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Consider an open quantum system governed by a Gorini, Kossakowski, Sudarshan, Lindblad (GKSL) master equation with two times-scales: a fast one, exponentially converging towards a linear subspace of quasi-equilibria; a slow one resulting from perturbations (small arbitrary decoherence and Hamiltonian dynamics). Usually adiabatic elimination is performed in the Schr\"odinger picture. We propose here an Heisenberg formulation where the invariant operators attached to the fast decay dynamics towards the quasi-equilibria subspace play a key role. Based on geometric singular perturbations, asympotic expansions of the Heisenberg slow dynamics and of the fast invariant linear subspaces are proposed. They exploit Carr's approximation lemma from center-manifold and bifurcation theory. Second-order expansions are detailed and shown to ensure preservation, up to second-order terms, of the trace and complete positivity for the slow dynamics on a slow time-scale. Such expansions can be exploited numerically., Comment: submitted. arXiv admin note: text overlap with arXiv:2303.05089

A numerical method is proposed for simulation of composite open quantum systems. It is based on Lindblad master equations and adiabatic elimination. Each subsystem is assumed to converge exponentially towards a stationary subspace, slightly impacted by some decoherence channels and weakly coupled to the other subsystems. This numerical method is based on a perturbation analysis with an asymptotic expansion. It exploits the formulation of the slow dynamics with reduced dimension. It relies on the invariant operators of the local and nominal dissipative dynamics attached to each subsystem. Second-order expansion can be computed only with local numerical calculations. It avoids computations on the tensor-product Hilbert space attached to the full system. This numerical method is particularly well suited for autonomous quantum error correction schemes. Simulations of such reduced models agree with complete full model simulations for typical gates acting on one and two cat-qubits (Z, ZZ and CNOT) when the mean photon number of each cat-qubit is less than 8. For larger mean photon numbers and gates with three cat-qubits (ZZZ and CCNOT), full model simulations are almost impossible whereas reduced model simulations remain accessible. In particular, they capture both the dominant phase-flip error-rate and the very small bit-flip error-rate with its exponential suppression versus the mean photon number., Comment: Submitted, minor release from the previous one

Cat qubits provide appealing building blocks for quantum computing. They exhibit a tunable noise bias yielding an exponential suppression of bit flips with the average photon number and a protection against the remaining phase errors can be ensured by a simple repetition code. We here quantify the cost of a repetition code and provide valuable guidance for the choice of a large scale architecture using cat qubits by realizing a performance analysis based on the computation of discrete logarithms on an elliptic curve with Shor's algorithm. By focusing on a 2D grid of cat qubits with neighboring connectivity, we propose to implement 2-qubit gates via lattice surgery and Toffoli gates with off-line fault-tolerant preparation of magic states through projective measurements and subsequent gate teleportations. All-to-all connectivity between logical qubits is ensured by routing qubits. Assuming a ratio between single- and two-photon losses of 1e-5 and a cycle time of 500 ns, we show concretely that such an architecture can compute a 256-bit elliptic curve logarithm in 9 h with 126133 cat qubits and on average 19 photons by cat state. We give the details of the realization of Shor's algorithm so that the proposed performance analysis can be easily reused to guide the choice of architecture for others platforms., Comment: 5+37 pages, 41 figures, 5 tables

Bosonic cat qubits stabilized by two-photon driven dissipation benefit from exponential suppression of bit-flip errors and an extensive set of gates preserving this protection. These properties make them promising building blocks of a hardware-efficient and fault-tolerant quantum processor. In this paper, we propose a performance optimization of the repetition cat code architecture using fast but noisy CNOT gates for stabilizer measurements. This optimization leads to high thresholds for the physical figure of merit, given as the ratio between intrinsic single-photon loss rate of the bosonic mode and the engineered two-photon loss rate, as well as a very interesting scaling below threshold of the required overhead, to reach an expected level of logical error rate. Relying on the specific error models for cat qubit operations, this optimization exploits fast parity measurements, using accelerated low-fidelity CNOT gates, combined with fast ancilla parity-check qubits. The significant enhancement in the performance is explained by: 1- the highly asymmetric error model of cat qubit CNOT gates with a major component on control (ancilla) qubits, and 2- the robustness of the error correction performance in presence of the leakage induced by fast operations. In order to demonstrate these performances, we develop a method to sample the repetition code under circuit-level noise that also takes into account cat qubit state leakage., Comment: 19 pages, 12 figures

As the COVID-19 virus spread over the world, governments restricted mobility to slow transmission. Public health measures had different intensities across European countries but all had significant impact on peoples daily lives and economic activities, causing a drop of CO2 emissions of about 10% for the whole year 2020. Here, we analyze changes in natural gas use in the industry and built environment sectors during the first half of year 2020 with daily gas flows data from pipeline and storage facilities in Europe. We find that reductions of industrial gas use reflect decreases in industrial production across most countries. Surprisingly, natural gas use in buildings also decreased despite most people being confined at home and cold spells in March 2020. Those reductions that we attribute to the impacts of COVID-19 remain of comparable magnitude to previous variations induced by cold or warm climate anomalies in the cold season. We conclude that climate variations played a larger role than COVID-19 induced stay-home orders in natural gas consumption across Europe.

