Performance of Quantum Approximate Optimization with Quantum Error Detection

TL;DR

The paper demonstrates a partially fault-tolerant QAOA using the Iceberg QED code on a trapped-ion device, achieving improved MaxCut performance for up to roughly 20 logical qubits and developing a predictive model to extrapolate Iceberg performance to future hardware. It shows that Iceberg protection yields gains at moderate circuit sizes but faces overhead limits as problem size grows, and it benchmarks against Pauli-Check Sandwiching while quantifying the trade-offs between syndrome measurements and post-selection. The authors validate their model against emulator data, calibrate error rates, and explore how hardware improvements and graph topology affect the breakeven point where QAOA with Iceberg can outperform classical baselines like the Goemans-Williamson algorithm. Collectively, the work provides a concrete path to evaluating fault-tolerant quantum optimization on near-term devices and highlights the continuing need for quantum error correction for scaling beyond current limits.

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.

Figures (13)

• Figure 1: Motivation: The Iceberg code is a performant method for error detection in the near-term.A The Iceberg code detects errors that occur in the execution of a $k$-qubit circuit by encoding it in $(k+2)$ physical qubits. B Shots containing a detected error are discarded, resulting in a post-selection overhead. C Performance of QAOA with 10 layers with and without the Iceberg code on the Quantinuum H2-1 quantum computer. Here, the logical fidelity (defined in Eq. \ref{['eq:noisy_ar']}) directly indicates the approximation ratio. The Iceberg code with 4 syndrome measurements improves performance on small problems, while being detrimental on larger ones. D An example of measured samples with and without the Iceberg code obtained from the H2-1 device. After detecting the errors, the probability of the higher energy states is amplified, reflecting the approximation to the noiseless QAOA performance. E The Iceberg code performs better than other commonly-used techniques for error detection in QAOA circuits like Pauli Check Sandwiching (PCS) gonzales2023quantum. Data are obtained from the H2-1 emulator. Error bars show the standard errors. All hardware data are labeled as H2-1, while all emulated data are labeled as H2-1E.
• Figure 2: Proposed model accurately reflects behavior of Iceberg encoding circuits observed in high-accuracy emulation. The fitted model matches the qualitative and quantitative behavior of logical fidelity and post-selection rate for both varying qubit count with a fixed $\ell=9$ (A,C) and varying number of syndrome measurements with a fixed $k=16$ and $\ell=11$ (B,D). The shaded regions represent the standard errors.
• Figure 3: Model prediction: Predicting the performance of QAOA with the number of logical qubits in the range $k \in [6, 48]$ and the number of QAOA layers in the range $\ell \in [1, 16]$. We use the model proposed in this work to estimate (A, B) the difference $\mathcal{F}_\mathrm{ice} - \mathcal{F}_\mathrm{une}$ in logical fidelity between the Iceberg and the unencoded circuits, and to estimate (C, D) the post-selection rate. In A and C, we use the model fitted error rates in Table \ref{['tab:fitted_para']} and vary the number of syndrome measurements. In B and D, we fix the number of syndrome measurements at 4 and scale down the model error rates by the indicated factors. The red lines in the top row (A, B) show where the logical fidelity of Iceberg code circuits equals that of unencoded circuits. The cyan lines in the bottom row (C, D) indicate where the Iceberg code circuits have a 10% post-selection rate.
• Figure 4: Example of using the model to bound the hardware improvement that is necessary but not sufficient for QAOA to become competitive with the Goemans-Williamson algorithm.A Solve $k=16$ MaxCut using different solvers. Each data is reported as the mean of approximation ratio over 100 $k=16$$3$-regular graphs. The standard errors are too small to be seen. B Scaling of model parameter to beat Goemans-Williamson (GW) algorithm for $k=16$ graphs. The Iceberg code helps the QAOA to beat GW earlier than an unencoded one. The Iceberg and unencoded QAOA data are obtained from the H2-1 emulator.
• Figure 5: Comparison between emulated data and model predictions on random Erdős--Rényi graphs with different numbers of edges. The prediction of Iceberg logical fidelity is less accurate compared to testing on 3-regular graphs, while the prediction of unencoded logical fidelity and the prediction of post-selection rate remain accurate.