Spatially parallel decoding for multi-qubit lattice surgery

TL;DR

This work tackles real-time decoding for surface-code quantum error correction under lattice-surgery operations that merge patches into larger regions. It introduces spatially parallel windows—overlapping, hardware-fixed decoder regions—to manage large patches while maintaining throughput and logical fidelity; the inner decoders can be any real-time algorithm, coordinated across layers to resolve seam corrections. Through simulations and analysis, it shows that buffer width and patch geometry critically influence logical error rates, with optimal width scaling roughly as $w\sim d/2$ and depending on the physical noise $p$. The results also quantify inter-window communication as a non-bottleneck and argue that inner-decoder scalability (and possibly ASICs) largely limits the maximum feasible window size, guiding practical hardware implementations for scalable, real-time decoding.

Abstract

Running quantum algorithms protected by quantum error correction requires a real time, classical decoder. To prevent the accumulation of a backlog, this decoder must process syndromes from the quantum device at a faster rate than they are generated. Most prior work on real time decoding has focused on an isolated logical qubit encoded in the surface code. However, for surface code, quantum programs of utility will require multi-qubit interactions performed via lattice surgery. A large merged patch can arise during lattice surgery -- possibly as large as the entire device. This puts a significant strain on a real time decoder, which must decode errors on this merged patch and maintain the level of fault-tolerance that it achieves on isolated logical qubits. These requirements are relaxed by using spatially parallel decoding, which can be accomplished by dividing the physical qubits on the device into multiple overlapping groups and assigning a decoder module to each. We refer to this approach as spatially parallel windows. While previous work has explored similar ideas, none have addressed system-specific considerations pertinent to the task or the constraints from using hardware accelerators. In this work, we demonstrate how to configure spatially parallel windows, so that the scheme (1) is compatible with hardware accelerators, (2) supports general lattice surgery operations, (3) maintains the fidelity of the logical qubits, and (4) meets the throughput requirement for real time decoding. Furthermore, our results reveal the importance of optimally choosing the buffer width to achieve a balance between accuracy and throughput -- a decision that should be influenced by the device's physical noise.

Figures (16)

• Figure 1: A merge operation in lattice surgery, specifically, a logical $Z\otimes Z$ measurement.
• Figure 2: (a) A square patch of surface code. The red and blue faces are the X and Z stabilizers, respectively. The blue and red lines mark $\hat{Z_L}$ and $\hat{X_L}$, the $Z$ and $X$ logical operators. (b) An example decoding graph for the X errors. The vertices are the $Z$ stabilizers, augmented by the virtual boundary node (in yellow). The stabilizers that are flipped are marked in red, and a minimum weight matching is denoted by the green edges.
• Figure 3: A segment from a longer patch of surface code, divided into 2 non-overlapping windows by an artificial boundary (dashed line). The optimal corrections are shown in green and the suboptimal ones in purple. (a) One flipped syndrome in window A. The suboptimal correction will be selected if matching to the artificial boundary is allowed. (b) An error flips two syndromes, one in each window. The suboptimal correction will be selected if matching to the artificial boundary is disallowed.
• Figure 4: Decoding graphs for $X$ errors, used by (a) window A and (b) window B. In both (a) and (b), the buffer is marked in pink, and the two nodes where artificial defects could be introduced are marked by solid black dots. In (a), the red edges are the ones that connect the nodes in A's commit region to the buffer. Corrections along the dotted edges are on nodes in the buffer, and are not committed by window A. In (b) the decoding graph has solid edges in the buffer, because the corrections in the buffer are finalized in this step.
• Figure 5: (a) A patch with a tree structure is 2-colorable. (b) A patch that arises during a $Y\otimes Y$ logical measurement. Not 2-colorable.