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arXiv:1204.1363 · Physical Review A

Mixed-state quantum transport in correlated spin networks

Ashok Ajoy, Paola Cappellaro

Published Apr 5, 2012• 18 citations
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Abstract

Quantum spin networks can be used to transport information between separated registers in a quantum-information processor. To find a practical implementation, the strict requirements of ideal models for perfect state transfer need to be relaxed, allowing for complex coupling topologies and general initial states. Here we analyze transport in complex quantum spin networks in the maximally mixed state and derive explicit conditions that should be satisfied by propagators for perfect state transport. Using a description of the transport process as a quantum walk over the network, we show that it is necessary to phase-correlate the transport processes occurring along all the possible paths in the network. We provide a Hamiltonian that achieves this correlation and use it in a constructive method to derive engineered couplings for perfect transport in complicated network topologies.

DOI
10.1103/physreva.85.042305

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Quantum spin networks can be used to transport information between separated registers in a quantum-information processor. To find a practical implementation, the strict requirements of ideal models for perfect state transfer need to be relaxed, allowing for complex coupling topologies and general initial states. Here we analyze transport in complex quantum spin networks in the maximally mixed state and derive explicit conditions that should be satisfied by propagators for perfect state transport. Using a description of the transport process as a quantum walk over the network, we show that it is necessary to phase-correlate the transport processes occurring along all the possible paths in the network. We provide a Hamiltonian that achieves this correlation and use it in a constructive method to derive engineered couplings for perfect transport in complicated network topologies.
Spin chains have been proposed as wires to transport information between distributed registers in a quantum information processor. Unfortunately, the challenges in manufacturing linear chains with engineered couplings has hindered experimental implementations. Here we present strategies to achieve perfect quantum information transport in arbitrary spin networks. Our proposal is based on the weak coupling limit for pure state transport, where information is transferred between two end spins that are only weakly coupled to the rest of the network. This regime allows ignoring the complex, internal dynamics of the bulk network and relying on virtual transitions or on the coupling to a single bulk eigenmode. We further introduce control methods capable of tuning the transport process and achieve perfect fidelity with limited resources, involving only manipulation of the end qubits. These strategies could be thus applied not only to engineered systems with relaxed fabrication precision, but also to naturally occurring networks; specifically, we discuss the practical implementation of quantum state transfer between two separated nitrogen vacancy (NV) centers through a network of nitrogen substitutional impurities.
Spin chains have been proposed as quantum wires in many quantum-information processing architectures. Coherent transmission of quantum information in spin chains over short distances is enabled by their internal dynamics, which drives the transport of single-spin excitations in perfectly polarized chains. Given the practical challenge of preparing the chain in a pure state, we propose to use a chain that is initially in the maximally mixed state. We compare the transport properties of pure and mixed-state chains and find similarities that enable the experimental study of pure-state transfer via mixed-state chains. We also demonstrate protocols for the perfect transfer of quantum information in these chains. Remarkably, mixed-state chains allow the use of Hamiltonians that do not preserve the total number of single-spin excitations and are more readily obtainable from the naturally occurring magnetic dipolar interaction. We discuss experimental implementations using solid-state nuclear magnetic resonance and defect centers in diamond.
We propose a method for Hamiltonian engineering that requires no local control but only relies on collective qubit rotations and field gradients. The technique achieves a spatial modulation of the coupling strengths via a dynamical construction of a weighting function combined with a Bragg grating. As an example, we demonstrate how to generate the ideal Hamiltonian for perfect quantum information transport between two separated nodes of a large spin network. We engineer a spin chain with optimal couplings starting from a large spin network, such as one naturally occurring in crystals, while decoupling all unwanted interactions. For realistic experimental parameters, our method can be used to drive almost perfect quantum information transport at room temperature. The Hamiltonian engineering method can be made more robust under decoherence and coupling disorder by a novel apodization scheme. Thus, the method is quite general and can be used to engineer the Hamiltonian of many complex spin lattices with different topologies and interactions.
Perfect transfer of a quantum state through a one-dimensional chain is now well understood, allowing one not only to decide whether a fixed Hamiltonian achieves perfect transfer but to design a suitable one. We are particularly interested in being able to design, or understand the limitations imposed upon, Hamiltonians subject to various naturally arising constraints such as a limited coupling topology with low connectivity (specified by a graph) and type of interaction. In this paper, we characterize the necessary and sufficient conditions for transfer through a network and describe some natural consequences such as the impossibility of routing between many different recipients for a large class of Hamiltonians and the limitations on transfer rate. We also consider some of the trade-offs that arise in uniformly coupled networks (both Heisenberg and XXXXXX models) between transfer distance and the size of the network as a consequence of the derived conditions.
Quantum spin systems carry information in the form of quantum states, and the propagation of these quantum states is described by a quantum walk. In order to construct an operational quantum computer, the task of reliably transmitting quantum states from one part of a quantum computer to another must be accomplished. Quantum walks are known to be powerful tools in achieving this task. Pure states are quantum states represented by unit complex vectors. In this thesis, we use algebraic and combinatorial techniques to develop the theory of perfect state transfer on pure states in quantum walks, with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a spin network. We prove basic results about eigenvalue supports, periodicity, and strong cospectrality on pure states. We investigate a special type of strong cospectrality called m-strong cospectrality, and show that m-strongly cospectral pure states admit desirable quantum state transfer properties. Several characterisations of perfect state transfer are also given, one for general pure states, one for m-strongly pure states, and another one for real pure states. These characterisations give rise to algebraic and analytic properties of pure states admitting perfect state transfer. Moreover, we determine all complete graphs, complete bipartite graphs, cycles, and paths that admit perfect state transfer between m-strongly pure states. Constructions of infinite families of graphs admitting perfect state transfer are also given. We also characterise perfect state transfer between vertex states in joins and blowup graphs. We determine when the join operation preserves or induces perfect state transfer in the resulting graph. We also use blow-up graphs to construct new families of regular graphs with perfect state transfer. We use these two graph operations to demonstrate that graphs with perfect state transfer can be constructed from graphs that do not exhibit such a property. Finally, we investigate perfect state transfer on s-pair states – pure states that represent entanglement between two vertices. We establish combinatorial and spectral properties of s-pair states with perfect state transfer, and characterise its existence in complete graphs, complete bipartite graphs, paths and distance-regular graphs. Further, we construct infinite families of graphs with s-pair state transfer.
We show that a perfect state transfer of an arbitrary unknown two-qubit state can be achieved via a discrete-time quantum walk with various settings of coin flips and extend this method to the distribution of an arbitrary unknown multiqubit entangled state between every pair of sites in the multidimensional network. Furthermore, we study the routing of quantum information on this network in a quantum-walk architecture, which can be used as quantum information processors to communicate between separated qubits.