Fast Quantum Algorithms for Trace Distance Estimation

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
$r \cdot \widetilde O \left ({1/\varepsilon ^{2}}\right)$
queries to the quantum circuits that prepare the purifications of quantum states. Then, we modify this quantum algorithm to obtain another algorithm using
$\widetilde O \left ({r^{2}/\varepsilon ^{5}}\right)$
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
$O \left ({\log \left ({N}\right)}\right)$
factor. In addition, we show that the decision version of low-rank trace distance estimation is
$\mathsf {BQP}$
-complete.