Extracting Equations of Motion from Superconducting Circuits

TL;DR

The paper develops a practical, first-principles framework for deriving the equations of motion of superconducting circuits by specializing to a class of circuits and employing irrotational flux coordinates to produce a transparent Lagrangian. By combining the circuit's kinetic and potential energies with a Rayleigh dissipation approach and fluxoid quantization constraints, the authors obtain Langevin-type dynamics in terms of physically interpretable degrees of freedom. They validate the method by reproducing the SQUID's Lagrangian and then extend it to inductively coupled SQUIDs, yielding a controllable 2-bit energy landscape with four metastable minima suitable for memory and logic operations. This approach enables rapid circuit prototyping, enables analysis of thermodynamic performance, and provides a pathway toward universal classical gates implemented with superconducting circuits. The work lays groundwork for analyzing energy landscapes and information processing in near-classical regimes of superconducting devices, with potential extensions to full thermodynamic studies and gate implementations.

Abstract

Alternative computing paradigms open the door to exploiting recent innovations in computational hardware to probe the fundamental thermodynamic limits of information processing. One such paradigm employs superconducting quantum interference devices (SQUIDs) to execute classical computations. This, though, requires constructing sufficiently complex superconducting circuits that support a suite of useful information processing tasks and storage operations, as well as understanding these circuits' energetics. First-principle circuit design, though, leads to prohibitive algebraic complications when deriving the effective equations of motion -- complications that to date have precluded achieving these goals, let alone doing so efficiently. We circumvent these complications by (i) specializing our class of circuits and physical operating regimes, (ii) synthesizing existing derivation techniques to suit these specializations, and (iii) implementing solution-finding optimizations which facilitate physically interpreting circuit degrees of freedom that respect physically-grounded constraints. This leads to efficient, practical circuit prototyping and access to scalable circuit architectures. The analytical efficiency is demonstrated by reproducing the potential energy landscape generated by the quantum flux parametron (QFP). We then show how inductively coupling two QFPs produces a device that is capable of executing 2-bit computations via its composite potential energy landscape. More generally, the synthesis methods detailed here provide a basis for constructing universal logic gates and investigating their thermodynamic performance.

Figures (3)

• Figure 1: A SQUID with $N = 5$ and $F = 2$. Slight adjustments are made to the physical construction of the circuit compared to Ref. Han_1992_rf.
• Figure 2: Two SQUIDs inductively coupled via $M$: A superconducting device that supports $2$-bit classical computations through the manipulation of its potential energy landscape.
• Figure 3: Effective energy landscape generated by Eq. (\ref{['eq:simplified potential']}) after making simplifications and assumptions that introduce symmetry into the potential. Additionally, all other external circuit parameters take on zero value. Instantiating each region surrounding an energy minimum as a computational memory state---$00$, $10$, $01$, and $11$---permits information storage. Information processing is accomplished by way of a control protocol that employs the mutual inductance and external flux parameters to transform the landscape.