Our recent work (Ayral et al., 2020 IEEE Computer Society Annual Symposium on VLSI (ISVLSI)) showed the first implementation of the Quantum Divide and Compute (QDC) method, which allows to break quantum circuits into smaller fragments with fewer qubits and shallower depth. QDC can thus deal with the limited number of qubits and short coherence times of noisy, intermediate-scale quantum processors. This article investigates the impact of different noise sources -- readout error, gate error and decoherence -- on the success probability of the QDC procedure. We perform detailed noise modeling on the Atos Quantum Learning Machine, allowing us to understand tradeoffs and formulate recommendations about which hardware noise sources should be preferentially optimized. We describe in detail the noise models we used to reproduce experimental runs on IBM's Johannesburg processor. This work also includes a detailed derivation of the equations used in the QDC procedure to compute the output distribution of the original quantum circuit from the output distribution of its fragments. Finally, we analyze the computational complexity of the QDC method for the circuit under study via tensor-network considerations, and elaborate on the relation the QDC method with tensor-network simulation methods.

The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations. Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state of the art quantum many-body methods. Upon cooling into its insulating antiferromagnetic ground-state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate temperature metallic regime and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations. We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo) serving as a benchmark. By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes. Based on this analysis, we explain how quasiparticles can coexist with increasingly long-range antiferromagnetic correlations, and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime. We also critically discuss whether imaginary time methods are able to capture the non-Fermi liquid singularities of this fully nested system., Comment: 55 pages, 39 figures, 288 references

We constructed a near-real-time daily CO2 emission dataset, namely the Carbon Monitor, to monitor the variations of CO2 emissions from fossil fuel combustion and cement production since January 1st 2019 at national level with near-global coverage on a daily basis, with the potential to be frequently updated. Daily CO2 emissions are estimated from a diverse range of activity data, including: hourly to daily electrical power generation data of 29 countries, monthly production data and production indices of industry processes of 62 countries/regions, daily mobility data and mobility indices of road transportation of 416 cities worldwide. Individual flight location data and monthly data were utilised for aviation and maritime transportation sectors estimates. In addition, monthly fuel consumption data that corrected for daily air temperature of 206 countries were used for estimating the emissions from commercial and residential buildings. This Carbon Monitor dataset manifests the dynamic nature of CO2 emissions through daily, weekly and seasonal variations as influenced by workdays and holidays, as well as the unfolding impacts of the COVID-19 pandemic. The Carbon Monitor near-real-time CO2 emission dataset shows a 7.8% decline of CO2 emission globally from Jan 1st to Apr 30th in 2020 when compared with the same period in 2019, and detects a re-growth of CO2 emissions by late April which are mainly attributed to the recovery of economy activities in China and partial easing of lockdowns in other countries. Further, this daily updated CO2 emission dataset could offer a range of opportunities for related scientific research and policy making.

Noisy, intermediate-scale quantum computers come with intrinsic limitations in terms of the number of qubits (circuit "width") and decoherence time (circuit "depth") they can have. Here, for the first time, we demonstrate a recently introduced method that breaks a circuit into smaller subcircuits or fragments, and thus makes it possible to run circuits that are either too wide or too deep for a given quantum processor. We investigate the behavior of the method on one of IBM's 20-qubit superconducting quantum processors with various numbers of qubits and fragments. We build noise models that capture decoherence, readout error, and gate imperfections for this particular processor. We then carry out noisy simulations of the method in order to account for the observed experimental results. We find an agreement within 20% between the experimental and the simulated success probabilities, and we observe that recombining noisy fragments yields overall results that can outperform the results without fragmentation., Comment: Accepted in ISVLSI 2020

Invasive alien species are a growing threat for environment and health. They also have a major economic impact, as they can damage many infrastructures. The Japanese knotweed (Fallopia japonica), present in North America, Northern and Central Europe as well as in Australia and New Zealand, is listed by the World Conservation Union as one of the world's worst invasive species. So far, most models have dealt with how the invasion spreads without management. This paper aims at providing a model able to study and predict the dynamics of a stand of Japanese knotweed taking into account mowing as a management technique. The model we propose is stochastic and individual-based, which allows us taking into account the behaviour of individuals depending on their size and location, as well as individual stochasticity. We set plant dynamics parameters thanks to a calibration with field data, and study the influence of the initial population size, the mean number of mowing events a year and the management project duration on mean area and mean number of crowns of stands. In particular, our results provide the sets of parameters for which it is possible to obtain the stand eradication, and the minimal duration of the management project necessary to achieve this latter.