On the other hand, a general fusion category can be realized with no mixing with the lattice translation on an anyonic chain [17, 18], which generally does not have a tensor product Hilbert space.

Note that, in $2+1\mathrm{D}$, the Ising lattice model is not self-dual under gauging; rather, it is mapped to the lattice ${\mathbb{Z}}_2$ gauge theory. The corresponding noninvertible symmetry in ${\mathbb{Z}}_2$ gauge theory coupled to Ising matter is constructed in Ref. [45] and generalized in Sec. 6f6. This is related to the fact that gauging a ${\mathbb{Z}}_2^{(q)}$ symmetry in $(d+1)\mathrm{D}$ returns a dual ${\mathbb{Z}}_2^{(d-q-1)}$ symmetry [50, 51]. [The superscript ($q$) denotes the form degree of a $q$-form global symmetry.] Hence, the dual symmetry can be isomorphic to the original one only if $q=0$, $d=1$ and $q=1$, $d=3$.

This includes states (or kets), which are linear maps with domain $\mathbb{C}$, and their adjoints (or bras), which are linear maps with codomain $\mathbb{C}$.

Here, a helpful mnemonic is that the GHZ state is constructed from the green $Z$ spider.

Unless otherwise specified, we place the inputs (outputs) on the left (right) so that “time” flows from left to right. Where this is not possible—for example, in higher-dimensional systems—we use ingoing (outgoing) arrows to represent the inputs (outputs).

One has to add more rules to the list above so that the $ZX$ calculus is complete. We do not go into the details here and point interested readers to Ref. [69].

In deriving the first expression using $ZX$ calculus, it is useful to note the Euler decomposition of the Hadamard gate,

Moreover, the model with $h=0$ has an additional (nontopological) magnetic 2-form symmetry. This is analogous to the (topological) 1-form and 2-form symmetries in the modified Villain (integer BF) Euclidean lattice model for the $3+1\mathrm{D}$ ${\mathbb{Z}}_2$ gauge theory [94].

Topological and nontopological 1-form symmetries are, respectively, referred to as relativistic or nonrelativistic 1-form symmetries in Ref. [95] and as nonfaithful or faithful in Ref. [96]. Here, we adopt the terminology topological (nontopological) symmetries as in Ref. [45] because it directly reflects the property of the operators.

The relation between $\mathsf{D}$ and quantum cellular automata will be discussed in Ref. [103].

Recall that $\mathfrak{t}(p)$ is the link obtained by translating from the plaquette $p$ by $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, and $\mathfrak{t}(\ell )$ is the plaquette obtained by translating from the link $\ell$ by $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$.

The Wilson line operator, which is charged under the ${\mathbb{Z}}_2$ 1-form symmetry, is well defined even when $\gamma$ is noncontractible.

The first line is obtained as follows: First, the $ZX$ diagram of $({\mathsf{D}}_{\mathrm{p}\leftarrow \mathrm{l}}{)}^{†}$ is obtained by reversing the arrows (exchanging inputs and outputs) in the diagram of ${\mathsf{D}}_{\mathrm{p}\leftarrow \mathrm{l}}$ (since there are no nontrivial phases in any of the spiders, flipping the signs of the phases does not change the diagram). It is clear that the resulting diagram is the same as that of ${\mathsf{D}}_{\mathrm{l}\leftarrow \mathrm{p}}$, so $({\mathsf{D}}_{\mathrm{p}\leftarrow \mathrm{l}}{)}^{†}={\mathsf{D}}_{\mathrm{l}\leftarrow \mathrm{p}}$. It follows that $$\begin{gathered} {\mathsf{D}}^{†}=({\mathsf{D}}_{\mathrm{p}\leftarrow \mathrm{l}}{)}^{†}(T_{\mathrm{l}\leftarrow \mathrm{p}}{)}^{†}={\mathsf{D}}_{\mathrm{l}\leftarrow \mathrm{p}}(T_{\mathrm{l}\leftarrow \mathrm{p}}{)}^{†} \\ ={\mathsf{D}}_{\mathrm{l}\leftarrow \mathrm{p}}{T}_{\mathrm{p}\leftarrow \mathrm{l}}(T_{\mathrm{p}\leftarrow \mathrm{l}}{)}^{†}(T_{\mathrm{l}\leftarrow \mathrm{p}}{)}^{†}=\mathsf{D}{T}_{1,1,1}^{-1}. \end{gathered}$$Similarly, it can be shown that ${\mathsf{D}}^{†}=T_{1,1,1}^{-1}\mathsf{D}$.

