Tailoring Surface Codes for Highly Biased Noise

The surface code, with a simple modification, exhibits ultrahigh error-correction thresholds when the noise is biased toward dephasing. Here, we identify features of the surface code responsible for these ultrahigh thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases and show how to exploit these features to achieve significant improvement in the logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is 50%, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time-decoding algorithm. We demonstrate that the subthreshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough or smooth open boundaries, it is controlled by the parameter $g=\gcd (j,k)$, where $j$ and $k$ are dimensions of the surface code lattice. We demonstrate a significant improvement in the logical failure rate with pure dephasing for coprime codes that have $g=1$ and closely related rotated codes, which have a modified boundary. The effect is dramatic: The same logical failure rate achievable with a square surface code and $n$ physical qubits can be obtained with a coprime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum-likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased toward dephasing. In particular, comparing with a square surface code, we observe a significant improvement in the logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.

Popular Summary

Quantum error correction is likely to be needed for large-scale quantum computers to operate in the presence of noise and faulty operations. The amount of noise that can be accommodated depends critically on not only the choice of quantum error correcting code but also the characteristics of the noise. Recent experiments in nascent quantum technologies have been able to probe these noise characteristics, allowing engineers to tailor the code to optimally address the specific noise properties afflicting the system and squeeze out the best performance. Here, we have shown how the well-studied “surface code” can be tailored to demonstrate exceptional performance in systems where the noise is highly biased towards dephasing, a common characteristic of many quantum devices.

In dephasing, a quantum system loses its coherence, or the clear phase relationship between states that is needed for quantum computation. To optimally correct dephasing errors, we determine the specific structure needed for a surface code, a family of error-correcting codes defined on a 2D lattice of quantum bits. This structure depends on the form of the check operators used to identify errors as well as the choice of boundary conditions. We identify boundary conditions that lead to optimal performance, with ultrahigh thresholds and exceptional subthreshold scaling of logical error rates.

The surface code is well studied because of its remarkable performance, but our study shows that very substantial gains remain to be found by tailoring this code to the specific noise of the system. Quantum computer architects have a new avenue to squeeze better performance out of future quantum devices.

Figure 1

Threshold error rate $p_c$ as a function of bias $\eta$. Points show threshold estimates for the surface code. Error bars indicate one standard deviation relative to the fitting procedure. The point at the smallest bias corresponds to $\eta =0.5$ or standard depolarizing noise. The point at infinite bias indicates the analytically proven 50% threshold value. The gray line is the hashing bound for the associated Pauli error channel.

Figure 2

Logical failure rates $f_{\text{square}}$ and $f_{\text{rotated}}$ as a function of physical error probability $p$ for small comparable square and rotated $9\times 9$ codes and the logarithm of the ratio of logical failure rates ${\log }_{10}(f_{\text{rotated}}/f_{\text{square}})$ with noise biases $\eta \in \{0.5,10,100,1000,10000,\infty \}$. Error bars indicate one standard deviation. Data points are sample means over 30 000 and 1 200 000 runs for the square and rotated codes, respectively, using approximate maximum-likelihood decoding converged to within half a standard deviation for both codes. Dotted lines connect successive data points for a given $\eta$.

Figure 3

Standard $4\times 5$ surface code, with logical operators given by a product of $X$ along the left edge and a product of $Z$ along the top edge. Stabilizer generators are shown at the right.

Figure 4

(a) Rotated $5\times 5$ surface code defined by drawing the boundary at 45° relative to the surface code lattice. Logical operators are given by a product of $X$ along the northeast edge and $Z$ along the northwest edge. As with the standard code, stabilizer generators consist of $X$ ($Z$) operators on edges around vertices (plaquettes). (b) Rotated $5\times 5$ surface code as it is usually, and equivalently, depicted, where shaded (blank) faces corresponding to $X$-type ($Z$-type) stabilizer generators.

Figure 5

A sample of $X$-, $Y$-, and $Z$-error strings, indicated by colored circles, with corresponding anticommuting syndrome locations, indicated by yellow stars.

Figure 6

Diagonals ${\delta }^i$ for the $4\times 4$ surface code. We consider the symmetry group $\mathcal{R}$ generated by reflections of the lattice against ${\delta }^1$ and ${\delta }^5$. Note that any diagonal ${\delta }^i$ is symmetric under reflections from $\mathcal{R}$.

