One of the first practical applications of quantum computers is expected to be molecular modelling. Performing this task would profoundly affect areas such as chemistry, materials science and drug synthesis. Modelling of molecules, which are classically intractable, can be achieved with just over $30$ qubits, whereas state of the art quantum computers already have more than $50$ qubits. The Variational Quantum Eigensolver (VQE) algorithm and VQE based protocols, are promising candidates to enable this task on emerging Noisy Intermediate-Scale Quantum (NISQ) computers. These protocols require short quantum circuits and short coherence times, and are particularly resilient to quantum errors. Nevertheless, there is still a significant gap between the accuracy and the coherence times of current NISQ computers, and the hardware requirements of VQE protocols to simulate practically interesting molecules. In this thesis, I present my contribution to narrowing this gap by developing VQE protocols for molecular modelling that are less demanding on quantum hardware.

The VQE relies on the Rayleigh-Ritz variational principle to estimate the eigenvalues of a Hamiltonian operator, by minimizing its expectation value with respect to a trial quantum state, prepared by an ansatz. A major challenge for the practical realisation of VQE protocols on NISQ computers is to construct an ansatz that: (1) can accurately approximate the eigenstates of the Hamiltonian; (2) is easy to optimize; and (3) can be implemented by a shallow circuit, within the capabilities of a NISQ computer. The most widely used, unitary coupled cluster (UCC), type of ans"atze mathematically correspond to a product of unitary evolutions of fermionic excitation operators. Owing to their fermionic structure, UCC ans"atze preserve the symmetries of electronic wavefunctions, and thus are accurate and easy to optimize. Nevertheless, UCC ans"atze are implemented by high depth circuits, which severely limit the size of the molecules that can be reliably simulated on NISQ computers. In this thesis, I begin by constructing efficient quantum circuits to perform evolutions of fermionic excitation operators. The circuits are optimized in the number of two-qubit entangling gates, which are the current bottleneck of NISQ computers. Compared to the standard circuits used to implement evolutions of fermionic excitation operators, the circuits derived in this thesis reduce the number of two-qubit entangling gates by more than $70\%$ on average. As an intermediate result, I also derive efficient circuits to perform evolutions of qubit excitation operators (excitation operators that account for qubit, rather than fermionic commutation relations).

Even with the fermionic-excitation-evolution circuits derived here, UCC ans"atze still require very long circuits, with a particularly large number of two-qubit entangling gates. In this thesis, I consider the use of alternative VQE ans"atze, based on evolutions of qubit excitation operators. Due to not accounting for fermionic anticommutation, evolutions of qubit excitation operators can be performed by circuits that require asymptotically fewer two-qubit entangling gates. Furthermore, qubit excitation operators preserve many of the physical properties of fermionic excitation operators. Performing a number of classical numerical VQE simulations for small molecules, I show that qubit-excitation-based ans"atze can approximate molecular electronic wavefunctions almost as accurately as fermionic-excitation-based ans"atze. Hence, I argue that evolutions of qubit excitation operators are more suitable to construct molecular ans"atze than evolutions of fermionic excitation operators, especially in the era of NISQ computers.

Motivated by the advantage of qubit-excitation-based ans"atze, I introduce the qubit-excitation-based adaptive variational quantum eigensolver (QEB-ADAPT-VQE). The QEB-ADAPT-VQE belongs to a family of ADAPT-VQE protocols for molecular modelling that grow a problem-tailored ansatz by iteratively appending unitary operators sampled from a predefined finite-size pool of operators. The operator at each iteration is sampled based on an ansatz-growing strategy, which aims to achieve the lowest estimate for the Hamiltonian expectation value at each iteration. In this way, ADAPT-VQE protocols construct shallow-circuit, few-parameter ans"atze tailored specifically to the molecular systems of interest. In the case of the QEB-ADAPT-VQE, the operator pool is defined by a set of evolutions of single and double qubit excitation operators. I benchmark the performance of the QEB-ADAPT-VQE, by performing classical numerical simulations. I demonstrate that it can construct ans"atze that are several orders of magnitude more accurate, and require significantly shallower circuits, than standard UCC ans"atze. I also compare the QEB-ADAPT-VQE against the original fermionic-ADAPT-VQE, which utilizes a pool of fermionic excitation evolutions, and the qubit-ADAPT-VQE, which utilizes a pool of Pauli-string evolutions. I demonstrate that, in terms of circuit efficiency and convergence speed, the QEB-ADAPT-VQE systematically outperforms the qubit-ADAPT-VQE, which to my knowledge was the previous most circuit-efficient, scalable VQE protocol for molecular modeling. The QEB-ADAPT-VQE protocol, therefore represents a significant improvement in the field of VQE protocols for molecular modelling and brings us closer to achieving practical quantum advantage.

Lastly, I outline a modified version of the QEB-ADAPT-VQE, the excited-QEB-ADAPT-VQE, designed to estimate energies of excited molecular states. The excited-QEB-ADAPT-VQE is more robust to initial simulation conditions, at the expense of increased computational complexity.