Quantum Process Tomography
Process tomography fully reconstructs what an operation does to every input state, giving a complete but expensive picture of a gate.
Characterizing a whole operation
State tomography reconstructs an unknown quantum state by measuring many copies in different bases. Process tomography goes further: it reconstructs the entire operation (channel) a gate performs, by feeding in a complete set of known input states and doing state tomography on each output. The result is a process matrix that predicts the output for any input.
The process matrix
A quantum operation is fully described by a chi matrix in a fixed operator basis. For a single qubit that is a 4-by-4 matrix; for n qubits it grows as 4^n by 4^n. From it one reads coherent errors (over- or under-rotations), incoherent errors (decoherence), and leakage, information a single fidelity number cannot convey.
The cost
- Number of settings grows exponentially with qubit count
- Results are contaminated by state-preparation and measurement (SPAM) errors
- Reconstruction can return unphysical matrices without careful fitting
Gate-set tomography
Gate-set tomography (GST) improves on standard process tomography by treating preparation, gates, and measurement as unknowns to be solved together, removing the SPAM bias that limits ordinary tomography. GST is self-consistent and highly accurate but even more resource-intensive, so it is used to deeply characterize a few gates rather than a whole processor.
In practice, tomography is a diagnostic tool for small systems and individual gates, complementing scalable methods like randomized benchmarking. It tells engineers not just how much error a gate has, but what kind, which is what guides the next calibration or design change.