According to one exemplary aspect of the present invention, there is provided a waveguide transmission line comprising at least 10 Josephson elements therein, each of the at least 10 Josephson elements comprising a loop, exactly in one half of the loop. there is one Josephson junction of a first size, there are at least two Josephson junctions of a second size in a second half of the loop, the second size comprising: a waveguide transmission line greater than the first size; and at least one of the loops; A traveling wave parametric amplifier is provided that includes a flux bias line configured to produce a threading magnetic flux, and a set of resistors coupled with the flux bias line.
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