Quantum computation of a complex system: The kicked Harper model

Abstract

The simulation of complex quantum systems on a quantum computer is studied, taking the kicked Harper model as an example. This well-studied system has a rich variety of dynamical behavior depending on parameters, displays interesting phenomena such as fractal spectra, mixed phase space, dynamical localization, anomalous diffusion, or partial delocalization, and can describe electrons in a magnetic field. Three different quantum algorithms are presented and analyzed, enabling to simulate efficiently the evolution operator of this system with different precision using different resources. Depending on the parameters chosen, the system is near-integrable, localized, or partially delocalized. In each case we identify transport or spectral quantities which can be obtained more efficiently on a quantum computer than on a classical one. In most cases, a polynomial gain compared to classical algorithms is obtained, which can be quadratic or less depending on the parameter regime. We also present the effects of static imperfections on the quantities selected, and show that depending on the regime of parameters, very different behaviors are observed. Some quantities can be obtained reliably with moderate levels of imperfection, whereas others are exponentially sensitive to imperfection strength. In particular, the imperfection threshold for delocalization becomes exponentially small in the partially delocalized regime. Our results show that interesting behavior can be observed with as little as 7-8 qubits, and can be reliably measured in presence of moderate levels of internal imperfections.

Quantum simulation

“…An application of DQS with a few qubits is the study of the dynamics of simple quantum maps (Georgeot, 2004;Levi and Georgeot, 2004;Schack, 1998;Terraneo et al, 2003). For instance, the quantum baker's map, a prototypical example in quantum chaos, has an efficient realization in terms of quantum gates (Schack, 1998).…”

“…The quantum baker's map has been experimentally realized in NMR (Weinstein et al, 2002) and with linear optics (Howell and Yeaze, 2000). Another example is the kicked Harper model (Levi and Georgeot, 2004). It has been shown that in some cases the quantum approach to the kicked Harper model can provide a polynomial gain as compared to classical algorithms.…”

Complexity of chaos and quantum computation

“…Numerical simulations for increasing n s show a good precision for relatively low n s (Lévi and Georgeot 2004). The implementation of both e −iK cos(θ)/~a nd e −iL cos(~n)/~b y n s iterations of the appropriate M(α, U) needs 4 + 2(n q − a) + (n s − 1)(7 + 2(n q − a)) gates; the quantum Fourier transforms need n q (n q + 1) gates.…”

Quantum computation and analysis of Wigner and Husimi functions: Toward a quantum image treatment

“…The localization length in this regime becomes quickly larger than the system size for small number of qubits, thus allowing to explore the complexity of a chaotic wave function. Indeed, in the localized regime, the most important information resides not so much in such distributions, but in the localization properties, and their measurement on a quantum computer was already analyzed in [13,30].…”

“…We focus on the phase space distribution (Wigner and Husimi functions) [10,11] These functions provide a two-dimensional picture of a one-dimensional wave function, and can be compared directly with classical phase space distributions. They have also been shown in [12,13] to be stable with respect to various quantum computer error models. Different phase space representation which can be implemented efficiently on a quantum computer will be explored, first the discrete Wigner transform, for which an original algorithm will be presented, and then a Husimilike transform, first introduced in this context in [14].…”