Entanglement versus relaxation and decoherence in a quantum algorithm for quantum chaos

Abstract

We study analytically and numerically the behavior of the concurrence (a measure of the entanglement of formation) of a pair of qubits in a quantum computer operating an efficient algorithm for quantum chaos. Our results show that in an ideal algorithm the entanglement decays exponentially with the diffusive relaxation rate induced by classical chaos. This decay reaches a residual level which drops exponentially with the number of qubits nq. Decoherence destroys the residual entanglement with a rate exponential in nq.PACS numbers: 03.67. Lx, 05.45.Mt Enormous interest into quantum information and computation [1] has generated serious efforts in characterizing and understanding quantum entanglement which is considered as the ultimate origin of quantum power (see a recent review [2]). A quantitative measure of the entanglement of formation, namely the concurrence C, was introduced and shown to be able to characterize an arbitrary state of two qubits [3,4]. Being closely related to the von Neumann entropy S of the reduced density matrix ρ of two qubits[31], this quantity was recently found to have interesting applications to quantum phase transitions in interacting spin systems [5]. In parallel, the properties of entanglement were investigated in a quantum model of coupled tops where it was shown that there exists a typical value of entanglement which is determined by the chaotic behavior of the dynamics of the model [6] and that the growth rate of entanglement of initially decoupled tops is increased by the underlying classical chaos [7]. On the same line, it was recently shown that contrary to intuition even a heat bath may create entanglement between two qubits [8].All these studies [5,6,7,8] clearly demonstrated how rich entanglement properties can be in interacting quantum systems. However, in the context of quantum computation it is much more crucial to analyze the evolution of entanglement in a specific algorithm performing an operational task. Indeed, it is expected that the entanglement is very sensitive to noise and decoherence [9,10,11] and the understanding of its behavior in an operating algorithm can lead to better strategies in the control of decoherence and imperfection effects. As for our knowledge such direct investigations have not been performed until now. Therefore, in this paper we study the behavior of the concurrence in an efficient algorithm for the quantum sawtooth map which has been proposed recently in [12]. The algorithm for this model has a number of important advantages: all n q qubits are used in an optimal way and no ancillae are required, one map iteration in * URL: http://www.quantware.ups-tlse.fr the Hilbert space of size N = 2 nq is performed in O(n 2 q ) quantum gates, and the algorithm is based on the quantum Fourier transform (QFT) which is one of the main elements of various quantum algorithms [1]. This allows to simulate a complex dynamics in the regime of quantum chaos with a small number of qubits. Since the entanglement can be efficiently measured experimentally (see e...

Dynamical Aspects of Quantum Entanglement for Coupled Mapping Systems

“…It is often believed that chaos enhances entanglement, but that can be the case for weakly chaotic regions, and such effects can be also explained by the factor D 0 in our formula, eq. (14). Note again that the above conclusion is basically for strongly chaotic regions where D 0 saturates.…”

“…Since the dynamical aspects of entanglement production have been studied in relation with quantum computing, 14) we hope that our analysis will be useful in the studies of quantum information processings. In ref.…”

Entanglement in many-body systems

“…If the imperfection strength increases, new phenomena occur and above a certain threshold the core of the computer can even "melt" due to the setting in of chaotic behavior (Benenti et al, 2001;Georgeot and Shepelyansky, 2000). In addition to the understanding of the behaviour of the fidelity as an indicator to measure the stability of the quantum memory (see Gorin et al, 2006 for a review), a more complete characterization has recently included the behavior of entanglement on approaching the transition to the chaos either by considering a dynamics of a disordered (Montangero et al, 2003;Montangero and Viola, 2006;Sen(De) et al, 2006) or time-dependent Ising model (Lakshminarayan and Subrahmayam, 2005;see Prosen, 2007 for a recent review on the dynamical complexity analysed on the kicked Ising model) and by studying the dynamics of a quantum map (Bandyopadhyay and Lakshminarayan, 2002;Bettelli and Shepelyansky, 2003;Ghose and Sanders, 2004;Miller and Sarkar, 1999;Rossini et al, 2004). In particular the disordered Ising model has been proposed (Georgeot and Shepelyansky, 2000) to describe the hardware of a quantum computer, in which system imperfections generate unwanted inter-qubit couplings and energy fluctuations.…”

Entangling power of the quantum baker s map

“…We therefore expect an intimate relationship between the dynamics and the multipartite entanglement induced amongst the qubits. Using different approaches, the dynamics of entanglement in qubit bases was recently investigated in [23,24].…”