F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits

Generative modelling is an important unsupervised task in machine learning. In this work, we study a hybrid quantum-classical approach to this task, based on the use of a quantum circuit born machine. In particular, we consider training a quantum circuit born machine using f-divergences. We first discuss the adversarial framework for generative modelling, which enables the estimation of any f-divergence in the near term. Based on this capability, we introduce two heuristics which demonstrably improve the training of the born machine. The first is based on f-divergence switching during training. The second introduces locality to the divergence, a strategy which has proved important in similar applications in terms of mitigating barren plateaus. Finally, we discuss the long-term implications of quantum devices for computing f-divergences, including algorithms which provide quadratic speedups to their estimation. In particular, we generalise existing algorithms for estimating the Kullback–Leibler divergence and the total variation distance to obtain a fault-tolerant quantum algorithm for estimating another f-divergence, namely, the Pearson divergence.

On the mixing time of coordinate Hit-and-Run

We obtain a polynomial upper bound on the mixing time $T_{CHR}(\epsilon)$ of the coordinate Hit-and-Run (CHR) random walk on an $n-$ dimensional convex body, where $T_{CHR}(\epsilon)$ is the number of steps needed to reach within $\epsilon$ of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex body that is bounded above by a constant). Our upper bound is polynomial in n, R and $\frac{1}{\epsilon}$ , where we assume that the convex body contains the unit $\Vert\cdot\Vert_\infty$ -unit ball $B_\infty$ and is contained in its R-dilation $R\cdot B_\infty$ . Whether CHR has a polynomial mixing time has been an open question.

Quantitative normal approximations for the stochastic fractional heat equation

In this article we present a quantitative central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space–time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius R converges, as R tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.

Analysis of Gas Content and Permeability Change Pattern of a Coal Reservoir in the Tectonic Positions Based on a THM Coupled Model

The theory of coalbed methane distribution controlled by tectonism is a hot issue in the field of geofluid-geotectonic interaction research. Taking the geological structure in the scale range of the 1302 working face of a Guojiahe wellfield in a Yonglong mining area as the background, this paper focuses on the basic research problem of the influence of geological structure on the control of coal reservoir gas content and uses a THM coupling model to analyze the change of coalbed gas content and distribution characteristics of different tectonic positions. The change of CBM content and permeability in the anticline, syncline, and faults is analyzed. Accordingly, the variation distance of gas content and reservoir permeability controlled by tectonism of different geological structures is quantified to provide guidance for the selection of CBM-favorable areas. The research results show that the gentle dip syncline hinge zone is a potential gas-rich area with heat preservation and low permeability, while the gentle dip anticline hinge zone is a gas-poor area with low temperature and low pressure and high permeability. The thick coal seam zone of the syncline hinge zone is the potential gas accumulation zone, and the high-permeability area is near the fault plane of a normal fault. The coal matrix near the normal fault is subjected to tensile tectonic stresses to form tensional fissures, and the coal seam in the fault plane area is susceptible to coalbed gas dispersion and increases the permeability of the coal reservoir. The variation distance of gas content and reservoir permeability controlled by the normal fault within the Guojiahe wellfield is 37 m and 54 m from the fault plane, respectively.

Simulation of a Random Variable and its Application to Game Theory

We provide a new tool for simulation of a random variable (target source) from a randomness source with side information. Considering the total variation distance as the measure of precision, this tool offers an upper bound for the precision of simulation, which is vanishing exponentially in the difference of Rényi entropies of the randomness and target sources. This tool finds application in games in which the players wish to generate their actions (target source) as a function of a randomness source such that they are almost independent of the observations of the opponent (side information). In particular, we study zero-sum repeated games in which the players are restricted to strategies that require only a limited amount of randomness. Let be the max-min value of the n stage game. Previous works have characterized [Formula: see text], that is, the long-run max-min value, but they have not provided any result on the value of vn for a given finite n-stage game. Here, we utilize our new tool to study how vn converges to the long-run max-min value.

Trading Locality for Time: Certifiable Randomness from Low-Depth Circuits

The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation and more recent proposals for randomness certification based on computational assumptions. Furthermore, to demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 362(6412):308–311, 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our results lead to a new proposal for a demonstrated quantum advantage that has some advantages compared to existing proposals. First, our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Second, success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.