Symmetry breaking and restoration for many-body problems treated on quantum computers

Abstract: This thesis explores the application of the Symmetry-Breaking/Symmetry-Restoration methodology on quantum computers to better approximate a Hamiltonian's ground state energy within a variational framework in many-body physics. This involves intentionally breaking and restoring the symmetries of the wave function ansatz at different stages of the variational search for the ground state. The Variational Quantum Eigensolver (VQE) is utilized for the variational component together with an ansatz inspired by the Bardeen-Cooper-Schrieffer (BCS) theory. The applications were demonstrated using the pairing and Hubbard Hamiltonians. Two approaches were identified with the VQE method: varying the symmetry-breaking ansatz parameters before or after symmetry restoration, termed Quantum Projection After Variation and Quantum Variation After Projection, respectively. The main contribution of this thesis was the development of a variety of symmetry restoration techniques based on the principles of the Quantum Phase Estimation algorithm, the notion of a Quantum "Oracle," and the Classical Shadow formalism. In the final part, hybrid quantum-classical techniques were introduced to extract an approximation of the low-lying spectrum of a Hamiltonian. Assuming accurate Hamiltonian moment extraction from their generating function with a quantum computer, two methods were presented for spectral analysis: the t-expansion method and the Krylov method, which provides, in particular, information about the evolution of the survival probability. Furthermore, the Quantum Krylov method was introduced, offering similar insights without the need to estimate Hamiltonian moments, a task that can be difficult on near-term quantum computers.