Quantum harmonic oscillator with time-dependent mass

We use the Fourier operator to transform a time-dependent mass quantum harmonic oscillator into a frequency-dependent one. Then we use Lewis–Ermakov invariants to solve the Schrödinger equation by using squeeze operators. Finally, we give two examples of time dependencies: quadratically and hyperbolically growing masses.

Storing the Quantum Fourier Operator in the QuIDD Data Structure

Quantum algorithms can be simulated using classical computers, but the typical time complexity of the simulation is exponential. There are some data structures which can speed up this simulation to make it possible to test these algorithms on classical computers using more than a few qubits. One of them is QuIDD by Viamontes et al., which is an extension of the Algebraic Decision Diagram. In this paper, we examine the matrix of Fourier operator and its QuIDD representation. To utilize the structure of the operator we propose two orderings (reversed column variables and even-odd order), both resulting in smaller data structure than the standard one. After that, we propose a new method of storing the Fourier operator, using a weighted decision diagram that further reduces its size. It should be the topic of subsequent research whether the basic operations can be performed efficiently on this weighted structure.

On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks

We consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on $\mathbb {R}$, and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent α∈ ]0,2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators Pt,Tt. The operator Pt is a Fourier operator and is considered here as a perturbation of the Markov operator P=P0 of the random walk. The operator Tt is related to Pt by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T0=T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.