  We propose a scheme to implement general quantum measurements, also known as
Positive Operator Valued Measures (POVMs) in dimension $d$ using only classical
resources and a single ancillary qubit. Our method is based on the
probabilistic implementation of $d$-outcome measurements which is followed by
postselection of some of the received outcomes. We conjecture that the success
probability of our scheme is larger than a constant independent of $d$ for all
POVMs in dimension $d$. Crucially, this conjecture implies the possibility of
realizing arbitrary nonadaptive quantum measurement protocol on a
$d$-dimensional system using a single auxiliary qubit with only a
\emph{constant} overhead in sampling complexity. We show that the conjecture
holds for typical rank-one Haar-random POVMs in arbitrary dimensions.
Furthermore, we carry out extensive numerical computations showing success
probability above a constant for a variety of extremal POVMs, including
SIC-POVMs in dimension up to 1299. Finally, we argue that our scheme can be
favourable for the experimental realization of POVMs, as noise compounding in
circuits required by our scheme is typically substantially lower than in the
standard scheme that directly uses Naimark's dilation theorem.
