Quantum doubles in symmetric blockade structures

TL;DR

This work presents a blockade-based framework to realize non-abelian topological order described by quantum doubles $\mathcal{D}(G)$ using two-body interactions on a honeycomb lattice. By encoding group elements on links and enforcing no-flux constraints at sites via blockade graphs, the authors map the ground state to the zero-flux sector of $\mathcal{D}(G)$ and prove topological order for weak transverse fields; flux-anyon excitations and braiding are then accessible through adiabatic protocols. They develop a comprehensive toolbox: adiabatic ground-state preparation, deterministic flux-anyon creation in a fixed fusion channel, adiabatic braiding, and Wilson-loop-based flux readout, with a detailed implementation for the non-abelian $\mathcal{D}(S_3)$ case. The approach is platform-agnostic and compatible with current quantum simulation technologies, offering a path to experimentally probe non-abelian statistics in artificial matter. The results advance the inverse-design paradigm for topological quantum matter and open avenues for exploring richer anyon models such as Fibonacci anyons.

Abstract

Exactly solvable models of topologically ordered phases with non-abelian anyons typically require complicated many-body interactions which do not naturally appear in nature. This motivates the "inverse problem" of quantum many-body physics: given microscopic systems with experimentally realistic two-body interactions, how to design a Hamiltonian that realizes a desired topological phase? Here we solve this problem on a platform motivated by Rydberg atoms, where elementary two-level systems couple via simple blockade interactions. Within this framework, we construct Hamiltonians that realize topological orders described by non-abelian quantum double models. We analytically prove the existence of topological order in the ground state, and present efficient schemes to prepare these states. We also introduce protocols for the controlled adiabatic braiding of anyonic excitations to probe their non-abelian statistics. Our construction is generic and applies to quantum doubles $\mathcal{D}(G)$ for arbitrary finite groups $G$. We illustrate braiding for the simplest non-abelian quantum double $\mathcal{D}(S_3)$.

Figures (12)

