ETD

• anti-flatness
• entanglement
• non-stabilizerness
• quantum complexity
• sawtooth map
• simulation

The search for quantum advantage, understood as the point at which quantum computation surpasses classical computation, has led to the study of the emergence of quantum complexity, and consequently to the investigation of what makes a quantum algorithm hard to simulate on a classical computer. Quantum complexity can be characterized in different ways: the first is given by the entanglement present within the state of the system, which characterizes the non-classical correlations among its subsystems. In general, entanglement provides a measure of how physically complex the system is, but it is not informative about the complexity of simulating the system. This latter kind of complexity is quantified by non-stabilizerness, which is generated, in particular, by all the operations forming a quantum algorithm that do not belong to the Clifford group. Recently, the anti-flatness has been introduced to provide a bridge between the entanglement present in a bipartition of the system and non-stabilizerness, a global property of the system's state.
This thesis considers these three complementary quantities in the context of the simulation of a specific quantum algorithm: the quantum sawtooth map, for which an efficient implementation on a quantum computer exists. The dynamics of the sawtooth map is governed by a kicking strength parameter $k$ and the period of the kicking $T$; depending on their values, the system exhibits a rich phenomenology, including integrable, diffusive, and localized regimes. Among them, this thesis focuses on dynamical localization, a phenomenon strictly analogous to Anderson localization in disordered systems, in which the evolution of momentum eigenstates suppresses momentum diffusion, causing the wavefunction to remain localized in the momentum basis around a given eigenvalue, over a characteristic localization length $\xi$.
The study investigates how dynamical localization affects bipartite entanglement, non-stabilizerness, and anti-flatness, and how these quantities vary as a function of the system size, the parameters characterizing the sawtooth map, and the choice of initial state. The main result is that, as the system is driven out of the localized regime and into the ergodic one, the onset of ergodicity is clearly and consistently marked by all three quantities, each approaching the corresponding theoretical prediction for Haar-random states. This behavior makes entanglement, non-stabilizerness, and anti-flatness reliable and complementary probes for distinguishing the two regimes: entanglement and anti-flatness capture the correlations that arise during the application of the algorithm, including features that depend on the logical position of the qubits within the register due to the encoding, while non-stabilizerness offers a basis-independent, global measure of computational complexity. Furthermore, this work promotes the use of the sawtooth map as a test and benchmarking algorithm for quantum hardware.