Quantum Process Tomography
Process tomography reconstructs the full channel a device implements by probing it with known states and measuring the outputs.
Learning a channel
Quantum process tomography characterizes an unknown quantum operation E completely. The idea is to prepare a set of known input states that span the operator space, send each through the channel, and perform state tomography on the outputs. From these input-output pairs one reconstructs the full description of E, for example its Choi matrix or its chi-matrix in a fixed operator basis.
The chi-matrix representation
Writing E(rho) = sum_{mn} chi_{mn} P_m rho P_n-dagger in a fixed basis of Pauli operators P_m, the channel is captured by the Hermitian, positive chi-matrix. For a single qubit chi is 4x4; for n qubits it is 4^n by 4^n. The diagonal entries give the probabilities of the corresponding Pauli errors, so chi directly exposes the device's error structure.
- Prepare a complete set of input states
- Apply the unknown channel to each
- Measure each output by state tomography
- Invert the data to obtain chi or the Choi matrix
Cost and alternatives
Standard process tomography scales badly: the number of experiments grows as roughly 16^n for n qubits, making full tomography impractical beyond a few qubits. It also conflates state-preparation and measurement (SPAM) errors with the channel itself. These limits motivate alternatives: gate-set tomography separates SPAM from gate errors self-consistently, and randomized benchmarking extracts an average error rate cheaply without full reconstruction.
Uses
Despite its cost, process tomography remains the reference method for fully characterizing small operations, validating new gates, and diagnosing specific error mechanisms during hardware development. The reconstructed Choi matrix can be checked for complete positivity and compared to the ideal gate, yielding process fidelity and revealing whether errors are coherent or incoherent, information that guides calibration and error mitigation.