Quantum low-density parity-check codes for erasure-biased atomic quantum processors

TL;DR

The paper addresses the resource demands of quantum error correction for near-term devices by evaluating high-rate quantum LDPC codes under erasure-biased noise enabled by erasure conversion in neutral-atom processors. It demonstrates that Clifford-deformed La-cross codes can achieve circuit-level thresholds comparable to the surface code under large erasure fractions and deliver orders of magnitude lower logical error probabilities, while Bivariate Bicycle codes offer strong sub-threshold suppression at lower overhead but with smaller thresholds. The work provides a detailed comparison of two LDPC families, outlines practical near-term implementation strategies (dynamic shuttling vs static long-range schemes), and discusses decoding approaches (BP+OSD) and the potential for offline decoding with heralded erasures. Overall, it shows that erasure-specific QEC resources can enable practical, low-overhead quantum memories in the near term and likely extend to other platforms beyond neutral atoms.

Abstract

Identifying the best families of quantum error correction (QEC) codes for near-term experiments is key to enabling fault-tolerant quantum computing. Ideally, such codes should have low overhead in qubit number, high physical error thresholds, and moderate requirements on qubit connectivity to simplify experiments, while allowing for high logical error suppression. Quantum Low-Density Parity-Check (LDPC) codes have been recently shown to provide a path towards QEC with low qubit overhead and small logical error probabilities. Here, we demonstrate that when the dominant errors are erasures -- as can be engineered in different quantum computing architectures -- quantum LDPC codes additionally provide high thresholds and even stronger logical error suppression in parameter regimes that are accessible to current experiments. Using large-scale QEC numerical simulations, we benchmark the performance of two families of high-rate quantum LDPC codes, namely Clifford-deformed La-cross codes and Bivariate Bicycle codes, under a noise model strongly biased towards erasure errors. Both codes outperform the surface code by offering up to orders of magnitude lower logical error probabilities. Interestingly, we find that this decrease in the logical error probability may not be accompanied by an increase in the code threshold, as different QEC codes benefit differently from large erasure fractions. While here we focus on neutral atom qubits, the results also hold for other quantum platforms, such as trapped ions and superconducting qubits, for which erasure conversion has been demonstrated.

Figures (7)

