Alladi Ramakrishnan Hall

IID and problem-specific samples of quantum states from Wishart distributions

Shrobona Bagchi

Center for Quantum Information, Korea Institute of Science and Technology, Seoul, South Korea

Random samples of quantum states are an important resource for
various tasks in quantum information science, and samples in accordance
with a problem-specific distribution can be indispensable ingredients. Some
algorithms generate random samples by a lottery that follows certain rules
and yield samples from the set of distributions that the lottery can
access. Other algorithms, which use random walks in the state space like
the Monte Carlo, can be tailored to any distribution, at the price of
autocorrelations in the sample and with restrictions to low-dimensional
systems in practical implementations. We present a two-step algorithm for
sampling from the quantum state space that overcomes some of these
limitations. We first produce a CPU-cheap large proposal sample, of
uncorrelated entries, by drawing from the family of complex
Wishartdistributions,
and then reject or accept the entries in the proposal sample such that the
accepted sample is strictly in accordance with the target distribution. We
establish the explicit form of the induced Wishart distribution for quantum
states. This enables us to generate a proposal sample that mimics the
target distribution and, therefore, the efficiency of the algorithm,
measured by the acceptance rate, can be many orders of magnitude larger
than that for a uniform sample as the proposal. We demonstrate that this
sampling algorithm is very efficient for one-qubit and two-qubit states,
and reasonably efficient for three-qubit states, while it suffers from the
"curse of dimensionality" when sampling from structured distributions of
four-qubit states.