Error-correction and noise-decoherence thresholds for coherent errors in planar-graph surface codes

We numerically study coherent errors in surface codes on planar graphs, focusing on noise of the form of $Z$ or $X$ rotations of individual qubits. We find that, similar to the case of incoherent bit and phase flips, a trade-off between resilience against coherent $X$ and $Z$ rotations can be made via the connectivity of the graph. However, our results indicate that, unlike in the incoherent case, the error-correction thresholds for the various graphs do not approach a universal bound. We also study the distribution of final states after error correction. We show that graphs fall into three distinct classes, each resulting in qualitatively distinct final-state distributions. In particular, we show that a graph class exists where the logical-level noise exhibits a decoherence threshold slightly above the error-correction threshold. In these classes, therefore, the logical level noise above the error-correction threshold can retain a significant amount of coherence even for large-distance codes. To perform our analysis, we develop a Majorana-fermion representation of planar-graph surface codes and describe the characterization of logical-state storage using fermion-linear-optics-based simulations. We thereby generalize the approach introduced for the square lattice by Bravyi et al. [npj Quantum Inf. 4, 55 (2018)] to surface codes on general planar graphs.

Figure 1

A small surface code on a square lattice. The white circles mark qubits. At each vertex of the lattice, a $Z$ stabilizer is placed, acting on all adjacent qubits. On each plaquette, that is an area surrounded (or at the boundary partially surrounded) by links, an $X$ stabilizer is placed, acting on all qubits that are on its boundary. Examples of $X$ and $Z$ stabilizers are indicated as grey boxes. The code patch has two rough (dashed bars) and two smooth boundaries (solid bars). For both types of boundaries, an appropriately truncated stabilizer is shown.

Figure 2

(a) The embedded planar graph of the $Z$ stabilizers for the setup equivalent to that in Fig. 1. Grey circles mark the nodes of the graph that represent $Z$ stabilizers. (b) The same patch with added virtual stabilizers (see text). (c) The dual of the graph, forming the graph of $X$ stabilizers. Light grey circles mark the nodes of the dual graph that represent $X$ stabilizers. (d) The final code, made up of both the $X$- and $Z$-stabilizer graphs and the qubits. The qubits are shown as white circles. (To avoid clutter, the stabilizer nodes are not shown.)

Figure 3

(a) An example of a general planar graph with two rough boundaries (left and right), defining the $Z$ stabilizers (dark grey circles) of the surface code patch. (b) The same graph with added virtual stabilizers to keep track of the code boundaries (see text). (c) In light grey, the dual graph, defining the $X$ stabilizers (light grey circles). (d) The qubits (white circles) placed on the resulting surface code patch.

Figure 4

(a) Representation of a qubit in the $C4$ encoding with four Majorana fermions $c_1,c_2,c_3,c_4$. The Pauli operators $X$ and $Z$ can each be formed by two equivalent dimers (pairs) of Majorana fermions. The order of the operators is indicated by arrows, e.g., $X=i{c}_1{c}_2$ is represented by an arrow from $c_1$ to $c_2$. (b) Majorana graph for four qubits. A stabilizer with support on different qubits can be represented in terms of the Pauli dimers (black) of the qubits, but also using link dimers (grey) between the qubits. Link dimer orientations must be chosen to ensure equivalence to Pauli dimerization of stabilizers; this results in a Majorana graph where edges (Pauli and link dimers) have Kasteleyn orientation.

Figure 5

(a) The construction of graph $\mathcal{G}$ with one face per stabilizer for the surface code in Fig. 3. (b) The edges of $\mathcal{G}$ can be used as Majorana dimers for the $C4$ encoding of the surface code. This results in four Majorana fermions per qubit except for the four qubits in the corners (grey). (c) A Kasteleyn orientation of the dimers. (d) Zoom from panel (c) showing the $C4$ encoding of the individual qubits (cf. Fig. 4).