• Figure 1: (a) Clifford-deformed La-cross quantum LDPC code ($k=3$ in this example) with two types of stabilizers mixing $X$ (red) and $Z$ (blue) Pauli operators. Logical operators are Pauli strings made of either all $X$ or all $Z$ operators (two examples are shown). The inset shows the syndrome extraction circuit for one stabilizer, which generalizes straightforwardly that of the XZZX surface code. (b) $[[72,12,6]]$ Bivariate Bicycle code, one $X$ (red) and one $Z$ (blue) stabilizer are shown.
• Figure 2: Cumulative logical error probability normalized by the number of QEC rounds for $k=2$(a), $k=3$(b) and $k=4$(c) La-cross codes (solid lines) under a fraction $R_e=0.98$ of unbiased erasure errors and $R_p=1-R_e=0.02$ of Pauli errors. A comparison with surface codes under the same noise model and at equal number of physical and logical qubits (dashed lines of the same color and marker style) is also shown. Codes sharing the same number of physical qubits, $N$, and logical qubits, $K$, are denoted with the same color. These results show that La-cross codes have high threshold (black stars) and significantly outperform the surface code in terms of QEC performance below physical error probabilities of $\sim10^{-2}$, for the code distances we have considered. The insets show how the thresholds of La-cross codes increase by increasing the unbiased erasure fraction for $R_e=0.40,0.75,0.90,0.98,1.0$. La-cross code instances shown in these plots are: $[[34,4,3]],[[52,4,4]],[[100,4,5]],[[130,4,6]]$ for $k=2$; $[[45,9,3]],[[65,9,4]],[[149,9,5]],[[225,9,6]]$ for $k=3$; $[[80,16,3]],[[106,16,4]],[[136,16,5]],[[208,16,6]]$ for $k=4$. Error bars on the data are standard deviations associated with the Monte Carlo error correction numerical simulations. For both La-cross and surface code, BP+OSD decoder was used.
• Figure 3: Cumulative logical error probability normalized by the number of QEC rounds for $k=2$(a), $k=3$(b) and $k=4$(c) La-cross code (colored lines) under a fraction $R_e=0.98$ of biased erasure errors and $R_p=1-R_e=0.02$ of Pauli errors. A comparison with La-cross codes under $R_e=0.98$ of unbiased erasure errors (pale lines) is shown, showing the benefit in the threshold (black stars) and in the logical error probability from a such a noise model. We also show a comparison with surface codes (dashed dark lines) with $R_e=0.98$ of biased erasure errors, at equal number of physical and logical qubits (dashed lines of the same color and marker style). Codes sharing the same number of physical qubits, $N$, and logical qubits, $K$, are denoted with the same color. La-cross codes start to outperform the surface code in terms of QEC performance below physical error probabilities of $\sim3\times10^{-2}$, for the code distances we have considered. La-cross code instances shown in these plots are: $[[34,4,3]],[[52,4,4]],[[100,4,5]],[[130,4,6]]$ for $k=2$; $[[45,9,3]],[[65,9,4]],[[149,9,5]],[[225,9,6]]$ for $k=3$; $[[80,16,3]],[[106,16,4]],[[136,16,5]],[[208,16,6]]$ for $k=4$. Error bars on the data are standard deviations associated with the Monte Carlo QEC simulations. For both La-cross and surface code, BP+OSD decoder was used.
• Figure 4: (a) Cumulative logical error probability normalized by the number of QEC rounds for Bivariate Bicycle (BB) codes under different fractions of unbiased erasure errors and Pauli errors, $R_e=0.00-0.98$. For each case, we study three code instances, namely $[[72,12,6]]$, $[[108,8,10]]$ and $[[144,12,12]]$, represented with different gradients of the same color and marker type. Thresholds (black stars) saturate at $\approx1.1-1.2\%$ for $R_e>0.10$, while the logical error probability at the threshold decreases. (b) Logical error probability at $R_e=0.98$ of unbiased erasures for Bivariate Bicycle (red) and La-cross codes (gray) at similar encoding rates, $R=K/N$. Codes considered are: $[[108,8,10]]$ ($R\approx0.074$) against $[[52,4,4]]$ ($R\approx0.077$), $[[149,9,5]]$ ($R\approx0.060$), and $[[208,16,6]]$ ($R\approx0.077$). In this case, BB codes are expected to quickly outperform La-cross codes below threshold due to the larger code distance at fixed qubit overhead ($D_{BB}=10$ against $D_{la-cross}=4,5,6$ in the shown example). (c) Logical error probability normalized by the number of logical qubits, $K$, at $R_e=0.98$ of unbiased erasures for BB (red) and La-cross codes (gray) at equal distance, $D=6$. La-cross codes slightly outperform BB codes due to the larger qubit overhead. In fact, La-cross encoding rates are lower than BB ones, i.e. $R_{BB}\approx0.167$, while $R_{k=2}\approx0.031$, $R_{k=3}=0.04$ and $R_{k=4}\approx0.077$ for $D=6$.
• Figure 5: Threshold probabilities for different fractions, $R_e$, of unbiased erasure errors of Bivariate Bicycle (red), $k=2,3,4$ La-cross (gray) and surface codes (black). For La-cross codes, despite the low values for depolarizing errors, the threshold rapidly increase with the increasing erasure fraction and for $R_e\gtrsim0.98$ they become comparable to the high surface code thresholds. Instead, for Bivariate Bicycle codes, the threshold saturates at $p_{th}^{BB}\approx1.1\%$ and does not display any significant improvement for increasing erasure fractions.