Properties of finite Gaussians and the discrete-continuous transition

Abstract

Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finitedimensional systems. Based on Weyl's approach, generalized by Schwinger, a selfconsistent theoretical framework describing physical systems characterised by a finitedimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension d of the system increases.

Time-independent quantum circuits with local interactions

“…where p is an integer between 0 to N − 1. A Gaussian wave packet G⟩ can be defined as [29] Figure 3: (Color Online) M qubits correspond to 2M spin chains. As we will see, for the implementation of entangling gates, it is better to arrange the chains in two layers, the 0 chains in one layer and the 1 chains in another parallel layer.…”

Time independent universal computing with spin chains: quantum plinko machine

“…Of independent interest is the approximate Heisenberg relation, and the approximately dispersion free result. While these concepts have been noted by other authors [7,28], we have provided a powerful new application for them. These may allow for the construction of other interesting schemes using propagating Gaussian packets on interconnected rings.…”

“…Then, we can extend this to a tensor product of m packets and finally to a superposition in the dual rail encoding. Note that the following proof is contained in [28]. We provide it here for completeness, and to verify that the proof still works for the case we are interested in Using the substitution…”

Four-dimensional Bloch sphere representation of qutrits using Heisenberg-Weyl Operators

“…HW operators are unitary operators with several useful properties which makes them useful in several applications [13][14][15][16]. One of the most significant applications of the HW operators has been in the construction of the discrete Wigner function in finite dimensions [17,18].…”