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Quantum error correction for long chains of trapped ions

Min Ye, Nicolas Delfosse

TL;DR

This work addresses fault-tolerant quantum computation on long trapped-ion chains by designing syndrome-extraction circuits and LDPC-like quantum codes tailored to the ion-chain hardware. It introduces a tunable syndrome-extraction protocol and demonstrates that BB5 codes with weight-5 stabilizers, notably the $[[48,4,7]]$ BB5 code, can outperform baseline BB6 codes and match distance-7 surface codes while using significantly fewer physical qubits. The findings argue that quantum LDPC codes are highly suitable for long ion chains, offering reduced qubit overhead and practical pathways for scalable trapped-ion QEC. The study also highlights the trade-offs between ancilla qubit count, syndrome extraction depth, and hardware-specific noise, with avenues for future refinement in biased/noisy measurement models and gate decomposition.

Abstract

We propose a model for quantum computing with long chains of trapped ions and we design quantum error correction schemes for this model. The main components of a quantum error correction scheme are the quantum code and a quantum circuit called the syndrome extraction circuit, which is executed to perform error correction with this code. In this work, we design syndrome extraction circuits tailored to our ion chain model, a syndrome extraction tuning protocol to optimize these circuits, and we construct new quantum codes that outperform the state-of-the-art for chains of about $50$ qubits. To establish a baseline under the ion chain model, we simulate the performance of surface codes and bivariate bicycle (BB) codes equipped with our optimized syndrome extraction circuits. Then, we propose a new variant of BB codes defined by weight-five measurements, that we refer to as BB5 codes and we identify BB5 codes that achieve a better minimum distance than any BB codes with the same number of logical qubits and data qubits, such as a $[[48, 4, 7]]$ BB5 code. For a physical error rate of $10^{-3}$, the $[[48, 4, 7]]$ BB5 code achieves a logical error rate per logical qubit of $5 \cdot 10^{-5}$, which is four times smaller than the best BB code in our baseline family. It also achieves the same logical error rate per logical qubit as the distance-7 surface code but using four times fewer physical qubits per logical qubit.

Quantum error correction for long chains of trapped ions

TL;DR

This work addresses fault-tolerant quantum computation on long trapped-ion chains by designing syndrome-extraction circuits and LDPC-like quantum codes tailored to the ion-chain hardware. It introduces a tunable syndrome-extraction protocol and demonstrates that BB5 codes with weight-5 stabilizers, notably the BB5 code, can outperform baseline BB6 codes and match distance-7 surface codes while using significantly fewer physical qubits. The findings argue that quantum LDPC codes are highly suitable for long ion chains, offering reduced qubit overhead and practical pathways for scalable trapped-ion QEC. The study also highlights the trade-offs between ancilla qubit count, syndrome extraction depth, and hardware-specific noise, with avenues for future refinement in biased/noisy measurement models and gate decomposition.

Abstract

We propose a model for quantum computing with long chains of trapped ions and we design quantum error correction schemes for this model. The main components of a quantum error correction scheme are the quantum code and a quantum circuit called the syndrome extraction circuit, which is executed to perform error correction with this code. In this work, we design syndrome extraction circuits tailored to our ion chain model, a syndrome extraction tuning protocol to optimize these circuits, and we construct new quantum codes that outperform the state-of-the-art for chains of about qubits. To establish a baseline under the ion chain model, we simulate the performance of surface codes and bivariate bicycle (BB) codes equipped with our optimized syndrome extraction circuits. Then, we propose a new variant of BB codes defined by weight-five measurements, that we refer to as BB5 codes and we identify BB5 codes that achieve a better minimum distance than any BB codes with the same number of logical qubits and data qubits, such as a BB5 code. For a physical error rate of , the BB5 code achieves a logical error rate per logical qubit of , which is four times smaller than the best BB code in our baseline family. It also achieves the same logical error rate per logical qubit as the distance-7 surface code but using four times fewer physical qubits per logical qubit.

Paper Structure

This paper contains 9 sections, 3 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The ion chain model
  3. Syndrome extraction circuits for the ion chain model
  4. Improved codes for the ion chain model
  5. Conclusion
  6. Acknowledgments
  7. Review of surface codes and BB codes
  8. Validation of the syndrome extraction tuning protocol
  9. Fitting formulas for surface codes and BB5 codes

Figures (4)

  • Figure 1: Comparison between BB5 codes and baseline codes. Each point on the blue curve is the minimum logical error rate per logical qubit for a BB6 code from the baseline family. More specifically, there are 11 BB6 codes in the baseline family, as listed in \ref{['tab:baseline_code_parameters']}. For each value of physical error rate $p$, we obtain the logical error rate per logical qubit for these 11 BB6 codes through Monte Carlo simulations, and each point on the blue curve is simply the minimum of these 11 values.
  • Figure 2: Comparison between BB5 codes and BB6 codes with the same parameters.
  • Figure 3: A distance-5 surface code and its stabilizer generators.
  • Figure 4: \ref{['algorithm:syndrome_extraction_tunning_protocol']} chooses $n_a=3$ ancillas for $[[30,4,4]]$ BB6 code and $n_a=6$ for $[[48,4,6]]$ BB6 code.