Fidelity threshold of the surface code beyond single-qubit error models

The surface code is a promising alternative for implementing fault-tolerant, large-scale quantum information processing. Its high threshold for single-qubit errors under stochastic noise is one of its most attractive features. We develop an exact formulation for the fidelity of the surface code that allows us to probe much further on this promise of strong protection. This formulation goes beyond the stochastic single-qubit error model approximation and can take into account both correlated errors and inhomogeneities in the coupling between physical qubits and the environment. For the case of a bit-flipping environment, we map the complete evolution after one quantum error correction cycle onto the problem of computing correlation functions of a two-dimensional Ising model with boundary fields. Exact results for the fidelity threshold of the surface code are then obtained for several relevant types of noise. Analytical predictions for a representative case are confirmed by Monte Carlo simulations.

Figure 1

Schematic representation of the surface code. The circles are physical qubits. The gray squares represent plaquettes ($B_{\square }$) and stars ($A_{♦}$). The product of single-qubit operators along the dash-dotted lines ${\Gamma }_X$ (vertical) and ${\Gamma }_Z$ (horizontal) define the logical operators ${\overline{X}}_{{\Gamma }_X}$ and ${\overline{Z}}_{{\Gamma }_Z}$, respectively.

Figure 2

Qubit variable ${\sigma }_{\mathbf{i}}$ and corresponding plaquette variables ${\mu }_{\mathbf{m}}$ and ${\mu }_{\mathbf{n}}$.

Figure 3

(a) Links and plaquettes involved in Eq. (27). (b) Links and plaquette involved in Eq. (32).

Figure 4

Phase diagram of the Ising model with homogeneous complex coupling constant $h$ and its significance to the one-cycle QEC fidelity threshold. Here $u=e^{-4h}$. In the gray regions quantum information cannot be recovered.

Figure 5

Phase diagram of the Ising model with random bond sign and its significance to the one-cycle QEC fidelity threshold.

Figure 6

Ratio $\langle X\rangle =|\frac{\mathcal{B}}{\mathcal{A}}|$ evaluated through Monte Carlo sampling for three system sizes: $L=20$ (761 qubits, circles), $L=30$ (1741 qubits, diamonds), and $L=40$ (3121 qubits, triangles). The solid lines are guides for the eye.