Symmetry-protected topological order at nonzero temperature

We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global onsite symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such onsite symmetries are not thermally stable. Second, we prove that the three-dimensional (3D) cluster-state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high-error tolerance of this 3D cluster-state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry.

Figure 1

(a) The triangular lattice and one of the terms $h_v$ belonging to $H_1$. The 1-link of the vertex $v$ is the set of thick blue edges. (b) A valid configuration has sinks (red) for each square region in ${\mathcal{P}}_l$, where a sink is a vertex $v$ with $k_v=0$.

Figure 2

The disentangler acts first on terms in $H\left(\mathit{k}\right)$ neighboring the sinks (red), then moves outward. The set $V\left(0\right)$ consists of the sinks, depicted in red and the successive shaded blue disks represent the sets $V\left(1\right),V\left(2\right)$, and $V\left(3\right)$.

Figure 3

In two dimensions, there are two distinct Pachner moves: (a) two triangles are replaced by two triangles and (b) three triangles are replaced by one triangle and the number of vertices changes by 1. The arrows represent the orientation of each edge. Notice that there are no oriented loops on any triangle.

Figure 4

The two principal steps to take the original triangulation $T^{\mathrm{\Delta }}$ in (a) to the triangulation $T^S$ in (c), whose vertices are all sinks (depicted in red). The first step is to renormalize the $T^{\mathrm{\Delta }}$, resulting in the triangulation in (b). The second step is a vertex shifting, resulting in the triangulation in (c). The grid ${\mathcal{P}}_l$ with side lengths ${[clog\left(L\right)]}^{\frac{1}{d}}$ is displayed by the black, dashed lines. Faded gray nodes denote ambient vertices no longer part of the triangulation, which correspond to decoupled spins in the $\left|+\right.〉$ state after the circuit $\mathcal{D}$ has been applied.

Figure 5

The first six parallel Pachner moves for a single renormalization step that scales the edge lengths of the triangular lattice by a factor of 2. Red indicates the new edges arising from Pachner moves. Notice that some of the edges are unchanged (namely, the diagonal ones), but this process can be repeated to rescale them too. Sinks are not displayed in this figure as they do not yet play a role. This process can be repeated $O[loglog(L\left)\right]$ times to rescale $T^{\mathrm{\Delta }}$ to the renormalized triangulation in Fig. 4.

Figure 6

(a) A unit cell of the cluster lattice $\mathcal{C}$ with a single cluster term $K\left({\sigma }_2\right)$. (b) An elementary 1-form operator $S\left(\partial {\sigma }_3\right)$. The primal qubits are depicted in green and the dual qubits are depicted in blue.

Figure 7

Examples of excitation operators. A 1-cycle is depicted in blue, while a dual-1-cycle is depicted in green.

Figure 8

The gauging map on computational basis states. (a) States on the dual sublattice map to states on the primal sublattice. (b) States on the primal lattice map to states on the dual sublattice. The sums are performed mod 2.

Figure 9

(a) Gauging ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry of the two-dimensional model (which has the 1D cluster model on its boundary) leads to a twisted gapped boundary where pointlike $e_1{m}_2$ and $e_2{m}_1$ particles may condense. This can be viewed as a nontrivial domain wall in the two-dimensional toric code. (b) Gauging the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ 1-form symmetry of the four-dimensional model (which has the three-dimensional RBH model on its boundary) leads to a nontrivial domain wall in the four-dimensional toric code, which exchanges electric and magnetic looplike excitations.

Figure 10

(a) The membrane operator $M\left({\mathrm{\Gamma }}_1\right)$ and (b) the membrane operator $M\left({\mathrm{\Gamma }}_2\right)$. The top and bottom boundaries are identified, as are the front and back boundaries. The primal qubits are shaded in blue, and the dual qubits are shaded in green. The restrictions of these membrane operators to either $L$ or $R$ anticommute. Note that length in the $\widehat{z}$ direction has been exaggerated.

Figure 11

Sketch of the phase diagram of the three-dimensional random-plaquette ${\mathbb{Z}}_2$ gauge theory [60]. The random-plaquette ${\mathbb{Z}}_2$ gauge theory has $\pm 1$ couplings, and the fraction of negative couplings is labeled $p$, the disorder strength. The disorder strength $p$ is on the horizontal axis, and temperature $T$ is on the vertical axis. The solid black line is the boundary between the ordered and disordered phases. The Nishimori point at $(p_c,T_0)$ lies at the intersection of the phase boundary and the Nishimori line $e^{-2\beta }=p/(1-p)$. The critical temperatures of $H\left(\lambda \right)$ in the limiting case of $\lambda =0,\infty$ are depicted on the vertical axis. For intermediate values $\lambda \in (0,\infty )$, if correlation between plaquette-random variables is ignored, the critical temperature is expected to interpolate between $T_0$ and $T_{\infty }$.