Hardware Efficient Quantum Simulation of Non-Abelian Gauge Theories with Qudits on Rydberg Platforms

Non-Abelian gauge theories underlie our understanding of fundamental forces in nature, and developing tailored quantum hardware and algorithms to simulate them is an outstanding challenge in the rapidly evolving field of quantum simulation. Here we take an approach where gauge fields, discretized in spacetime, are represented by qudits and are time evolved in Trotter steps with multiqudit quantum gates. This maps naturally and hardware efficiently to an architecture based on Rydberg tweezer arrays, where long-lived internal atomic states represent qudits, and the required quantum gates are performed as holonomic operations supported by a Rydberg blockade mechanism. We illustrate our proposal for a minimal digitization of SU(2) gauge fields, demonstrating a significant reduction in circuit depth and gate errors in comparison to a traditional qubit-based approach, which puts simulations of non-Abelian gauge theories within reach of NISQ devices.

Figure 1

Gauge field dynamics on qudit vs qubit quantum simulator: (a) Our proposal employs Rydberg atoms trapped in optical tweezers, arranged on the links $\ell$ of a hypercubic lattice. Each atom encodes a qudit using $d$ internal levels, where single-qudit gates are realized holonomically. To implement the entangling two-qudit gate ${\mathrm{\Theta }}_{\ell |{\ell }^{\prime }}$ we first bring pairs of atoms within the Rydberg blockade radius $R_{\mathrm{b}}$. (b) First order decomposition of a Trotter step, including the four-qudit plaquette interaction, into the native atomic gates ${\mathcal{U}}_{\ell }^{(E/B)}$ and ${\mathrm{\Theta }}_{\ell |{\ell }^{\prime }}$. (c) For comparison, we show a qubit-based circuit decomposition of ${\mathrm{\Theta }}_{\ell |{\ell }^{\prime }}$ for the $Q_8$ group, where the number of required atoms is increased by a factor ${\log }_{2}8=3$, leading to much lower gate fidelities, while our qudit approach enables a faithful simulation [see Figs. 2 and 2].

Figure 2

Qudit encoding and native gates: (a) Atomic level structure serving as a $Q_8$ register, where the $d=8$ group basis elements are encoded into different hyperfine manifolds. Single and two-qudit gates are performed holonomically using the auxiliary ground-state level $|p⟩$ and excited states $|e⟩$ and $|r⟩$, with corresponding decay rates ${\gamma }_{e/r}$. (b) Controlled-permutation gate $C_{\theta (g)}^{{\ell }^{\prime }\to \ell }(g)$ as a sequence of four single qudit gates in the blockade regime. Quantified by the maximally entangled state obtained after applying ${\mathrm{\Theta }}_{\ell |{\ell }^{\prime }}$ to $|{\mathrm{\Psi }}_0⟩$, implemented through the (c) qudit and (d) qubit protocols [see the circuit in Fig. 1] using the same experimental parameters (see main text), we obtain a state fidelity of 99.6% and 21.4%, respectively, demonstrating the clear advantage of a qudit-based decomposition.

Figure 3

Fidelity of the simulation: (a) Infidelity $1-\mathcal{F}$ of the digital simulation at final evolution time $t{\lambda }_B=1$ as a function of the Trotter step $\delta t{\lambda }_B$, where an exact second-order Trotter decomposition (crosses) results in the scaling $1-\mathcal{F}\sim (\delta t{)}^4$ (dashed-dotted lines). Including gate errors, the infidelity increases again at small $\delta t{\lambda }_B$. Correcting for atomic losses due to decoherence, we find that optimal step size $\delta t{\lambda }_B=1/3$ at the minimum of $1-\mathcal{F}\approx 6.7\%$ (red circle). (b) We measure the losses included in the simulated gates by a decay of the norm $⟨\psi |\psi ⟩$ of the time evolved state. As illustrated for the optimal and a very small Trotter step (symbols), the decay is consistent with a constant loss of $\approx 0.75\%$ per Trotter step (solid lines). (c) Trotterized quench dynamics of a non-Abelian $Q_8$ LGT on a single plaquette for ${\lambda }_E/{\lambda }_B=2.88$. The Trotter step corresponds to the “sweet spot” indicated in (a). Here, the observables are corrected by a multiplicative time-dependent factor due to the decay shown in (b).