We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasienergy which is defined only modulo 2π. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative quantum field theory. We use this formalism to compare the scattering amplitudes of a continuous-time model and of the corresponding discretized one. We give a rigorous assessment of the comparison for the case of bounded free Hamiltonian, as in a lattice theory with a bounded number of particles. Our framework can be applied to a wide class of quantum simulators, like quantum walks and quantum cellular automata. As a case study, we analyze the scattering properties of a one-dimensional cellular automaton with locally interacting fermions.
Discrete time quantum lattice systems recently have come into the focus of quantum computation because they provide a versatile tool for many different applications and they are potentially implementable in current experimental realizations. In this thesis we study the fundamental structures of such quantum lattice systems as well as consequences of experimental imperfections. Essentially, there are two models of discrete time quantum lattice systems, namely quantum cellular automata and quantum walks, which are quantum versions of their classical counterparts, i.e., cellular automata and random walks. In both cases, the dynamics acts locally on the lattice and is usually also translationally invariant. The main difference between these structures is that quantum cellular automata can describe the dynamics of many interacting particles, where quantum walks describe the evolution of a single particle. The first part of this thesis is devoted to quantum cellular automata. We characterize one-dimensional quantum cellular automata in terms of an index theory up to local deformations. Further, we characterize in detail a subclass of quantum cellular automata by requiring that Pauli operators are mapped to Pauli operators. This structure can be understood in terms of certain classical cellular automata. The second part of this thesis is concerned with quantum walks. We identify a quantum walk with the one-particle sector of a quantum cellular automaton. We also establish an index theory for quantum walks and we discuss decoherent quantum walks, i.e., the behavior of quantum walks with experimental imperfections.
In the thesis, models of quantum systems characterized by discrete time dynamics are presented and applied to different problems. More specifically, aspects of excitation transport, quantum state transfer and thermalization are investigated mainly within the framework of quantum cellular automata (QCA), which are discrete time evolution quantum systems defined on a regular lattice of, in this case, qubits. The peculiarities of QCA, together with the restriction to the first excitation sector of the Hilbert space, make it possible to define a transport dynamics on a one dimensional lattice that encompasses all classical Markov chains, as well as maps where quantum coherence between sites can build up over time. This allows for the possibility to make a direct comparison between a fully classical dynamics and a dynamics affected by quantum effects, which is of relevance both in the field of quantum biology, especially regarding the transport of electronic excitations in photosynthetic systems, and in quantum computing when dealing with the transport of an unknown quantum state. In a quantum thermodynamics context, instead, QCA can serve the purpose to validate arguments of tipicality that have been brought forward in recent years in order to demonstrate the thermalization of closed quantum systems (i.e. the tendency of any small subsystems to evolve towards the maximally mixed state). Numerical results concerning this issue are presented. Finally, a modelization of scattering-like processes, where the scattering consists of random unitary interactions between an environment and part of a quantum bipartite system is discussed. Relevant properties of the resulting state such as its mean purity and the correlations between the interacting and the “sheltered” part of the system can be analitically evaluated, and their fluctuations bounded.
Given two Hamiltonians
H0,H1=H0+V corresponding respectively to a free particle Hamiltonian and the free particle Hamiltonian plus its interaction energy with a scattering centre, we can compute the S-matrix, i.e., the matrix that determines the quantum mechanical amplitude of the process involving a particle coming from a free particle state at an infinite distance from the scattering centre at time
t=−∞, entering into an initial scattered state, interacting with the scattering centre, entering into a final scattered state, and then escaping away to infinity at time
t→∞ to a final free particle state. Such amplitudes were first studied by Lippmann and Schwinger (Steven Weinberg, The Quantum Theory of Fields, Vol. 1, W.O. Amrein, Hilbert Space Methods in Quantum Mechanics) and then analyzed in a mathematically precise way by [T. Kato], [W. Amrein], and [K.B. Sinha]. In this paper, we first derive the Lippmann-Schwinger equation for the input and output scattered states in terms of the input and output free particle states using intuitive arguments originally due to Lippmann and Schwinger, and then, using rigorous operator theoretic arguments revolving around the spectral theorem for unbounded self-adjoint operators in a Hilbert space, we obtain an elegant formula for the scattering matrix elements in terms of the interaction potential and the free particle Hamiltonian between two free particle states corresponding to the same measure. We then discuss a computationally more efficient method based on the Dyson series expansion (Steven Weinberg, The Quantum Theory of Fields, Vol. 1), which works even in the case when the interaction Hamiltonian is time varying, in contrast to the Lippmann-Schwinger method, for calculating the S-matrix. We assume that the scattering potential, which can be time varying, can be controlled by incorporating control parameters and explain how the resulting unitary S-matrix can be made to approximate a given unitary gate in infinite-dimensional Hilbert space by optimizing the Hilbert-Schmidt/Frobenius norm distance between the given gate and the S-matrix gate, in which the latter is approximated by a truncated Dyson series. Of course, the S-matrix given by the Dyson series is a Taylor series functional expansion in the interaction potential and hence is a highly nonlinear function of the control parameters, and hence we propose numerical methods for such optimization. We then observe that if the control parameters are taken as random variables with joint moments, then the TPCP map involving statistical averaging of the adjoint action of the S-matrix acting on an initial mixed state w.r.t. the probability distribution of the control parameters will result in a final mixed state that is linear in the statistical moments of the control parameters, and hence optimization w.r.t. these moments can be easily carried out. Our method of TPCP map design is to optimize the sum of Frobenius distance squares between a sequence of desired output mixed states and the S-matrix based formula for the TPCP map acting on the corresponding input mixed states w.r.t. the statistical moments of the control parameters. We then generalize this method to include the case of TPCP maps obtained from the Hudson-Parthasarathy quantum noisy Schrödinger equation. In summary, we simulate the S-matrix gate for independent realizations of the control parameters having a probability distribution defined by their joint moments obtained by the above optimization procedure and apply the adjoint action of these independent S-matrices on a given input state, and take their ensemble average to obtain a good approximation to the action of the desired TPCP map on any given input state. This suggests that TPCP maps of arbitrarily large size can be realized using scattering experiments in the laboratory.
It is shown that discrete-time quantum walks can be used to digitize, i.e., to time discretize fermionic models of continuous-time lattice gauge theory. The resulting discrete-time dynamics is thus not only manifestly unitary, but also ultralocal, i.e., the particle's speed is upper bounded, as in standard relativistic quantum field theories. The lattice chiral symmetry of staggered fermions, which corresponds to a translational invariance, is lost after the requirement of ultralocality of the evolution; this fact is an instance of Meyer's 1996 no-go results stating that no nontrivial scalar quantum cellular automaton can be translationally invariant [D. A. Meyer, J. Stat. Phys. 85, 551 (1996); Phys. Lett. A 223, 337 (1996)]. All results are presented in a single-particle framework and for a (1+1)-dimensional space-time.
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.