Figure 7

A set of qubits $\mathcal{O}$ such that each orbit of $\mathcal{R}$ contains exactly one qubit from $\mathcal{O}$. In this example, the group $\mathcal{R}$ has ten orbits of size 1, 2, and 4.

Figure 8

Restrictions of the diagonal ${\delta }^i$ onto $\mathcal{O}$ define a basis set of codewords for the top-level code.

Figure 9

Partition of the $8\times 12$ surface code into $4\times 4$ tiles. Solid red circles: The extended diagonal ${\mathrm{\Delta }}^1$ alternating between ${\delta }^1$ and ${\delta }^5$; see Fig. 6.

Figure 10

Extended diagonal ${\mathrm{\Delta }}^i$.

Figure 11

Logical failure rate $f$ as a function of physical error probability $p$ for surface code families: square, coprime, and rotated, subject to pure $Y$ noise. Data points are sample means over 60 000 runs using exact maximum-likelihood decoding. Dotted lines connect successive data points for a given code size.

Figure 12

Exponential decay of the logical failure rate $f$ with respect to code distance $d_Y$ to pure $Y$ noise in the regime of physical error probability $p$ at and below the error threshold for surface code families: square, coprime, and rotated, subject to pure $Y$ noise. Data points are sample means over 60 000 runs using exact maximum-likelihood decoding. Dotted lines indicate a least-squares fit to data for a given $p$, and error bars indicate one standard deviation.

Figure 13

Decoder convergence for the rotated $33\times 33$ surface code, represented by shifted logical failure rate $f_{\chi }-f_{{\chi }_{\max }}$, as a function of $\chi$ at a physical error probability $p$ near the threshold for the given bias $\eta$. Each data point corresponds to 30 000 runs with identical errors generated across all $\chi$ for a given bias.

Figure 14

(a) Tensor network representing a coset probability for a rotated code. It consists of qubit tensors (circles) and check tensors (stars). The layout in (a) is obtained by applying the construction of Ref. [9] to the rotated code without modification. (b) Splitting a check tensor into multiple check tensors. This splitting is possible because check tensors take the value one if all indices are identical and zero otherwise. (c) A modified tensor network representing a coset probability where a single cell is outlined by a dashed box. This network is obtained from (a) by splitting check tensors. (d) Final modified tensor network obtained by contracting tensors in cells together to form merged tensors (squares). In the discussion of the contraction of this tensor network, we imagine rotating the network anticlockwise by 45° and contracting from left to right. Note that this tensor network is not isotropic: In this rotated frame, the bond dimension is 2 for horizontal edges and is 4 for most vertical edges (except on the boundary).

Figure 15

(a) Check coordinates are assigned to each check in the rotated layout. (b) The tensor network is defined such that each horizontal edge corresponds to a specific check. The $\alpha$ indices correspond to $X$ checks, and the $\beta$ indices correspond to $Z$ checks, with the subscripts indicating the check coordinates. Each tensor corresponds to a specific qubit. The bond dimension of horizontal edges is 2, while the bond dimension of the vertical edges is 4.

Figure 16

(a) A boundary state obtained by contracting the network up to a given column and leaving the right-going indices of that column uncontracted. (b) An example check and error configuration illustrated for calculating the boundary state for the third column $j=3$ of a $5\times 5$ code with rotated layout. A bulk configuration ${\alpha }^B;{\beta }^B$ is represented by the red dots, and a boundary configuration ${\alpha }^R;{\beta }^R$ is represented by the blue dots (where a dot on a check indicates that the check variable ${\alpha }_{i,k}$ or ${\beta }_{i,k}$ is 1). An error $f^{\prime }$ is represented by green letters. Note that for this calculation we consider only the action of the checks and error on qubits in the first three columns $Q_3$, inside the dashed box. So the action on the boundary checks on qubits outside the box is ignored. The quantity $T^{\prime }({\alpha }^B;{\beta }^B;{\alpha }^R;{\beta }^R)$ is the probability of the product of all dotted checks and the error $f^{\prime }$ in the dashed box. In the example configuration depicted above, this product contains four $X$, six $Y$, and two $Z$ errors, giving $T^{\prime }({\alpha }^B;{\beta }^B;{\alpha }^R;{\beta }^R)=p_X^4{p}_Y^6{p}_Z^2{p}_I^3$. In order to calculate the boundary state, all possible configurations of bulk checks must be summed over. In the special case of pure $Y$ noise, where $p_X=p_Z=0$, only at most one term in this sum is nonzero for any boundary configuration.

