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Alternative computing paradigms open the door to exploiting recentinnovations in computational hardware. Dynamical computing is one such paradigmthat synthesizes momentum computing -- an extremely energy-efficient designframework -- with nanoscale thermodynamic computing. This synthesis can beimplemented with Josephson junction technology, giving a testbed with tunablecoupling to a thermal environment. Investigating the dynamics andthermodynamics of these superconducting quantum interference devices (SQUIDs),though, requires (i) constructing physically-realizable superconductingcircuits, (ii) thoroughly understanding circuit energetics, and (iii) designingsufficiently complex circuits that support a suite of useful operations.First-principle circuit design leads to prohibitive algebraic complicationsthat to-date precluded achieving these goals quickly and efficiently. Wecircumvent these complications by (i) specializing our class of circuits andoperating regime, (ii) synthesizing existing methods to suit thesespecializations, and (iii) implementing solution-finding optimizations thatfacilitate physically interpreting circuit degrees of freedom and that respectphysically-grounded constraints. Practically, this leads to efficient circuitprototyping and analysis and access to scalable circuit architectures. Theanalytical efficiency is demonstrated by directly reproducing the thermodynamicpotential induced by the variable $\beta$ rf (v$\beta$-rf) SQUID. We then showhow inductively couple two v$\beta$-rf SQUIDs yielding a device capable ofperforming 2-bit computations. The methods detailed here provide a basis toconstruct universal logic gates and investigate their thermodynamicperformance.

Dynamical Computing on the Nanoscale: Superconducting Circuits for Thermodynamically-Efficient Classical Information Processing