High-threshold decoding of non-Pauli codes for 2D universality

Abstract

Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit $X$ corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of $\approx 2.5\,\%$, close to the $\approx 2.9\,\%$ of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for $Z$ errors is required and the non-Clifford gates in the circuit reduce the threshold from $\approx 2.9\,\%$ to $\approx 1.8\,\%$ with a naive decoder. We show how $Z$ decoding can be improved using knowledge of the $X$ corrections, pushing the threshold to $\approx 2.2\,\%$. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose $X$ stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by $CCZ$, $CS$, and $T$ gates.

Figures (10)

• Figure 1: The TQD code can be defined on the square lattice, with three qubits assigned to each edge, which we label in three colors, $\{\texttt{r}, \texttt{g}, \texttt{b}\}\eqqcolon \mathcal{C}$. The code is defined by its stabilizers: For each color $\lambda\in\mathcal{C}$ and face $f\in F$ we define a stabilizer $B_{\lambda f} = \prod_{e\in f} Z_{\lambda e}$ acting with a product of $Z$ operators on the qubits of color $\lambda$ that are adjacent to $f$. For each color $\lambda\in\mathcal{C}$ and each vertex $v\in V$ and we define a Clifford stabilizer $A_{\lambda v} = CZ_{(\lambda v)}\prod_{e\ni v}X_{\lambda e}$ that acts with a product of Pauli $X$ operators on adjacent qubits of color $\lambda$ and two $CZ$ operators that we together denote as $CZ_{(\lambda v)}$. They each act on two neighboring qubits of the two color different from $\lambda$. The figure shows a set of stabilizer generators -- all others are obtained from spatial translation. Each bent line segment corresponds to one $CZ$ gate and the color on the endpoints indicates which qubit it acts on.
• Figure 2: Twisted errors in the TQD circuit. Figure (a) shows how to obtain the twisted errors from the flux configuration $b$ identified as a set of faces. There are three types of twisted errors per cubical unit cell, corresponding to $\widetilde{M}^b (\lambda,p)$ for any color-vertex pair $(\lambda, p)$. Each twisted error consists of a subset of 6 color-edge pairs, which are marked in the drawing as colored edges. Each of these 6 color-edge pairs is part of the subset iff $b_{\lambda f}=1$ for a specific color-face pair $(\lambda, f)$ depending on the color-edge pair. These color-face pairs are marked by a colored dot at the center of $f$ and connected to their color-edge pair with a thin line. In particular, the twisted errors are trivial (the empty subset) if $b=0$ on all faces nearby. Figure (b) displays a zoomed-out picture of an example $b$ configuration (solid lines) and an example $c$ configuration (dashed lines) that fulfills the twisted constraints. We see that the twisted constraints allow some charge lines $c$ to "terminate" in the vicinity of the flux lines $b$ without being detected by the twisted detectors. Figure (c) illustrates how a non-trivial flux $b\neq 0$ can reduce the fault distance for the $Z$ decoding. A non-trivial cycle $c_{\texttt{r}}$ in the $Z$ decoding graph in the presence of flux loops $b_{\texttt{b}}$ can have lower weight than a corresponding non-trivial cycle in the absence of fluxes since the $c_{\texttt{r}}$ is allowed to "jump between" the loops. Figure (d) shows an exemplary flux configuration $b$ consisting of red and green faces ($b_\texttt{r}$ and $b_\texttt{g}$ with $b_\texttt{b}=0$). Each connected component forms a loop on the Poincaré dual lattice. Each color-edge pair that is part of a twisted error is marked by a small ball of the corresponding color on the edges. For each flux configuration of one color, there are twisted errors of the other two colors on edges in its geometric neighborhood. For an isolated flux loop of one color, the set of edges that host all the twisted errors are obtained by shifting the dual lattice onto the direct lattice in a specific way. In the example, there are green and blue twisted errors along the red $b$ loop and red and blue twisted errors along the green $b$ loop. Most twisted errors correspond to a single color-edge pair. Some twisted errors are also correlated weight-2 errors. These are indicated by a connecting line between the two corresponding balls. When multiple $b$ loops coincide at a cube, the situation can get more complicated. For example, at the top front there is a twisted error consisting of both a red and a green edge, so the non-trivial $b$ leads to a coupling of the red, green, and blue $Z$ decoding problems. For general $b$ configurations, twisted errors can consist of up to 6 color-edge pairs, even though the example above shows only twisted errors with 1 or 2 edges (which are the most common for a sparse $b$ configuration). We would like to note that this analysis directly applies to the 3-copy-cup gate on 3D toric codes on the cubic lattice breuckmann2025cupsgatesicohomology.
• Figure 3: A JIT decoder is a decoder that decides on a correction at time $t$ given access to syndrome information acquired until $t$ only. We devise a JIT decoder from a global decoder $\mathsf{D}$ defined on the full decoding graph. The resulting JIT decoder calls $\mathsf{D}$ two times on the decoding graph with modified weights. In the first step a current estimate is obtained from $\mathsf{D}$ on the currently available syndrome information. In the second step previous corrections and the new best estimate are merged by solving a matching problem on a single time-slice of the decoding graph. The figure illustrates the workings of this JIT matching decoder step-by-step on a representative error configuration in a 2D toy picture of the actual decoding graph. At each timestep the JIT decoder first performs an estimate of the error based on the currently available syndrome information and then merges this with the endpoints of the corrections done in the previous step. This guarantees that all syndromes are paired up after the execution of the circuit. We see that while the JIT decoder fails to identify the minimum-weight correction, it matches up the syndromes within each connected component of the error chain.
• Figure 4: Estimated logical error rates $p_{\mathrm{log}}$ as a function of the physical error rate $p_{\mathrm{phys}}$ for various system sizes $L$. The JIT-decoder yields the logical error rates shown in red and indicate a threshold value around $2.5\,\%$. We compare that to a global decoder, whose logical error rates are shown in blue. We find a threshold for the global decoder of around $2.9\,\%$, in alignment with the literature Wang2003statmech. The dashed vertical lines indicate the thresholds $p_{\mathrm{th}}$ as extracted via the FSS ansatz. Note that the logical error-rate approaches 0.75 for $p\gg p_{\mathrm{th}}$ since there are two independent non-trivial cycles in the decoding graph.
• Figure 5: We estimate the effective distance of the JIT decoder $\mathsf{D}_{\mathrm{JIT}}$ and compare it to the global minimum-weight decoder $\mathsf{D}$ in the regime where $p_{\mathrm{phys}}$ is an order of magnitude below $p_{\mathrm{th}}$ by fitting the ansatz in Equation \ref{['eq:ansatz-deff']}. The data is obtained for $L=7$. The fitted parameters suggest that both the JIT and the global decoder successfully correct all errors up to weight $L/2$. While this is expected behavior for the global decoder, the JIT-decoder could in principle fail on errors of lower weight. Our simulations suggest that this is not the case and one should hence expect the JIT-decoder to lead to the same asymptotic error suppression for low $p_{\mathrm{phys}}$ as the global decoder.

Theorems & Definitions (11)

• Remark 1
• Definition 1
• Remark 2
• Definition 2
• Remark 3
• Definition 3
• Definition 4: Commit region
• Definition 5
• Definition 6
• Theorem 1: Validity of corrections