Coherent error threshold for surface codes from Majorana delocalization

Statistical mechanics mappings provide key insights on quantum error correction. However, existing mappings assume incoherent noise, thus ignoring coherent errors due to, e.g., spurious gate rotations. We map the surface code with coherent errors, taken as $X$- or $Z$-rotations (replacing bit or phase flips), to a two-dimensional (2D) Ising model with complex couplings, and further to a 2D Majorana scattering network. Our mappings reveal both commonalities and qualitative differences in correcting coherent and incoherent errors. For both, the error-correcting phase maps, as we explicitly show by linking 2D networks to 1D fermions, to a $\mathbb{Z}_2$-nontrivial 2D insulator. However, beyond a rotation angle $\phi_\text{th}$, instead of a $\mathbb{Z}_2$-trivial insulator as for incoherent errors, coherent errors map to a Majorana metal. This $\phi_\text{th}$ is the theoretically achievable storage threshold. We numerically find $\phi_\text{th}\approx0.14\pi$. The corresponding bit-flip rate $\sin^2(\phi_\text{th})\approx 0.18$ exceeds the known incoherent threshold $p_\text{th}\approx0.11$.Comment: 9 pages, 7 figure

Local integrals of motion detection of localization-protected topological order

Many-body-localized (MBL) phases can be topologically distinct, but distinguishing these phases using order parameters can be challenging. Here we show how topologically distinct local integrals of motion, variationally parametrized by quantum circuits, can be used to numerically demonstrate the topological inequivalence of MBL phases. We illustrate our approach on a fermionic chain where both topologically distinct MBL phases and benchmark comparisons to order parameters are possible. We also use our approach, augmented by the DMRG-X algorithm, to extract high-energy topological doublets. We describe applying our methods to higher dimensions to identify MBL topological order and topological multiplets hidden by the dense many-body spectrum.Comment: 8 pages, 7 figures; v2: accepted manuscript; v3: typo fixe

Aharonov-Bohm oscillations and magnetic focusing in ballistic graphene rings

We present low-temperature magnetotransport measurements on graphene rings encapsulated in hexagonal boron nitride. We investigate phase-coherent transport and show Aharonov-Bohm (AB) oscillations in quasi-ballistic graphene rings with hard confinement. In particular, we report on the observation of $h/e$, $h/2e$ and $h/3e$ conductance oscillations. Moreover we show signatures of magnetic focusing effects at small magnetic fields confirming ballistic transport. We perform tight binding calculations which allow to reproduce all significant features of our experimental findings and enable a deeper understanding of the underlying physics. Finally, we report on the observation of the AB conductance oscillations in the quantum Hall regime at reasonable high magnetic fields, where we find regions with enhanced AB oscillation visibility with values up to $0.7$%. These oscillations are well explained by taking disorder into account allowing for a coexistence of hard and soft-wall confinement.Comment: 28 pages, 8 figure

Global maps of soil temperature

Research in global change ecology relies heavily on global climatic grids derived from estimates of air temperature in open areas at around 2 m above the ground. These climatic grids do not reflect conditions below vegetation canopies and near the ground surface, where critical ecosystem functions occur and most terrestrial species reside. Here, we provide global maps of soil temperature and bioclimatic variables at a 1-km2 resolution for 0–5 and 5–15 cm soil depth. These maps were created by calculating the difference (i.e. offset) between in situ soil temperature measurements, based on time series from over 1200 1-km2 pixels (summarized from 8519 unique temperature sensors) across all the world\u27s major terrestrial biomes, and coarse-grained air temperature estimates from ERA5-Land (an atmospheric reanalysis by the European Centre for Medium-Range Weather Forecasts). We show that mean annual soil temperature differs markedly from the corresponding gridded air temperature, by up to 10°C (mean = 3.0 ± 2.1°C), with substantial variation across biomes and seasons. Over the year, soils in cold and/or dry biomes are substantially warmer (+3.6 ± 2.3°C) than gridded air temperature, whereas soils in warm and humid environments are on average slightly cooler (−0.7 ± 2.3°C). The observed substantial and biome-specific offsets emphasize that the projected impacts of climate and climate change on near-surface biodiversity and ecosystem functioning are inaccurately assessed when air rather than soil temperature is used, especially in cold environments. The global soil-related bioclimatic variables provided here are an important step forward for any application in ecology and related disciplines. Nevertheless, we highlight the need to fill remaining geographic gaps by collecting more in situ measurements of microclimate conditions to further enhance the spatiotemporal resolution of global soil temperature products for ecological applications

Global maps of soil temperature

Research in global change ecology relies heavily on global climatic grids derived from estimates of air temperature in open areas at around 2 m above the ground. These climatic grids do not reflect conditions below vegetation canopies and near the ground surface, where critical ecosystem functions occur and most terrestrial species reside. Here, we provide global maps of soil temperature and bioclimatic variables at a 1-km² resolution for 0–5 and 5–15 cm soil depth. These maps were created by calculating the difference (i.e., offset) between in-situ soil temperature measurements, based on time series from over 1200 1-km² pixels (summarized from 8500 unique temperature sensors) across all the world’s major terrestrial biomes, and coarse-grained air temperature estimates from ERA5-Land (an atmospheric reanalysis by the European Centre for Medium-Range Weather Forecasts). We show that mean annual soil temperature differs markedly from the corresponding gridded air temperature, by up to 10°C (mean = 3.0 ± 2.1°C), with substantial variation across biomes and seasons. Over the year, soils in cold and/or dry biomes are substantially warmer (+3.6 ± 2.3°C) than gridded air temperature, whereas soils in warm and humid environments are on average slightly cooler (-0.7 ± 2.3°C). The observed substantial and biome-specific offsets emphasize that the projected impacts of climate and climate change on near-surface biodiversity and ecosystem functioning are inaccurately assessed when air rather than soil temperature is used, especially in cold environments. The global soil-related bioclimatic variables provided here are an important step forward for any application in ecology and related disciplines. Nevertheless, we highlight the need to fill remaining geographic gaps by collecting more in-situ measurements of microclimate conditions to further enhance the spatiotemporal resolution of global soil temperature products for ecological applications

Global maps of soil temperature.

