Beyond the thermodynamic limit: finite-size corrections to state interconversion rates

Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.Comment: 31 pages, 10 figures. Final version, to be published in Quantu

Energy storage-boiler tank

Activities performed in an effort to demonstrate heat of fusion energy storage in containerized salts are reported. The properties and cycle life characteristics of a eutectic salt having a boiling point of about 385 C (NaCl, KCl, Mg Cl2) were determined. M-terphenyl was chosen as the heat transfer fluid. Compatibility studies were conducted and mild steel containers were selected. The design and fabrication of a 2MWh storage boiler tank are discussed

A new R package and web application for detecting bilateral asymmetry in parasitic infections.

When parasites invade paired structures of their host non-randomly, the resulting asymmetry may have both pathological and ecological significance. To facilitate the detection and visualisation of asymmetric infections we have developed a free software tool, Analysis of Symmetry of Parasitic Infections (ASPI). This tool has been implemented as an R package (https://cran.r-project.org/package=aspi) and a web application (https://wayland.shinyapps.io/aspi). ASPI can detect both consistent bias towards one side, and inconsistent bias in which the left side is favoured in some hosts and the right in others. Application of ASPI is demonstrated using previously unpublished data on the distribution of metacercariae of species of Diplostomum von Nordmann, 1832 in the eyes of ruffe Gymnocephalus cernua (Linnaeus). Invasion of the lenses appeared to be random, with the proportion of metacercariae in the left and right lenses showing the pattern expected by chance. However, analysis of counts of metacercariae from the humors, choroid and retina revealed asymmetry between eyes in 38% of host fish

Tailoring surface codes for highly biased noise

The surface code, with a simple modification, exhibits ultra-high error correction thresholds when the noise is biased towards dephasing. Here, we identify features of the surface code responsible for these ultra-high thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases, and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is $50\%$, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time decoding algorithm. We demonstrate that the sub-threshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough/smooth open boundaries, it is controlled by the parameter $g=\gcd(j,k)$, where $j$ and $k$ are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for co-prime codes that have $g=1$, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: the same logical failure rate achievable with a square surface code and $n$ physical qubits can be obtained with a co-prime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased towards dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.Comment: 18+4 pages, 24 figures; v2 includes additional coauthor (ASD) and new results on the performance of surface codes in the finite-bias regime, obtained with beveled surface codes and an improved tensor network decoder; v3 published versio

Education Reform for the Digital Era

Will the digital-learning movement repeat the mistakes of the charter-school movement? How much more successful might today's charter universe look if yesterday's proponents had focused on the policies and practices needed to ensure its quality, freedom, and resources over the long term? What mistakes might have been avoided? Damaging scandals forestalled? Missed opportunities seized

Tailoring three-dimensional topological codes for biased noise

Tailored topological stabilizer codes in two dimensions have been shown to exhibit high storage threshold error rates and improved subthreshold performance under biased Pauli noise. Three-dimensional (3D) topological codes can allow for several advantages including a transversal implementation of non-Clifford logical gates, single-shot decoding strategies, parallelized decoding in the case of fracton codes as well as construction of fractal lattice codes. Motivated by this, we tailor 3D topological codes for enhanced storage performance under biased Pauli noise. We present Clifford deformations of various 3D topological codes, such that they exhibit a threshold error rate of $50\%$ under infinitely biased Pauli noise. Our examples include the 3D surface code on the cubic lattice, the 3D surface code on a checkerboard lattice that lends itself to a subsystem code with a single-shot decoder, the 3D color code, as well as fracton models such as the X-cube model, the Sierpinski model and the Haah code. We use the belief propagation with ordered statistics decoder (BP-OSD) to study threshold error rates at finite bias. We also present a rotated layout for the 3D surface code, which uses roughly half the number of physical qubits for the same code distance under appropriate boundary conditions. Imposing coprime periodic dimensions on this rotated layout leads to logical operators of weight $O(n)$ at infinite bias and a corresponding $\exp[-O(n)]$ subthreshold scaling of the logical failure rate, where $n$ is the number of physical qubits in the code. Even though this scaling is unstable due to the existence of logical representations with $O(1)$ low-rate Pauli errors, the number of such representations scales only polynomially for the Clifford-deformed code, leading to an enhanced effective distance.Comment: 51 pages, 34 figure