Quantum Overlapping Tomography

It is now experimentally possible to entangle thousands of qubits, and efficiently measure each qubit in parallel in a distinct basis. To fully characterize an unknown entangled state of $n$ qubits, one requires an exponential number of measurements in $n$, which is experimentally unfeasible even for modest system sizes. By leveraging (i) that single-qubit measurements can be made in parallel, and (ii) the theory of perfect hash families, we show that all $k$-qubit reduced density matrices of an $n$ qubit state can be determined with at most $e^{\mathcal{O}(k)} \log^2(n)$ rounds of parallel measurements. We provide concrete measurement protocols which realize this bound. As an example, we argue that with current experiments, the entanglement between every pair of qubits in a system of 1000 qubits could be measured and completely characterized in a few days. This corresponds to completely characterizing entanglement of nearly half a million pairs of qubits.