A milliohm resistor is fabricated as a Josephson junction device that contains ferromagnetic or antiferromagnetic material of sufficient thickness to render the device entirely resistive between terminals. The device can have a resistance on the order of milliohms and can be consume a much smaller chip footprint than resistors of the same resistance fabricated using conventional resistive materials. Because the device can be fabricated without modification to processes used to fabricate reciprocal quantum logic (RQL) circuitry, it can easily be incorporated in RQL circuits to mitigate flux trapping or to perform other functions where very small resistances are needed. In particular, the device can burn off circulating currents induced by trapped flux without affecting the transmission of SFQ pulses through RQL circuitry.