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Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D -dimensional hyperbolic space. 

Abstract This study uncovers a correlation between the mid-infrared emissivity of butterfly wings and the average air temperature of their habitats across the world. Butterflies from cooler climates have a lower mid-infrared emissivity, which limits heat losses to surroundings, and butterflies from warmer climates have a higher mid-infrared emissivity, which enhances radiative cooling. The mid-infrared emissivity showed no correlation with other investigated climatic factors. Phylogenetic independent contrasts analysis indicates the microstructures of butterfly wings may have evolved in part to regulate mid-infrared emissivity as an adaptation to climate, rather than as phylogenetic inertia. Our findings offer new insights into the role of microstructures in thermoregulation and suggest both evolutionary and physical constraints to butterflies’ abilities to adapt to climate change. 

While surface microstructures of butterfly wings have been extensively studied for their structural coloration or optical properties within the visible spectrum, their properties in infrared wavelengths with potential ties to thermoregulation are relatively unknown. The midinfrared wavelengths of 7.5 to 14 µm are particularly important for radiative heat transfer in the ambient environment, because of the overlap with the atmospheric transmission window. For instance, a high midinfrared emissivity can facilitate surface cooling, whereas a low midinfrared emissivity can minimize heat loss to surroundings. Here we find that the midinfrared emissivity of butterfly wings from warmer climates such asArchaeoprepona demophoon(Oaxaca, Mexico) andHeliconius sara(Pichincha, Ecuador) is up to 2 times higher than that of butterfly wings from cooler climates such asCelastrina echo(Colorado) andLimenitis arthemis(Florida), using Fourier-transform infrared (FTIR) spectroscopy and infrared thermography. Our optical computations using a unit cell approach reproduce the spectroscopy data and explain how periodic microstructures play a critical role in the midinfrared. The emissivity spectrum governs the temperature of butterfly wings, and we demonstrate thatC. echowings heat up to 8 °C more thanA. demophoonwings under the same sunlight in the clear sky of Irvine, CA. Furthermore, our thermal computations show that butterfly wings in their respective habitats can maintain a moderate temperature range through a balance of solar absorption and infrared emission. These findings suggest that the surface microstructures of butterfly wings potentially contribute to thermoregulation and provide an insight into butterflies' survival.