of increased performance in CPU and GPU based computation. The prediction of reaction properties of varying chemical compounds in silico promises to speed up development in, e.g., new catalytic processes to reduce energy demand of varying known industrial used reactions. Theoretical chemistry has found ways to approximate the complexity of the underlying intractable quantum many-body problem to various degrees to achieve chemically accurate ab initio calculations for various, experimentally verified systems. Still, in theory limited by fundamental complexity theorems accurate and reliable predictions for large and/or highly correlated systems elude computational chemists today. As solving the Schrödinger equation is one of the main use cases of quantum computation, as originally envisioned by Feynman himself, computational chemistry has emerged as one of the applications of quantum computers in industry, originally motivated by potential exponential improvements in quantum phase estimation over classical counterparts. As of today, most rigorous speed ups found in quantum algorithms are only applicable for so called error-corrected quantum computers, which are not limited by local qubit decoherence in the length of the algorithms possible. Over the last decade, the size of available quantum computing hardware has steadily increased and first proof of concepts of error-correction codes have been achieved in the last year, reducing error rates below the individual error rates of qubits comprising the code. Still, fully error-corrected quantum computers in sizes that overcome the constant factor in speed up separating classical and quantum algorithms in increasing system size are a decade or more away. Meanwhile, considerable efforts have been made to find potential quantum speed ups of non-error corrected quantum systems for various applications in the noisy intermediate-scale quantum (NISQ) era. In chemistry, the variational quantum eigensolver (VQE), a family of classical-quantum hybrid algorithms, has become a topic of interest as a way of potentially solving computational chemistry problems on current quantum hardware. The main contributions of this work are: extending the VQE framework with two new potential ansätze, (1) a maximally dense first-order trotterized ansatz for the paired approximation of the electronic structure Hamiltonian, (2) a gate fabric with many favourable properties like conserving relevant quantum numbers, locality of individual operations and potential initialisation strategies mitigating plateaus of vanishing gradient during optimisation. (3) Contributions to one of largest and most complex VQE to date, including the aforementioned ansatz in paired approximation, benchmarking different error-mitigation techniques to achieve accurate results, extrapolating performance to give perspective on what is needed for NISQ devices having potential in competing with classical algorithms and (4) Simulations to find optimal ways of measuring Hamiltonians in this error-mitigated framework. (5) Furthermore a simulation of different purification error mitigation techniques and their combination under different noise models and a way of efficiently calibrating for coherent noise for one of them is part of this manuscript. We discuss the state of VQE after almost a decade after its introduction and give an outlook on computational chemistry on quantum computers in the near future