Scalable topological quantum computing based on Sine-Cosine chain models

Abstract

This work proposes a scalable framework for topological quantum computing using Matryoshka-type Sine-Cosine chains. These chains support high-dimensional qudit encoding within single systems, reducing the physical resource overhead compared to conventional qubit arrays. We describe how these chains can be used in Y-junction braiding protocols for gate operations and in extended memory architectures capable of storing multiple qubits simultaneously. Fidelity analysis shows partial topological protection against disorder, suggesting this approach is a possible pathway toward low-overhead quantum hardware.

Figures (20)

• Figure 1: Lattice representation of the first orders of $P$ in the Matryoshka model. The figure illustrates how the unit cell evolves for $P = 0, 1, 2$, becoming progressively larger as we increase $P$, with hopping terms given by sine and cosine functions of angles $\theta_j$.
• Figure 2: Representation of the resulting structures $H_A$ and $H_B$ after applying the squaring operation to the parent Hamiltonian. The diagram illustrates the separation into two decoupled sublattices, with effective hoppings dependent on the parameters $\theta_1$ and $\theta_2$.
• Figure 3: Band structure of a) $P=0$, b) $P=1$ and c) $P=2$ Matryoshka chain. To calculate the base lattice ($P=0$) energies, $\sin{\theta_1}=0.588$ was used. To obtain the next lattice's band structure $t^{(0)}=0.9/\sqrt{2}$ and $t^{(1)}=0.8/\sqrt{2}$ were used. The yellow line represents the edge states relative to the weak links of $P=0$, which double after each squaring. In the case that $P=0$ does not present weak links, none of the yellow lines can be seen in the spectra. This same logic applies to the degenerate blue states in $P=1$, which unfold and form two in $P=2$. The green line corresponds to the weak links for $P=2$.
• Figure 4: In this figure, we reproduce the transferring protocol of Boross2019. (a) Base system. (b) Adiabatic sweep of the couplings $v(t)$ and $w(t)$. (c) Transfer protocol illustration. The yellow squares indicate finite state amplitudes. When $v = 0$ we have $\ket{\Psi(t_i)} = \ket{0}$, and $\ket{\Psi (t_f)} = -\ket{2}$ if $w = 0$. For the case where $v = w$, the state will be equally distributed between $\ket{0}$ and $\ket{2}$.
• Figure 5: Consecutive square root of the initial Hamiltonian in the quantum transfer process. The highlighted defect state ($H_1$) corresponds to the parent Hamiltonian's full chain ($H_0$), where single-site, dimer, and 4-site cluster configurations are interchangeable under adiabatic evolution. The highlight shows the defect part in each model and, curiously, the defect in the square-root chain is the previous full chain (note that the ordering of the single site, the dimer and 4‑site cluster can be exchanged under an adiabatic evolution). $H_{(1)}(t=0)$ corresponds to $\theta_{1,i} = \pi/2$, $\theta_{2,i} = 0$, and $\theta_{3,i} = \lambda$. In the first chain, we have one angle, in the second , we have three, and in the third, we have seven (see Appendix A for details). The amplitudes of the original 3-site SSH chain eigenstates will be present in the blue sites of all chains, except for eigenstates that result from the zero energy eigenstates of $H_{(1)}$, $H_{(2)}$, etc.