Rydberg atom arrays natively encode the maximum-weight independent set (MWIS) problem through the blockade mechanism, so the Boolean circuit satisfiability problem (Circuit-SAT) can be brought onto the platform once it is reduced to MWIS. The conventional encoding of Circuit-SAT in the Rydberg atom array proceeds through conjunctive normal form (CNF) and incurs a substantial atom overhead. We introduce CAMERA (Circuit-SAT Atom-efficient MWIS Encoding for Rydberg Arrays), a method that provides MWIS encodings of Circuit-SAT instances on the king subgraph geometry of the array. CAMERA represents each logic gate as a compact weighted gadget and assembles the gadgets with a placement and routing compiler inspired by very large scale integration (VLSI) design. On random multi-gate benchmarks, the direct encoding route lowers the atom cost relative to the CNF route by an average factor of $22.4 \pm 1.8$. To demonstrate that the encoding extends from individual weighted gadgets to multi-gate arithmetic blocks, we compile a full adder and a multiplier, verifying each against its complete truth table by exact classical ground state calculations. We further showcase solving a representative Circuit-SAT instance end-to-end, from gate level compilation through a closed-system tensor-network simulation of a hardware-compatible annealing protocol on the encoded 30-atom instance to readout of a satisfying assignment. These results establish a complete encoding and simulation workflow as a proof of principle, and a concrete route toward solving a broader family of combinatorial problems on Rydberg atom arrays.
Rydberg atom arrays natively encode the maximum-weight independent set (MWIS) problem through the blockade mechanism, so the Boolean circuit satisfiability problem (Circuit-SAT) can be brought onto the platform once it is reduced to MWIS. The conventional encoding of Circuit-SAT in the Rydberg atom array proceeds through conjunctive normal form (CNF) and incurs a substantial atom overhead. We introduce CAMERA (Circuit-SAT Atom-efficient MWIS Encoding for Rydberg Arrays), a method that provides MWIS encodings of Circuit-SAT instances on the king subgraph geometry of the array. CAMERA represents each logic gate as a compact weighted gadget and assembles the gadgets with a placement and routing compiler inspired by very large scale integration (VLSI) design. On random multi-gate benchmarks, the direct encoding route lowers the atom cost relative to the CNF route by an average factor of 22.4±1.8. To demonstrate that the encoding extends from individual weighted gadgets to multi-gate arithmetic blocks, we compile a full adder and a multiplier, verifying each against its complete truth table by exact classical ground state calculations. We further showcase solving a representative Circuit-SAT instance end-to-end, from gate level compilation through a closed-system tensor-network simulation of a hardware-compatible annealing protocol on the encoded 30-atom instance to readout of a satisfying assignment. These results establish a complete encoding and simulation workflow as a proof of principle, and a concrete route toward solving a broader family of combinatorial problems on Rydberg atom arrays.
Neutral atom arrays provide a versatile platform to implement coherent quantum annealing as an approach to solving hard combinatorial optimization problems. Here we present and experimentally demonstrate an efficient encoding scheme based on chains of Rydberg-blockaded atoms, which we call quantum wires, to natively embed maximum weighted independent set (MWIS) and quadratic unconstrained binary optimization (QUBO) problems on a neutral atom architecture. For graphs with quasi-unit-disk connectivity, in which only a few long-range edges are required, our approach requires a significantly lower overhead in the number of ancilla qubits than previous proposals, facilitating the implementation on currently available hardware. To demonstrate the approach, we perform annealing of weighted graphs on a programmable atom array using local light shifts to encode problem-specific weights across graphs of varying sizes. This approach successfully identifies the solutions to the original MWIS and QUBO graph instances. Our work expands the operational toolkit of near-term neutral atom arrays, enhancing their potential for scalable quantum optimization.
We present a hardware-native gadget framework for solving constraint satisfaction problems on Rydberg quantum computing architectures. Our approach introduces a compact xor1 gadget that enforces exactly-one constraints, ubiquitous in combinatorial optimization, directly through geometric embedding and blockade interactions. A key advantage of the xor1 gadget is its fixed, problem-size-independent detuning requirements: enforcing constraints through blockade interactions eliminates the need for large penalty terms, thereby substantially reducing the detuning range compared to Quadratic Unconstrained Binary Optimization (QUBO) formulations and improving experimental feasibility. By tailoring the construction to the geometric connectivity of Rydberg atom arrays, the framework bypasses the all-to-all physical couplings often assumed in logical encodings. This enables embeddings compatible with planar layouts and avoids highly connected arrangements. We develop scalable implementations that reduce atom count and connectivity overhead while avoiding extensive classical preprocessing, making them compatible with near-term neutral-atom hardware. As illustrations, we apply our framework to the gate-assignment and N-queens problems, highlighting its practicality, resource efficiency, and hardware compatibility. In these examples, we observe reductions in detuning range of up to 99% and savings in atom count and connectivity overhead of up to 54% compared to the QUBO method. These results establish a route toward implementing large-scale combinatorial optimization on Rydberg platforms beyond the limits of existing encodings.
