Tailored XZZX codes for biased noise

Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.

Figure 1

Map from the $\mathcal{S}(13,2,1)$ code (a) to a GTC with ${\vec{L}}_1=(3,2),{\vec{L}}_2=(-2,3)$ [(b),(c)]. The qubits are represented by black dots and labeled along the solid-grey string. The stabilizer generators are represented by green plaquettes (only one generator is plotted and others are obtained by shifting along the grey string). [(b),(c)] Obtained by wrapping the grey string around the torus. (c) The 2D layout of (b), with opposite sides of the parallelogram enclosed by ${\vec{L}}_1, {\vec{L}}_2$ identified.

Figure 2

The construction of doubled graph for a non-two-colorable GTC with ${\vec{L}}_1=(0,2),{\vec{L}}_2=(3,0)$. (a) The original non-two-colorable graph $G$ that defines the GTC. There is a XZZX stabilizer on each of the shaded plaquettes. The red cycle depicts a logical operator $Y_4{Y}_5{Y}_6$ that cannot wrap back on itself after a single loop due to odd periodicity in the horizontal direction. (b) The doubled graph $G_d$ obtained by taking two copies of $G$ and gluing them horizontally. Now $G_d$ becomes two colorable and the stabilizers are only put on the shaded plaquettes. Consequently, a single loop is sufficient for the logical operator to wrap back on itself. (c) The equivalent graph $G_d^{\prime }$ of $G_d$ where qubits are placed on the edges and stabilizers are placed on the vertices. Now the logical operator corresponds to a well-defined cycle that is homologically nontrivial on the doubled torus.

Figure 3

Geometric representation of the logical operators of the $\left[\right[13,1,5\left]\right]$ GTC with ${\vec{L}}_1=(-1,5)=2\widehat{x}+3\widehat{z}, {\vec{L}}_2=(-3,2)=-\frac{1}{2}\widehat{x}+\frac{5}{2}\widehat{z}$ for (a) $\omega =1$ and (b) $\omega =3$. The horizontal and vertical axes are aligned with the $X$ and $Z$ axes, respectively. The doubled graph is obtained by doubling along ${\vec{L}}_2$: ${\vec{L}}_{1,d}={\vec{L}}_1,{\vec{L}}_{2,d}=2{\vec{L}}_2=-\widehat{x}+5\widehat{z}$. The blue (red) cycle represents the logical operator associated with ${\vec{L}}_{1,d}$ (${\vec{L}}_{2,d}$). The shortest nontrivial cycles ${\vec{L}}_m$ in the modified 1-norm determined by Eq. (3) are thickened. For (a) $\omega =1, {\vec{L}}_m={\vec{L}}_{1,d}$ and $d^{\prime }={\parallel {\vec{L}}_{1,d}\parallel }_{xz,1}=5$; For (b) $\omega =3, {\vec{L}}_m={\vec{L}}_{2,d}$ and $d^{\prime }={\parallel {\vec{L}}_{2,d}\parallel }_{xz,1}=8$.

Figure 4

Verification of the effective code distance Eq. (3) as a good performance metrics for the GTCs. Given a bias parameter $\omega$, we numerically obtain the logical error rate $p_L$ for a set of randomly chosen GTCs using the MWPM decoders over a range of physical error rate $p$ (below the threshold), and fit the logical error rate by $p_L\propto p^r$. There is a good agreement between $r$ and $\lfloor (d^{\prime }+1)/2\rfloor$ for both (a) $\omega =1$ and (b) $\omega =3$. The range of the physical error rate $p$ used for the fitting is (a) $p\in [0.02,0.08]$ and (b) $p\in [0.06,0.1]$.

Figure 5

The thresholds $p_c$ of GTCs using MWPM decoders as functions of the bias parameter $\omega$, obtained by doing the critical exponent fit [38] on numerical data from MC simulations. (a) Code threshold assuming perfect syndrome measurements. The dashed line indicates the hashing bound. (b) Code threshold under a phenomenological error model, in which each syndrome measurement fails with a probability that equals the total error probability $p$ of the data qubits.

Figure 6

The densest packing of diamonds that correspond to the optimal choice of the GTCs given a bias parameter $\omega$. The diamonds to pack (a) have aspect ration $\omega$ and the densest packing pattern is the regular tiling (b).

Figure 7

The scatter plots of the required size $n$ of different codes for achieving a target effective distance $d^{\prime }$ under (a) $\omega =1$ and (b) $\omega =3$. The black-dashed line indicates the bound [Eq. (5)] $n=d^{\prime 2}/2\omega$ for the GTCs. The green-dashed line indicates the standard scaling $n=d^{\prime 2}$ for $d^{\prime }$ by $d^{\prime }$ rotated planar surface codes. The grey-dashed line indicates the scaling $n=max[2d^{\prime 2}/\omega -d^{\prime }(1+1/\omega ),3d^{\prime }-2]$ for the unrotated planar surface codes with optimized aspect ratio. All the codes are of the XZZX type.

Figure 8

The flag circuit to extract a weight-4 stabilizer $X_1{X}_2{Z}_3{Z}_4$. We prepare a syndrome qubit s in $|+\rangle$ state, apply a sequence of $CX$ (a,b) and $CZ$ (c,d) gates between the s and the data qubits, and measure s in the $X$ basis to obtain the syndrome associated with $X_1{X}_2{Z}_3{Z}_4$. In addition, we apply two $CX$ gates between s and an extra flag qubit f to catch the bad (bit-flip) errors on the syndrome qubit. Without the flag qubit the bad errors after gates $b$ and $c$ can propagate to the data qubits with larger (effective) weight. With an extra flag qubit, the bad errors (e.g., a Pauli $X$ error depicted by the red star) can be detected and corrected adaptively.

Figure 9

Performance of the close-to-optimal GTCs under the correlated Pauli $X$ and $Z$ noise for (a) $\omega =1$ and (b) $\omega =3$. The dashed line indicates the bound Eq. (5). The orange bars indicate the effective code distance bounds given in Eq. (7). The green dots are given by $2r-1$, where $r$ is the exponent in the expression $p_L\propto p^r$ that is extracted from fitting the MC simulations using tensor network decoders. The green bars indicate the standard numerical error. The range of the physical error rate $p$ used for the fitting is (a) $p\in [0.01,0.03]$ and (b) $p\in [0.04,0.07]$.

Figure 10

The propagation of bad gate errors that are caught by the flag qubit in Fig. 8 during the measurement of a stabilizer $X_1{X}_2{Z}_3{Z}_4$. We use the convention that the errors happen after an ideal gate. Then only the failure of gates $\mathrm{b}$ and $\mathrm{c}$ can propagate into errors with larger effective weights on the data qubits (modulo the stabilizer to be measured). For each table, in the left column, we list the possible failures of the gate $\mathrm{b}$ (or $\mathrm{c}$) that can trigger the flag, where $AB$ indicates an error $A$ on the control (syndrome qubit) and an error $B$ on the target (data qubits). While in the right column we list the induced errors on the four data qubits (from top to down in Fig. 8) on which the measured stabilizer is supported. The set of listed gate errors are denoted by ${\mathit{X}}_{\mathit{s}}$ and the resulted data errors are denoted by $\mathit{\xi }$.