🔬 mesoscale physics

Gyrators for superconducting circuit design

This paper presents a flux-charge symmetric Hamiltonian formalism for non-linear circuits containing gyrators, inductors, and capacitors that enables symplectic reduction, defines a new circuit duality, and provides transformations to extend Tellegen's duality principle to non-linear elements like Josephson junctions while identifying conditions for converting non-reciprocal circuits into dynamically equivalent reciprocal ones.

Original authors: Ciro Salcedo, Sasha Cocquyt, Andrew Osborne, Andrew A. Houck

Published 2026-10-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ciro Salcedo, Sasha Cocquyt, Andrew Osborne, Andrew A. Houck

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to build the next generation of quantum computers, scientists are turning to circuits made of superconducting materials. These are not the copper wires found in a home appliance, but microscopic loops of metal that carry electricity with zero resistance when cooled to temperatures near absolute zero. Within these loops, engineers place tiny components like capacitors and inductors to create artificial atoms, known as qubits, which can hold and process information. For decades, the design of these circuits has relied on a set of rules that treat electricity as a flow of charge or a magnetic field of flux, but these rules often stumble when the circuit includes components that break the usual symmetry of time. Specifically, many advanced designs require non-reciprocal elements—devices that allow signals to pass in one direction but block them in the other, much like a one-way street for electricity. These components are essential for protecting delicate quantum information from noise, yet they are notoriously difficult to model mathematically and even harder to build with the precision required for quantum computing.

A team of researchers at Princeton University has now developed a new mathematical framework that brings clarity to these complex, one-way circuits. Their work focuses on a specific theoretical component called a gyrator, a device that couples voltage and current in a way that breaks time-reversal symmetry. While real-world devices like circulators and isolators perform this function, they are often bulky and difficult to integrate onto the tiny chips needed for quantum processors. The researchers asked a fundamental question: can we describe the behavior of these non-reciprocal circuits with the same precision and elegance as we do for standard, two-way circuits? They found that by treating the magnetic flux and electric charge as equal partners in their equations, they could construct a universal description that works for any circuit containing capacitors, inductors, and gyrators, regardless of how complex or non-linear the components are.

The core of their discovery is a new way of writing the energy equations for these circuits. In traditional physics, describing a circuit often requires choosing a single perspective, such as looking only at the flow of charge or only at the magnetic fields. This choice can lead to mathematical dead ends, especially when non-reciprocal elements are involved. The Princeton team instead created a balanced approach that keeps both perspectives visible at once. This symmetry allows them to prove that a valid energy description always exists for any well-behaved circuit, effectively removing the mathematical roadblocks that have plagued the field. They demonstrated that this framework can handle the most difficult cases, including circuits with "pathological" topologies that previously defied standard analysis, by systematically identifying and resolving the hidden constraints that govern the system's behavior.

Beyond simply describing these circuits, the researchers uncovered a powerful set of rules for transforming them. They showed that a gyrator can act as a mirror, swapping the roles of capacitors and inductors in a way that preserves the fundamental energy levels of the system. This means that a circuit designed with a difficult-to-build component, such as a quantum phase slip junction, can be mathematically converted into an equivalent circuit using only standard, easy-to-fabricate parts like Josephson junctions and capacitors. This is not just a theoretical trick; it provides a concrete recipe for engineers to design circuits that behave as if they contain exotic, non-reciprocal elements, even if those elements are physically replaced by combinations of standard parts and gyrators. The team proved that if the connections between different parts of a circuit follow a specific tree-like structure, the entire non-reciprocal system can be replaced by a fully reciprocal one, meaning the one-way behavior can be simulated without any actual one-way components.