Figure 17

All allowed bulk and boundary configurations for $Y$ noise illustrated for the boundary state for the third column $j=3$ of a $5\times 5$ code on the rotated layout for the trivial coset $f^{\prime }=I$. The product of the dotted checks must result in only $I$ and $Y$ errors on $Q_3$ (and no $X$ or $Z$ errors). Blue dots represent the boundary configuration, while red dots represent the corresponding bulk configuration. Blue dots must be connected to a two-qubit check on the left boundary by a string of red dots. The fact that these strings never overlap and are absorbed at the left boundary implies that the boundary variables are uncorrelated, and, therefore, there is no entanglement in the boundary state.

Figure 18

Some examples of boundary and bulk configurations for the standard layout of the surface code with three-qubit checks on the boundary for the trivial coset $f^{\prime }=I$. The lineons travel in straight lines through the four-qubit bulk checks, but the three-qubit boundary checks have the effect of reflecting them by 90°, such that they emerge on the right boundary on the exact opposite side. Therefore, each boundary variable is perfectly correlated with the boundary variable on the exact opposite side. Separate lineons can also cross paths, resulting in a cancellation of the bulk variables. Also, for neighboring pairs of lineons, $Y$ on the qubits shared between them cancel. These all result in correlations between the boundary variables and, therefore, entanglement in the boundary MPS.

Figure 19

Threshold error rate $p_c$ as a function of bias $\eta$. Red inverted triangles show threshold estimates for the triangular 6.6.6 color code. For comparison, blue triangles show threshold estimates for the surface code (reproduced from Sec. 5a), with the point at infinite bias, i.e., only $Y$ errors, indicating the analytically proven 50% threshold. Error bars indicate one standard deviation relative to the fitting procedure. The gray line is the hashing bound for the associated Pauli error channel.

Figure 20

(a) Distance five triangular 6.6.6 color code with logical operators given by a product of $Z$ along the bottom edge and a product of $X$ along the left edge. (b) Color-code stabilizers. (c) Tensor network corresponding to the coset probability of a distance five color code; gray disks represent qubit tensors; white stars represent stabilizer tensors; and lines represent bonds. (d) Equivalent tensor network as a square lattice.

Figure 21

Decoder convergence for the distance $d=19$ triangular 6.6.6 color code, represented by shifted logical failure rate $f_{\chi }-f_{36}$, as a function of $\chi$ at a physical error probability $p$ near the threshold for the given bias $\eta$. Each data point corresponds to 60 000 runs with identical errors generated across all $\chi$ for a given bias.

Figure 22

Examples of $Y$-type stabilizer and logical operator construction by applying $Y$ operators along the indicated path until a corner is encountered or the path cycles. Minimum-weight $Y$-type logical operators (a) and (b) for square $4\times 4$ and coprime $3\times 4$ codes, respectively, are constructed by starting at the top-left qubit. Generators of the group of $Y$-type stabilizers (c) for the square $4\times 4$ code are constructed by starting at each of the next $\gcd (j,k)-1=3$ qubits of the top row. (For coprime codes, there are no $Y$-type stabilizers other than the identity.)

Figure 23

Example of $Y$-type destabilizer construction for a coprime code. (a) Single syndrome location. (b) A partial recovery operator is constructed by applying $Y$ operators, from below the syndrome location along a diagonal to any boundary, then from that diagonal along perpendicular diagonals, until the bottom boundary is encountered. (c) Residual recovery operators are constructed by applying $Y$ operators, from right of each residual boundary syndrome location along a diagonal away, until a corner is encountered. (d) The destabilizer is a product of partial and residual recovery operators.

Figure 24

Example of candidate $Y$-type recovery operator construction for a square code using partial recovery operators. (a) Original error and complete syndrome. (b) Partial recovery operators with residual boundary syndrome locations. (c) The candidate recovery operator is the product of all partial recovery operators, since residual boundary syndrome locations cancel in the case of square codes.