Michael Perk

Solving systems of linear equations is a key subroutine in many quantum algorithms. In the last 15 years, many quantum linear solvers (QLS) have been developed, competing to achieve the best asymptotic worst-case complexity. Most QLS assume fault-tolerant quantum computers, so they cannot yet be benchmarked on real hardware. Because an algorithm with better asymptotic scaling can underperform on instances of practical interest, the question of which of these algorithms is the most promising remains open. In this work, we implement a method to partially address this question. We consider four well-known QLS algorithms which directly implement an approximate matrix inversion function: the Harrow-Hassidim-Lloyd algorithm, two algorithms utilizing a linear combination of unitaries, and one utilizing the quantum singular value transformation (QSVT). These methods, known as functional QLS, share nearly identical assumptions about the problem setup and oracle access. Their computational cost is dominated by query calls to a matrix oracle encoding the problem one wants to solve. We provide formulas to count the number of queries needed to solve specific problem instances; these can be used to benchmark the algorithms on real-life instances, under benevolent, idealized assumptions, without requiring access to quantum hardware to implement the procedure. We select three data sets: random generated instances that obey the assumptions of functional QLS, linear systems from simplex iterations on MIPLIB, and Poisson equations. Our methods can be easily extended to other data sets and provide a high-level guide to evaluate the performance of a QLS algorithm. In particular, our work shows that HHL underperforms in comparison to the other methods across all data sets, often by orders of magnitude, while the Chebyshev-approximation based methods typically have the best performance.

We present significant progress for scalable and fine-grained computation of large flooding events. As a first step, we describe (I) breakthroughs for simulating time-dependent, high-resolution runoff in large-scale heavy-rain scenarios, based on two algorithmic speedup techniques. The first is (a) a flow-based subdivision technique of the terrain that enables independent, parallel simulation of different regions, while ensuring consistency of the overall simulation. The second is (b) a feature-based algorithmic grid refinement technique. Using this refinement, runtimes for high-resolution hydrodynamic simulations are reduced by 85–90% compared to uniform 1m grids, which can result in better than real-time performance for realistic inputs in medium-sized regions. In combination with state-of-the-art commercial software and powerful computing devices, these techniques make it possible to carry out simulations for large-scale scenarios that were previously infeasible due to memory requirements, and provide considerable speedup for medium- or small-scale scenarios. Combining both techniques, the time-dependent runoff during a five-hour heavy-rain event at 1 m output resolution for an area of 370 km2 in the mountainous Harz region (i.e., for 370,000,000 terrain points with considerable elevation differences and ensuing dynamic flows) can be simulated within three days; without our techniques, such a simulation would not even be possible due to the unmanageable memory requirements. This progress enables us to demonstrate how to address a second challenge: How can we deal with the instability of precipitation events, which are notoriously difficult to predict with good accuracy? We use our new computational capabilities to conduct (II) a large-scale, multifaceted sensitivity analysis, providing insights into different scenarios for possible time-dependent runoff and flooding when varying the input precipitation for our study area. This makes it possible to identify both regions of stability (for which the outcomes are largely unchanged under input variations) as well as critically unstable locations (for which the outcome may change significantly under input variations). In particular the latter is of crucial value for effective disaster mitigation: Knowing where relatively small changes in the input can make a huge difference in the outcome is invaluable when deciding where to invest limited resources with the greatest impact. We demonstrate the viability of this approach by identifying such a critical location for our study area.

We present significant progress for scalable and fine-grained computation of large flooding events. As a first step, we describe (I) breakthroughs for simulating time-dependent, high-resolution runoff in large-scale heavy-rain scenarios, based on two algorithmic speedup techniques. The first is (a) a flow-based subdivision technique of the terrain that enables independent, parallel simulation of different regions, while ensuring consistency of the overall simulation. The second is (b) a feature-based algorithmic grid refinement technique. Using this refinement, runtimes for high-resolution hydrodynamic simulations are reduced by 85–90% compared to uniform 1m grids, which can result in better than real-time performance for realistic inputs in medium-sized regions. In combination with state-of-the-art commercial software and powerful computing devices, these techniques make it possible to carry out simulations for large-scale scenarios that were previously infeasible due to memory requirements, and provide considerable speedup for medium- or small-scale scenarios. Combining both techniques, the time-dependent runoff during a five-hour heavy-rain event at 1 m output resolution for an area of 370 km2 in the mountainous Harz region (i.e., for 370,000,000 terrain points with considerable elevation differences and ensuing dynamic flows) can be simulated within three days; without our techniques, such a simulation would not even be possible due to the unmanageable memory requirements. This progress enables us to demonstrate how to address a second challenge: How can we deal with the instability of precipitation events, which are notoriously difficult to predict with good accuracy? We use our new computational capabilities to conduct (II) a large-scale, multifaceted sensitivity analysis, providing insights into different scenarios for possible time-dependent runoff and flooding when varying the input precipitation for our study area. This makes it possible to identify both regions of stability (for which the outcomes are largely unchanged under input variations) as well as critically unstable locations (for which the outcome may change significantly under input variations). In particular the latter is of crucial value for effective disaster mitigation: Knowing where relatively small changes in the input can make a huge difference in the outcome is invaluable when deciding where to invest limited resources with the greatest impact. We demonstrate the viability of this approach by identifying such a critical location for our study area.

