Exponential learning advantages with conjugate states and minimal quantum memory

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.