Self-correcting quantum memory with a boundary

We study the two-dimensional toric-code Hamiltonian with effective long-range interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow an arbitrary increase in the lifetime of the stored quantum information by making $L$, the linear size of the memory, larger [Chesi et al., Phys. Rev. A 82, 022305 (2010)]. We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough $L$. This influence can be either beneficial or detrimental. In particular, we study an effective Hamiltonian proposed by Pedrocchi et al. [Phys. Rev. B 83, 115415 (2011)] that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in $L^2$ for both periodic and open boundary conditions, although the exponent in the latter case is found to be less favorable. However, $L$ is upper bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.

Figure 1

A planar code of size $L=8$. Depicted are a four-qubit plaquette operator $n_p$, a three-qubit plaquette operator $n_{p^{\prime }}$, a four-qubit star operator $n_s$, and a three-qubit star operator $n_{s^{\prime }}$. The logical operator $X$ ($Z$) is given by any chain of Pauli operators ${\sigma }_x$ (${\sigma }_z$) connecting the top and bottom (left and right) boundaries.

Figure 2

The plot shows the autocorrelation function $C_{\mathrm{corr}}^X$ as a function of the fraction of physical spins $p_z$ that have suffered spin-flip errors. Circles give the numerical results; the lines are guides to the eye. The curves correspond to different lattice sizes $L=32$ (blue), 64 (magenta), and 128 (red), bottom to top at small $p_z$. We anticipate that in the limit $L\to \infty$ error correction is unambiguously possible for any $p_z<p_c$, with $p_c\gtrsim 0.102$.

Figure 3

The code has suffered ${\sigma }_x$ errors at the black qubits in (a). The syndrome measurement detects plaquette anyons $A$, $B$, and $C$. For the error-correction procedure, we first perform a Delaunay triangulation of the full graph with edges connecting all real anyons (in the case of only three real anyons, the full graph and the triangulation coincide). Then, virtual anyons $A^{\prime }$, $B^{\prime }$, and $C^{\prime }$ are added. The graph (b) is then used for the perfect-matching algorithm, where the obtained matching is highlighted with arrows. Error correction fails, because anyons $B$ and $C$ are fused and anyon $A$ is moved to the upper boundary, thereby performing a logical $X$ operator on the qubit stored in ${\mathcal{K}}_0$.

Figure 4

Ratio of the number of anyons created on the boundary to the number of anyons created in the bulk if the anyon population approaches its equilibrium value. The blue circles show numerical results sampled over a time ${10}^5\times {\left({\kappa }_1\Delta \right)}^{-1}$. The red curve shows the analytical prediction $\exp \left(\beta e_{\mathrm{eq}}\right)/4L$. We have used parameters $\beta \Delta =1/0.3$ and $A/\Delta =0.1$ and a bath $\gamma \left(\omega \right)$ as in (15).

Figure 5

Temporal decay of the autocorrelation functions $C_{\mathrm{corr}}^Z\left(t\right)$ for different lattice sizes and boundary conditions, for the Hamiltonian (17) and the bath (15). Times are in units of ${\left({\kappa }_1\Delta \right)}^{-1}$ and the physical parameters are $\beta \Delta =1/0.3$ and $A/\Delta =0.1$. From left to right (at height $0.6$, say) we have (grid type size $L$) toric 32, planar 32, toric 64, planar 64, toric 128, planar 128, toric 256, planar 256, toric 512, and planar 512. The intersection of $C_{\mathrm{corr}}^Z\left(t\right)$ with the horizontal line at height $0.9$ is used to determine the lifetimes $\tau \left(0.1\right)$ given in Fig. 6.

Figure 6

Lifetimes $\tau \left(0.1\right)$ of the quantum information stored in the systems described in the caption of Fig. 5. The blue crosses represent planar codes, the red circles toric codes. The sizes $L$ of the codes are 8, 16, 32, 64, 128, 256, 512, and 1024. The red line shows the analytical prediction for the lifetime in the toric case obtained from (18) with $c\left(0.1\right)=5.1\%$. The authors of Ref. [5] find with an analogous plesot (Fig. 6) $c\left(0.1\right)=4.4\%$. The increased lifetime is due to the choice of the square of the Euclidean distance rather than the Manhattan length as the weight of an error chain.

Figure 7

An effective planar code obtained from a honeycomb lattice with a boundary. The black dots represent effective qubits emerging from the underlying honeycomb model. The four isolated effective qubits in the corners are depicted for completeness but not needed for information storage. Blue (dark) squares (like $p$) are plaquette operators, and red (light) squares (like $s$) are star operators. There are two nonequivalent ways of defining a pair of macroscopic Pauli-like observables, the “big” logical operators $X$ and $Z$ and the “small” logical operators $X^{\prime }$ and $Z^{\prime }$. However, an undetectable logical error consisting of only three single-qubit errors can be performed on all of these operators. For instance, a sequence of ${\sigma }_x$ errors on the three qubits along path 1 (2) causes an error on the logical $Z^{\prime }$ ($Z$) operator.

Figure 8

A planar code consisting only of four- and one-qubit stabilizers. We consider only ${\sigma }_x$ errors and the corresponding plaquette operators here. As in Fig. 7 ${\sigma }_z$ measurements are performed on the top and bottom boundaries. The top and bottom boundaries are thus able to store and detect the presence of plaquette anyons. The logical $Z$ operator is a path of single-qubit ${\sigma }_z$ operators connecting the top and bottom boundaries (cf. the operator $Z^{\prime }$ in Fig. 7). The dashed line lies midway between the left and right boundaries. All error paths in the left part of the figure will not lead to a logical error after error correction, while for the error paths in the right part of the figure a logical $X$ operator is performed. Whether an error path of type ($*$) is correctable depends not only on its length, but also on its position relative to the boundaries. However, any error path of type ($*$) and of horizontal length below $L/2$ $\mathit{is}$ correctable.

Figure 9

A planar code of size $L=64$ consisting only of four- and one-qubit stabilizers in contact with a bath with $\gamma \left[\right(N-1\left)\Delta \right]/\gamma \left(0\right)=40$, where $N=2L(L+1)$. The vertical axis shows the number $n$ of single anyons that have been created. The green (rightmost) curve shows the error-corrected autocorrelation function $C_{\mathrm{corr}}\left(n\right)$ and the blue (leftmost) curve the uncorrected autocorrelation function [(7) without ${\Phi }_{\mathrm{corr}}$]. Both curves are sampled over $1.2\times {10}^4$ experiments. The red curve (reaching negative values) shows the analytical prediction (29) with $c=1.12$. We see that performing a single error-correction step before the readout of the stored quantum information allows enhancement of its lifetime [$\tau \left(0.1\right)$, say] by more than an order of magnitude.

Figure 10

The number of single anyons created during the lifetime $\tau \left(\varepsilon \right)$ of the memory as a function of the lattice size $L$. The points show numerical results obtained for a bath with $\gamma \left[\right(N-1\left)\Delta \right]/\gamma \left(0\right)=40$, where $N=2L(L+1)$, the lines show the analytical prediction (30) with $c^{\prime }=1.12$. From bottom to top we have $\varepsilon =0.1,0.2,0.3$.