We present significant progress for scalable and fine-grained computation of large flooding events. As a first step, we describe (I) breakthroughs for simulating time-dependent, high-resolution runoff in large-scale heavy-rain scenarios, based on two algorithmic speedup techniques. The first is (a) a flow-based subdivision technique of the terrain that enables independent, parallel simulation of different regions, while ensuring consistency of the overall simulation. The second is (b) a feature-based algorithmic grid refinement technique. Using this refinement, runtimes for high-resolution hydrodynamic simulations are reduced by 85–90% compared to uniform 1m grids, which can result in better than real-time performance for realistic inputs in medium-sized regions. In combination with state-of-the-art commercial software and powerful computing devices, these techniques make it possible to carry out simulations for large-scale scenarios that were previously infeasible due to memory requirements, and provide considerable speedup for medium- or small-scale scenarios. Combining both techniques, the time-dependent runoff during a five-hour heavy-rain event at 1 m output resolution for an area of 370 km² in the mountainous Harz region (i.e., for 370,000,000 terrain points with considerable elevation differences and ensuing dynamic flows) can be simulated within three days; without our techniques, such a simulation would not even be possible due to the unmanageable memory requirements. This progress enables us to demonstrate how to address a second challenge: How can we deal with the instability of precipitation events, which are notoriously difficult to predict with good accuracy? We use our new computational capabilities to conduct (II) a large-scale, multifaceted sensitivity analysis, providing insights into different scenarios for possible time-dependent runoff and flooding when varying the input precipitation for our study area. This makes it possible to identify both regions of stability (for which the outcomes are largely unchanged under input variations) as well as critically unstable locations (for which the outcome may change significantly under input variations). In particular the latter is of crucial value for effective disaster mitigation: Knowing where relatively small changes in the input can make a huge difference in the outcome is invaluable when deciding where to invest limited resources with the greatest impact. We demonstrate the viability of this approach by identifying such a critical location for our study area.

We present significant progress for scalable and fine-grained computation of large flooding events. As a first step, we describe (I) breakthroughs for simulating time-dependent, high-resolution runoff in large-scale heavy-rain scenarios, based on two algorithmic speedup techniques. The first is (a) a flow-based subdivision technique of the terrain that enables independent, parallel simulation of different regions, while ensuring consistency of the overall simulation. The second is (b) a feature-based algorithmic grid refinement technique. Using this refinement, runtimes for high-resolution hydrodynamic simulations are reduced by 85–90% compared to uniform 1m grids, which can result in better than real-time performance for realistic inputs in medium-sized regions. In combination with state-of-the-art commercial software and powerful computing devices, these techniques make it possible to carry out simulations for large-scale scenarios that were previously infeasible due to memory requirements, and provide considerable speedup for medium- or small-scale scenarios. Combining both techniques, the time-dependent runoff during a five-hour heavy-rain event at 1 m output resolution for an area of 370 km² in the mountainous Harz region (i.e., for 370,000,000 terrain points with considerable elevation differences and ensuing dynamic flows) can be simulated within three days; without our techniques, such a simulation would not even be possible due to the unmanageable memory requirements. This progress enables us to demonstrate how to address a second challenge: How can we deal with the instability of precipitation events, which are notoriously difficult to predict with good accuracy? We use our new computational capabilities to conduct (II) a large-scale, multifaceted sensitivity analysis, providing insights into different scenarios for possible time-dependent runoff and flooding when varying the input precipitation for our study area. This makes it possible to identify both regions of stability (for which the outcomes are largely unchanged under input variations) as well as critically unstable locations (for which the outcome may change significantly under input variations). In particular the latter is of crucial value for effective disaster mitigation: Knowing where relatively small changes in the input can make a huge difference in the outcome is invaluable when deciding where to invest limited resources with the greatest impact. We demonstrate the viability of this approach by identifying such a critical location for our study area.

