Topological order and memory time in marginally-self-correcting quantum memory

We examine two proposals for marginally-self-correcting quantum memory: the cubic code by Haah and the welded code by Michnicki. In particular, we prove explicitly that they are absent of topological order above zero temperature, as their Gibbs ensembles can be prepared via a short-depth quantum circuit from classical ensembles. Our proof technique naturally gives rise to the notion of free energy associated with excitations. Further, we develop a framework for an ergodic decomposition of Davies generators in CSS codes which enables formal reduction to simpler classical memory problems. We then show that memory time in the welded code is doubly exponential in inverse temperature via the Peierls argument. These results introduce further connections between thermal topological order and self-correction from the viewpoint of free energy and quantum circuit depth.

Figure 1

White dots represent missing vertex terms which act as sinks of excitations. Red dots represents charge excitations that can be absorbed into a sink. The entire system is split into a grid of $\sqrt{clog\left(L\right)}\times \sqrt{clog\left(L\right)}$ qubits such that there is at least one sink per patch.

Figure 2

An elimination of excitations and the Peierls argument. Grey circles represent excitations that evolved from the isolated single charge at the center. Dotted circles represent sinks of excitations. In this example, two excitations surrounded by sinks can be absorbed but the other two remain.

Figure 3

(a) The Sierpinski triangle from $f=1+x$. (b) The model from $f=1+x+x^2$.

Figure 4

Propagations of excitations according to a fractal geometry.

Figure 5

Graphical representations of $B,B_t,L_t$.

Figure 6

Evolutions of probability distributions of $d_t$.

Figure 7

Graphical representations of $B,B_t,L_t$ for $m=2$.

Figure 8

Ergodic decomposition for the toric code. Qubits live on edges of the lattice.

Figure 9

Two-dimensional sparse lattice. The lattice has length $L$ and each small square has dimensions $\sqrt{L}\times \sqrt{L}$. A Peierls droplet of frustration is shown in red, marking, for example, a cluster of spin down sites in a sea of spin up sites (blue). This contour has length $\ell =6$, or energy cost $12J$.

Figure 10

Complexity of the Gibbs ensemble and free energy as a function of the system size $L$.

Figure 11

Rescaled memory time $\tau$ as a function of system size $L$ at different inverse temperatures $\beta$ for the classical ferromagnetic sparse Ising model. The line is added to guide the eye along each temperature curve. At each temperature, there is an optimal system size, and the temperature at which $L$ is the optimal size goes to zero as $L\to \infty$.

Figure 12

Maximum memory time ${\tau }_{\text{max}}$ as a function of $\beta$ for the classical ferromagnetic sparse Ising model. Coefficient of determination $R^2=0.999567$ for this fit, which suggests that the double exponential prediction is accurate.

Figure 13

Ergodic decomposition for the toric code in a polynomial representation.

Figure 14

Welding as a gapped boundary.

Figure 15

(a) Terms near the welding surface. (b) The ergodic decomposition for the 2D welded code.

Figure 16

The ergodic decomposition for the three-dimensional toric code. The figure shows only two layers of $(x,y)$ planes in the three-dimensional lattice. For the full decomposition, one needs to stack these layers in the $z$ direction.

Figure 17

A lattice gauge theory $H$ defined on a homogeneous graph. Composite spins live on edges of the graph. Red lines represent a vertex operator on a vertex $v$ and blue lines represent a plaquette operator on a plaquette $p$.

Figure 18

(a) Immune spheres at vertices on a cubic grid. Shaded dots represent spheres at half-integer sites. We split the entire system into octahedral objects and view them as composite spins. (b) A construction of a lattice gauge theory. Composite spins live on faces of a cubic lattice.