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Mapping the Ising model onto two-dimensional Rubidium atom arrays

Abhishek Anand

Published Jun 18, 2020• 0 citations
OpenAlex

Abstract

We describe an architecture to map the Ising model problem onto the dynamics of two-dimensional neutral trapped atom arrays. Specifically, we show that the NP-complete Ising model decision problem on $n$ variables can be reduced to deciding whether, for $O(n^2)$ Rubidium atoms in two dimensions, interacting via Rydberg excitations, there exists a state with energy below a certain threshold. This provides an alternative proof of the NP-hardness of two dimensional Rydberg dynamics with a single species in the regime where long-range interactions are negligible. Therefore, we can encode NP-complete problems onto the ground state of Rydberg atom arrays and use quantum optimization techniques such as quantum adiabatic algorithms and variational algorithms to estimate the ground state. Furthermore, as the Ising model describes many-body phenomena in statistical physics, we can probe a plethora of emergent phenomena via quantum simulators. Our result is based on the Lechner-Hauke-Zoller (LHZ) scheme that enables the encoding of an all-to-all connected Ising model onto systems with local-only interactions. We improve the scheme in the following ways to make the it feasible for near-term Rydberg simulators: (1) Implementations of the scheme in literature that use qubits suffer from a double-counting error causing inconsistencies in the strength of the interactions. Identical interactions must have different strengths based on their spatial location. We describe and resolve this error without impacting the $O(n^2)$ scaling of the atom array by introducing additional ancillary qubits. (2) We introduce new formalism and simpler proofs via novel approaches regarding the validity of the LHZ scheme and its variants. These proofs lend more easily to complexity-theoretic arguments regarding the hardness of Rydberg dynamics. (3) Finally, we circumvent the LHZ scheme constraint of requiring four-body interactions, qutrits, or multiple species of atoms by providing an implementation that uses only Rubidium atoms to encode both physical and ancillary qubits. To engineer the required interactions, the atoms are coupled to different Rydberg energy levels. We also provide a brief review of classical and quantum computation, quantum computing platforms, near-term quantum experiments, and Rydberg physics.

