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arXiv:2607.14229

Nonplanar qubit with tunable gauge symmetry

Muqing Yu, Han Bi, Hengli Lo, Vishvesha Sridhar, Guilherme Delfino, Dmitry Green, Claudio Chamon, Nadya Mason, Andrew P. Higginbotham

quant-phcond-mat.mes-hallSubmitted Jul 15, 2026• Announced Jul 17, 2026• 0 citations
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Abstract

Circuit quantum electrodynamics embeds Josephson junction qubits within superconducting cavities, and has emerged as a leading approach to quantum computing and quantum simulation. Despite the many permutations of circuit geometry that have been explored, Josephson connectivities have so far been planar, making them effectively low-dimensional. Here we show that a non-planar qubit -- a $3\times3$ crossbar Josephson array -- gives rise to flux-tunable $\mathbb{Z}_3$ combinatorial gauge symmetry (CGS), potentially enabling spin-liquid behavior when networked into a lattice. The observed excitation spectrum shows excellent agreement with predictions from a neural network trained to generate variational quantum states, demonstrating that we have predictive power over our high-dimensional quantum system. Fine-structure splittings near the CGS point are compatible with weak tunneling or symmetry breaking due to experimental imperfections. We additionally use the superconducting cavity to externally induce symmetry breaking, observing a restoration of symmetry at the CGS point where ground states differ only by a $\mathbb{Z}_3$ phase. This work initiates a general program exploring lattice gauge theories using the toolbox of circuit quantum electrodynamics. More broadly, introducing non-planar Josephson connectivities opens a vast space for experimental and theoretical exploration of structures in almost any imaginable dimensionality and geometry.