The continuum duality operator is self-adjoint, $\mathcal{D}(M)=\mathcal{D}(M{)}^{†}$, but on the lattice ${\mathsf{D}}^{†}$ and $\mathsf{D}$ differ by a lattice translation $T_{1,1,1}$. In the continuum, the duality operator associated with a ${\mathbb{Z}}_N^{(1)}$ symmetry with $N>2$ is not self-adjoint; $\mathcal{D}$ and ${\mathcal{D}}^{†}$ differ by a charge conjugation.

Technically, the rotational part of this group exists only when $L_x=L_y=L_z$ or when the lattice is infinite.

One can think of $\mathit{ϵ}$ as labeling the space diagonal along that direction. Note that $\mathit{ϵ}$ and $-\mathit{ϵ}$ label the same space diagonal.

The half-translation map $T_{\mathrm{p}\leftarrow \mathrm{l}}$ transforms into $T_{\mathrm{p}\leftarrow \mathrm{l}}^{\mathit{ϵ}}$ under the rotation $R$. Here, $T_{\mathrm{p}\leftarrow \mathrm{l}}^{\mathit{ϵ}}$ maps the physical qubit on link $\ell$ to the auxiliary qubit on plaquette ${\mathfrak{t}}_{\mathbf{\epsilon }}(\ell )$, where ${\mathfrak{t}}_{\mathbf{\epsilon }}(\mathit{x})=\mathit{x}+\frac{1}{2}\mathit{ϵ}$ is the half-translation in the $\mathit{ϵ}$ direction. The desired relation then follows from the simple observation $T_{\mathrm{p}\leftarrow \mathrm{l}}^{\mathit{ϵ}}=T_{\mathrm{p}\leftarrow \mathrm{l}}{T}_{\frac{1}{2}(\mathit{ϵ}-\mathbf{1})}$. Similar statements apply to the other half-translation map $T_{\mathrm{l}\leftarrow \mathrm{p}}$.

Indeed, starting with the Hamiltonian $H$ Eq. (4.5), which has the same ground states as $\tilde{H}$ Eq. (4.2) for $g>0$, and setting $h=0$ gives the Hamiltonian of the $3+1\mathrm{D}$ toric code. The normalization of $|\xi ⟩$ is chosen such that $⟨\xi |{\xi }^{\prime }⟩={\delta }_{\xi ,{\xi }^{\prime }}$.

Comparing with the discussion in Sec. 3d, the basis $|\xi ⟩$ is the analog of $|0\dots 0⟩,|1\dots 1⟩$ in $1+1\mathrm{D}$, while the basis $|\zeta ⟩$ is the analog of $|{\mathrm{GHZ}}^{\pm }⟩$.

On the other hand, the state $|\xi =0⟩$ can be obtained by acting a condensation operator for a 2-form symmetry on the product state $|+\cdots +⟩$. See Appendix pp1-s4, in particular, Eq. (a28).

A noninvertible global symmetry is sometimes said to have a ’t Hooft anomaly if it is incompatible with a trivially gapped phase [53, 120, 121] (see also Refs. [108, 122, 123, 124, 125, 126]). See Refs. [127, 128, 129] for the relation between this definition of anomalies and the obstruction to gauging.

To see this, note that $\ell \perp p\iff \mathfrak{t}(\ell )\perp \mathfrak{t}(p)$.

Generally, a bosonic $3+1\mathrm{D}$ ${\mathbb{Z}}_N^{(1)}$ SPT phase is labeled by an integer $p$ modulo $2N$ if $N$ is even, and $p$ modulo $N$ if $N$ is odd [50, 133]. Under gauging the ${\mathbb{Z}}_N^{(1)}$ symmetry, the phase remains invertible if $\gcd (p,N)=1$. Under this condition, gauging maps $p\to -1/p$ for even $N$, and $p\to -1/4p$ for odd $N$ [53]. In particular, when $N=2$, the $p=1$ and $p=3$ SPT phases are exchanged under gauging ${\mathbb{Z}}_2^{(1)}$, and there is no bosonic SPT phase invariant under the gauging. In Ref. [134], the authors further show that, for some values of $N$, there is no fermionic topological quantum field theory (TQFT) (with a unique local operator) that is invariant under gauging ${\mathbb{Z}}_N^{(1)}$. However, their result does not exclude the possibility of a bosonic TQFT invariant under gauging ${\mathbb{Z}}_2^{(1)}$.

We thank Nathan Seiberg for discussions on this point.