• Figure 1: Conventions and Construction. (a) We construct quantum doubles \ref{['eq:QD']} for a finite group $G$ on the trivalent and bipartite honeycomb lattice. By convention, the links of the lattice are assigned an orientation (solid arrows). Every link $l$ is associated with a $N=|G|$-dimensional quantum system with one state $\mathinner{|{g}\rangle}_l$ for each group element $g\in G$. Each plaquette $p$ is assigned a counter-clockwise orientation (dotted arrows). With this convention, a plaquette $p$ is on the left (right) of a bounding link $l$ [write $l\in{}_p\!\!\uparrow$ ($l\in\uparrow_{\!p}$)], if the link's orientation is parallel (antiparallel) to the orientation of the plaquette. On each site $s$, we define a projector $B_s$ that singles out states that satisfy the no-flux condition $g_1g_2g_3=\mathrm{e}$, where $\mathrm{e}\in G$ denotes the identity and the multiplication sequence depends on the sublattice (two green sites). On each plaquette, there are operators $A_p(h)$ that act by left/right multiplication on the group elements on the bounding links (blue/red). $A_p(h)$ acts by left (right) multiplication if the plaquette orientation aligns (counter-aligns) with the link orientation. This construction ensures that $A_p(h)$ commutes with all site constraints $g_1g_2g_3=\mathrm{e}$ enforced by $B_s$. Summing over all group elements yields the projector $A_p$. (b) Microscopically, the quantum double is realized by a blockade Hamiltonian \ref{['eq:H']} encoded by a blockade graph $\mathcal{G}=(V,E,W)$. Depicted is an example for $N=6$ with group elements $\mathrm{g_i}$ for $i=1,\ldots,6$ and exemplary group product $\mathrm{g_2}=(\mathrm{g_6}\mathrm{g_1})^{-1}$. Note that roman symbols like $\mathrm{g_2}$ label specific group elements, whereas italic symbols like $g_3$ are used as variables. For the construction it is convenient to mark one of the three edges at each site (crosses); this choice has no physical consequence. Then one places $N$ two-level systems (vertices) on each link $l$ which are not in blockade which each other; each is assigned a group element and labeled by $v_l^{g}$. Additionally, there are $N^2$ two-level systems on the site labeled by pairs of group elements and denoted by $w_s^{g_1g_2}$; these site systems are all in blockade which each other (blockades not shown). The crucial part is how the link vertices are connected by edges (blockades) with the site vertices (solid arcs, only a few are shown). This construction is explained in the text and depends on the orientation of the site (= the sublattice) and the marked edge (crosses). The inset shows an exemplary classical ground state $\mathinner{|{\mathrm{g_1g_4g_2}}\rangle}\mathinner{|{w_s^{\mathrm{g_1g_4}}}\rangle}$ that satisfies all blockades and realizes the state with constraint $\mathrm{g_1g_4g_2}=\mathrm{e}$. Note that there is only one two-level system excited on the site and all but one on each link. Although the construction seems to break the three-fold rotation symmetry of a site (via the edge marked with a cross), the cyclic symmetry of the constraint $g_1g_2g_3=\mathrm{e}$ ensures that the constructed blockade graph is completely symmetric under rotations by $120^\circ$. To obtain the blockade graph on the other sublattice, one can rotate the shown site by $180^\circ$ and swap two of the three edges, thereby inverting the orientation of the multiplication around the vertex.
• Figure 2: Plaquette automorphisms. The blockade graph $\mathcal{G}$ (shaded gray) constructed in \ref{['fig:fig1']} allows for local automorphisms $\Theta_p(h)$ for each $h\in G$ that affect only the vertices on links and sites bounding a single plaquette $p$. The automorphism decomposes into a product of permutations of vertices $\varphi_l$ and $\phi_s$ on links $l\in p$ and sites $s\in p$, respectively. The link permutations depend on whether the link orientation is parallel (anti-parallel) to the orientation of the plaquette (blue and red arrows). The definition of these permutations is given in the text. Describing the site permutations $\phi_s$ is most convenient if the site vertices are labeled by the pairs of group elements of the two adjacent edges that bound the plaquette. (To apply the construction detailed in \ref{['sec:model']}, the radial edge on every site of the plaquette is identified with $l_3$ [marked by a cross]; this is indicated by rotated and mirrored arrays of site vertices.) As a blockade graph automorphism, $\Theta_p(h)$ induces a symmetry of the blockade Hamiltonian $H_\mathcal{G}$ on the full Hilbert space. As such, the ground state manifold remains invariant, and the representation induced by the link permutations acts by left and right group multiplications on the ground states $\mathinner{|{{\boldsymbol{g}}}\rangle}$ in $\mathcal{H}^0_\mathcal{G}$. The insets show exemplary actions of the permutations on states (black vertices: $\mathinner{|{0}\rangle}$, orange vertices: $\mathinner{|{1}\rangle}$). Note that links (and sites) that are not adjacent to $p$ are unaffected by the automorphism (gray).
