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Quantum state transfer with energy potential

Yujia Shi

Published Jan 1, 2024• 0 citations
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Abstract

This dissertation investigates continuous-time quantum walks and asymptotic state transfer in weighted graphs, with a particular focus on the impact of starting and target node weights on transfer fidelity and readout time. Through analysis of the eigenvectors and eigenvalues of the Hamiltonian, we derive quantitative results applicable to generalized graph structures. Our findings focus on the relationship between a graph's maximum degree and the minimum weights required to achieve specific transfer fidelities. This research advances the understanding of quantum state transfer in spin networks, with potential applications in quantum communication and computing.--Author's abstract

DOI
10.17760/d20697856

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Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. Quantum walks, the quantum analogues of classical random walks, provide one potential method for resolving this challenge. \n \nIn this thesis, we use techniques from algebraic graph theory and number theory to analyze the mathematical model for continuous time quantum walks on graphs. For the continuous time quantum walk model, we define a transition operator, which is a function of a Hamiltonian. We focus on the cases where the adjacency matrix or the Laplacian of a graph act as the Hamiltionian. We mainly consider quantum walks on paths as a model for spin chains, which are the underlying basis of a quantum communication protocol. \n \nFor communication to be efficient, we desire states to be transferred with high fidelity, a measure of the amount of similarity between the transmitted state and the received state. At the maximum fidelity of 1, we say we have achieved perfect state transfer. Examples of perfect state transfer are relatively rare, so the concept of pretty good state transfer was introduced as a natural relaxation, which exists if fidelities arbitrarily close to 1 are obtained. \n \nOur first main result is to characterize pretty good state transfer on paths. Previously, pretty good state transfer on paths was considered mainly for the end vertices, though results for both models indicated that if there was pretty good state transfer between the end vertices, then there was pretty good state transfer between internal vertices equidistant from each end. We complete the characterization by demonstrating, for the adjacency matrix model, a family of paths where pretty good state transfer exists between internal vertices but not between end vertices, and verifying that no other example exists. For the Laplacian model, we show that there are no paths with pretty good state transfer between internal vertices but not between the end vertices. \n \nOur second main result considers initial states involving multiple vertices. Under the adjacency matrix model, we provide necessary and sufficient conditions for pretty good state transfer in a particular family of paths in terms of the eigenvalue support of the initial state. We also discuss recent results on fractional revival, which is another form of multiple qubit state transfer.

•Aug 23, 2019•3 citations
This dissertation investigates continuous-time quantum walks and asymptotic state transfer in weighted graphs, with a particular focus on the impact of starting and target node weights on transfer fidelity and readout time. Through analysis of the eigenvectors and eigenvalues of the Hamiltonian, we derive quantitative results applicable to generalized graph structures. Our findings focus on the relationship between a graph's maximum degree and the minimum weights required to achieve specific transfer fidelities. This research advances the understanding of quantum state transfer in spin networks, with potential applications in quantum communication and computing.--Author's abstract
Quantum walks generated by the adjacency matrix or the Laplacian are known to exhibit low transfer fidelity on general graphs. In this paper, we study continuous-time quantum walks governed by the generalized Laplacian operator L_k = A+kD, where A is the adjacency matrix, D is the degree matrix, and k is a real-valued parameter. Recent work of Duda, McLaughlin, and Wong showed that in the single-excitation Heisenberg (XYZ) spin model, one can realize walks generated by this family of operators on signed weighted graphs. Motivated by earlier studies on vertex-weighted graphs, we demonstrate that for certain graphs, tuning the parameter k can significantly enhance the fidelity of state transfer between endpoints.
We quantify the effect of weighted loops at the source and target nodes of a graph on the strength of quantum state transfer between these vertices. We give lower bounds on loop weights that guarantee strong transfer fidelity that works for any graph where this protocol is feasible. By considering local spectral symmetry, we show that the required weight size depends only on the maximum degree of the graph and, in some less favorable cases, the distance between vertices. Additionally, we explore the duration for which transfer strength remains above a specified threshold.
We introduce and study peak state transfer, a notion of high state transfer in qubit networks modeled by continuous-time quantum walks. Unlike perfect or pretty good state transfer, peak state transfer does not require fidelity arbitrarily close to 1, but crucially allows for an explicit determination of the time at which transfer occurs. We provide a spectral characterization of peak state transfer, which allows us to find many examples of peak state transfer, and we also establish tight lower bounds on fidelity and success probability. As a central example, we construct a family of weighted path graphs that admit peak state transfer over arbitrarily long distances with transfer probability approaching π/4≈0.78π/4 \approx 0.78π/4≈0.78. These graphs offer exponentially improved sensitivity over known perfect state transfer examples such as the weighted paths related to hypercubes, making them practical candidates for efficient quantum wires.
Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. Quantum walks, the quantum analogues of classical random walks, provide one potential method for resolving this challenge. \n \nIn this thesis, we use techniques from algebraic graph theory and number theory to analyze the mathematical model for continuous time quantum walks on graphs. For the continuous time quantum walk model, we define a transition operator, which is a function of a Hamiltonian. We focus on the cases where the adjacency matrix or the Laplacian of a graph act as the Hamiltionian. We mainly consider quantum walks on paths as a model for spin chains, which are the underlying basis of a quantum communication protocol. \n \nFor communication to be efficient, we desire states to be transferred with high fidelity, a measure of the amount of similarity between the transmitted state and the received state. At the maximum fidelity of 1, we say we have achieved perfect state transfer. Examples of perfect state transfer are relatively rare, so the concept of pretty good state transfer was introduced as a natural relaxation, which exists if fidelities arbitrarily close to 1 are obtained. \n \nOur first main result is to characterize pretty good state transfer on paths. Previously, pretty good state transfer on paths was considered mainly for the end vertices, though results for both models indicated that if there was pretty good state transfer between the end vertices, then there was pretty good state transfer between internal vertices equidistant from each end. We complete the characterization by demonstrating, for the adjacency matrix model, a family of paths where pretty good state transfer exists between internal vertices but not between end vertices, and verifying that no other example exists. For the Laplacian model, we show that there are no paths with pretty good state transfer between internal vertices but not between the end vertices. \n \nOur second main result considers initial states involving multiple vertices. Under the adjacency matrix model, we provide necessary and sufficient conditions for pretty good state transfer in a particular family of paths in terms of the eigenvalue support of the initial state. We also discuss recent results on fractional revival, which is another form of multiple qubit state transfer.