Analog quantum simulation of partial differential equations

Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular - Schrodinger's equation. Can quantum simulators also efficiently simulate other PDEs? While most computational methods for PDEs - both classical and quantum - are digital (PDEs must be discretised first), PDEs have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrodingerisation, we show how to directly map D-dimensional linear PDEs onto a (D+1)-qumode quantum system where analog or continuous-variable Hamiltonian simulation on D+1 qumodes can be used. This very simple methodology does not require one to discretise PDEs first, and it is not only applicable to linear PDEs but also to some nonlinear PDEs and systems of nonlinear ODEs. We show some examples using this method, including the Liouville equation, heat equation, Fokker-Planck equation, Black-Scholes equations, wave equation and Maxwell's equations. We also devise new protocols for linear PDEs with random coefficients, important in uncertainty quantification, where it is clear how the analog or continuous-variable framework is most natural. This also raises the possibility that some PDEs may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems

Quantum simulation of partial differential equations via Schrodingerisation

We present a simple new way - called Schrodingerisation - to simulate general linear partial differential equations via quantum simulation. Using a simple new transform, referred to as the warped phase transformation, any linear partial differential equation can be recast into a system of Schrodinger's equations - in real time - in a straightforward way. This can be seen directly on the level of the dynamical equations without more sophisticated methods. This approach is not only applicable to PDEs for classical problems but also those for quantum problems - like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit

Quantum simulation of Maxwell's equations via Schr\"odingersation

We present quantum algorithms for electromagnetic fields governed by Maxwell's equations. The algorithms are based on the Schr\"odingersation approach, which transforms any linear PDEs and ODEs with non-unitary dynamics into a system evolving under unitary dynamics, via a warped phase transformation that maps the equation into one higher dimension. In this paper, our quantum algorithms are based on either a direct approximation of Maxwell's equations combined with Yee's algorithm, or a matrix representation in terms of Riemann-Silberstein vectors combined with a spectral approach and an upwind scheme. We implement these algorithms with physical boundary conditions, including perfect conductor and impedance boundaries. We also solve Maxwell's equations for a linear inhomogeneous medium, specifically the interface problem. Several numerical experiments are performed to demonstrate the validity of this approach. In addition, instead of qubits, the quantum algorithms can also be formulated in the continuous variable quantum framework, which allows the quantum simulation of Maxwell's equations in analog quantum simulation

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schr\"odinger framework to solve the Liouville equation for the HJE, since it can be recast as the semiclassical limit of the Wigner transform of the Schr\"odinger equation. Comparsion between the Schr\"odinger and the Liouville framework will also be made.Comment: quantum algorithms,linear representations,noninea

Dilation theorem via Schr\"odingerisation, with applications to the quantum simulation of differential equations

Nagy's unitary dilation theorem in operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block-encoding techniques, based on finite-dimensional scenarios. In this study, we delve into the recently devised Schr\"odingerisation approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of $V(t)=\exp(-At)$, which arises in wide-ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schr\"odingerisation approach is adaptable to both finite and infinite-dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous-variable version of the Schr\"odingerisation procedure which makes it suitable for analog quantum computing. Furthermore, by discretising continuous variables, the Schr\"odingerisation method can also be effectively employed in finite-dimensional scenarios suitable for qubit-based quantum computing

Synthetical Analysis on Geological Factors Ccontrolling Coalbed Methane

AbstractThe gas-controlling property is the important content for the coalbed methane (CBM) theoretical research, and it has the important role for guiding the CBM exploration and development. The evolution features of the coal-bearing strata and structure and the current CBM preservation condition are the keys determining the CBM enrichment and reservoir formation. In the case that the earth curst is stable in the sedimentary period of the coal-bearing strata and after the coal-bearing strata are deposited, the coal seam deposited by the coal-accumulation has the large and stable thickness, the earth curst is stably subsided after the coal-accumulation period or the strength of the structural movement is low and the uplifted amplitude is little, then it is favorable for the CBM enrichment. In the area there the coal-bearing strata have the simple structure, the enclosing rock of coal seam is stable and compact, the seam buried depth is deep, and in the stagnant area with the simple hydrogeological condition, the CBM-controlling property is well. The research on the CBM-controlling property is restricted by the exploration degree, with respect to the area with the low exploration degree, the research on the CBM-controlling property could be combined with the exploration results of the area with the high exploration degree, on the basis of analysing the CBM distribution features and control factors of the area with the high exploration degree, adopting the analysis method such as the geological analogy and so on, it conducts the research work from the evolution features of the coal-bearing strata andstructure and the current CBM preservation condition

Dense MoS2 Micro‐Flowers Planting on Biomass‐Derived Carbon Fiber Network for Multifunctional Sulfur Cathodes

