A simple algorithm to reflect through eigenspaces of unitaries

Abstract: Reflections are omnipresent tools in quantum algorithms. We consider the task of reflecting through the eigenspace of an implementable unitary. Such reflections are generally designed using phase estimation or linear combination of unitaries. These methods have size and depth that scale favorably with the desired precision and the spectral gap of the unitary. However, they require a number of ancilla qubits that grows with both parameters. Here, we present a simple algorithm with the same size and depth scaling but requiring only a single ancilla qubit for all problem instances. As such, this algorithm is expected to become the reference method to reflect through eigenspaces of unitaries.