Exploiting symmetries to improve variational quantum algorithms

Quantum computation could give an exponential speed-up to quantum simulations. Actually, in the so-called noisy-intermediate scale quantum (NISQ) era, the available devices are composed of a few qubits, are affected by noise and are prone to decoherence. Finding a way to simulate quantum systems that can be run on these devices is not a trivial task and, to this scope, many quantum algorithms have been developed. In particular, the variational quantum eigensolver (VQE) is one of the most promising algorithms to simulate quantum systems on NISQ devices. Nevertheless, this algorithm is affected by some limitations that do not allow us to use it for practical scope. The goal of this thesis is to exploit the symmetries of the systems under consideration to find new strategies to implement the VQE algorithm. This work is divided into two parts: the first one is composed of introductory material while the second one shows the original results achieved in this work. In particular, in chapter 1 we give an overview of the most important concepts of quantum computation and quantum information while 2 illustrate some algorithms used in classical and quantum simulations. In chapter 3 we define a method that optimizes the Hamiltonian to adapt it to the wavefunction through orbital rotations. Chapter 4 improve this method by reducing the required quantum computational resources while 5 show that, for small molecules, the algorithm converges to the natural orbitals single-particle basis set. In chapter 6 the algorithm is generalized to include quasiparticles states and it is applied to SU(N) fermionic systems. Chapter 7 define a strategy to create optimal hardware-efficient ans¨atze starting from the correlations caught by an approximated groundstate. In chapter 8 we show how to exploit the spin symmetry of the Hamiltonian to build ansatz preserving it. Finally, in chapter 9 we summarize the obtained results and show how they open the way to future developments.