The significant challenge in lithium‐sulfur batteries (LSBs) arises from low conductivity of sulfur cathode, loss of active sulfur species due to less anchoring sites and sluggish redox kinetics of lithium polysulfides (LPSs). Herein, the dense MoS2 micro‐flowers assembled by cross‐linked 2D MoS2 nanoflakes planting on biomass‐derived carbon fiber (CF) network (MoS2/CFs) are fabricated as multifunctional sulfur cathodes of LSBs. The 2D MoS2 nanoflakes supported on CF provide abundant anchoring sites for strong adsorption, while the 3D flowerlike structure prevents lamellar aggregation of 2D MoS2 nanoflakes. Importantly, the dense MoS2 micro‐flowers planting on the network weaved by biomass‐derived CFs ensures the high electronic conductivity of the MoS2/CFs composite, sufficient electrode/electrolyte interaction, fast electron and Li+ transportation. Moreover, the CF network weaved from cost‐effective tissue paper reduces the cost of LSBs. Thus, the S‐MoS2/CFs cathode exhibits a high rate capability (1149 and 608 mA h g−1 are obtained at 0.2 C and 4 C, respectively), excellent cyclic performance with ∼75% capacity retention and 99% Coulombic efficiency at 2 C after 500 cycles, corresponding to ∼0.05% capacity fading per cycle only, as well as structure integrity during the discharge/charge process.800 Dong Chuan Road, Minhang District, Shanghai 200240, ChinaA novel, cost‐effective, dense 3 D MoS2 micro‐flowers assembled by cross‐linked 2D MoS2 nanoflakes planting on biomass‐derived carbon fiber (CF) network (MoS2/CFs) are fabricated as multifunctional sulfur cathodes of LSBs. The 2D MoS2 nanoflakes provide abundant anchoring sites for strong adsorption, while the 3D flowerlike structure prevents lamellar aggregation of 2D MoS2 nanoflakes. Significantly, the dense MoS2 micro‐flowers supported on carbon fibers ensures the high electronic conductivity of the MoS2/CFs composite, sufficient electrode/electrolyte interaction, fast electron and Li+ transportation.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/155938/1/slct202001729-sup-0001-misc_information.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/155938/2/slct202001729_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/155938/3/slct202001729.pd

Electrocatalytic conversion of lithium polysulfides by highly dispersed ultrafine Mo2C nanoparticles on hollow N‐doped carbon flowers for Li‐S batteries

The significant challenge in exploring novel nanostructured sulfur host materials for Li‐S batteries is to simultaneously mitigate the notorious shuttle effect and catalytically enhance the redox kinetics of lithium polysulfides (LPSs). Herein, a novel ultrafine Mo2C nanoparticles uniformly distributed on 2D nanosheet‐assembled 3D hollow nitrogen‐doped carbon flowers (HNCFs) is designed. The Mo2C/HNCFs architecture with unique flower‐like morphologies not only efficiently suppressed the aggregation of 2D nanosheets but also highly distributed the ultrafine Mo2C nanoparticles that act as catalytic active sites for efficient adsorption and conversion of LPSs. Furthermore, the 3D hierarchical arrangement can afford ample internal space to accommodate sulfur species, large volume expansion, 3D electron pathway, and physical/chemical blockage of LPSs to reduce the loss of active materials. The Mo2C/HNCFs composite exhibits a high rate capability, unprecedented capacity retention of 92% over 100 cycles at 0.5 C placing Mo2C/HNCFs one of the best LPSs adsorbents and electrocatalysts.Ultrafine Mo2C nanoparticles on hollow N‐doped carbon flowers have been employed as efficient catalytic active sites for conversion of LPSs, which can not only enhance the LPSs‐adsorption ability but also accelerate the redox kinetics of polysulfide conversion. Besides, the unique architecture of 2D nanosheets assembled 3D hollow N‐doped carbon flowers contributes to Li+ transportation and electrolyte infiltration.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/155989/1/eom212020.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/155989/2/eom212020_am.pd

Ocean acidification increases the accumulation of toxic phenolic compounds across trophic levels

Increasing atmospheric CO2 concentrations are causing ocean acidification (OA), altering carbonate chemistry with consequences for marine organisms. Here we show that OA increases by 46–212% the production of phenolic compounds in phytoplankton grown under the elevated CO2 concentrations projected for the end of this century, compared with the ambient CO2 level. At the same time, mitochondrial respiration rate is enhanced under elevated CO2 concentrations by 130–160% in a single species or mixed phytoplankton assemblage. When fed with phytoplankton cells grown under OA, zooplankton assemblages have significantly higher phenolic compound content, by about 28–48%. The functional consequences of the increased accumulation of toxic phenolic compounds in primary and secondary producers have the potential to have profound consequences for marine ecosystem and seafood quality, with the possibility that fishery industries could be influenced as a result of progressive ocean change