  The ability of quantum computers to directly manipulate and analyze quantum
states stored in quantum memory allows them to learn about aspects of our
physical world that would otherwise be invisible given a modest number of
measurements. Here we investigate a new learning resource which could be
available to quantum computers in the future -- measurements on the unknown
state accompanied by its complex conjugate $\rho \otimes \rho^\ast$. For a
certain shadow tomography task, we surprisingly find that measurements on only
copies of $\rho \otimes \rho^\ast$ can be exponentially more powerful than
measurements on $\rho^{\otimes K}$, even for large $K$. This expands the class
of provable exponential advantages using only a constant overhead quantum
memory, or minimal quantum memory, and we provide a number of examples where
the state $\rho^\ast$ is naturally available in both computational and physical
applications. In addition, we precisely quantify the power of classical shadows
on single copies under a generalized Clifford ensemble and give a class of
quantities that can be efficiently learned. The learning task we study in both
the single copy and quantum memory settings is physically natural and
corresponds to real-space observables with a limit of bosonic modes, where it
achieves an exponential improvement in detecting certain signals under a noisy
background. We quantify a new and powerful resource in quantum learning, and we
believe the advantage may find applications in improving quantum simulation,
learning from quantum sensors, and uncovering new physical phenomena.
