Surface code quantum computing with error rates over 1%

Large-scale quantum computation will only be achieved if experimentally implementable quantum error correction procedures are devised that can tolerate experimentally achievable error rates. We describe an improved decoding algorithm for the Kitaev surface code, which requires only a two-dimensional square lattice of qubits that can interact with their nearest neighbors, that raises the tolerable quantum gate error rate to over 1%. The precise maximum tolerable error rate depends on the error model, and we calculate values in the range 1.1–1.4% for various physically reasonable models. These values represent a very high threshold error rate calculated in a constrained setting.

Figure 1

(a) Two-dimensional lattice of data qubits (squares) and syndrome qubits (circles) and examples of the data qubit stabilizers. (b) Sequence of cnot gates permitting simultaneous measurement of all stabilizers. Numbers indicate the relative timing of gates. The highlighted gates can be tiled to fill the plane.

Figure 2

Circuits determining the sign of a stabilizer (a) $S=\mathit{XXXX}$ or (b) $S=\mathit{ZZZZ}$ without explicit initialization gates.

Figure 3

Observed syndrome changes as a result of a single error. The vertical axis is time, the horizontal grid represents Fig. 1a, and only relevant qubit world lines are shown. A cnot gate suffering an error is drawn in full; otherwise it is represented by a tick mark protruding in the direction of application. Syndrome changes occur in pairs. Pairs of changes (shaded circles connected by lines) resulting from (a–d) specific two-qubit errors on specific cnot gates, (e) a syndrome qubit measurement error, and (f) a data qubit memory error.

Figure 4

Numbered error processes from Fig. 3 contributing to specific links. Superscripts 1, 2, and 3 indicate error processes occurring with probability $8p_2/15$, $2p_I/3$, and $p_M$, respectively. All others occur with probability $4p_2/15$.

Figure 5

Average number of rounds of error correction before logical $X$ failure as a function of the gate error rate $p$ when using $d_{\max }(s_1,s_2)$ and the standard error model.

Figure 6

Average number of rounds of error correction before logical $X$ failure as a function of the gate error rate $p$ when using $d_0(s_1,s_2)$ and the standard error model. The threshold error rate $p_{\mathrm{th}}$ remains 1.1%.