We present significant progress for scalable and fine-grained computation of large flooding events. As a first step, we describe (I) breakthroughs for simulating time-dependent, high-resolution runoff in large-scale heavy-rain scenarios, based on two algorithmic speedup techniques. The first is (a) a flow-based subdivision technique of the terrain that enables independent, parallel simulation of different regions, while ensuring consistency of the overall simulation. The second is (b) a feature-based algorithmic grid refinement technique. Using this refinement, runtimes for high-resolution hydrodynamic simulations are reduced by 85–90% compared to uniform 1m grids, which can result in better than real-time performance for realistic inputs in medium-sized regions. In combination with state-of-the-art commercial software and powerful computing devices, these techniques make it possible to carry out simulations for large-scale scenarios that were previously infeasible due to memory requirements, and provide considerable speedup for medium- or small-scale scenarios. Combining both techniques, the time-dependent runoff during a five-hour heavy-rain event at 1 m output resolution for an area of 370 km² in the mountainous Harz region (i.e., for 370,000,000 terrain points with considerable elevation differences and ensuing dynamic flows) can be simulated within three days; without our techniques, such a simulation would not even be possible due to the unmanageable memory requirements. This progress enables us to demonstrate how to address a second challenge: How can we deal with the instability of precipitation events, which are notoriously difficult to predict with good accuracy? We use our new computational capabilities to conduct (II) a large-scale, multifaceted sensitivity analysis, providing insights into different scenarios for possible time-dependent runoff and flooding when varying the input precipitation for our study area. This makes it possible to identify both regions of stability (for which the outcomes are largely unchanged under input variations) as well as critically unstable locations (for which the outcome may change significantly under input variations). In particular the latter is of crucial value for effective disaster mitigation: Knowing where relatively small changes in the input can make a huge difference in the outcome is invaluable when deciding where to invest limited resources with the greatest impact. We demonstrate the viability of this approach by identifying such a critical location for our study area.

We present a simple approach to laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) U-Pb dating of carbonate minerals from isotopic image maps, made possible using a high repetition rate femtosecond laser ablation system. The isotopic ratio maps are built from 25-µm-height linear scans, at a minimal repetition rate of 100 Hz. The analysis of ²³⁸U, ²³²Th, ²⁰⁸Pb, ²⁰⁷Pb and ²⁰⁶Pb masses by a sector field ICP-MS is set to maximize the number of mass sweeps, and thus of pixels on the produced image maps (∼8 to 19 scans s^-1). After normalization by sample standard bracketing using the Iolite 4 software, the isotopic image maps are discretized into squares. The squares correspond to virtual spots of chosen dimension, for which the mean and its uncertainty are calculated, allowing to plot corresponding concordia diagrams commonly used to obtain an absolute age. Because the ratios can vary strongly at the pixel scale, the values obtained from the virtual spots display higher uncertainties compared to static spots of similar size. However, their size, and thus the number of virtual spots can be easily adapted. A low size will result in higher uncertainty of individual spots, but their higher number and potentially larger spread along the isochron can result in a more precise age. Reliability of this approach is improved by using a mobile grid for the virtual spot dataset of a set size, returning numerous concordia allowing to select the more statistically robust result. One can also select from all the possible spot locations on the image map, those that will enable regression to be obtained with the best goodness of fit. We present examples of the virtual spot approach, for which in the most favorable cases (U > 1 ppm, ²³⁸U/²⁰⁶Pb >> 1, and highly variable U/Pb ratios) a valid age can be obtained within reasonable uncertainty (< 5–10 %) from maps as small as 100 µm x 100 µm, i.e. the size of a single spot in common in situ approaches. Although the method has been developed on carbonates, it should be applicable to other minerals suited to U-Pb geochronology.

We present a simple approach to laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) U-Pb dating of carbonate minerals from isotopic image maps, made possible using a high repetition rate femtosecond laser ablation system. The isotopic ratio maps are built from 25-µm-height linear scans, at a minimal repetition rate of 100 Hz. The analysis of ²³⁸U, ²³²Th, ²⁰⁸Pb, ²⁰⁷Pb and ²⁰⁶Pb masses by a sector field ICP-MS is set to maximize the number of mass sweeps, and thus of pixels on the produced image maps (∼8 to 19 scans s^-1). After normalization by sample standard bracketing using the Iolite 4 software, the isotopic image maps are discretized into squares. The squares correspond to virtual spots of chosen dimension, for which the mean and its uncertainty are calculated, allowing to plot corresponding concordia diagrams commonly used to obtain an absolute age. Because the ratios can vary strongly at the pixel scale, the values obtained from the virtual spots display higher uncertainties compared to static spots of similar size. However, their size, and thus the number of virtual spots can be easily adapted. A low size will result in higher uncertainty of individual spots, but their higher number and potentially larger spread along the isochron can result in a more precise age. Reliability of this approach is improved by using a mobile grid for the virtual spot dataset of a set size, returning numerous concordia allowing to select the more statistically robust result. One can also select from all the possible spot locations on the image map, those that will enable regression to be obtained with the best goodness of fit. We present examples of the virtual spot approach, for which in the most favorable cases (U > 1 ppm, ²³⁸U/²⁰⁶Pb >> 1, and highly variable U/Pb ratios) a valid age can be obtained within reasonable uncertainty (< 5–10 %) from maps as small as 100 µm x 100 µm, i.e. the size of a single spot in common in situ approaches. Although the method has been developed on carbonates, it should be applicable to other minerals suited to U-Pb geochronology.