Muhammad Usman Farooq

arXiv2024

Matrix majorization in large samples

One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Mu et al, we show that if certain monotones - namely multivariate extensions of Rényi divergences - are strictly ordered between the two tuples, then for sufficiently large $n$, there exists a stochastic matrix taking the $n$-fold Kronecker power of each input distribution to the $n$-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

math.ST↗

arXiv2021

Generic detection-based error-mitigation using quantum autoencoders

Efficient error-mitigation techniques demanding minimal resources is key to quantum information processing. We propose a generic protocol to mitigate quantum errors using detection-based quantum autoencoders. In our protocol, the quantum data are compressed into a latent subspace while leaving errors outside, the latter of which is then removed by a measurement and post-selection. Compared to previously developed methods, our protocol on the one hand requires no extra qubits, and on the other hand has a near-optimal denoising power, in which under reasonable requirements all errors detected outside of the latent subspace can be removed, while those inside the subspace cannot be removed by any means. Our detection-based quantum autoencoders are therefore particularly useful for near-term quantum devices in which controllable qubits are limited while noise reduction is important.

quant-ph↗