Encoding computationally hard problems in triangular Rydberg atom arrays

TL;DR

The paper demonstrates that triangular-lattice subgraphs (TLSGs) provide a more robust encoding of NP-hard MIS/MWIS problems for Rydberg-atom quantum optimization than King’s subgraphs. By developing an automated gadget-search framework and a crossing-lattice encoding, the authors achieve an $O(8n^2)$ TLSG overhead with a 12-vertex crossing gadget and demonstrate a two-orders-of-magnitude reduction in constraint violations compared to KSGs in numerical simulations. The approach substantially mitigates the need for post-processing and is implemented in open-source Julia packages, enabling broader application to other NP-hard problems and platforms. The work offers a practical path toward scalable, interpretable quantum optimization using TLSG embeddings on Rydberg platforms.

Abstract

Rydberg atom arrays are a promising platform for quantum optimization, encoding computationally hard problems by reducing them to independent set problems with unit-disk graph topology. In Nguyen et al., PRX Quantum 4, 010316 (2023), a systematic and efficient strategy was introduced to encode multiple problems into a special unit-disk graph: the King's subgraph. However, King's subgraphs are not the optimal choice in two dimensions. Due to the power-law decay of Rydberg interaction strengths, the approximation to unit-disk graphs in real devices is poor, necessitating post-processing that lacks physical interpretability. In this work, we develop an encoding scheme that can universally encode computationally hard problems on triangular lattices, based on our innovative automated gadget search strategy. Numerical simulations demonstrate that quantum optimization on triangular lattices reduces independence-constraint violations by approximately two orders of magnitude compared to King's subgraphs, substantially alleviating the need for post-processing in experiments.

Figures (9)

• Figure 1: Procedure for encoding a MWIS/MIS problem on an arbitrary graph into MWIS on a TLSG, enabling optimization using programmable Rydberg atom arrays. (a) Example problem: finding MIS of $K_{2,3}$ graph. (b) Extend each vertex to a copy gadget and form a crossing lattice; numbers label equivalent copies of the same source vertex, and red edges indicate the original-graph connections. (c) Replace the substructures that violate the unit-disk constraint by logically equivalent gadgets. The gadgets and their composition rule are detailed in Fig. \ref{['fig:show_gadget']}. (d) Final encoding of $K_{2,3}$; the solution to the original problem is inferred from the ground state configuration of the numbered nodes.
• Figure 2: (a-c) Three essential gadgets for TLSG encoding. In each subfigure, the left column is the source graph, and the right column is the mapped graph on a triangular lattice. The red-framed vertices on the boundary are pin vertices, and only pin vertices can connect to external vertices. The full list of gadgets is provided in the Supplemental Material. (d) The composition of a copy gadget and a T-connection gadget. This requires summing the weights of the connected pin vertices at the junction.
• Figure 3: Quantum annealing simulations comparing TLSG and KSG encodings for the $K_{2,3}$ problem. (a) The KSG encoding of $K_{2,3}$ for comparison. The TLSG encoding is shown in Fig. \ref{['fig:main']}(d). (b) Piecewise-linear annealing pulses for Rabi frequency $\Omega(t)$ and detuning $\Delta(t)$. The shaded regions indicate the operational range $\frac{C_6}{R_{\rm min}^6} <\Omega <\frac{C_6}{r_{\rm max}^6}$ with $C_6=2\pi\times 862690 \,\text{MHz} \, \mu\text{m}^6$. The lattice unit is chosen so that $\Omega_{\rm max}$ equals the geometric mean of the interaction bounds. (c) Violation rate of independence constraints versus total annealing time for both encodings. Dashed lines show exponential fits, and shaded regions denote $95\%$ confidence intervals. The TLSG encoding reduces the violation rate by nearly two orders of magnitude compared to the KSG encoding across most annealing regimes.
• Figure S1: (a) King's subgraph (KSG) on a square lattice. (b) Triangular lattice subgraph (TLSG) and the transformation process between the original triangular lattice and its reshaped square-grid representation used in the code implementation.
• Figure S2: Illustration of how logical conjunction translates to gadget composition in the MWIS framework. (a) Two NOT gadgets encode the constraints $a = \bar{b}$ and $b = \bar{c}$, and are combined by merging the vertices representing the shared variable $b$, ensuring both constraints are enforced simultaneously. (b) shows an extended version where an odd-length vertex wire is constructed from a sequence of NOT gadgets. Red-framed vertices are equivalent in the MWIS sense, allowing local information to propagate over long distances.

Theorems & Definitions (11)

• Theorem 1
• proof
• Definition 1: Logical equivalence
• Definition 2: Gadget
• Proposition 1: MWIS is maximal
• proof
• Proposition 2: Gadget composition principle
• Proposition 3
• Definition 3: Maximal Independent Set
• Proposition 4: MWIS is maximal