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In this paper, we present an efficient weak measurement-based scheme fordirect quantum state tomography (DQST) and direct quantum process tomography(DQPT), and experimentally implement it on an NMR ensemble quantum informationprocessor without involving any projective measurements. We develop ageneralized quantum circuit that enables us to directly measure selectedelements of the density matrix and process matrix which characterize unknownquantum states and processes, respectively. This generalized quantum circuituses the scalar J-coupling to control the interaction strength between thesystem qubits and the metre qubit. We experimentally implement these weakmeasurement-based DQST and DQPT protocols and use them to accuratelycharacterize several two-qubit quantum states and single-qubit quantumprocesses. An extra qubit is used as a metre qubit to implement the DQSTprotocol, while for the DQPT protocol, two extra qubits (one as a metre qubitand the other as an ancilla qubit) are used.

Direct tomography of quantum states and processes via weak measurements of Pauli spin operators on an NMR quantum processor