Performance of surface codes in realistic quantum hardware

Surface codes are generally studied based on the assumption that each of the qubits that make up the surface code lattice suffers noise that is independent and identically distributed (i.i.d.). However, real benchmarks of the individual relaxation $\left(T_1\right)$ and dephasing $\left(T_2\right)$ times of the constituent qubits of state-of-the-art quantum processors have recently shown that the decoherence effects suffered by each particular qubit actually vary in intensity. In consequence, in this paper we introduce the independent nonidentically distributed (i.n.i.d.) noise model, a decoherence model that accounts for the nonuniform behavior of the decoherence parameters of qubits. Additionally, we use the i.n.i.d. model to study how it affects the performance of a specific family of quantum error correction codes known as planar codes. For this purpose we employ data from four state-of-the-art superconducting processors: ibmq_brooklyn, ibm_washington, Zuchongzhi, and Rigetti Aspen-M-1. Our results show that the i.i.d. noise assumption overestimates the performance of surface codes, which can suffer up to $95\%$ performance decrements in terms of the code pseudothreshold when they are subjected to the i.n.i.d. noise model. Furthermore, we consider and describe two methods which enhance the performance of planar codes under i.n.i.d. noise. The first method involves a so-called reweighting process of the conventional minimum weight perfect matching (MWPM) decoder, while the second one exploits the relationship that exists between code performance and qubit arrangement in the surface code lattice. The optimum qubit configuration derived through the combination of the previous two methods can yield planar code pseudothreshold values that are up to $650\%$ higher than for the traditional MWPM decoder under i.n.i.d. noise.

Figure 1

Graphical representation of a $7\times 7$ planar code. The data qubits, the measure-$X$ qubits, and the measure-$Z$ qubits are depicted by white, yellow, and green dots, respectively. Data qubits that suffer $X$, $Z$, and $XZ$ operators are portrayed by light green, light yellow, and lighter green dots. The action of various stabilizer elements is highlighted using different color patterns: (a) action of a measure-$Z$ qubit, (b) action of a measure-$X$ qubit, (c) combination of two adjacent plaquette-plaquette stabilizers, (d) combination of two adjacent vertex-vertex stabilizers, (e) combination of two adjacent vertex-plaquette stabilizers, (f) $Z_L$ operator, and (g) $X_L$ operator.

Figure 2

Logical error rates of planar codes for the i.i.d. and i.n.i.d. models. (a) Planar codes operating over the i.i.d. noise model. (b) Planar codes operating over the i.n.i.d. noise model constructed with data from the ibm_washington hardware [20]. The green line represents the performance of the uncoded system, i.e., $P_L\left(p_{\mathrm{PT}}\right)=p_{\mathrm{PT}}$. Note that the $x$ axis represents the probability that a Pauli error occurs $p=p_X+p_Y+p_Z$ and represents the value of $p$ obtained for the mean values of $T_1$ and $T_2$ (for the i.n.i.d. channel, since each value of $p$ is different for each qubit, we use the overall mean for the plot).

Figure 3

Example of a MWPM and rMWPM performance towards an error in a $5\times 5$ planar code. Diagram 1 shows the proposed error composed by data qubits (white circles), data qubits experiencing nontrivial error operators (red circles), measure-$z$ qubits, and measure-$x$ qubits (green and yellow circles). One-syndrome measurement qubits are labeled with an exclamation mark. Diagrams 2 and 3 represent the two CSS subgraphs, where pink and blue denote the qubits with lower and higher relaxation parameters, respectively. Diagram 4 shows the overall graph. Diagrams 5 and 6 show the minimum weight perfect matching of the graphs considering taxicab distance and reweighted distance, and diagrams 7 and 8 show the recovery operators proposed by the MWPM and rMWPM decoders.

Figure 4

The $5\times 5$ planar code codifying the information of a logical qubit. The data qubits of both sublattices are highlighted with identifiable blue and orange qubits. A $Z$ chain of weight 4 in the orange lattice is detected by two measuring qubits highlighted in yellow with an exclamation mark. Moreover, a $Z$ chain in the blue sublattice commutes with all the measuring qubits.

Figure 5

Code pseudothresholds obtained for the qubit data of four quantum processors in different distance square planar codes. From left to right: ibm_washington, ibmq_brooklyn, Zuchongzhi, and Aspen-M-1 Rigetti. The blue dots represent the results under i.i.d., and the yellow dots reflect the performance of the code under i.n.i.d. using the conventional MWPM decoder. The green dots represent the performance of the code under the rMWPM decoder, and the red dots represent the performance of the conventional MWPM decoder but with the code undergoing the architecture optimizing method. Lastly, the purple lines indicate the performance of the code under both architecture optimizing and the rMWPM decoder. The bars correspond to the standard deviation of all the considered configurations of the specific code.

Figure 6

Change in $p_{\mathrm{PT}}$ when considering the optimized architecture and the rMWPM compared to two different scenarios with respect to the variance coefficient of $T_2$. At top, the change in $p_{\mathrm{PT}}$ is compared with the i.n.i.d. case under the conventional MWPM decoder. At bottom, it is compared with the i.i.d. case.

Figure 7

Considered relaxation and dephasing times for the i.n.i.d. decoherence model. The $T_1$ and $T_2$ values associated to the individual qubits of each of the systems are presented using histograms. (a) Data for the ibmq_brooklyn machine with 65 qubits. Timestamp: 16/11/2021 10:41 (CET) [21]. (b) Data for the ibm_washington machine with 127 qubits. Timestamp: 16/11/2021 12:21 (CET) [20]. (c) Data for the Zuchongzhi machine with 66 qubits. Data are from [22]. (d) Data for the Rigetti Aspen-M-1 machine with 79 qubits. Data are obtained using the Strangeworks platform [23]. Timestamp: 30/05/2022 11:37 (CET). Note that the Aspen-M-1 quantum computer is made up of 80 qubits, but when we requested the corresponding data we were only able to obtain measurements related to its qubits.

Figure 8

Graphical representation of the data qubits of $3\times 3$, $5\times 5$, and $7\times 7$ planar codes. Each data qubit is represented by a gray-shaded dot and a number representing its index.