forest.benchmarking.direct_fidelity_estimation.estimate_dfe(results: List[forest.benchmarking.observable_estimation.ExperimentResult], kind: str) → Tuple[float, float]

Analyse data from experiments to obtain a direct fidelity estimate (DFE).

State fidelity between the experimental state σ and the ideal (pure) state ρ (both states represented by density matrices) is defined as F(σ,ρ) = tr σ⋅ρ [Joz].

The direct fidelity estimate for a state is given by the average expected value of the Pauli operators in the stabilizer group of the ideal pure state (i.e., Eqn. 1 of [DFE1]).

The average gate fidelity between the experimental process ℰ and the ideal (unitary) process 𝒰 is defined as F(ℰ,𝒰) = (tr ℰ 𝒰⁺ + d)/(d^2+d) where the processes are represented by linear superoperators acting of vectorized density matrices, and d is the dimension of the Hilbert space ℰ and 𝒰 act on (and where ⁺ represents the Hermitian conjugate). See [Nie] for details.

The average gate fidelity can be re-written a F(ℰ,𝒰)=(d^2 tr J(ℰ)⋅J(𝒰) + d)/(d^2+d) where J() is the Choi-Jamiolkoski representation of the superoperator in the argument. Since the Choi-Jamiolkowski representation is given by a density operator, the connection to the calculation of state fidelity becomes apparent: F(J(ℰ),J(𝒰)) = tr J(ℰ)⋅J(𝒰) is the state fidelity between Choi-Jamiolkoski states.

Noting that the Choi-Jamiolkoski state is prepared by acting on half of a maximally entangled state with the process in question, the direct fidelity estimate of the Choi-Jamiolkoski state is given by the average expected value of a Pauli operator resulting from applying the ideal unitary 𝒰 to a Pauli operator Pᵢ, for the state resulting from applying the ideal unitary to a stabilizer state that has Pᵢ in its stabilizer group (one must be careful to prepare states that have both +1 and -1 eigenstates of the operator in question, to emulate the random state preparation corresponding to measuring half of a maximally entangled state).

[Joz]Fidelity for Mixed Quantum States. Jozsa. Journal of Modern Optics. 41:12, 2315-2323 (1994). DOI: 10.1080/09500349414552171
[Nie]A simple formula for the average gate fidelity of a quantum dynamical operation. Nielsen. Phys. Lett. A 303 (4): 249-252 (2002).
  • results – A list of ExperimentResults from running a DFE experiment
  • kind – A string describing the kind of DFE data being analysed (‘state’ or ‘process’)

the estimate of the mean fidelity along with the associated standard err