Research in global change ecology relies heavily on global climatic grids derived from estimates of air temperature in open areas at around 2 m above the ground. These climatic grids do not reflect conditions below vegetation canopies and near the ground surface, where critical ecosystem functions occur and most terrestrial species reside. Here, we provide global maps of soil temperature and bioclimatic variables at a 1-km2 resolution for 0-5 and 5-15 cm soil depth. These maps were created by calculating the difference (i.e. offset) between in situ soil temperature measurements, based on time series from over 1200 1-km2 pixels (summarized from 8519 unique temperature sensors) across all the world's major terrestrial biomes, and coarse-grained air temperature estimates from ERA5-Land (an atmospheric reanalysis by the European Centre for Medium-Range Weather Forecasts). We show that mean annual soil temperature differs markedly from the corresponding gridded air temperature, by up to 10°C (mean = 3.0 ± 2.1°C), with substantial variation across biomes and seasons. Over the year, soils in cold and/or dry biomes are substantially warmer (+3.6 ± 2.3°C) than gridded air temperature, whereas soils in warm and humid environments are on average slightly cooler (-0.7 ± 2.3°C). The observed substantial and biome-specific offsets emphasize that the projected impacts of climate and climate change on near-surface biodiversity and ecosystem functioning are inaccurately assessed when air rather than soil temperature is used, especially in cold environments. The global soil-related bioclimatic variables provided here are an important step forward for any application in ecology and related disciplines. Nevertheless, we highlight the need to fill remaining geographic gaps by collecting more in situ measurements of microclimate conditions to further enhance the spatiotemporal resolution of global soil temperature products for ecological applications

The Many Phases of the Surface Code: Coherent Errors and Many-Body Localisation

This thesis investigates the far-from-ground-state physics of the surface code, in particular its quantum error correction applications and formulations. We contribute to this field via two lines of research: we study the behaviour of the surface code under coherent errors, which create superpositions of excited states, and we probe topological many body localization (MBL) which protects topological order for all eigenstates. In the first strand, we develop an interpretation of the error correction threshold for coherent error rotations as a phase transition. For this, we first generalize a numerical method for the simulation of coherent errors in surface codes on square lattices to work with surface codes on general planar graphs. This method is based on a mapping to a free fermion model which allows calculating the expectation values using fermion linear optics. Using this method, we show that the connectivity of the graph can shift the error correcting performance between resilience against *X*- and *Z*-rotations. Building on this work, we further explore the relationship between coherent errors in surface codes and free fermion models. We develop a formalism to map the surface code under coherent errors to a complex Ising model and from there to a Majorana fermion scattering model. We analyze its conductivity and find that for rotations below the error correction threshold the resulting model is an insulator, and it becomes a metal above the threshold. By estimating the position of this phase transition, we obtain the achievable error correction threshold for coherent errors. The second line of research is focused on the disordered and perturbed toric code. We implement a recently proposed method that numerically approximates the local integrals of motion that are present in (topological) MBL phases using sets of stabilizers that are dressed by optimized quantum circuits. First, we apply this method to the disordered Kitaev chain as a benchmark. Then, we proceed by adapting it to the toric code. We show how it can be used to distinguish topological and trivial MBL and how it can be combined with exact diagonalization to obtain an approximate phase diagram.ER

Surface codes, quantum circuits, and entanglement phases

Surface codes–leading candidates for quantum error correction (QEC)–and entanglement phases–a key notion for many-body quantum dynamics–have heretofore been unrelated. Here, we establish a link between the two. We map two-dimensional (2D) surface codes under a class of incoherent or coherent errors (bit flips or uniaxial rotations) to $(1+1)$D free-fermion quantum circuits via Ising models. We show that the error-correcting phase implies a topologically nontrivial area law for the circuit's 1D long-time state $\ket{\Psi_\infty}$. Above the error threshold, we find a topologically trivial area law for incoherent errors and logarithmic entanglement in the coherent case. In establishing our results, we formulate 1D parent Hamiltonians for $\ket{\Psi_\infty}$ via linking Ising models and 2D scattering networks, the latter displaying respective insulating and metallic phases and setting the 1D fermion gap and topology via their localization length and topological invariant. We expect our results to generalize to a duality between the error-correcting phase of ($d+1$)D topological codes and $d$-dimensional area laws; this can facilitate assessing code performance under various errors. The approach of combining Ising models, scattering networks, and parent Hamiltonians can be generalized to other fermionic circuits and may be of independent interest.This work was supported by EPSRC grant EP/V062654/1, a Leverhulme Early Career Fellowship, the Newton Trust of the University of Cambridge, and in part by the ERC Starting Grant No. 678795 TopInSy. Our simulations used resources at the Cambridge Service for Data Driven Discovery operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using EPSRC Tier-2 funding via grant EP/T022159/1, and STFC DiRAC funding (www.dirac.ac.uk)