Compound poisson approximation via information functionals

An information-theoretic development is given for the problem of compound Poisson approximation, which parallels earlier treatments for Gaussian and Poisson approximation. Nonasymptotic bounds are derived for the distance between the distribution of a sum of independent integer-valued random variables and an appropriately chosen compound Poisson law. In the case where all summands have the same conditional distribution given that they are non-zero, a bound on the relative entropy distance between their sum and the compound Poisson distribution is derived, based on the data-processing property of relative entropy and earlier Poisson approximation results. When the summands have arbitrary distributions, corresponding bounds are derived in terms of the total variation distance. The main technical ingredient is the introduction of two "information functionals,'' and the analysis of their properties. These information functionals play a role analogous to that of the classical Fisher information in normal approximation. Detailed comparisons are made between the resulting inequalities and related bounds

On Deterministically Approximating Total Variation Distance

Total variation distance (TV distance) is an important measure for the difference between two distributions. Recently, there has been progress in approximating the TV distance between product distributions: a deterministic algorithm for a restricted class of product distributions (Bhattacharyya, Gayen, Meel, Myrisiotis, Pavan and Vinodchandran 2023) and a randomized algorithm for general product distributions (Feng, Guo, Jerrum and Wang 2023). We give a deterministic fully polynomial-time approximation algorithm (FPTAS) for the TV distance between product distributions. Given two product distributions $\mathbb{P}$ and $\mathbb{Q}$ over $[q]^n$, our algorithm approximates their TV distance with relative error $\varepsilon$ in time $O\bigl( \frac{qn^2}{\varepsilon} \log q \log \frac{n}{\varepsilon \Delta_{\text{TV}}(\mathbb{P},\mathbb{Q}) } \bigr)$. Our algorithm is built around two key concepts: 1) The likelihood ratio as a distribution, which captures sufficient information to compute the TV distance. 2) We introduce a metric between likelihood ratio distributions, called the minimum total variation distance. Our algorithm computes a sparsified likelihood ratio distribution that is close to the original one w.r.t. the new metric. The approximated TV distance can be computed from the sparsified likelihood ratio. Our technique also implies deterministic FPTAS for the TV distance between Markov chains

Bounds for Approximation in Total Variation Distance by Quantum Circuits

It was recently shown that for reasonable notions of approximation of states and functions by quantum circuits, almost all states and functions are exponentially hard to approximate [Knill 1995]. The bounds obtained are asymptotically tight except for the one based on total variation distance (TVD). TVD is the most relevant metric for the performance of a quantum circuit. In this paper we obtain asymptotically tight bounds for TVD. We show that in a natural sense, almost all states are hard to approximate to within a TVD of 2/e-\epsilon even for exponentially small \epsilon. The quantity 2/e is asymptotically the average distance to the uniform distribution. Almost all states with probability amplitudes concentrated in a small fraction of the space are hard to approximate to within a TVD of 2-\epsilon. These results imply that non-uniform quantum circuit complexity is non-trivial in any reasonable model. They also reinforce the notion that the relative information distance between states (which is based on the difficulty of transforming one state to another) fully reflects the dimensionality of the space of qubits, not the number of qubits.Comment: uuencoded compressed postscript, LACES 68Q-95-3