Figure 6

Majorana encoding of logical operators. A logical operator of the code translates to an additional face in $\mathcal{M}$ bounded by a link dimer formed by two of the four initially unpaired corner Majorana fermions. (a) shows $\mathcal{M}$ with the two link dimers that form both realizations of $Z_{\text{L}}$ (each dimer corresponds to one of the two virtual $Z$ stabilizers of Sec. 2). The additional faces of $\mathcal{M}$ for including these dimers are shown in grey. (b) Link dimers for $X_{\text{L}}$, showing again the two additional faces.

Figure 7

The logical error rate $p_{\text{L}}$ for the coherent and Pauli-twirled error models versus the angle $\eta$ for three different code sizes of each lattice. (The argument of $\mathcal{O}$ indicates the approximate number $N$ of qubits; the concrete value of $N$ depends on the lattice.) The dashed lines indicate the thresholds obtained by fitting a finite-size scaling ansatz [15]. Above the threshold, we sketch the graphs of the $X$ stabilizers. They are, from left to right, dual of trihex, dual of hexagonal, dual of kagome, square, kagome, hexagonal, and trihex.

Figure 8

The thresholds for $X$ rotations versus the thresholds for $Z$ rotations for both the coherent (green) and Pauli-twirled (orange) error models, for the lattices indicated in Fig. 7. (The thresholds obtained in Ref. [31] are shown in red.) The blue line indicates the bound Eq. (42).

Figure 9

A comparison between the logical error rate $p_{\text{L}}$ for the doubly odd and square lattice. (A small patch of the $Z$-stabilizer graph for the doubly odd lattice shown as inset.) The upper graph shows $p_{\text{L}}$ for coherent errors, the lower shows $p_{\text{L}}$ for incoherent errors. The simulations for the square lattice are the same used for Fig. 7. The doubly odd lattice is simulated for system sizes of 437, 1365, and 2805 qubits.

Figure 10

Panels (a), (c), (d): The projection of 100 000 sampled states $|{\psi }_s\rangle$ resulting after error and correction operation with initial state $|{+}_{\text{L}}\rangle$. Distribution for a surface code on the (a) hexagonal lattice, (c) square lattice, (d) kagome lattice. The size of the system is for all examples chosen such that the number of qubits is around 500. (b) The Bloch sphere and an example of an initial state (red dot). The green (blue) circle marks the possible final states in lattices with even-weight $Z$ stabilizers and odd-weight (even-weight) $Z_{\text{L}}$.

Figure 11

Distributions of 100 000 samples of ${\langle {\delta }_s^2\rangle }_{\mathrm{\Omega }}$ for the seven lattices in Fig. 7, and the doubly odd lattice (on the right of the square lattice). We give a sketch of the corresponding $Z$-stabilizer graph above the distributions. (These are the duals of the graphs in Fig. 7.) The values of the coherent noise parameter $\eta$ are shown above the distributions; they are chosen as $\eta \approx {\eta }_{\text{th}}$, i.e., approximately at the threshold (middle row), $\eta \approx {\eta }_{\text{th}}-0.01\pi$ (top row), and $\eta \approx {\eta }_{\text{th}}+0.01\pi$ (bottom row). Samples are taken for three different systems sizes $\mathcal{O}\left(500\right)$ qubits (red), $\mathcal{O}\left(1500\right)$ qubits (green), and $\mathcal{O}\left(2500\right)$ qubits (blue).

Figure 12

The syndrome average $\mathrm{\Delta }$ [cf. Eq. (61)] for a decoder that chooses a Pauli correction optimizing ${\langle {\delta }_s^2\rangle }_{\mathrm{\Omega }}$ for each $s$. Different colors correspond to different $Z$-stabilizer lattices. (D denotes the dual of a lattice.) The value of $\mathrm{\Delta }$ for strong rotations depends on the graph class; the estimate Eq. (62) [Eq. (63)] for lattices with even- and odd-weight stabilizers (even-weight stabilizer and odd-weight logical operators) is indicated by ${\mathrm{\Delta }}_{\text{e-o}}$ (${\mathrm{\Delta }}_{\text{unitary}}$). The system sizes are the same as those considered in Fig. 7.