Nonplanar qubit with tunable gauge symmetry

Circuit quantum electrodynamics (cQED) has traditionally relied on planar Josephson junction geometries, which inherently limit the connectivity and dimensionality of the quantum systems that can be simulated. This paper introduces a non-planar "waffle" architecture—a 3x3 crossbar array of Josephson junctions—to overcome these constraints. By threading the array with a specific magnetic flux, the researchers engineer a Z3 combinatorial gauge symmetry (CGS), a critical ingredient for creating long-range entangled phases like quantum spin liquids.

The waffle consists of three gauge wires and three matter wires intersecting at nine junctions. At a critical magnetic flux of Φ/Φ0 = 1/3, the system's potential energy landscape develops six degenerate minima. To validate this, the team capacitively coupled the waffle to a superconducting readout resonator. They performed microwave spectroscopy to map the excitation spectrum as a function of external magnetic field, comparing their results against predictions from a neural-network variational Monte Carlo (NN-VMC) approach.

The experimental excitation spectrum shows excellent agreement with the NN-VMC simulations, confirming the predicted potential landscape. The researchers observed a characteristic evolution of the system's spectral features as the magnetic field tuned the number of degenerate minima (from one to four, then six, and finally two). Near the CGS point, the restoration of symmetry—evidenced by the merging of transmission features in the readout resonator—provides direct proof of the Z3 gauge structure. This work establishes the waffle as a versatile, scalable building block for simulating complex many-body models and topological phases of matter.