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Quantum state tomography, which aims to estimate quantum states that are described by density matrices, plays an important role in quantum science and quantum technology. This paper examines the eigenspace estimation and the reconstruction of large low-rank density matrix based on Pauli measurements. Both ordinary principal component analysis (PCA) and iterative thresholding sparse PCA (ITSPCA) estimators of the eigenspace are studied, and their respective convergence rates are established. In particular, we show that the ITSPCA estimator is rate-optimal. We present the reconstruction of the large low-rank density matrix and obtain its optimal convergence rate by using the ITSPCA estimator. A numerical study is carried out to investigate the finite sample performance of the proposed estimators. 

This paper introduces a unified approach for modeling high-frequency financial data that can accommodate both the continuous-time jump–diffusion and discrete-time realized GARCH model by embedding the discrete realized GARCH structure in the continuous instantaneous volatility process. The key feature of the proposed model is that the corresponding conditional daily integrated volatility adopts an autoregressive structure, where both integrated volatility and jump variation serve as innovations. We name it as the realized GARCH-Itô model. Given the autoregressive structure in the conditional daily integrated volatility, we propose a quasi-likelihood function for parameter estimation and establish its asymptotic properties. To improve the parameter estimation, we propose a joint quasi-likelihood function that is built on the marriage of daily integrated volatility estimated by high-frequency data and nonparametric volatility estimator obtained from option data. We conduct a simulation study to check the finite sample performance of the proposed methodologies and an empirical study with the S&P500 stock index and option data. 

We present FireSim, an open-source simulation platform that enables cycle-exact microarchitectural simulation of large scale-out clusters by combining FPGA-accelerated simulation of silicon-proven RTL designs with a scalable, distributed network simulation. Unlike prior FPGA-accelerated simulation tools, FireSim runs on Amazon EC2 F1, a public cloud FPGA platform, which greatly improves usability, provides elasticity, and lowers the cost of large-scale FPGA-based experiments. We describe the design and implementation of FireSim and show how it can provide sufficient performance to run modern applications at scale, to enable true hardware-software co-design. As an example, we demonstrate automatically generating and deploying a target cluster of 1,024 3.2 GHz quad-core server nodes, each with 16 GB of DRAM, interconnected by a 200 Gbit/s network with 2 microsecond latency, which simulates at a 3.4 MHz processor clock rate (less than 1,000x slowdown over real-time). In aggregate, this FireSim instantiation simulates 4,096 cores and 16 TB of memory, runs ~ 14 billion instructions per second, and harnesses 12.8 million dollars worth of FPGAs-at a total cost of only ~ $100 per simulation hour to the user. We present several examples to show how FireSim can be used to explore various research directions in warehouse-scale machine design, including modeling networks with high-bandwidth and low-latency, integrating arbitrary RTL designs for a variety of commodity and specialized datacenter nodes, and modeling a variety of datacenter organizations, as well as reusing the scale-out FireSim infrastructure to enable fast, massively parallel cycle-exact single-node microarchitectural experimentation.