Dynamics of Entanglement in Quantum Computers with Imperfections

Abstract

The dynamics of the pairwise entanglement in a qubit lattice in the presence of static imperfections exhibits different regimes. We show that there is a transition from a perturbative region, where the entanglement is stable against imperfections, to the ergodic regime, in which a pair of qubits becomes entangled with the rest of the lattice and the pairwise entanglement drops to zero. The transition is almost independent of the size of the quantum computer. We consider both the case of an initial maximally entangled and separable state. In this last case there is a broad crossover region in which the computer imperfections can be used to create a significant amount of pairwise entanglement.PACS numbers: 03.67. Lx, 03.67.Mn, 24.10.Cn Any practical implementation of a quantum computer will have to face errors, due to the inevitable coupling to the environment [1,2] or to device imperfections [3]. The effect of the environment is to introduce decoherence which sets the time scale over which quantum computation is no longer possible. The presence of static imperfections, although not leading to any decoherence, may be also decremental for the implementation of any quantum computational task. A small inaccuracy in the coupling constants induces errors in gating or an unwanted time evolution in the case in which the Hamiltonian cannot be switched exactly to zero. 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 [3]. Previous investigations of the stability of quantum information processing in the presence of such effects mainly studied the fidelity of the quantum evolution as an indicator of the quality of the computation [3,4,5]. In particular, in Ref.[3] the fidelity was used to measure the stability of the quantum memory, that is of a state loaded on a quantum computer with imperfections. A more complete characterization of the stability of a quantum computation requires a deeper investigation of the state of the system. Fidelity is one of the ways to characterize it. In this Letter we discuss the behavior of entanglement on approaching the transition to quantum chaos. Entanglement is not only one of the most intriguing features predicted by the quantum theory but also a fundamental resource for quantum computation and communication [6]. Therefore studies of the stability of entanglement under decoherence and imperfection effects are, in our opinion, of interest.We model the quantum computer as a lattice of interacting spins (qubits) where, due to the unavoidable presence of imperfections, the spacing between the up and down states (external field) and the couplings between the qubits (exchange interactions) are both random. The entanglement properties in spin systems has recently attracted attention and we refer to [7] for a more detailed introduction. In this work we consider, for the first time, the effect of disorder in the couplings.We consider n qubits on a two-dimensional lattice, d...

Dynamics of entanglement in one-dimensional spin systems

“…In this work we consider exclusively ordered systems, and the role of imperfections is studied elsewhere [45]. In this appendix, in order to make the presentation self-contained, we review the results obtained in [36] where the out of equilibrium correlation functions are calculated exactly, expressed as an expansion of pfaffians.…”

Dynamical imperfections in quantum computers

“…Moreover, the role of static imperfections depends on the regime, chaotic or not, of the system under consideration [3]. The stability of a quantum computation in the presence of static imperfections has been already analyzed both in terms of fidelity [3,4,5] and entanglement [6].…”

“…1. The system (1) is characterized by two distinct dynamical regimes depending on the critical value J c ∼ δ/n: the Fermi Golden Rule (FGR) (J < J c ) and the ergodic regime (J > J c ) [3,6]. The FGR is characterized by a Lorentzian local density of states with width Γ FGR .…”

Generalized entanglement as a natural framework for exploring quantum chaos

“…In general, fidelity decay is connected to other signatures of quantum chaos, such as the shape of the local density of states [6] and eigenvector statistics [8]. Montangero et al [33] demonstrate the similarity of behavior between fidelity decay and the decay of (bi-partite) entanglement of an initial Bell pair. Here, it is the generalized purity which decays and, as shown, behaves qualitatively similarly to fidelity decay -complementing the results obtained for local purity and fidelity in [22].…”