Low-cost noise reduction for Clifford circuits (2407.06583v1)

Abstract: We propose a Clifford noise reduction (CliNR) scheme that provides a reduction of the logical error rate of Clifford circuit with lower overhead than error correction and without the exponential sampling overhead of error mitigation. CliNR implements Clifford circuits by splitting them into sub-circuits that are performed using gate teleportation. A few random stabilizer measurements are used to detect errors in the resources states consumed by the gate teleportation. This can be seen as a teleported version of the CPC scheme, with offline fault-detection making it scalable. We prove that CliNR achieves a vanishing logical error rate for families of $n$-qubit Clifford circuits with size $s$ such that $nsp^2$ goes to 0, where $p$ is the physical error rate, meaning that it reaches the regime $ns = o(1/p^2)$ whereas the direct implementation is limited to $s = o(1/p)$. Moreover, CliNR uses only $3n+1$ qubits, $2s + o(s)$ gates and has zero rejection rate. This small overhead makes it more practical than quantum error correction in the near term and our numerical simulations show that CliNR provides a reduction of the logical error rate in relevant noise regimes.

• The paper presents the CliNR scheme that segments Clifford circuits via gate teleportation and uses random stabilizer measurements to detect errors.
• Theoretical analysis shows that the method achieves a vanishing logical error rate under the condition nsp² → 0, enabling circuits with size s = o(1/p²).
• Simulation results indicate a 2x reduction in error rates for 25-qubit systems and a 4x reduction for 60-qubit systems, demonstrating low overhead and practical efficiency.

Low-cost Noise Reduction for Clifford Circuits

The paper "Low-cost Noise Reduction for Clifford Circuits" by Nicolas Delfosse and Edwin Tham presents an innovative technique for decreasing the logical error rate in Clifford circuits. The proposed approach, Clifford Noise Reduction (CliNR), offers a compelling trade-off between overhead costs and error reduction capabilities, circumventing the extensive overheads typically associated with quantum error correction (QEC) and the exponential sampling overhead of quantum error mitigation (QEM).

Summary of Contributions

The primary contributions of the paper are:

1. Clifford Noise Reduction (CliNR) Scheme:
   • The CliNR method segments Clifford circuits into sub-circuits implemented via gate teleportation.
   • A selected few random stabilizer measurements are utilized to detect errors in the resource states crucial for gate teleportation.
   • This strategy is akin to a modified, teleported version of the Coherent Parity Check (CPC) scheme.
2. Theoretical Performance Bound:
   • The authors provide theoretical analysis showing that the CliNR achieves a vanishing logical error rate for families of $n$-qubit Clifford circuits of size $s$, given that $nsp^2 \to 0$, where $p$ is the physical error rate.
   • This suggests that the CliNR is capable of handling circuits with $s = o(1/p^2)$, surpassing the direct implementation bound of $s = o(1/p)$.
3. Efficient Resource Utilization:
   • CliNR implementation requires only $3n+1$ qubits, $2s + o(s)$ gates, and boasts zero rejection rate.
   • This approach results in significantly reduced overhead compared to quantum error correction methods.
4. Performance in Simulated Noise Regimes:
   • Numerical simulations indicate that the CliNR provides a significant reduction in logical error rates within practical noise regimes, demonstrating a 2x reduction for 25-qubit systems with $p_2 = 10^{-3}$ and a 4x reduction for 60-qubit systems with $p_2 = 10^{-4}$.

Implications and Future Directions

Practical Implications

• Near-term Quantum Devices:

CliNR’s low overhead makes it particularly appealing for near-term quantum devices, which are constrained by the number of qubits and the achievable gate fidelities. It provides a feasible route to reduce logical errors without necessitating the extensive overheads of QEC.

• Reduced Qubit Requirements:

By only requiring 3 physical qubits per logical qubit, this technique is highly efficient in terms of qubit resource management, which is crucial for the current generation of quantum hardware.

Theoretical Implications

• Advancing FTQC:

The CliNR approach elegantly demonstrates that practical fault tolerance can be achieved with techniques that lie between the conventional QEC and QEM paradigms. This could stimulate new explorations into hybrid noise reduction strategies leveraging both error detection and correction.

• Implications for Circuit Design:

The trade-off between circuit size and teleportation frequency (parameter $t$) provides a new dimension in quantum algorithm design, allowing more flexible strategies that can adapt to the specific error profiles of quantum devices.

Speculative Future Developments

• Integration with Non-Clifford Operations:

While the current CliNR scheme is designed for Clifford circuits, future work could explore extensions or adaptations for circuits with non-Clifford gates, potentially involving more intricate forms of state preparation and error detection.

• Hybrid Error Mitigation Techniques:

Combining CliNR with other emerging scalable QEM techniques could yield robust, yet low-overhead, quantum computational frameworks. This may involve synergizing methods like dynamical decoupling, post-selection, and other post-processing strategies.

• Hardware-Specific Optimizations:

Adapting the CliNR approach to various quantum hardware architectures (e.g., superconducting qubits, trapped ions) could further enhance its effectiveness, tailoring error detection mechanisms to the specific noise profiles and connectivity constraints of different systems.

Conclusion

The work by Delfosse and Tham presents a significant advancement in noise reduction techniques for Clifford circuits, showcasing an efficient and practical approach that bridges the gap between full-fledged quantum error correction and simpler mitigation strategies. The innovative use of gate teleportation and offline fault detection positions CliNR as a promising candidate for enhancing the reliability of near-term quantum computations, while also providing valuable insights for the ongoing development of robust fault-tolerance methodologies.