  In quantum information, trace distance is a basic metric of
distinguishability between quantum states. However, there is no known efficient
approach to estimate the value of trace distance in general. In this paper, we
propose efficient quantum algorithms for estimating the trace distance within
additive error $\varepsilon$ between mixed quantum states of rank $r$.
Specifically, we first provide a quantum algorithm using $r \cdot \widetilde
O(1/\varepsilon^2)$ queries to the quantum circuits that prepare the
purifications of quantum states. Then, we modify this quantum algorithm to
obtain another algorithm using $\widetilde O(r^2/\varepsilon^5)$ samples of
quantum states, which can be applied to quantum state certification. These
algorithms have query/sample complexities that are independent of the dimension
$N$ of quantum states, and their time complexities only incur an extra $O(\log
(N))$ factor. In addition, we show that the decision version of low-rank trace
distance estimation is $\mathsf{BQP}$-complete.
