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Quantum process tomography

Characterization

The textbook protocol for fully characterizing a quantum gate: apply it to an informationally complete set of input states, tomograph the outputs, and reconstruct the complete channel.

Measures
The complete quantum channel (process matrix) of a gate or operation, from which figures of merit like average gate fidelity and diamond distance are derived
Introduced by
Chuang & Nielsen, Poyatos, Cirac & Zoller (1996)
Also known as
QPT, Process tomography, Standard quantum process tomography, SQPT

Quantum process tomography (QPT) reconstructs the complete quantum channel implemented by a gate or operation: every coherent and incoherent way it deviates from the ideal. It was proposed independently in late 1996 by Chuang & Nielsen, who framed it as determining the dynamics of a “quantum black box,” and by Poyatos, Cirac & Zoller (both published 1997). It is a characterization protocol rather than a benchmark: it yields a full mathematical description, and comparative scores like average gate fidelity or diamond distance are computed afterward from the reconstruction.

How it works

Prepare each state in an informationally complete input set (for each qubit, e.g., |0⟩, |1⟩, |+⟩, and |+i⟩), apply the process to every input, and run quantum state tomography on each output. Because the inputs span the space of density matrices, the input–output data determine the channel, typically expressed as a process (χ) matrix via linear inversion or maximum-likelihood fitting. An n-qubit channel has of order 16^n parameters, so full QPT is practical only for one to three qubits (Mohseni, Rezakhani & Lidar analyze the resource costs of the main variants).

Strengths and limitations

QPT’s payoff is completeness: it distinguishes coherent miscalibrations from stochastic noise and shows what kind of error a gate makes, not just how much. Its central flaw is that it is not SPAM-robust: it assumes perfectly known input states and measurements, so state-preparation and measurement errors are silently absorbed into the reconstructed channel. That self-consistency failure is what motivated gate set tomography, and for scalable average error rates QPT has largely ceded to randomized benchmarking. “QPT” also names a family (ancilla-assisted, compressed-sensing, and self-consistent variants) of which this entry covers the standard prepare-and-measure scheme.

Notable results

O’Brien et al.’s 2004 full QPT of an optical CNOT gate was a landmark two-qubit demonstration. QPT remains the default for reconstructing one- and two-qubit channels: Qiskit Experiments maintains a ProcessTomography experiment, and it is still routinely applied to entangling gates in current hardware papers.

Key papers

Reference implementations

  • Quantum state tomography : The baseline characterization protocol: reconstructs the full density matrix of a prepared state from repeated measurements in an informationally complete set of bases.
  • Gate set tomography : Calibration-free tomography that reconstructs every gate, state preparation, and measurement in a gate set simultaneously and self-consistently, yielding predictive error models rather than a single score.