PMC

In the verification, input states are under control of the verifier, and noisy commuting quantum gates are verified by using the measurement outcomes.

(a) A unit cell of the RHG lattice (a graph of degree four) represents a depth-four commuting quantum circuit. (b) A non-constant depth commuting quantum circuit for a direct magic state injection.

(a–c) The comparison of the distribution of commuting distance, the distribution of the number of commuters between O-D pairs, and the comparison of the census commuting flow T_ij with the expanded cell phone users' commuting flow . The close fitting in all three figures shows that we can recover the commuting patterns of the whole population from the seed matrix provided by the cell phone records. (d–h) Comparison of the distributions of commuting distance for Lisbon, Santo Domingo, Kigali, Portugal, and Rwanda. The extended radiation model can be successfully applied to all these cases and the corresponding α versus l relationship is marked in .

Gross commuting flows in Seoul and Daegu Provinces, in units of average number of commuting people. Notes: The figure shows gross commuting flows, in units of number of commuting people, averaged across weekdays and weekends, in Seoul and Daegu provinces in April 2019 and April 2020. Seoul has 25 municipalities and Daegu has 9 municipalities.