Sachdev-Ye-Kitaev Circuits for Braiding and Charging Majorana Zero Modes

The Sachdev-Ye-Kitaev (SYK) model is an all-to-all interacting Majorana fermion model for many-body quantum chaos and the holographic correspondence. Here we construct fermionic all-to-all Floquet quantum circuits of random four-body gates designed to capture key features of SYK dynamics. Our circuits can be built using local ingredients in Majorana devices, namely, charging-mediated interactions and braiding Majorana zero modes. This offers an analog-digital route to SYK quantum simulations that reconciles all-to-all interactions with the topological protection of Majorana zero modes, a key feature missing in existing proposals for analog SYK simulation. We also describe how dynamical, including out-of-time-ordered, correlation functions can be measured in such analog-digital implementations by employing foreseen capabilities in Majorana devices.

Figure 1

(a) Our model as an all-to-all quantum circuit with four-body gates (maroon) acting on Majorana fermion lines. Shown is a Floquet operator ${\mathcal{F}}_3$ for $k=9$. (b) Implementing the circuit layers $j-1$ (bottom) and $j$ (top) via superconducting islands (gray rectangles). Each island hosts four Majorana zero modes (blue disks, ${\gamma }_{{\pi }_{j-1}(4i)}$ and ${\gamma }_{{\pi }_j(4i)}$ in black). Charging effects, controlled via coupling (cyan) to a bulk superconductor [46, 47, 48], induce the four-body gate $u_i$ [Eq. (3)] for island $i$. Braiding (blue lines), e.g., via ancilla Majoranas [47, 49] generates the permuted indices ${\pi }_j(i)$.

Figure 2

Protocol for measuring Majorana monomial correlations with respect to the fermionic vacuum $|\mathrm{\Omega }⟩=|{\mathrm{\Omega }}_{\mathrm{sys}}⟩\otimes |{\mathrm{\Omega }}_{\overline{j}}⟩\otimes |{\mathrm{\Omega }}_{\overline{k}}⟩$. Here $|{\mathrm{\Omega }}_{\mathrm{sys}}⟩$ and $|{\mathrm{\Omega }}_{\overline{j},\overline{k}}⟩$ are vacua of system and ancilla fermions, respectively. (a) For $C_{ab}$ in Eq. (4), we apply the generalized exchange $W_{b,2\overline{j}}$ (cyan), $m$ Floquet operators ${\mathcal{F}}_n$ (beige), and $\mathrm{\Lambda }({\mathrm{\Upsilon }}_a{\gamma }_{2\overline{k}})$ [Eq. (9), pink], on the initial state. Each ${\mathcal{F}}_n$ consists of $2n-1$ operators $f_j$ (light blue) for the $j$th circuit layer; cf. Eq. (2). (b) For OTOCs, we generate ${\mathcal{F}}_n^{†}$ for backward time evolution via double braids (bottom).

Figure 3

Numerically evaluated $q$-body correlations $C_q(m)={\sum }_{a|q_a=q}\mathrm{Re}{C}_{a,a}(m)/(\genfrac{}{}{0ex}{}{k}{q})$, rescaled by the Hilbert space dimension $M$ and averaged over up to $2^{16}$ realizations of $J_{j,i}$ taken from a box distribution $J_{j,i}\in [-20,20]$. Statistical error bars are smaller than the linewidth. Panels (a)–(b) show $q=1$ and panels (c)–(d) show $q=3$ for $k=18$ (cross-parity correlations set by absence of $T_{+}$) and $k=19$ Majoranas (cross-parity correlations set by $T_{+}^2=-1$). The dashed lines show plateau estimates based on random eigenstates subject only to symmetry constraints—the same estimates hold for the plateaus in Hamiltonian SYK evolution [27].

Figure 4

Numerically evaluated single-Majorana OTOC $F={\sum }_{i\ne i^{\prime }}\mathrm{Re}{F}_{i{i}^{\prime }}/[k(k-1)]$ averaged over up to $2^{11}$ realizations of box-distributed $J_{j,i}\in [-0.01,0.01]$. The number $k$ of Majoranas is shown by the colors. All curves are for $n=35$; the OTOCs do not appreciably change upon increasing $n$ further. Solid (dashed) lines show $\beta J_{\mathrm{eff}}=3$ ($\beta J_{\mathrm{eff}}=20$); the statistical error bars are smaller than the linewidth.