Research ArticlePHYSICS

Simulating complex quantum networks with time crystals

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Science Advances  16 Oct 2020:
Vol. 6, no. 42, eaay8892
DOI: 10.1126/sciadv.aay8892
  • Fig. 1 Obtaining the associated graph of a 2T-DTC.

    (A) On the left, diagram of the 2T-DTC dynamics with no rotation error, ϵ. The initial state (∣Ψ(0)〉 = ∣ ↑ ↑ ↑ … ↑ 〉), represented with the green arrows pointing up, is recovered after two periods of the driving protocol. From the first period, we obtain the unitary, U(T), that will be used as the Floquet operator, F̂ϵ=0, to derive the effective Hamiltonian, Ĥϵ=0,Teff. On the right, fidelity of evolving state of an n = 8 sites 2T-DTC against its initial state, F(t) = ∣〈Ψ(0)∣ Ψ(t)〉∣2, showing the 2T periodicity of the dynamics. (B) The effective Hamiltonian, Ĥϵ=0,Teff, represented as a tight-binding matrix. ℰi and Kij are the energies of the ∣i〉 configuration (see the main text for full description) and transition energy between configurations ∣i〉 and ∣j〉, respectively. The right panel shows the entries of the effective Hamiltonian matrix, where only the diagonal and antidiagonal entries are nonzero. (C) After applying the percolation rule to the effective Hamiltonian, we obtain an adjacency matrix, which is, in turn, represented as a graph with the nodes being each of the 2n configuration basis set of the Hilbert space. In the right, for no rotation error, the crystal order of the 2T-DTC can be observed as 2n−1 decoupled dimers. Here, the elements of each dimer are the configurations ∣i〉 and ∣2n − 1 − i〉, which are related by a global π rotation along the x axis. All the nodes with the same color have the same number of domain walls with the color map gradient going from dark blue (0 domain walls) to dark red (n − 1 domain walls).

  • Fig. 2 Melting of a 2T-DTC using Ĥϵ,Teff.

    (A) Graph representation obtained from Ĥϵ,Teff with n = 8 sites and ϵ = 0.012. Nodes are color-coded according to the number of domain walls of the corresponding configuration (see color map). For moderate levels of error, the nodes attach to each other according to their number of domain walls. This can also be observed in the effective Hamiltonian matrix of (B) with the basis ordered in increasing number of domain walls and delimited by the colored squares. As the nodes start to cluster due to the presence of error, some nonzero off-diagonal terms appear in the center of the matrix. For this level of error, some dimers survive [see the top right corner of (A)], serving as good indication of the robustness of the system and meaning that the crystal order is still present. In (C), the degree distributions of the corresponding graph of (A) with different system sizes (n = 8–12) averaged over 100 realizations of disorder are shown in both linear and logarithmic scale (see the inset), which display heavy-tailed distributions. The distributions are fitted with a power law curve (solid lines in the inset), indicating the presence of large degree hub nodes. (D) Graph representation obtained from Ĥϵ,Teff with n = 8 sites and ϵ = 0.1 and (E) its associated effective Hamiltonian matrix. As we increase the error, the system forms a single cluster. This can be seen from the appearance of many new off-diagonal entries in the Hamiltonian matrix as well as the presence of a giant component in the graph. In this scenario, the time crystal has melted completely and no crystal order is left (no presence of isolated dimers). The degree distributions, shown in (F), also indicate that the heavy-tailed nature is destroyed and approximate to a normal distribution.

  • Fig. 3 Melting of a 2T-DTC using Ĥϵ,2Teff.

    (A) Graph representation obtained from Ĥϵ,2Teff with n = 8 sites and a rotation error of ϵ = 0.012 with the nodes corresponding to the configuration basis set of the Hilbert space color-coded according to the number of domain walls of the corresponding state (see color map). For no or moderate levels of error, the nodes are decoupled, as shown in the bottom part of the graph, or forming clusters of small size. In (B), we present the effective Hamiltonian Ĥϵ,2Teff that presents mainly diagonal terms. For moderate levels of error, the nodes form small clusters. Again, they attach to each other according to their number of domain walls. In (C), the degree distributions of the corresponding graph of (A) with different system sizes (n = 8–12) averaged over 100 realizations of disorder are shown in both linear and logarithmic scale (see the inset), fitted with a Poisson distribution. This indicates that the corresponding graph is a random graph with low connectivity. (D) Graph representation obtained from Ĥϵ,2Teff with n = 8 and a rotation error of ϵ = 0.1. As we increase the error, the system forms a larger, highly connected cluster. (E) This can be seen from the increasing magnitude of the off-diagonal entries in the effective Hamiltonian matrix as well as the presence of a giant component in the graph. In this scenario, the time crystal has melted completely. (F) The degree distributions are well approximated to a Poisson distribution in the lower degree region, which is the characteristic of highly connected random networks.

  • Fig. 4 The average degree of the graph obtained from Ĥϵ,Teff with n = 8 sites, plotted against the error ϵ.

    The nodes are distinguished by their number of domain walls and averaged individually for each value of domain walls, as well as averaging over 100 realizations of disorder. Notice that some of the curves exhibit a local maximum approximately between 0.01 < ϵ < 0.03. This is exactly the range in which we observe the scale-free behavior emerging in our system. Comparing each curve also confirms that the states with a lower number of domain walls are more robust against the error in a sense that they have lower degrees.

Supplementary Materials

  • Supplementary Materials

    Simulating complex quantum networks with time crystals

    M. P. Estarellas, T. Osada, V. M. Bastidas, B. Renoust, K. Sanaka, W. J. Munro, K. Nemoto

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    The PDF file includes:

    • I. A roadmap through the supplementary materials
    • II. Level statistics of the quasienergies and many-body localization
    • III. Power spectrum of time series: Stability of the time-crystalline phase
    • IV. Semiclassical description of a DTC: a stability analysis of the fixed points in the absence of error ϵ = 0
    • V. Quantum walks in the configurations space: Experimental protocol to simulate complex quantum networks
    • Figs. S1 to S4
    • References

    Other Supplementary Material for this manuscript includes the following:

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