Published in Quantum 3, 123 (2019)

These entanglement purification circuits were generated by a discreet optimization algorithm. The circuits employ three types of gates: CNOT, CPHASE and CPNOT (the product of the first two). No single-qubit gates are employed. Three types of coincidence measurements are employed: coincidence in the X or Z basis or an anticoincidence in the Y basis. Gate and measurement infidelity are taken into account during the optimization. The fidelity of the final pair is the main component of the cost function. Different error models are possible, but here we consider only depolarization and flips during measurement without any memory errors. Additional gates and measurement types can be added to the optimization.

Code available at https://github.com/Krastanov/qevo/

## Select two circuits to compare:"Ops" is the number of gates and measurements in the circuit. "Steps" is the number of time steps assuming each operation takes one time step, but operations on different qubits can be performed at the same time. "Raw Pairs" is the number of raw Bell pairs consumed by the circuit. | |

## Leading Order BehaviorThe table represents the leading order behavior of the purification circuit. \(3q\) is the infidelity of the raw Bell pairs. \(\varepsilon_{p2}\) and \(\varepsilon_{\eta}\) are the two-qubit-gate and measurement infidelities. The second table is a simplification where \(\varepsilon=\varepsilon_{p2}=\varepsilon_{\eta}\). The third and last table is for \(\varepsilon=0\). \(q\) is highlighted as it is the main source of error. | |

## CircuitBob's side of the circuit. Alice performs the same operations. Shading is "contagious", i.e. it represents which ancillas are currently entangled with the memory qubit. Each operation takes one step, even if operations can be performed in parallel. The small white circles represent the generation of raw Bell pairs. "Ops" is the number of gates and measurements in the circuit. "Steps" is the number of time steps assuming each operation takes one time step, but operations on different qubits can be performed at the same time. "Raw Pairs" is the number of raw Bell pairs consumed by the circuit. Legend: Conditional CNOT and CPHASE gates use the standard notation. CPNOT, the product of those two is represented by overlaying the signs for both gates. The measurements are either coincidence measurements in X (coinX) or Z (coinZ) or anticoincidence measurements in Y (antiY). The small empty circles denote the generation of a new raw Bell pair. | |

## Circuit with ParallelismThe same circuit with operations performed in parallel when possible. The diagram might be confusing because of multiple operations drawn on top of each other. | |

## Average Case Resource UsageA coincidence measurement is not guaranteed to succeed and in such a case a purification procedure might need to backtrack and restart from an earlier point. The left side of the plots contain histograms of Monte Carlo runs of the circuit, where such backtracking is taken into account. Number of steps and number of raw Bell pairs used is tracked. The average value for these quantities is provided as well. The right side of the plots tracks how many raw Bell pairs are used for a fixed probability of success if we backtrack the circuit on failed measurements (the number of raw Bell pairs is a good proxy for how long a circuit will take, as Bell pair production is typically the slowest operation). | |

## Obtainable Fidelity vs Starting Fidelity and Success Probability vs Starting FidelityThe x-axis is always the fidelity of the raw bell pairs. The three lines correspond to three different values for the gate and measurement fidelities. | |

## Compare against 3-qubit Circuits (red triangle marks current circuit)Compare against the entire familly of generated 3-qubit circuits. A number of named 3-qubit circuits from the literature are included as well. Probabilities and Fidelities evaluated at p2=η=0.99 (gate and measurement fidelity) and F0=0.9 (raw Bell pair fidelity, Werner state type). | |