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This book concerns a Josephson device for supercomputers which has extremely low heat dissipation (about 106 times less than semiconductor devices and 103 times less than voltage-based Josephson devices). In the previous book on Quantum Flux Parametrons (QFPs), DC Flux Parametron, the basic device operation are described. This book deals in much greater depth on the problems which are faced by the QFP. The device characteristics are worked out in detail showing clearly the analysis methods used. A new logic gate using the QFP is described with respect to its basic scheme, operation, and ways for forming logic circuits. The problems faced by the basic QFP are much reduced in the new logic gate. As the QFP operates near the Heisenberg and Boltzmann limits for computing devices, we also show the relationship between speed and stability. The book contains the latest analytical results on QFPs.The material presented in the book can be understood with very little mathematical training or knowledge about superconducting physics. It is also self-contained and does not require reading of other material. Most of the device characteristics can be reproduced from the equations given using simple programs. A circuit simulator is not needed except for high speeds when transient behavior becomes important.