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arXiv:1408.3765 · Quantum Information Processing

Information transfer fidelity in spin networks and ring-based quantum routers

EDMOND A. JONCKHEERE, Frank C. Langbein, S. G. Schirmer

Published Oct 8, 2015• 16 citations
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Abstract

Spin networks are endowed with an information transfer fidelity (ITF), which defines an absolute upper bound on the probability of transmission of an excitation from one spin to another. The ITF is easily computable, but the bound can be reached asymptotically in time only under certain conditions. General conditions for attainability of the bound are established, and the process of achieving the maximum transfer probability is given a dynamical model, the translation on the torus. The time to reach the maximum probability is estimated using the simultaneous Diophantine approximation, implemented using a variant of the Lenstra–Lenstra–Lovász (LLL) algorithm. For a ring with uniform couplings, the network can be made into a metric space by defining a distance (satisfying the triangle inequality) that quantifies the lack of transmission fidelity between two nodes. It is shown that transfer fidelities and transfer times can be improved significantly by means of simple controls taking the form of nondynamic, spatially localized bias fields, opening up the possibility for intelligent design of spin networks and dynamic routing of information encoded in them, while being more flexible than engineering fixed couplings to favor some transfers, and less demanding than control schemes requiring fast dynamic controls.

DOI
10.1007/s11128-015-1136-4

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Spin networks are endowed with an information transfer fidelity (ITF), which defines an absolute upper bound on the probability of transmission of an excitation from one spin to another. The ITF is easily computable, but the bound can be reached asymptotically in time only under certain conditions. General conditions for attainability of the bound are established, and the process of achieving the maximum transfer probability is given a dynamical model, the translation on the torus. The time to reach the maximum probability is estimated using the simultaneous Diophantine approximation, implemented using a variant of the Lenstra–Lenstra–Lovász (LLL) algorithm. For a ring with uniform couplings, the network can be made into a metric space by defining a distance (satisfying the triangle inequality) that quantifies the lack of transmission fidelity between two nodes. It is shown that transfer fidelities and transfer times can be improved significantly by means of simple controls taking the form of nondynamic, spatially localized bias fields, opening up the possibility for intelligent design of spin networks and dynamic routing of information encoded in them, while being more flexible than engineering fixed couplings to favor some transfers, and less demanding than control schemes requiring fast dynamic controls.
Propagation of information encoded in spin degrees of freedom through networks of coupled spins enables important applications in spintronics and quantum information processing. We study control of information propagation in networks of spin-1/2 particles with uniform nearest neighbor couplings forming a ring with a single excitation in the network as simple prototype of a router for spin-based information. Specifically optimizing spatially distributed potentials, which remain constant during information transfer, simplifies the implementation of the routing scheme. However, the limited degrees of freedom makes finding a control that maximizes the transfer probability in a short time difficult. We show that the structure of the eigenvalues and eigenvectors must fulfill a specific condition to be able to maximize the transfer fidelity, and demonstrate that a specific choice among the many potential structures that fulfill this condition significantly improves the solutions found by optimal control.
This data set contains energy landscape controllers for information propagation in networks of spin-1/2 particles (qubits) with a ring topology and uniform nearest neighbor couplings of XXZ type. The controllers are optimized to maximize the fidelity of excitation transfer between nodes of ring in the single excitation subspace. The network can be regarded as simple prototype of a router for spin-based information. The dataset contains controllers for rings ranging in size from 3 to 20 spins with excitation transfer from node 1 to nodes 2, 3, ... ceil(N/2) considered. Controllers optimized for both instantaneous readout and time-windowed (finite bandwidth) readout are included.
Control laws for selective transfer of information encoded in excitations of a quantum network, based on shaping the energy landscape using time-invariant, spatially-varying bias fields, can be successfully designed by using numerical optimization. Such control laws, already departing from classicality by replacing closed-loop asymptotic stability with alternative notions of localization, have the intriguing property that for all practical purposes they achieve the upper bound on the fidelity, yet the (logarithmic) sensitivity of the fidelity to such structured perturbation as spin coupling errors and bias field leakages is nearly vanishing. Here, these differential sensitivity results are extended to large structured variations using μ-design tools to reveal a crossover region in the space of controllers where objectives usually thought to be conflicting are actually concordant.
This data set contains energy landscape controllers for information propagation in networks of spin-1/2 particles (qubits) with a ring topology and uniform nearest neighbor couplings of XXZ type. The controllers are optimized to maximize the fidelity of excitation transfer between nodes of ring in the single excitation subspace. The network can be regarded as simple prototype of a router for spin-based information. The dataset contains controllers for rings ranging in size from 3 to 20 spins with excitation transfer from node 1 to nodes 2, 3, ... ceil(N/2) considered. Controllers optimized for both instantaneous readout and time-windowed (finite bandwidth) readout are included.
Information encoded in networks of stationary, interacting spin-1/2 particles is central for many applications ranging from quantum spintronics to quantum information processing. Without control, however, information transfer through such networks is generally inefficient. Currently available control methods to maximize the transfer fidelities and speeds mainly rely on dynamic control using time-varying fields and often assume instantaneous readout. We present an alternative approach to achieving efficient, high-fidelity transfer of excitations by shaping the energy landscape via the design of time-invariant feedback control laws without recourse to dynamic control. Both instantaneous readout and the more realistic case of finite readout windows are considered. The technique can also be used to freeze information by designing energy landscapes that achieve Anderson localization. Perfect state or superoptimal transfer and localization are enabled by conditions on the eigenstructure of the system and signature properties for the eigenvectors. Given the eigenstructure enabled by superoptimality, it is shown that feedback controllers that achieve perfect state transfer are, surprisingly, also the most robust with regard to uncertainties in the system and control parameters.
Efficient routing of information between end-nodes is a key enabler for secure quantum networks and quantum secret key sharing, which rely on creating and sustaining entangled states over time. However, such pairwise entanglements degrade due to channel loss and the storage of the entangled photons at the network nodes. The state of entanglement in turn impacts fidelity, a metric which quantifies the degree of similarity between a pair of quantum states. In this paper, we propose a routing solution that satisfies threshold fidelity requirements imposed by a receiver on the quantum information received from multiple transmitter nodes. Our solution selects intermediate repeaters from a pool of such nodes within the network to maximize the sum-rate of quantum information transfer. To this extent, we first provide expressions for the fidelity loss between adjacent nodes as well as for the end-to-end quantum data rate. Then, we propose a novel two-stage routing solution that (i) identifies the k-shortest paths for each transmitter using fidelity as cost metric and (ii) (heuristically) assigns a path for each transmitter depending on whether the repeater nodes have a single or multiple available memory units. Simulation results demonstrate that our proposed fidelity-based routing solution satisfies a wide range of fidelity requirements [0.6-0.79] while maximizing the quantum information transfer rate, outperforming the existing distance- and hop-based routing approaches.