The implications of this work extend to the very architecture of future quantum computers. By providing a rigorous method to model and simplify non-reciprocal circuits, the researchers have given designers a new toolkit to create protected qubits and high-fidelity gates. They showed that many circuits currently thought to require exotic, hard-to-engineer components can instead be realized using standard superconducting elements combined with gyrators. This shifts the burden of engineering from the impossible to the manageable. The paper establishes that the presence of a gyrator does not necessarily mean a circuit is fundamentally non-reciprocal in a way that cannot be tamed; rather, it often represents a choice of representation. If the circuit's connections form a specific pattern, the non-reciprocal nature can be removed entirely, leaving a system that is easier to build and analyze.

The researchers also addressed the limits of this approach. They identified specific conditions under which a circuit cannot be simplified or simulated by reciprocal elements alone. If the connections between the circuit's components form a loop that includes a gyrator in a certain way, the non-reciprocal behavior becomes intrinsic and cannot be eliminated. This distinction is crucial for engineers, as it tells them exactly when they must fight to build a difficult component and when they can simply rearrange their design to avoid the difficulty. The work does not claim to have solved every problem in quantum circuit design, but it provides a definitive map for navigating the landscape of non-reciprocal systems. It clarifies which circuits are fundamentally unique and which are merely complex variations of simpler, reciprocal systems.

In the end, this research offers a unifying language for a field that has been fragmented by the difficulty of modeling one-way electrical flows. By treating charge and flux with equal respect and introducing a set of transformation rules that preserve the system's energy, the authors have turned a chaotic collection of difficult problems into a structured, solvable framework. They have shown that the mystery of non-reciprocal circuits can be unraveled through symmetry and careful mathematical reduction. For the scientists building the quantum computers of tomorrow, this means that the path forward is clearer. They can now design circuits with confidence, knowing that if a component seems too exotic to build, there may be a mathematical equivalent waiting to be discovered, or a way to restructure the design so that the exotic part is no longer needed. The work stands as a testament to the power of rethinking fundamental assumptions, proving that even the most stubborn electrical puzzles can yield to a fresh perspective.

Technical Summary: Gyrators for Superconducting Circuit Design

Problem Statement
Superconducting circuits are a primary platform for quantum information processing, yet systematic Hamiltonian formalisms for circuits containing non-reciprocal elements remain challenging to establish. Traditional quantization methods (node flux or loop charge) often fail for "pathological" topologies, such as those with rank-deficient kinetic matrices or ill-posed constitutive relations. While recent geometric approaches (e.g., Faddeev–Jackiw) address non-reciprocal circuits, they often obscure physical insights by reducing pre-symplectic forms before defining a Lagrangian. Furthermore, existing frameworks struggle to handle arbitrary non-linearities in non-reciprocal circuits or to provide a unified description of circuit duality that extends beyond linear elements. Specifically, there is a need for a formalism that can rigorously describe the dynamics of circuits containing gyrators (linear, non-reciprocal two-port elements) alongside non-linear reciprocal elements like Josephson junctions (JJs) and quantum phase slips (QPS), and determine when such non-reciprocal dynamics can be simulated by reciprocal circuits.

Methodology
The authors extend the flux-charge symmetric formalism (previously applied to reciprocal LC circuits) to include gyrators, creating a framework for "LCG circuits" (containing inductors, capacitors, and gyrators). The methodology relies on the following pillars:

Flux-Charge Symmetric Lagrangian: The authors define a Lagrangian that treats node fluxes ( $\phi$ ) and loop charges ( $q$ ) on equal footing. For gyrators, they introduce a specific term $L_g$ that couples the flux of one port to the charge of the other, satisfying Kirchhoff's laws and constitutive relations without requiring parasitic regularization.
Symplectic Reduction: The formalism constructs a pre-symplectic matrix $\Omega$ . The authors demonstrate that for any nonsingular, well-posed LCG circuit, the kernel of $\Omega$ can be systematically identified and solved. This reduces the redundant configuration space to a valid phase space, guaranteeing the existence of a Hamiltonian regardless of the non-linearity of the reciprocal elements.
Circuit Duality and Transformations: The paper defines a global duality transformation that exchanges flux and charge, swapping capacitors and inductors while inverting gyrator ratios ( $G_g \to -G_g^{-1}$ ). Crucially, they introduce Partial Dual Transformations, which allow a reciprocal sub-circuit connected to one side of a gyrator to be moved to the other side. This operation transforms the sub-circuit into its dual (with rescaled parameters) and converts the connecting gyrator into an ideal transformer or removes it entirely.
Gyrator Reduction Rules: The authors derive canonical transformations (point transformations on the Lagrangian) that simplify circuits:
- Series/Parallel Deletion: Two gyrators in specific configurations can be replaced by a single effective gyrator.
- Cascade-to-Transformer: Two gyrators cascaded through a reciprocal-free junction reduce to a single ideal transformer.
- Open/Closed-Terminated Deletion: Gyrators connected to open or short circuits can be eliminated.
Graph-Theoretic Classification: To classify circuits based on their non-reciprocal nature, the authors define the Non-Reciprocal Coupling Graph (NCG). In this multigraph, vertices represent "really-connected components" (RCCs) of the circuit, and edges represent gyrators connecting them.

Key Contributions and Results

General Hamiltonian Formalism: The paper presents a flux-charge symmetric Hamiltonian formalism that admits a symplectic reduction for any nonsingular, well-posed LCG circuit with arbitrary non-linearity. This resolves the issue of pathological topologies that obstruct quantization in traditional methods.
Extension of Tellegen's Duality: The authors extend Tellegen's classic result—that gyrators make elements behave as their electromagnetic dual—to non-linear elements. They show that a Josephson junction coupled via a gyrator behaves as a Quantum Phase Slip (and vice versa), with the energy potential rescaled by the gyration ratio.
Isospectral Families: Through partial dual transformations, the paper constructs continuous families of circuits that are dynamically equivalent (isospectral) to a given reciprocal circuit but contain non-reciprocal elements. This allows for the realization of circuit dynamics in parameter regimes that might be physically inaccessible with standard components (e.g., simulating large inductances with capacitors and gyrators).
Reciprocal Simulability Criteria: The paper establishes rigorous conditions under which a non-reciprocal circuit can be transformed into a dynamically equivalent reciprocal circuit:
- If the NCG is a forest (a graph with no cycles), the circuit is reciprocally simulable by a circuit containing only capacitive and inductive elements alone (with no gyrators or transformers).
- If the NCG is bipartite, the circuit is reciprocally simulable using capacitive elements, inductive elements, and ideal transformers.
- If the NCG contains odd cycles, the non-reciprocal dynamics are intrinsic and cannot be removed by these transformations.

Significance and Claims
The authors claim that this work provides a systematic, graph-theoretic toolkit for the design and analysis of superconducting quantum circuits involving non-reciprocal elements. By formalizing the conditions for "reciprocal simulability," the paper distinguishes between circuits where non-reciprocity is a mere representational choice (and can be removed) and circuits where non-reciprocal elements are dynamically necessary.

The significance lies in:

Unification: Providing a single Lagrangian framework that handles linear and non-linear, reciprocal and non-reciprocal elements without ad-hoc regularization.
Design Flexibility: Enabling the synthesis of protected qubits and other architectures using standard components (JJs, capacitors) and gyrators to mimic exotic elements (QPS) or achieve specific parameter regimes.
Classification: Offering a clear topological criterion (the structure of the NCG) to determine if a complex non-reciprocal circuit can be simplified to a reciprocal one, thereby guiding the engineering of future quantum devices.

The paper concludes that while gyrators are often used to synthesize non-reciprocal behavior, their presence in a circuit does not automatically imply intrinsic non-reciprocity; rather, the topological connectivity of the gyrators determines whether the non-reciprocal dynamics are fundamental or reducible.

Technical Summary: Gyrators for Superconducting Circuit Design

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