Rydberg atom quantum optimizers naturally encode the maximum independent set problem on geometric graphs. However, to solve general combinatorial optimization problems on more complex topologies, an embedding is required to overcome the limited-range native interactions. Such embeddings use gadgets, specific arrangements of atoms, which inevitably introduce ancillary atoms. Here, we show that this apparent overhead instead provides an intrinsic resource for error correction. Specifically, we demonstrate that the Parity Architecture-based Rydberg embedding naturally forms a classical error-correcting code, a principle that extends to other Rydberg blockade-based embeddings. Within superatom substructures, exact Rydberg-excitation-number constraints provide local error syndromes at no additional hardware cost and give rise to a strictly nonlinear code. To correct bit-flip errors, we introduce the precisely tailored Cluster decoder, which exploits these local constraints and the underlying gadget structure. Numerically, this approach achieves an error threshold of pth≳26% across different layout geometries. Experimentally, on the QuEra Aquila QPU, the Cluster decoder recovers exact optimization solutions with an average success rate of ≳8% for systems of up to 107 atoms, even when bare hardware and standard greedy post-processing find none. A Rydberg atom array embedding is therefore not merely a hardware mapping of an optimization problem, but a co-designed architecture for quantum optimization in which embedding, computation, and error correction emerge from the same physical structure.
Programmable quantum systems based on Rydberg atom arrays have recently been used for hardwareefficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] with hundreds of qubits.In particular, the maximum independent set problem on so-called unit-disk graphs, was shown to be efficiently encodable in such a quantum system.Here, we extend the classes of problems that can be efficiently encoded in Rydberg arrays by constructing explicit mappings from a wide class of problems to maximum-weighted independent set problems on unit-disk graphs, with at most a quadratic overhead in the number of qubits.We analyze several examples, including maximum-weighted independent set on graphs with arbitrary connectivity, quadratic unconstrained binary optimization problems with arbitrary or restricted connectivity, and integer factorization.Numerical simulations on small system sizes indicate that the adiabatic time scale for solving the mapped problems is strongly correlated with that of the original problems.Our work provides a blueprint for using Rydberg atom arrays to solve a wide range of combinatorial optimization problems with arbitrary connectivity, beyond the restrictions imposed by the hardware geometry.
We propose and implement a comprehensive quantum compilation toolkit for solving the maximum independent set (MIS) problem on quantum hardware based on Rydberg atom arrays. Our end-to-end pipeline involves three core components to efficiently map generic MIS instances onto Rydberg arrays with unit-disk connectivity, with modules for graph reduction, hardware compatibility checks, and graph embedding. The first module (reducer) provides hardware-agnostic and deterministic reduction logic that iteratively reduces the problem size via lazy clique removals. We find that real-world networks can typically be reduced by orders of magnitude on subsecond timescales, thus significantly cutting down the eventual load for quantum devices. Moreover, we show that reduction techniques may be an important tool in the ongoing search for potential quantum speedups, given their ability to identify hard problem instances. In particular, for Rydberg-native MIS instances, we observe signatures of an easy-hard-easy transition and quantify a critical degree indicating the onset of a hard problem regime. The second module (compatibility checker) implements a hardware compatibility checker that quickly determines whether a given input graph may be compatible with the restrictions imposed by Rydberg quantum hardware. The third module (embedder) describes hardware-efficient graph embedding routines to generate (approximate) encodings with controllable overhead and optimized ancilla placements. We exemplify our pipeline with experiments run on the QuEra Aquila device available on Amazon Braket. In aggregate, our work provides a set of tools that extends the class of problems that can be tackled with near-term Rydberg atom arrays.
Programmable quantum systems based on Rydberg atom arrays have recently emerged as a promising test bed for combinatorial optimization. Indeed, the maximum weighted independent set problem on unit-disk graphs can be efficiently mapped to such systems due to their geometric constraints. However, extending this capability to arbitrary graph instances typically necessitates the use of reduction gadgets, which introduce additional experimental overhead and complexity. Here, we analyze the complexity\penalty1000-\hskip0pttheoretic limits of polynomial reductions from arbitrary graphs to unit\penalty1000-\hskip0ptdisk instances. We prove any such reduction incurs a quadratic blow up in vertex count and degrades solution approximation guarantees. As a practical alternative, we propose a divide\penalty1000-\hskip0ptand\penalty1000-\hskip0ptconquer heuristic with only linear overhead, which leverages precalibrated atomic layouts. We benchmark it on Erd\"os-R\'enyi graphs, and demonstrate feasibility on the Orion Alpha processor.