Quantum Computing and Information Extraction for Dynamical Quantum Systems

Abstract

We discuss the simulation of a complex dynamical system, the so-called quantum sawtooth map model, on a quantum computer. We show that a quantum computer can be used to efficiently extract relevant physical information for this model. It is possible to simulate the dynamical localization of classical chaos and extract the localization length of the system with quadratic speed up with respect to any known classical computation. We can also compute with algebraic speed up the diffusion coefficient and the diffusion exponent both in the regimes of Brownian and anomalous diffusion. Finally, we show that it is possible to extract the fidelity of the quantum motion, which measures the stability of the system under perturbations, with exponential speed up.

Localization in the quantum sawtooth map emulated on a quantum-information processor

“…The state of the quantum system is represented by a density matrix ρ expressed in the momentum basis. A detailed description and insightful discussion of the sawtooth map can be found in reference [5].…”

“…The quantum circuit for the QFT is given in [14], an implementation of the QFT is described in [15], and an analysis of the quantum process tomography of this implementation can be found in [10]. Circuits for U J and U Θ are conveniently found in realizing that the diagonal operators can be decomposed into a series of single-qubit z rotations and two-qubit controlled z rotations; the details of this decomposition are given in [5]. This circuit for the quantum sawtooth map is computationally efficient, in that the number of fundamental quantum gates required to implement the algorithm depends polynomially on the number of qubits [3].…”

“…Efficient algorithms have been developed for simulating the dynamics of localization in the kicked rotator model [2], the quantum sawtooth map [3], and the kicked Harper model [4] on a QIP. Although localization can in principle be observed in an ideal emulation using as few as three qubits [5], the sensitivity of this phenomenon to noise effects [3,4,5,6,7,8,9] poses a rigorous challenge in the task of creating and maintaining a localized state on a noisy QIP.…”

Observability of fidelity decay at the Lyapunov rate in few-qubit quantum simulations

“…Therefore a natural approach for classical Hamiltonian systems is to simulate their quantized versions and find ways to extract classically-relevant information. In fact, it was recognized early on that this technique can accelerate the computation of useful dynamical quantities, such as the Lyapunov exponent [21,22]. This is possible even without using many qubits to approach the limit of classical dynamics, raising the possibility of observing one of these quantities in few-qubit quantum simulations.…”

Quantum computation of multifractal exponents through the quantum wavelet transform

“…Thus a quantum algorithm should also include specification of how information is extracted from the final wavefunction at the end of the simulation. Several proposals have been made in order to extract efficiently information from a quantum simulation, looking at the fidelity [15], the spectral statistics [16], the localization length [17], the Wigner function [10], or the diffusion constants [11,18]. It has been found that the final gain compared to classical simulation depends on the choice of the observable and on the measurement procedure, and can dramatically change the efficiency of the quantum simulation.…”