Gauging Modulated Symmetries via Multiple Gauge Symmetry Operators and Adaptive Quantum Circuits

We introduce an extended framework for the simultaneous gauging of modulated symmetries in $(d+1)$ dimensions, employing {\it multiple} gauge symmetry operators whose corresponding gauging procedures 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 $(2+1)$D 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.

💡 Research Summary

The paper introduces a comprehensive framework for the simultaneous gauging of modulated symmetries in (d + 1) dimensions, extending the conventional notion of gauging a single global symmetry. In the traditional approach, multiple symmetries are gauged sequentially, each step promoting a global generator to a local gauge constraint and yielding a dual model. However, many modern symmetry concepts—higher‑form, subsystem, modulated, and non‑invertible symmetries—are intertwined in such a way that sequential gauging cannot capture all possible dualities. To address this, the authors define “n‑simultaneous gauging,” where n independent gauge symmetry operators are introduced and must be applied at the same time. The key requirement is that any non‑empty proper subset of these operators shares gauge degrees of freedom with its complement, ensuring that the combined action reproduces exactly the intended set of symmetry generators without generating spurious operators.

The paper first illustrates the idea with a 1‑simultaneous gauging example on a square lattice. A single gauge operator h_r simultaneously gauges a global charge symmetry and an x‑direction dipole symmetry. Under the resulting duality, a trivial paramagnet (H = ∑ X_r) maps to an anisotropic dipolar toric code, where electric (e) and magnetic (m) excitations carry dipole moments along the x‑axis. This demonstrates that even the simplest simultaneous gauging can produce a dual model that is inaccessible via any sequence of single‑symmetry gauging steps.

The authors then move to a more intricate 2‑simultaneous gauging scenario. Two qudits per unit cell reside on horizontal and vertical edges, and three symmetries are gauged together: a horizontal charge symmetry, a vertical charge symmetry, and a global dipole symmetry that acts differently on the two edge orientations. Two gauge operators h_eh and h_ev are constructed to share a plaquette‑center gauge field (˜Z). The simultaneous gauging yields the rank‑2 toric code (R2TC), a stabilizer model whose excitations are fractonic (i.e., immobile unless combined). Importantly, the intermediate state generated during the gauging process is identified as a “dipolar cluster state” (dCS). The dCS is shown to be a symmetry‑protected topological (SPT) phase protected simultaneously by two charge Z_N symmetries, a dipole‑bundle symmetry, and three 1‑form symmetries—the first explicit construction of a dipolar SPT in two dimensions.

Generalizing further, the authors present n‑simultaneous gauging for an arbitrary number n of qudits per unit cell. They introduce 2n − 2 distinct gauge fields and define gauge operators such that each shares fields with the others, satisfying the simultaneity constraint. This construction works for any modulated symmetry whose modulation functions are periodic with the lattice, and it extends straightforwardly to higher spatial dimensions.

A major contribution of the work is the translation of these dualities into adaptive quantum circuits. Starting from an initial product state |+⟩ (all X = +1), a unitary U is designed so that U X_r U† = h_r for every matter site, thereby turning the trivial stabilizers into the desired gauge operators. Ancilla qudits placed on gauge sites are similarly transformed. A subsequent measurement (projection) of the matter qudits onto X = +1 implements the duality: the measured qudits are eliminated, leaving only the gauge degrees of freedom that constitute the dual model’s ground state. In the concrete R2TC example, the circuit first creates the dCS by applying controlled‑Z (CZ) gates according to a prescribed pattern, then measures the edge qudits to project onto the R2TC stabilizer space. The authors argue that this procedure incurs no additional overhead compared with sequential gauging, and it can be generalized to arbitrary n and (d + 1) dimensions.

Finally, the paper leverages the duality to study the phase diagram of the rank‑2 toric code under transverse fields. Because the paramagnet ↔ R2TC mapping is exact, the critical point of the transverse‑field Ising model directly informs the confinement–deconfinement transition of the R2TC. The analysis demonstrates that the simultaneous gauging framework not only provides new theoretical insights into exotic gauge theories and fracton phases but also offers practical routes for preparing highly entangled states on near‑term quantum devices.

In summary, the authors develop a unified theory of simultaneous gauging of multiple modulated symmetries, reveal novel dualities (including a dipolar SPT intermediate state), and present concrete adaptive circuit protocols for realizing these dualities experimentally. This work opens avenues for exploring richer gauge‑theoretic phenomena, designing new quantum error‑correcting codes, and implementing complex many‑body states on quantum hardware.