Perfect quantum state transfer on diamond fractal graphs
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Perfect quantum state transfer on diamond fractal graphs Maxim Derevyagin1 · Gerald V. Dunne1,2 · Gamal Mograby1 · Alexander Teplyaev1,2 Received: 22 September 2019 / Accepted: 17 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In the quest for designing novel protocols for quantum information and quantum computation, an important goal is to achieve perfect quantum state transfer for systems beyond the well-known one- dimensional cases, such as 1D spin chains. We use methods from fractal analysis and probability to find a new class of quantum spin chains on fractal-like graphs (known as diamond fractals) which support perfect quantum state transfer and which have a wide range of different Hausdorff and spectral dimensions. The resulting systems are spin networks combining Dyson hierarchical model structure with transverse permutation symmetries of varying order. Keywords Quantum state transfer · Quantum computer · Spin chain · Hierarchical graphs · Diamond fractal · Hamiltonians with engineered couplings · Quantum channels
1 Introduction The study of state transfer was initiated by Bose [1,2], who considered a 1D chain of N qubits coupled by the time-independent Hamiltonian. The main idea is to transport a quantum state from one end of the chain to the other. The transport of the quantum state from one location to another is called perfect if it is realized with probability 1, that
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Alexander Teplyaev [email protected] Maxim Derevyagin [email protected] Gerald V. Dunne [email protected] Gamal Mograby [email protected]
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Mathematics Department, University of Connecticut, Storrs, CT 06269, USA
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Physics Department, University of Connecticut, Storrs, CT 06269, USA 0123456789().: V,-vol
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Fig. 1 Most standard diamond hierarchical fractal graphs, levels 3 and 4, with the similarity dimension dim = 2, [35, Section 7] and [20–30]
is, without dissipation. In addition to its fundamental interest, this means that perfect quantum state transfer also has potential applications to the design of sub-protocols for quantum information and quantum computation [3–5]. A number of one- dimensional cases, when perfect transmission can be achieved, have been found in some X X chains with inhomogeneous couplings, see [2,3,6–16, and references therein]. These models have the advantage that the perfect transfer can be done without the need for active control. Recently, there has been active interest to generalize these results to graphs with potentials and to graphs that are not one dimension [5,17–19]. These works illustrate the fact that perfect state transfer is a rare phenomenon, for which the construction of explicit examples remains rather non-trivial. The main result of our paper is to show that perfect quantum state transfer is possible on the large and diverse class of fractal-type diamond graphs, which have different geometrical properties including a wide range of dimensions. These graphs have provide
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