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arXiv:2207.12189

Encoded State Transfer: Beyond the Uniform Chain

Alastair Kay

Published Jul 25, 2022• 0 citations
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Abstract

In a recent work (arXiv:2207.01954), we showed that a uniformly coupled chain could be symmetrically extended by engineered spin chains in such a way that we could choose part of the spectrum of the overall system. When combined with an encoding that avoids the uncontrolled eigenvalues, this resulted in the possibility of achieving a range of tasks such as perfect quantum state transfer. In this paper, we apply the same strategy to a much broader range of initial systems - arbitrary chains, and even coupled networks of spins - while providing guarantees on the existence of solutions.

DOI
10.48550/arxiv.2207.12189

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In a recent work (arXiv:2207.01954), we showed that a uniformly coupled chain could be symmetrically extended by engineered spin chains in such a way that we could choose part of the spectrum of the overall system. When combined with an encoding that avoids the uncontrolled eigenvalues, this resulted in the possibility of achieving a range of tasks such as perfect quantum state transfer. In this paper, we apply the same strategy to a much broader range of initial systems - arbitrary chains, and even coupled networks of spins - while providing guarantees on the existence of solutions.
The XX model with uniform couplings represents the most natural choice for quantum state transfer through spin chains. Given that it has long been established that single-qubit states cannot be transferred with perfect fidelity in this model, the notion of pretty good state transfer has been recently introduced as a relaxation of the constraints on fidelity. In this paper, we study the transfer of multi-qubit entangled and unentangled states through unmodulated spin chains, and we prove that it is possible to have pretty good state transfer of any multi-particle state. This significantly generalizes the previous results on single-qubit state transfer and opens the way to using uniformly coupled spin chains as short-distance quantum channels for the transfer of arbitrary states of any dimension. Our results could be tested with current technology.
The transmission of quantum information between different parts of a quantum computer is of fundamental importance. Spin chains have been proposed as quantum channels for transferring information. Different configurations for the spin couplings were proposed in order to optimize the transfer. As imperfections in the creation of these specific spin-coupling distributions can never be completely avoided, it is important to find out which systems are optimally suited for information transfer by assessing their robustness against imperfections or disturbances. We analyze different spin coupling distributions of spin chain channels designed for perfect quantum state transfer. In particular, we study the transfer of an initial state from one end of the chain to the other end. We quantify the robustness of different coupling distributions against perturbations and we relate it to the properties of the energy eigenstates and eigenvalues. We find that the localization properties of the systems play an important role for robust quantum state transfer.
In this paper we build on the ideas presented in previous works for perfectly transferring a quantum state between opposite ends of a spin chain using a fixed Hamiltonian. While all previous studies have concentrated on nearest-neighbor couplings, we demonstrate how to incorporate additional terms in the Hamiltonian by solving an inverse eigenvalue problem. We also explore issues relating to the choice of the eigenvalue spectrum of the Hamiltonian, such as the tolerance to errors and the rate of information transfer.
Perfect state transfer between qubits on a uniformly coupled network, with interactions specified by a graph, has advantages over an engineered chain, such as much faster transfer times (independent of the distance between the input and output vertices). This is achieved by many couplings working in parallel. The trade-offs seem to be the need for increasing connectivity between qubits, and a large number of vertices in the graph. The size of existing graph constructions scale exponentially in the transfer distance, making these schemes impractical over anything but the shortest distances. This prompts the question of "How low can you go?" for the size of the graph achieving a particular transfer distance. In this paper, we present reductions in the vertex count required, although the overall scaling with transfer distance remains exponential. We also tighten existing bounds on the required degree of the vertices of the graph.
We demonstrate that perfect state transfer can be achieved using an engineered spin chain and clean local end-chain operations, without requiring the initialization of the state of the medium nor fine-tuning of control pulses. This considerably relaxes the prerequisites for obtaining reliable transfer of quantum information across interacting-spin systems. Moreover, it allows us to shed light on the interplay among purity, entanglement, and operations on a class of many-body systems potentially useful for quantum information processing tasks.
We present an introductory overview of the use of spin chains as quantum wires, which has recently developed into a topic of lively interest. The principal motivation is in connecting quantum registers without resorting to optics. A spin chain is a permanently coupled 1D system of spins. When one places a quantum state on one end of it, the state will be dynamically transmitted to the other end with some efficiency if the spins are coupled by an exchange interaction. No external modulations or measurements on the body of the chain, except perhaps at the very ends, is required for this purpose. For the simplest (uniformly coupled) chain and the simplest encoding (single qubit encoding), however, dispersion reduces the quality of transfer. We present a variety of alternatives proposed by various groups to achieve perfect quantum state transfer through spin chains. We conclude with a brief discussion of the various directions in which the topic is developing.