Comments
7+14 pages, 5+7 figures

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Circuit quantum electrodynamics embeds Josephson junction qubits within superconducting cavities, and has emerged as a leading approach to quantum computing and quantum simulation. Despite the many permutations of circuit geometry that have been explored, Josephson connectivities have so far been planar, making them effectively low-dimensional. Here we show that a non-planar qubit -- a 3×33\times33×3 crossbar Josephson array -- gives rise to flux-tunable Z3\mathbb{Z}_3Z3​ combinatorial gauge symmetry (CGS), potentially enabling spin-liquid behavior when networked into a lattice. The observed excitation spectrum shows excellent agreement with predictions from a neural network trained to generate variational quantum states, demonstrating that we have predictive power over our high-dimensional quantum system. Fine-structure splittings near the CGS point are compatible with weak tunneling or symmetry breaking due to experimental imperfections. We additionally use the superconducting cavity to externally induce symmetry breaking, observing a restoration of symmetry at the CGS point where ground states differ only by a Z3\mathbb{Z}_3Z3​ phase. This work initiates a general program exploring lattice gauge theories using the toolbox of circuit quantum electrodynamics. More broadly, introducing non-planar Josephson connectivities opens a vast space for experimental and theoretical exploration of structures in almost any imaginable dimensionality and geometry.
We propose a superconducting quantum circuit based on a general symmetry principle-combinatorial gauge symmetry-designed to emulate topologically ordered quantum liquids and serve as a foundation for the construction of topological qubits.The proposed circuit exhibits rich features: in the classical limit of large capacitances its ground state consists of two superimposed loop structures; one is a crystal of small loops containing disordered U(1) degrees of freedom, and the other is a gas of loops of all sizes associated with Z 2 topological order.We show that these classical results carry over to the quantum case, where phase fluctuations arise from the presence of finite capacitances, yielding Z 2 quantum topological order.A key feature of the exact gauge symmetry is that amplitudes connecting different Z 2 loop states arise from paths having zero classical energy cost.As a result, these amplitudes are controlled by dimensional confinement rather than tunneling through energy barriers.We argue that this effect may lead to larger energy gaps than previous proposals that are limited by such barriers, potentially making it more likely for a topological phase to be experimentally observable.Finally, we discuss how our superconducting circuit realization of combinatorial gauge symmetry can be implemented in practice.
Z3\mathbb{Z}_3Z3​ quantum double in a superconducting wire array
We show that a Z3\mathbb{Z}_3Z3​ quantum double can be realized in an array of superconducting wires coupled via Josephson junctions. With a suitably chosen magnetic flux threading the system, the inter-wire Josephson couplings take the form of a complex Hadamard matrix, which possesses combinatorial gauge symmetry -- a local Z3\mathbb{Z}_3Z3​ symmetry involving permutations and shifts by ±2π/3\pm 2π/3±2π/3 of the superconducting phases. The sign of the star potential resulting from the Josephson energy is inverted in this physical realization, leading to a massive degeneracy in the non-zero flux sectors. A dimerization pattern encoded in the capacitances of the array lifts up these degeneracies, resulting in a Z3\mathbb{Z}_3Z3​ topologically ordered state. Moreover, this dimerization pattern leads to a larger effective vison gap as compared to the canonical case with the usual (uninverted) star term. We further show that our model maps to a quantum three-state Potts model under a duality transformation. We argue, using a combination of bosonization and mean field theory, that altering the dimerization pattern of the capacitances leads to a transition from the Z3\mathbb{Z}_3Z3​ topological phase into a quantum XY-ordered phase. Our work highlights that combinatorial gauge symmetry can serve as a design principle to build quantum double models using systems with realistic interactions.
Integrating two-dimensional (2D) materials into circuit quantum electrodynamics (cQED) devices is an emerging field in recent years. This integration not only facilitates the exploration of potential applications in quantum information processing but also enables the study of material's fundamental properties using microwave techniques. While most studies employ 2D coplanar architectures with scalability potential, three-dimensional (3D) cavity-based cQED devices, due to their simpler design, offer the advantage of a quicker turnaround to probe the composite Josephson junctions (JJs). Here, we construct the first flux-tunable, 3D cavity-compatible superconducting quantum circuit based on 2D materials, featuring a graphene superconducting quantum interference device (SQUID) shunted by a capacitor that is accessible by both dc and microwave probes. We have shown how flux-modulated cavity frequency can be linked to the SQUID critical current under the influence of Fraunhofer's pattern. In addition, we extracted the symmetry information of the SQUIDs based on dc analysis, and correlated this with the flux-modulated cavity frequency observed in microwave measurements. Our platform can extend to topological materials, holding the prospect of establishing valid topological JJs with a dc probe while allowing a fast microwave probe to avoid quasiparticle poisoning.
Fluxonium superconducting circuits were originally proposed to realize highly coherent qubits. In this work, we explore how these circuits can be used to implement and harness qutrits, by tuning their energy levels and matrix elements via an external flux bias. In particular, we investigate the distinctive features of arrays of fluxonium qutrits, and their potential for the quantum simulation of exotic quantum matter. We identify four different operational regimes, classified according to the plasmon-like versus fluxon-like nature of the qutrit excitations. Highly tunable on-site interactions are complemented by correlated single-particle hopping, pair hopping and non-local interactions, which naturally emerge and have different weights in the four regimes. Dispersive corrections and decoherence are also analyzed. We investigate the rich ground-state phase diagram of qutrit arrays and propose practical dynamical experiments to probe the different regimes. Altogether, fluxonium qutrit arrays emerge as a versatile and experimentally accessible platform to explore strongly correlated bosonic matter beyond the Bose-Hubbard paradigm, and with a potential toward simulating lattice gauge theories and non-Abelian topological states.
A quantum simulator of [Formula: see text] lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.
We show that a Z 3 quantum double can be realized in an array of superconducting wires coupled via Josephson junctions.With a suitably chosen magnetic flux threading the system, the interwire Josephson couplings take the form of a complex Hadamard matrix, which possesses combinatorial gauge symmetry-a local Z 3 symmetry involving permutations and shifts by ±2π/3 of the superconducting phases.The sign of the star potential resulting from the Josephson energy is inverted in this physical realization, leading to a massive degeneracy in the nonzero flux sectors.A dimerization pattern encoded in the capacitances of the array lifts up these degeneracies, resulting in a Z 3 topologically ordered state.Moreover, this dimerization pattern leads to a larger effective vison gap as compared to the canonical case with the usual (uninverted) star term.We further show that our model maps to a quantum three-state Potts model under a duality transformation.We argue, using a combination of bosonization and mean field theory, that altering the dimerization pattern of the capacitances leads to a transition from the Z 3 topological phase into a quantum XY-ordered phase.Our work highlights that combinatorial gauge symmetry can serve as a design principle to build quantum double models using systems with realistic interactions.