Discrete-time quantum walks can be regarded as quantum dynamical simulators since they can simulate spatially discretized Schrödinger, massive Dirac, and Klein-Gordon equations. Here, two different types of Fibonacci discrete-time quantum walks are studied analytically. The first is the Fibonacci coin sequence with a generalized Hadamard coin and demonstrates six-step periodic dynamics. The other model is assumed to have three-or six-step periodic dynamics with the Fibonacci sequence. We analytically show that these models have ballistic transportation properties and continuous limits identical to those of the massless Dirac equation with coin basis change. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and are important in many fields, from fundamental quantum physics [15,[21][22][23]] to quantum algorithm [24,25] and condensed matter physics [26][27][28][29][30]. Previously, it has been shown that several DTQWs on a line admit a continuous limit identical to the propagation equations of a massive Dirac fermion [31][32][33][34][35] and those of massless Dirac fermion equations [21,32]. Furthermore, the relationship between DTQWs and artificial electric and gravitational fields has been shown [21,36]. Thus, DTQWs can be regarded as quantum dynamical simulators [37]. Additionally, it is well known that the classical random walk leads to a diffusive behavior characterized by the time evolution of the standard deviation, with σ(t) ∼ t 1/2 , while the standard DTQW leads to ballistic behavior, as σ(t) ∼ t. Further, the standard DTQW can be considerably enriched by generalizing the quantum coin operator and arranging it along different sequences. It has already been shown that quasi-periodic coin sequences induced by the Fibonacci sequence lead to sub-ballistic behavior, whereas random sequences lead to diffusive spreading [38]. Here, we consider two different Fibonacci DTQWs with periodic coin sequences. The first model (FDTQW-I) considers a time-dependent quantum coin following the Fibonacci sequence, while the second model (FDTQW-II) considers a modified version of the unitary operator first defined in Ref. [38], where the Fibonacci sequence is applied to the step operator. We show numerically and analytically that the continuous limit of these models reduces to a massless Dirac equation in (1 + 1) dimensions.Let us consider the two dimensional spin state Ψ m,j ∈ C 2 , spanned by the orthonormal basis (b u , b d ), and defined by its discrete one dimensional position m ∈ Z and discrete time j ∈ N 0 . The standard DTQW's time evolution is given by the application of the quantum coin op-d m,j , followed by the chiral-dependent translation operatorT , which is defined asHere, we introduce the simplest quantum coin, the generalized Hadamard coin, which is expressed aŝ C(θ) = cos(θ) sin(θ) sin(θ) − cos(θ) ,where θ ∈ [0, 2π]. The one-step discrete time evolution is then given by
On finalizing the publication process of the original article, we had the following mistakes and make the corrections.
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