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The surface code is a quantum error-correcting code for one logical qubit, protected by spatially localized parity checks in two dimensions. Due to fundamental constraints from spatial locality, storing more logical qubits requires either sacrificing the robustness of the surface code against errors or increasing the number of physical qubits. We bound the minimal number of spatially nonlocal parity checks necessary to add logical qubits to a surface code while maintaining, or improving, robustness to errors. We saturate the lower limit of this bound, when the number of added logical qubits is a constant, using a family of hypergraph product codes, interpolating between the surface code and constant-rate low-density parity-check codes. Fault-tolerant protocols for logical gates in the quantum code can be inherited from its classical parent codes. We provide near-term practical implementations of this code for hardware based on trapped ions or neutral atoms in mobile optical tweezers. Long-range-enhanced surface codes outperform conventional surface codes using hundreds of physical qubits, and they represent a practical strategy to enhance the robustness of logical qubits to errors in near-term devices.
Hong et al. (Thu,) studied this question.