Molecular Energy estimation with Quantum Variational Algorithm
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docs: contains documentation.
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notebooks: contains jupyter notebooks, further file's description available in folder's readme.
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classic_simulations: python scripts for HF, FCI, MP, CCSD with different basis
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manual_VQE: explicit VQE function, useful for hardware runs
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tapering: script to search for symmetries
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data: simulation and measurement data
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example.py: commented VQE example
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example_QiskitNature_015.py: commented VQE example with qiskit nature
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requirements.txt: list of installed packages used for the virtual python environment (python version: 3.8.5).
The idea of quantum computers was first introduced by Richard Feynman in 19821. Since then,in just 40 years, we have learned to exploit this new computational paradigm for problems ranging from nuclear physics to machine learning. In this thesis we will use quantum computing methods to study the dissociation energy of the H+3 molecule. To tackle this problem, both with classical and quantum computers, we must solve the molecular problem. First, we separate the electronic problem from the nuclear one through the Born-Oppenheimer approximation: this allows us to neglect the kinetic energy of nuclei and their movement. To describe the electronic wave functions, which belong to an infinite dimensional space,it is necessary to choose a finite basis that has good approximation capabilities while keeping the computational complexity to the minimum. There are several possible basis with different degrees of complexity: starting from STO functions (Slater Type Orbitals), that are very complex from the computational point of view, up to the Gaussian basis and ”minimal basis” (functions that often lead to inaccurate results, but which are extremely simple and will therefore will be used for measurements on quantum computers). There are also some more complex basis such as ”Pople’s basis”or correlated basis: both are much more effective than minimal basis, but difficult to use on quantum computers. Finally, we need the Hartree-Fock approximation which allows to solve the electronic problem without considering the motion of every electron at the same time, but by introducing a mean field which greatly simplifies the description. This approximation does not consider cor-relations between different electrons and therefore it is necessary to introduce corrective methods.Existing classical methods must balance precision and computational cost: the FCI method(FullConfiguration Interaction) solves the system in an exact way although it has a high computational cost. Other methods are introduced in order to reduce this cost such as MP (Møller-Plesset) or CC(Coupled Cluster) which introduce further approximations.At this point, we can introduce the fundamental elements of quantum computing. Instead of classical bits, in quantum computers the units of information are constituted by qubits: two-state quantum systems that can be superimposed in infinite combinations. Operations on these qubits are performed by quantum gate which are the equivalents of boolean gates. Most of these operations consider a single qubit, but there are gates operating on multiple qubits: for example, CNOT(Controlled NOT) which is needed to establish entangled systems that are fundamental to quantum computing. A set of gates applied to a certain number of qubits is called a circuit. The first step in tackling the electronic problem on quantum computers is to translate the fermionic system into a quantum circuit. To do this, different isomorphisms can be used, the main ones being Jordan-Wigner, parity and Bravyi-Kitaev isomorphisms.Currently existing quantum computers (even those that can be accessed freely via IBMQ) have some fundamental limitations: they have limited numbers of qubits and are also very noisy as they are subject to decoherence and other errors both in the application of gates either in measures or in the preparation of states. Therefore, we can say that we are in the NISQ era (Noisy Intermediate-ScaleQuantum) where it is not yet possible to build noiseless quantum computers. For this reason, is essential to minimize the number of qubits and implement errors mitigation techniques. The VQE (Variational Quantum Eigensolver) is an algorithm designed for NISQ quantum computers that exploits the variational principle to find the minimum energy of a system. It is a hybrid algorithm that uses both classical and quantum processors: it starts from a state (possibly obtained from the Hartree-Fock approximation), that is varied through a specific variational form and the energy corresponding to the final state is calculated; a classical optimizer takes care of varying the parameters of the variational form with the aim of minimizing the energy.The central part of this thesis focuses on the application of VQE to the trihydrogen cationH+3. First,classical computations are applied in order to obtain reference results for quantum measurements.VQE is then used on quantum computer simulators obtaining results from noiseless simulations. For simulations with noise, on the other hand, it is necessary to carry out a more in-depth study on the variational form to reduce the number of gates (main source of noise) and to better implement error mitigation techniques. Finally, measurements are performed on a real quantum computer.The dissociation energy found with FCI and the most accurate basis (0.164±0.005EH) is perfectly compatible with the experimental tabulated value (0.161EH). In quantum computing simulations,however, it was necessary to adopt a minimal basis in order not to use more qubits than those available, so the value calculated with FCI is: 0.145±0.005EH. In noise simulations the value deviates a little from the exact result: 0.124±0.005EH. Noise simulations represent the best execution for a real quantum computer, so the result (0.119±0.005EH) was expected to get even worse (although it is compatible with the simulated noise).Deviations from the expected value indicate that VQE is an algorithm still being perfected. In contemporary research various possible developments are being explored: one of these, the ADAPT-VQE6, introduces a variational form dependent on the analyzed problem that allows to use a smaller number of qubits (addressing one of the main NISQ problems). The main goal of research on quantum computers is to find an algorithm capable of solving a problem in an extremely more efficient way than any solution on classic computers: this is the quantum advantage (or quantum supremacy). The most promising algorithms, in this sense, are VQE and its derivatives