Throughout this section, we use $V$ to denote a set of vertices that host physical qubits. This is not to be confused with $V=L_x{L}_y{L}_z$, the number of sites on the cubic lattice in Sec. 5.

In the quantum information literature, $\sigma$ and $\mathcal{G}$ are called the parity-check matrix and the corresponding Tanner graph, respectively.

An automorphism $\alpha$ of a graph is a bijection from its vertex set to itself such that $\alpha (v)$ and $\alpha (w)$ have an edge between them if and only if $v$ and $w$ have an edge between them. For a bipartite graph, an automorphism is said to be preserving if it fixes the two parts of the vertex set, i.e., $\alpha (V)=V$ and $\alpha (\widehat{V})=\widehat{V}$, whereas it is said to be reversing if it exchanges the two parts of the vertex set, i.e., $\alpha (V)=\widehat{V}$ and $\alpha (\widehat{V})=V$.

Two necessary conditions for the existence of a reversing automorphism are (i) $|V|=|\widehat{V}|$ and (ii) the degree sequences of vertices in $V$ and $\widehat{V}$ match. However, they are not sufficient—a bipartite graph that satisfies these conditions but does not have a reversing automorphism can be found in Ref. [159].

The diagram in the second line is well defined because, for each $e$, there is only one $v\in V$ and one $\widehat{v}\in \widehat{V}$ that it is incident to. In other words, there are no dangling wires and no junctions of wires at any $e\in E$.

It is also equivalent to the ${\mathbb{Z}}_2$ dipole Ising model [167], so one can interpret this duality as the dipole KW duality [47, 168].

Given two graphs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, the tensor (Kronecker) product ${\mathcal{G}}_1\times {\mathcal{G}}_2$ is defined as the graph with vertex set $V({\mathcal{G}}_1)\times V({\mathcal{G}}_2)$ such that there is an edge between $(v_1,v_2)$ and $(w_1,w_2)$ if and only if there is an edge between $v_i$ and $w_i$ in $\mathcal{}{\mathcal{G}}_i$ for $i=1$, 2. The name is inspired by the fact that the adjacency matrix of ${\mathcal{G}}_1\times G_2$ is the tensor (Kronecker) product of the adjacency matrices of ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$.

An isomorphism $\varphi$ from ${\mathcal{G}}_1$ to ${\mathcal{G}}_2$ is a bijective map $\varphi :V({\mathcal{G}}_1)\to V({\mathcal{G}}_2)$ such that $v_1$ and $w_1$ have an edge between them if and only if $\varphi (v_1)$ and $\varphi (w_1)$ have an edge between them. Disjoint union of graphs ${\mathcal{G}}_0+{\overline{\mathcal{G}}}_0$ is also known as graph sum, which explains this notation. The canonical isomorphism from ${\mathcal{G}}_0+{\overline{\mathcal{G}}}_0$ to $\mathcal{G}$ is given by the obvious inclusions, $$\begin{gathered} V_0\hookrightarrow V_0\times \{0\},{\overline{V}}_0={\widehat{V}}_0\hookrightarrow {\widehat{V}}_0\times \{0\}, \\ {\widehat{V}}_0\hookrightarrow {\widehat{V}}_0\times \{1\},{\widehat{\overline{V}}}_0=V_0\hookrightarrow V_0\times \{1\}. \end{gathered}$$

While ${\mathcal{G}}_0$ does not necessarily have a duality symmetry, it always has a condensation operator ${\mathsf{C}}_0$, which is simply the sum of all the ${\mathbb{Z}}_2$ symmetry operators up to a normalization.

P. G. would like to thank Richard Hammack for pointing out a related graph.

The matrix element $⟨+\cdots +|\mathsf{C}|+\cdots +⟩$ can be interpreted as the partition function of the 3D (classical) topological ${\mathbb{Z}}_2$ gauge theory on $T^3$. This is the lattice counterpart of the continuum result in the second line of Eq. (5.32), where the fusion coefficient $({\mathcal{Z}}_2{)}_0$ is the partition function of the 3D ${\mathbb{Z}}_2$ gauge theory.

Since $W_k$ is multiplied by ${\prod }_p[(1+{\prod }_{\ell \in p}{Z}_{\ell })/2]$ in Eq. (a23), the choice of ${\gamma }_k$ does not matter.

The $1+1\mathrm{D}$ counterpart of this equation is where ${\mathsf{C}}_0=1+\eta$ can be viewed as the condensation operator for an ordinary ${\mathbb{Z}}_2$ symmetry. The higher quantum symmetry is ${\mathsf{C}}_1=1-\eta$.