• Figure 3: Wilson loops. Wilson loop operators $W^C(\gamma)$ are defined on closed, oriented loops $\gamma$ (red) on the dual lattice (dotted). For the evaluation of $W^C(\gamma)$ in the product basis $\mathinner{|{{\boldsymbol{g}}}\rangle}\in \mathcal{H}_G$, the product of all group elements $g_l^{\sigma_l}$ on links $l$ crossed by $\gamma$ is needed (with multiplication order from left to right along the loop's orientation); here $\sigma_l=+1$ ($\sigma_l=-1$) if the crossed edge $l$ points to the left (right) of $\gamma$ when following its orientation. This construction ensures that $W^C(\gamma)$ commutes with the plaquette operators $A_p(h)$ (plaquette in the bottom right corner).
• Figure 4: Adiabatic preparation of ground states and flux anyons. (a,b) The topological fixpoint ground state can be adiabatically "grown" starting from a completely de-excited array of two-level systems (i.e., $\Delta_s<0$ and $\Delta_l<0$ for all sites and links, white filling). One then iterates the ramping procedure (white → blue → purple → red) shown in the inset of panel (b) for sites and adjacent links repeatedly. Note that transverse fields $\Omega_{s,l}$ and detunings $\Delta_{s,l}$ are adiabatically modified uniformly for all two-level systems on a site or link, thereby respecting the local symmetries of the model at all times. A possible (non-optimal) initialization sequence is shown in (b) where numbers label time steps. Note that a site (link) can only be initialized in the ground state if there is at least one adjacent link (site) uninitialized (see four cases in time step 9). Since $\Omega_s=0=\Omega_l$ after initialization (red area), the prepared ground state is the fixpoint wave function of the quantum double $\tilde{H}_\mathcal{G}(0,\omega)$, without admixtures from classical excited states in $\mathcal{H}^\perp_\mathcal{G}$. (c) Once a patch is prepared in the topological ground state, one can use the generalized site graph $\mathcal{G}_{s_0}[\bm\Delta]$ to adiabatically inject flux anyons into the system [labelled $\bm\Delta$ in panel (b), see \ref{['sec:fluxlattice']} for its construction]. The special site $s_0$ is initialized with $\Delta_{C_\mathrm{e}}=-\Delta \nearrow 4\Delta$ (and all other $\Delta_{C}=-\Delta=\textrm{const}$) to prepare the no-flux constraint $g_1g_2g_3=\mathrm{e}$. Then this site, together with an adjacent link $l$ and site $s$ are de-excited again [following the inverse protocol in panel (a)]. Subsequently, the special site is re-initialized with $\Delta_{C_r}=-\Delta\nearrow 4\Delta$ (green boundary, red filling) so that it enforces the constraint $g_1g_2g_3\in C_r$ for some non-trivial conjugacy class $C_r$. This protocol prepares two flux anyons in the vacuum fusion channel: $[C_r,E]$ localized on the "flux factory" site $s_0$, and the corresponding antiparticle $[\bar{C}_r,E]$ in the hole of de-excited sites and links. The hole (carrying its anyon) can then be adiabatically moved by sequences of (de-/re-)initializations of links and sites (purple link).
• Figure 5: Probing non-abelian statistics. Protocol for the preparation, braiding and measurement (fusion) of anyons. The splitting/fusion diagrams (time evolution) are shown in the top row, the corresponding spatial configurations and manipulations in the bottom row. (a) As an initial step, two pairs $(1,2)$ and $(3,4)$ of $\text{\bfseries\sffamily{D}}=[C_\sigma,E]$ flux anyons (solid green lines) are created and separated within a topological domain (red area) following the protocol described in \ref{['sec:braiding']} and \ref{['fig:fig5']} (c). We denote splitting/fusion states of this form as $\mathinner{|{xx}\rangle}\equiv\mathinner{|{\text{\bfseries\sffamily{DD}}\rightarrow x,\text{\bfseries\sffamily{DD}}\rightarrow x}\rangle}$. With this nomenclature, the system is initialized in the state $\mathinner{|{\text{\bfseries\sffamily{AA}}}\rangle}$ where both pairs fuse into the vacuum $\text{\bfseries\sffamily{A}}$ (black dashed lines). (b) As a consistency check, the Wilson loop $W^C(\gamma)$ can be computed from measuring the link states $\mathinner{|{g}\rangle}_l$ along the depicted dual loop $\gamma$ (solid red line). Only for $C=C_\mathrm{e}$ the expectation value should be non-zero since the enclosed anyons are in the vacuum channel. (c) Alternatively, the two anyons 2 and 3 from different pairs can be exchanged with a half-braid (again using the protocol for adiabatically moving holes) to produce the new state $B_{23}\mathinner{|{\text{\bfseries\sffamily{AA}}}\rangle}$ shown in (c-i). Here, $B_{23}$ denotes the unitary braiding matrix for exchanging flux 2 and 3. Using the $F$- and $R$- matrices of the unitary braided fusion category that describes the quantum double $\mathcal{D}(S_3)$ (see \ref{['app:braiding']}), this state can be expanded in the basis $\mathinner{|{xx}\rangle}$, where the fusion rules allow $x\in\{\text{\bfseries\sffamily{A}},\text{\bfseries\sffamily{C}},\text{\bfseries\sffamily{F}},\text{\bfseries\sffamily{G}},\text{\bfseries\sffamily{H}}\}$. Here, $\{\text{\bfseries\sffamily{A}},\text{\bfseries\sffamily{C}}\}$ carry no flux ($C_\mathrm{e}$) and $\{\text{\bfseries\sffamily{F}},\text{\bfseries\sffamily{G}},\text{\bfseries\sffamily{H}}\}$ carry non-zero flux of type $C_R$. These fluxes can be measured again by the same Wilson loop $W^C(\gamma)$, where now the expectation value for $C=C_R$ is finite. This demonstrates the non-abelian nature of $\mathcal{D}(S_3)$ in that the two states depicted in the lower left and right corners are locally indistinguishable while being linearly independent.

Theorems & Definitions (16)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 1: Cui et al. Cui_2020
• Proposition 5
• proof
• Lemma 1
• proof
• Lemma 2