Dynamic local single-shot checks for toric codes

TL;DR

This work tackles the time overhead of quantum error correction by introducing local single-shot checks for the toric code and employing a dynamic measurement schedule to increase the decoding window's time distance $d_t$. Patch-based local checks bound check weight and, with alternating patch partitions, achieve a larger $d_t$ for a fixed number of syndrome rounds, reducing the required rounds by a factor dependent on the patch size $l$, as quantified by $d_t(t_i,t_i+W)$. Across phenomenological and circuit-level noise models, the authors demonstrate that local single-shot checks can improve decoding thresholds and, in some regimes, outperform standard toric-code checks, especially when using sliding-window decoding with smaller windows; however, performance is sensitive to internal errors and the chosen hyperedge decomposition. The approach provides a new design direction for reducing measurement overhead in fault-tolerant quantum computation, with potential extensions to other code families and decoding strategies on time-aware decoding graphs.

Abstract

Quantum error correction typically requires repeated syndrome extraction due to measurement noise, which results in substantial time overhead in fault-tolerant computation. Single-shot error correction aims to suppress errors using only one round of syndrome extraction. However, for most codes, it requires high-weight checks, which significantly degrade, and often eliminate, single-shot performance at the circuit level. In this work, we introduce local single-shot checks, where we impose constraints on check weights. Using a dynamic measurement scheme, we show that the number of required measurement rounds can be reduced by a factor determined by this constraint. As an example, we show through numerical simulation that our scheme can improve decoding performance compared to conventional checks when using sliding-window decoding with a reduced window size under circuit-level noise models for toric codes. Our work provides a new direction for constructing checks that can reduce time overhead in large-scale fault-tolerant quantum computation.

Figures (15)

• Figure 1: Illustration of different checks. In this representation, each link represents a qubit. We only show $Z$-type checks as $X$-type checks can be similarly obtained. The red lines in each subplot mark the qubits in support of each check. (a) Single-shot checks for an $L=4$ toric code. There are $L^2-1$ single-shot checks in total. Within each check, we use a red dot to mark the data qubit onto which we map the measurement error of the corresponding single-shot check. (b) Variable-width checks defined on a $l=4$ patch. (c) Fixed-width checks defined on a $l=4$ patch. These are two choices of local single-shot checks defined by different partitions of the torus. We use a red dot to mark the data qubits that are checked only by one variable-width check within the patch. If the data qubit is on the boundary of the patch, we map the measurement error of the corresponding local single-shot checks onto a time edge. Otherwise, we map the measurement error to a data qubit error on that data qubit instead. Note that the fixed-width checks all have the same width $1$, while the variable-width checks can have various widths.
• Figure 2: Illustration of the alternating partitions. (a) The alternating square partition used for the variable-width checks. (b) The alternating rectangular partition used for the fixed-width checks. The gray grids represent the torus, with links representing data qubits. The blue and pink lines mark the partition used for odd syndrome extraction rounds and even syndrome extraction rounds respectively.
• Figure 3: Illustration of the decoding graph used for different measurement schemes under a phenomenological noise model. Each plot shows two consecutive time layers of the decoding graph: an even layer and an odd layer. Black lines denote the detectors within the first layer and the time edges connecting them to the second layer. Green lines similarly denote the detectors within the second layer and the time edges connecting them to future layers. Each black dot represents a detector. Edges within each layer are space edges representing a data qubit error, or effective data qubit errors resulting from a mapped measurement error. Vertical edges connecting the layers are time edges representing measurement errors. The red line shows the shortest path to go from a detector in the first layer to the time boundary of the given decoding volume. The length of the path gives the time distance $d_t$ of the decoding volume. (a) Decoding graph for local checks. Each detector is defined by the product of consecutive measurements of the same check. Within each layer, the space edges form a torus. Across layers, we have time edges between each pair of consecutive detectors located at the same spatial position in different layers. The length of the shortest path connecting to the time boundary equals the number of layers. (b) Decoding graph for the fixed-width checks defined on $l=2$ patches with an aligned measurement scheme. Here, the detectors are constructed by first converting the local single-shot check syndrome to the local check syndrome and then taking the products of the consecutive local check syndrome. In the decoding graph, the space edges have the same structure as the local checks. The time edges are sparser and only occur in certain rows of each layer. However, the length of the shortest path to the boundary is still equal to the number of layers as the time edges are aligned. (c) Decoding graph for the fixed-width checks defined on $l=2$ patches with an offset measurement scheme. The detectors are defined as in the previous case. The time edges of each layer do not align in this case, but they can only be located at certain rows, depending on whether the time layer is odd or even. This offset alignment of time edges increases the length of the shortest path to the time boundary in the decoding graph, which is why the optimal decoding window can contain fewer syndrome extraction rounds than the local checks.
• Figure 4: Examples of syndrome extraction CNOT orderings for the local single-shot checks. Each colored data qubit indicates the order in which the ancilla interacts with the data via CNOT gates. (a) Square check. (b–c) Narrow rectangular checks, where the zigzag pattern prevents hook-error propagation.
• Figure 5: Illustration of different types of hyperedge decomposition. The vertical direction represents time, and the horizontal direction represents space. (a) A hyperedge (represented by a circle) connecting four detectors (blue dots). The detectors are located on two neighboring time layers (represented by two rows). A measurement error on any local single-shot check can cause this hyperedge, unless the check spans all boundary qubits of the patch. Hyperedges cannot be directly processed by MWPM algorithms and therefore need decomposition. (b) Space-edge-first decomposition: The hyperedge is decomposed into two space edges (red lines) on neighboring time layers, which corresponds to two data qubit errors happening at consecutive rounds of syndrome extraction. This decomposition is also used for the phenomenological noise model. (c) Time-edge-first decomposition: The hyperedge is decomposed into two time edges (blue lines). This effectively maps a measurement error on the local single-shot checks to two measurement errors of local checks. (d) A space-time edge representing a circuit error mechanism whose starting or ending detector is not contained within the allowed sets defined by the phenomenological decoding graph. Such edges are typically caused by an internal error. When using the time-edge-first decomposition, we directly add such an edge into the decoding graph. (e) In space-edge-first decomposition, since the original space-time edge in (d) is not allowed, we introduce an " auxiliary" detector (red dashed-line circle) by projecting the detector on time layer $t+1$ to time layer $t$. The original space-time edge is decomposed into a time edge connecting the detector in layer $t+1$ and the auxiliary detector, and a space edge connecting the auxiliary detector and the other detector on layer $t$.