Realization of an Error-Correcting Surface Code with Superconducting Qubits

Quantum error correction is a critical technique for transitioning from noisy intermediate-scale quantum devices to fully fledged quantum computers. The surface code, which has a high threshold error rate, is the leading quantum error correction code for two-dimensional grid architecture. So far, the repeated error correction capability of the surface code has not been realized experimentally. Here, we experimentally implement an error-correcting surface code, the distance-three surface code which consists of 17 qubits, on the Zuchongzhi 2.1 superconducting quantum processor. By executing several consecutive error correction cycles, the logical error can be significantly reduced after applying corrections, achieving the repeated error correction of surface code for the first time. This experiment represents a fully functional instance of an error-correcting surface code, providing a key step on the path towards scalable fault-tolerant quantum computing.

Figure 1

Layout and circuit implementation. (a) Structure schematic of distance-three surface code. 17 qubits are choosen from Zuchongzhi 2.1 superconducting quantum processor, with 8 data qubits (gray dots), $4Z$-type ancilla qubits (green dots), and $4X$-type ancilla qubits (red dots). Each pair of qubits is connected with a coupler (black rectangle). Connecting lines are colored according to their involvement in two-qubit gate layers as shown in (b). (b) Circuit for one error correction cycle. Dots on the left are in one-to-one correspondence to those in (a). Squares with $Y-$ and $Y+$ represent $Y$ rotation by an angle of $-\pi /2$ and $\pi /2$. Line with two dots denotes a controlled-phase gate. All gates in one color block are applied simultaneously. Gray rectangles with $Z$ denote a measurement in the $Z$ basis. Block labeled DD are for dynamical decoupling operators.

Figure 2

System calibration. (a) Integrated histogram of single qubit $R_Y(\pi /2)$ rotation (1Q XEB), CZ gate (2Q XEB) and readout error (readout). (b) Cross entropy benchmarking fidelity from random sampling tasks for 9 instances of random circuits. Solid dots represent average fidelity over repetitions of sampling task with the same random seed. The average fidelity is 0.030. Error bars describes $\pm 5\sigma$ statistical deviation, color band is for $\pm 3\sigma$ with $\sigma =1/\sqrt{N_{\text{sample}}}$, where $N_{\text{sample}}=1440000$.

Figure 3

Error detection rate. (a) Detection event fraction curve for logical $|0_L⟩$ state as a function of surface code cycles. Various lines describe the fraction of samples with an error detected on the corresponding Z-type ancilla qubit. (b) Detection event fraction for logical $|{-}_L⟩$ state.

Figure 4

Error detection correlation. (a) Correlation matrix for logical $|0_L⟩$ state. The finer scale is used for the marking of ancilla qubits and each block has a definite cycle index. Color scheme is presented in the side color bar with the dark side for low correlation and yellowish side for strong correlation. (b) Correlation matrix for logical $|{-}_L⟩$ state.

Figure 5

Fidelity of logical state with error detection and/or error correction. (a) Fidelity of the postselected logical $|0_L⟩$ state by error detection as function of clock cycles, i.e., the portion of samples that retain the logical state through some clock cycles. Various lines correspond to different postselection schemes. The purple (pentagon) lines are obtained by discarding all data with error detected from the measurements of either data qubits or ancilla qubits. The green (hexagon) line corresponds to the dropping scheme based on data qubits only while the navy (octagon) line is only affected by the measurement of ancilla qubits. The red (triangle) line describes the result with no postselection. The dotted line depicts the prediction based on relaxation time $T_1$ of the best physical qubit among all used physical qubits. Logical error rates ${\epsilon }_L$ are extracted from the curve and logical fidelities $T_L$ are calculated. These values are listed by each line. Inset describes the retained rate for the three postprocessing schemes as a function of rounds. (b) Results for the postselected logical $|{-}_L⟩$ state by error detection. (c) Fidelity of logical $|0_L⟩$ state with the number of surface code cycles with error correction (blue line with square) or without (red line with triangular). (d) Same quantity for the logical $|{-}_L⟩$ state after error correction.