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We describe an architecture to map the Ising model problem onto the dynamics of two-dimensional neutral trapped atom arrays. Specifically, we show that the NP-complete Ising model decision problem on nnn variables can be reduced to deciding whether, for O(n2)O(n^2)O(n2) Rubidium atoms in two dimensions, interacting via Rydberg excitations, there exists a state with energy below a certain threshold. This provides an alternative proof of the NP-hardness of two dimensional Rydberg dynamics with a single species in the regime where long-range interactions are negligible. Therefore, we can encode NP-complete problems onto the ground state of Rydberg atom arrays and use quantum optimization techniques such as quantum adiabatic algorithms and variational algorithms to estimate the ground state. Furthermore, as the Ising model describes many-body phenomena in statistical physics, we can probe a plethora of emergent phenomena via quantum simulators. Our result is based on the Lechner-Hauke-Zoller (LHZ) scheme that enables the encoding of an all-to-all connected Ising model onto systems with local-only interactions. We improve the scheme in the following ways to make the it feasible for near-term Rydberg simulators: (1) Implementations of the scheme in literature that use qubits suffer from a double-counting error causing inconsistencies in the strength of the interactions. Identical interactions must have different strengths based on their spatial location. We describe and resolve this error without impacting the O(n2)O(n^2)O(n2) scaling of the atom array by introducing additional ancillary qubits. (2) We introduce new formalism and simpler proofs via novel approaches regarding the validity of the LHZ scheme and its variants. These proofs lend more easily to complexity-theoretic arguments regarding the hardness of Rydberg dynamics. (3) Finally, we circumvent the LHZ scheme constraint of requiring four-body interactions, qutrits, or multiple species of atoms by providing an implementation that uses only Rubidium atoms to encode both physical and ancillary qubits. To engineer the required interactions, the atoms are coupled to different Rydberg energy levels. We also provide a brief review of classical and quantum computation, quantum computing platforms, near-term quantum experiments, and Rydberg physics.
There is a significant ongoing effort in realizing quantum annealing with different physical platforms. The challenge is to achieve a fully programmable quantum device featuring coherent adiabatic quantum dynamics. Here we show that combining the well-developed quantum simulation toolbox for Rydberg atoms with the recently proposed Lechner-Hauke-Zoller (LHZ) architecture allows one to build a prototype for a coherent adiabatic quantum computer with all-to-all Ising interactions and, therefore, a platform for quantum annealing. In LHZ an infinite-range spin-glass is mapped onto the low energy subspace of a spin-1/2 lattice gauge model with quasi-local four-body parity constraints. This spin model can be emulated in a natural way with Rubidium and Caesium atoms in a bipartite optical lattice involving laser-dressed Rydberg-Rydberg interactions, which are several orders of magnitude larger than the relevant decoherence rates. This makes the exploration of coherent quantum enhanced optimization protocols accessible with state-of-the-art atomic physics experiments.
Programmable quantum systems provide insights into many-body quantum phenomena and enable new methods for computation and metrology. The frontier challenge for these systems is scaling up in the number of quantum particles while retaining a high degree of quantum control. In recent years arrays of neutral atoms have emerged as a scalable and powerful platform for tackling these challenges, thanks to their large number of qubits, flexible geometry, and a well-established toolbox for preparation, detection, coherent control. At the forefront of quantum science and engineering, these systems have been used for quantum simulation of various spin models and have also demonstrated building blocks of quantum computation. In this thesis I will present advances in development of large two-dimensional atom arrays, where hundreds of individual atoms are trapped, rearranged, and initialized in various defect-free arrangements in one or two dimensions. Afterwards, interactions between atoms is controlled via coherent excitation to highly excited Rydberg states, resulting in strong long-range van der Waals forces. We utilize these interaction for quantum simulation of many-body spin Hamiltonians by demonstrating preparation of several exotic phases of matter, study phase transitions, and probe non-equilibrium dynamics. Further, we study the quantum speedup in solving a hard combinatorial optimization problem via efficiently mapping the problem onto the Rydberg atom array. We also utilize Rydberg interactions to demonstrate high-fidelity two-qubit gate operations in a reconfigurable atom array architecture. Lastly I will demonstrate recent technical improvements that have enabled larger system sizes, higher fidelity rydberg excitations, and local control of individual atoms, paving the way for novel approaches for studying quantum many-body phenomena.
We discuss the computational complexity of finding the ground state of the two-dimensional array of quantum bits that interact via strong van der Waals interactions. Specifically, we focus on systems where the interaction strength between two spins depends only on their relative distance xxx and decays as 1/x61/x^61/x6 that have been realized with individually trapped homogeneously excited neutral atoms interacting via the so-called Rydberg blockade mechanism. We show that the solution to NP-complete problems can be encoded in ground state of such a many-body system by a proper geometrical arrangement of the atoms. We present a reduction from the NP-complete maximum independent set problem on planar graphs with maximum degree three. Our results demonstrate that computationally hard optimization problems can be naturally addressed with coherent quantum optimizers accessible in near term experiments.
Optical tweezer arrays of neutral atoms have emerged as a versatile platform for analog quantum simulation and digital quantum computation, combining scalability, reconfigurable geometry, and flexible microscopic control. In the first part of this work, we develop a 133^{133}133Cs Rydberg simulator and demonstrate coherent Rydberg excitation in a rearrangeable array of more than 100 atoms. We then use this platform to probe critical many-body physics, extracting the scaling dimensions of the Ising universality class from an adiabatically prepared critical state. Dual-species arrays extend these capabilities by introducing spectroscopic selectivity and independently tunable interspecies interactions. Motivated by these opportunities, we develop a dual-species Rydberg array of 23^{23}23Na and 133^{133}133Cs atoms. The large separation between the optical transitions of the two species allows qubits encoded in one species to be cooled, trapped, imaged, and excited while the other species remains largely undisturbed. At the same time, strong van der Waals interactions provide a programmable coupling between the two arrays. We use this platform to demonstrate a proof-of-principle multi-qubit stabilizer readout based on an interaction-compensated protocol. In this setting, spectroscopic separation becomes an architectural advantage: it enables in-situ ancilla control and readout without requiring atom transport between separate interaction and measurement zones. Consequently, a multi-qubit entangling operation that would otherwise be decomposed into a sequence of two-qubit gates can be implemented as a native, effectively constant-depth primitive. These results point toward faster and more scalable syndrome extraction in neutral-atom quantum error correction, and establish dual-species Rydberg arrays as a promising architecture connecting programmable quantum simulation with fault-tolerant quantum computation.
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%p_{\text{th}}\gtrsim26 \%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%\gtrsim8\%≳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.
Quantum adiabatic optimization seeks to solve combinatorial problems using quantum dynamics, requiring the Hamiltonian of the system to align with the problem of interest. However, these Hamiltonians are often incompatible with the native constraints of quantum hardware, necessitating encoding strategies to map the original problem into a hardware-conformant form. While the classical overhead associated with such mappings is easily quantifiable and typically polynomial in problem size, it is much harder to quantify their overhead on the quantum algorithm, e.g., the transformation of the adiabatic timescale. In this work, we address this challenge on the concrete example of the encoding scheme proposed in [Nguyen , PRX Quantum , 010316 (2023)], which is designed to map optimization problems on arbitrarily connected graphs into Rydberg atom arrays. We consider the fundamental building blocks underlying this encoding scheme and determine the scaling of the minimum gap with system size along adiabatic protocols. Even when the original problem is trivially solvable, we find that the encoded problem can exhibit an exponentially closing minimum gap. We show that this originates from a quantum coherent effect, which gives rise to an unfavorable localization of the ground-state wave function. On the QuEra Aquila neutral atom machine, we observe such localization and its effect on the success probability of finding the correct solution to the encoded optimization problem. Finally, we propose quantum-aware modifications of the encoding scheme that avoid this quantum bottleneck and lead to an exponential improvement in the adiabatic performance. This highlights the crucial importance of accounting for quantum effects when designing strategies to encode classical problems onto quantum platforms.