The Hadamard test is one of the pillars on which quantum algorithm development rests and, at the same time, is naturally suited for the intermediate regime between the current era of noisy quantum devices and complete fault tolerance. Its applications use measurements of the auxiliary qubit to extract information but disregard the system register completely. Concomitantly, but independently of this development, advances in quantum learning theory have enabled the efficient representation of quantum states via classical shadows. This Letter shows that, strikingly, putting both lines of thought into a new context results in substantial improvements to the Hadamard test on a single auxiliary readout qubit, by suitably exploiting classical shadows on the remaining 𝑛 -qubit work register. We argue that this combination inherits the best of both worlds and discuss statistical phase estimation as a vignette application. At the same time, the framework is more general and applicable to a wide range of other algorithms. There, we can use the Hadamard test to estimate energies on the auxiliary qubit, while classical shadows on the system register provide access to additional features such as (i) the fidelity of the initial state with certain pure quantum states, (ii) the initial state’s energy, and (iii) how pure and how close the initial state is to an eigenstate of the Hamiltonian. Finally, we also discuss how anticontrolled unitaries can further augment this framework and explain how this Letter settles the exploitation of the Hadamard test for intermediate applications.
