UC Berkeley

The quantized torus is a finite-dimensional Hilbert space that represents quantum mechanicswith periodic phase space. The space can act as a toy model for many quantum effects, and it has the benefit of admitting numerical illustrations. Taking the dimension to infinity corresponds to taking a semiclassical limit, and allows us to visualize the quantum-classical correspondence in a simple setting. In this thesis, we examine several instances of such a semiclassical limit. We begin in the setting of quantum dynamics, and consider eigenvalues of a quantized cat map (i.e. hyperbolic symplectic integer matrix), cut off in phase space to include a fixed point as its only periodic orbit on the torus. We prove a simple formula for the eigenvalues on both the quantized real line and the quantized torus in the semiclassical limit as h → 0. We then consider the case with no fixed points, and prove a superpolynomial decay bound on the eigenvalues. We then study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between mea- surements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by propagation of a semiclassical defect mea- sure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is “less chaotic”. Finally, we apply semiclassical analysis to the field of quantum simulation, and improve bounds on the basic Trotterization algorithm in the setting of the semiclassical Schr ̈odinger equation. We show that the number of Trotter steps used for the observable evolution can be O(1). We then apply the theory of the quantized torus to extend our results to the discretized case, which is amenable to quantum computation models.