We introduce an extended framework for the simultaneous gauging of modulated symmetries in $(d+1)$ dimensions, employing multiple gauge symmetry operators the corresponding gauging procedures of which must be carried out simultaneously. Simultaneous gauging can capture a broader class of dualities than sequential gauging, the latter corresponding to the conventional gauging applied in successive steps. In general, performing simultaneous gauging and conventional gauging in sequence constitutes the most general framework for gauging modulated symmetries. We further show that the associated duality transformations can be implemented via adaptive state preparation protocols. As a concrete example, we consider a dipole symmetry in $(2+1)\text{dimensions}$ and illustrate both the simultaneous gauging procedure and the adaptive preparation protocol. Interestingly, we find that the intermediate state of the simultaneous gauging/adaptive circuit corresponds to a symmetry-protected topological phase protected by the dipole bundle symmetry. Finally, we utilize the duality to analyze the phase diagram of the rank-2 toric code under transverse fields.

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• Phase diagrams
• Quantum circuits
• Symmetry protected topological states
• Topological phases of matter
• Duality
• Gauge symmetries
• Spin lattice models