Learning marginals suffices!

Abstract: Beyond computer science, quantum complexity theory can potentially revolutionize multiple branches of physics, ranging from quantum many-body systems to quantum field theory. In this paper, we investigate the relationship between the sample complexity of learning a quantum state and the circuit complexity of the state. The circuit complexity of a quantum state refers to the minimum depth of the quantum circuit necessary to implement it. We show that learning its marginals for the quantum state with low circuit complexity suffices for state tomography, thus breaking the exponential barrier of the sample complexity for quantum state tomography. Our proof is elementary and overcomes difficulties characterizing short-range entanglement by bridging quantum circuit complexity and ground states of gapped local Hamiltonians. Our result, for example, settles the quantum circuit complexity of the multi-qubit GHZ state exactly.