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

TL;DR

The paper introduces $n$-simultaneous gauging, a framework to gauge multiple modulated symmetries with $n$ gauge operators, enabling simultaneous dualities beyond sequential gauging. It shows these dualities can be realized through adaptive state preparation, with concrete $(2+1)$D examples: 1-simultaneous gauging yields the anisotropic dipolar toric code and 2-simultaneous gauging yields the rank-2 toric code (R2TC). The intermediate dipolar cluster state (dCS) is identified as a dipolar SPT protected by dipole bundle symmetry and 1-form symmetries, featuring boundary anomalies and a rich ground-state structure. Using the dualities, the authors analyze phase structure of perturbed R2TC and outline a path to exploring higher-dimensional and non-Abelian generalizations, highlighting potential experimental relevance in tilted lattice platforms.

Abstract

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.

Figures (10)

• Figure 1: The duality transformation of operators for the $2$-simultaneous gauging of $g_{\rm d}$, $g_{e_{\rm h}}$, and $g_{e_{\rm v}}$ is depicted. A gauge field $\tilde{Z}$ is defined at the centers of the plaquettes. Two additional gauge fields, $\bar{Z}$ and $\check{Z}$, are assigned to the lattice vertices.
• Figure 2: (a) Three stabilizers of the R2TC. Red squares and lines indicate the stabilizer location. (b) Qudits and the quantum circuit arrangements for the preparation of dCS. Two qudits are placed at the vertices and labeled $v_{l}$ and $v_r$, respectively. Single qudits are placed at the horizontal and vertical edges ($e_{\rm h}$, $e_{\rm v}$) as well as plaquette centers ($p$). The entanglers $(CZ)_{ab}$ and $(CZ)_{ab}^{\dagger}$ act on the two qudits connected by red and black lines, respectively, in accordance with the rule that the $\textrm{CZ}$ operation involving one of the vertex qudits takes place between qudits of the same color. (c) Five stabilizers that realize the dCS. Red squares indicate the stabilizer location.
• Figure 3: Phase diagram of perturbed R2TC at (a) $\lambda_1 = 0$, and (b) $\lambda_3 = 0$. First- and second-order phase transitions are depicted by dotted and solid lines, respectively.
• Figure 4: (a) Examples of the $g_Z^1$, $g_Z^2$, and $g_Z^3$ when $N=3$. (b) Examples of $g_X^1$, $g_X^2$, and $g_X^3$ when $N=3$.
• Figure 5: $g_X^1(1)$, $g_X^2$, and $g_X^3$ defined along the $y$-axis with a smooth horizontal boundary for $N=3$