Proposal for a minimal surface code experiment

Current quantum technology is approaching the system sizes and fidelities required for quantum error correction. It is therefore important to determine exactly what is needed for proof-of-principle experiments, which will be a major step towards fault-tolerant quantum computation. Here we propose a surface code based experiment that is the smallest, both in terms of code size and circuit depth, that would allow errors to be detected and corrected for both the $X$ and $Z$ bases of a qubit. This requires 17 physical qubits initially prepared in a product state, on which 16 two-qubit entangling gates are applied before a final measurement of all qubits. A platform agnostic error model is applied to give some idea of the noise levels required for success. It is found that a true demonstration of quantum error correction will require fidelities for the preparation and measurement of qubits and the entangling gates to be above $99\%$.

Figure 1

Two alternative definitions of stabilizers in a surface code. Stabilizers are associated with plaquettes, including the four semioval plaquettes on the edges. Code qubits reside on the vertices of the lattices, and ancilla qubits reside on the center of each plaquette.

Figure 2

Code qubits are shown with filled circles, and ancilla qubits with empty circles. The entangling gates are shown with colored lines. The red gates are performed first, followed by orange, then green, and finally blue.

Figure 3

Graph of the threshold for gate noise, parametrized by the probability $g$, against preparation and measurement noise, parametrized by $p=m$.

Figure 4

Graph of the ratio $f$ achieved at the threshold value $g$ for different preparation and measurement noise.

Figure 5

Graph of the ratio $f$ for different values of the noise rate $p=m=g$. Results are shown for both the optimal decoder (a lookup table populated by experimental data) and Tomita-Svore (a lookup table based on a set of rules).