Performance of Quantum Approximate Optimization with Quantum Error Detection (2409.12104v3)

Abstract: Quantum algorithms must be scaled up to tackle real-world applications. Doing so requires overcoming the noise present on today's hardware. The quantum approximate optimization algorithm (QAOA) is a promising candidate for scaling up, due to its modest resource requirements and documented asymptotic speedup over state-of-the-art classical algorithms for some problems. However, achieving better-than-classical performance with QAOA is believed to require fault tolerance. In this paper, we demonstrate a partially fault-tolerant implementation of QAOA using the $[[k+2,k,2]]$ ``Iceberg'' error detection code. We observe that encoding the circuit with the Iceberg code improves the algorithmic performance as compared to the unencoded circuit for problems with up to $20$ logical qubits on a trapped-ion quantum computer. Additionally, we propose and calibrate a model for predicting the code performance. We use this model to characterize the limits of the Iceberg code and extrapolate its performance to future hardware with improved error rates. In particular, we show how our model can be used to determine the necessary conditions for QAOA to outperform the Goemans-Williamson algorithm on future hardware. To the best of our knowledge, our results demonstrate the largest universal quantum computing algorithm protected by partially fault-tolerant quantum error detection on practical applications to date, paving the way towards solving real-world applications with quantum computers.

• The paper demonstrates that implementing the Iceberg error detection code with QAOA on a trapped-ion quantum computer enhances algorithmic performance over unencoded circuits.
• It develops a predictive model based on specific error rates to estimate logical fidelity and post-selection, clarifying operational benefits.
• The study extrapolates conditions under which QAOA can outperform classical methods, underscoring the potential of partially fault-tolerant approaches for future hardware.

Performance of Quantum Approximate Optimization with Quantum Error Detection

In the effort to scale quantum algorithms for practical applications, the challenge of dealing with the noise inherent in current quantum hardware remains significant. The Quantum Approximate Optimization Algorithm (QAOA) is a promising candidate for such scaling due to its efficiency in resource requirements and its theoretical capability to provide asymptotic speedup for certain problems compared to classical algorithms. To achieve better-than-classical performance, it is widely believed that fault tolerance is necessary. This paper presents an investigation into a partially fault-tolerant implementation of QAOA using the $[[k+2,k,2]]$ "Iceberg" error detection code and demonstrates its advantages in improving algorithmic performance.

Summary of Contributions

The paper provides the following key contributions:

1. Implementation and Performance Improvement: The authors demonstrate the implementation of QAOA protected by the Iceberg error detection code on a Quantinuum H2-1 trapped-ion quantum computer. For problem instances up to 20 logical qubits, they show that the Iceberg code improves QAOA’s performance over unencoded circuits. This is especially noteworthy as it represents the largest instance of a universal quantum computing algorithm protected by a partially fault-tolerant quantum error detection scheme to date.
2. Model Development and Calibration: A predictive model was developed to estimate the performance of the Iceberg code, based on three key error rates related to noise from physical two-qubit gates. The model was calibrated against data from the emulator of the H2-1 quantum computer, allowing predictions about circuit fidelity and post-selection rates. The model assists in identifying the operational regimes and conditions under which the Iceberg code is beneficial.
3. Extrapolation to Future Hardware: The model was also used to predict the performance of the Iceberg code on future quantum hardware with improved error rates. The authors show exact conditions on effective error rates required for QAOA to outperform the Goemans-Williamson (GW) algorithm on small graphs, highlighting the potential of the Iceberg code for practical applications in enhanced future hardware setups.

Experimental Details and Findings

The experimental results are based on running QAOA circuits with and without Iceberg encoding on a 32-qubit trapped-ion quantum computer. Using up to 24 logical qubits and a depth of up to 11 QAOA layers, the authors measured the logical fidelity and post-selection rates, demonstrating better performance for Iceberg-encoded circuits in scenarios with up to 20 logical qubits.

For understanding the protectiveness of the Iceberg code, a simplified noise model was proposed assuming a global white noise distribution. This model facilitated a deeper analysis resulting in fitted parameters which were used to comprehensively predict logical fidelity and post-selection overhead.

Implications and Future Directions

Practical Implications:

The findings suggest that partially fault-tolerant error detection codes such as Iceberg can significantly improve the performance of quantum algorithms on near-term hardware. This improvement would not only make larger instance sizes feasible but also offer enhanced algorithmic outcomes, pushing the boundary closer to the quantum advantage.

Theoretical Implications:

The model established and validated in this paper provides a tool for understanding and predicting the behavior of QAOA under different noise regimes. It paves the way for theoretical investigation into other error detection and correction schemes, their practical implementations, and scalability.

Speculations on Future Developments:

As quantum hardware continues to improve, reducing error rates further, the efficiency and robustness of error detection schemes are expected to become more pivotal. The paper indicates that with improved hardware, the breakeven point where quantum methods such as Iceberg-protected QAOA outperform classical algorithms can be reached more swiftly. This has profound implications for the applicability of quantum computing in solving real-world combinatorial optimization problems.

In summary, the paper demonstrates that employing quantum error detection codes provides a significant enhancement in the performance of quantum algorithms like QAOA, particularly in the transitional phase towards fully fault-tolerant quantum computing. The insights and models developed here could be instrumental in guiding future quantum algorithm designs and hardware improvements.