Iceberg Beyond the Tip: Co-Compilation of a Quantum Error Detection Code and a Quantum Algorithm

TL;DR

The paper tackles the challenge of running fault-tolerant quantum circuits for near-term algorithms by co-optimizing a quantum error-detection code (Iceberg) with the QAOA circuit. It introduces new fault-tolerant gadgets, gadget resynthesis, and symmetry-based optimizations, organized within a tree-search compilation framework that minimizes encoded circuit depth while preserving fault-tolerance. Hardware demonstrations on the Quantinuum H2-1 show improved QAOA success probability and post-selection rates, extending the break-even point to 34 logical qubits, and achieving beyond-unencoded performance for sizeable instances. The work also demonstrates that Iceberg QED can serve as a useful tool for benchmarking QAOA energy populations under noise, highlighting practical pathways to near-term quantum advantage. Overall, the methodology provides a scalable approach to co-compile error-detection gadgets with quantum algorithms, with clear implications for hardware-aware quantum optimization and beyond-classical benchmarking.

Abstract

The rapid progress in quantum hardware is expected to make them viable tools for the study of quantum algorithms in the near term. The timeline to useful algorithmic experimentation can be accelerated by techniques that use many noisy shots to produce an accurate estimate of the observable of interest. One such technique is to encode the quantum circuit using an error detection code and discard the samples for which an error has been detected. An underexplored property of error-detecting codes is the flexibility in the circuit encoding and fault-tolerant gadgets, which enables their co-optimization with the algorthmic circuit. However, standard circuit optimization tools cannot be used to exploit this flexibility as optimization must preserve the fault-tolerance of the gadget. In this work, we focus on the $[[k+2, k, 2]]$ Iceberg quantum error detection code, which is tailored to trapped-ion quantum processors. We design new flexible fault-tolerant gadgets for the Iceberg code, which we then co-optimize with the algorithmic circuit for the quantum approximate optimization algorithm (QAOA) using tree search. By co-optimizing the QAOA circuit and the Iceberg gadgets, we achieve an improvement in QAOA success probability from $44\%$ to $65\%$ and an increase in post-selection rate from $4\%$ to $33\%$ at 22 algorithmic qubits, utilizing 330 algorithmic two-qubit gates and 744 physical two-qubit gates on the Quantinuum H2-1 quantum computer, compared to the previous state-of-the-art hardware demonstration. Furthermore, we demonstrate better-than-unencoded performance for up to 34 algorithmic qubits, employing 510 algorithmic two-qubit gates and 1140 physical two-qubit gates.

Figures (12)

• Figure 1: Overview of our proposed co-compilation pipeline for quantum algorithms and quantum error detection. By leveraging improved fault-tolerant QED gadgets, the flexibility of these gadgets, the properties of the problem structure, and a tree search method, we achieve a more compact circuit while retaining the partial fault-tolerant properties of the encoded circuit. An example of co-optimized and baseline syndrome measurement is shown on the leftmost side of the panel. Hardware experiments of Iceberg-encoded QAOA for a $k=22$ 3-regular graph instance on Quantinuum H2-1 Moses2023 demonstrate an improvement in QAOA success probability from $44\%$ to $65\%$ and an increase in the post-selection rate from $4\%$ to $33\%$ compared to the previous state-of-the-art hardware demonstration he2024performance.
• Figure 2: The circuit encoded in the Iceberg code from the prior works contains numerous opportunities for further optimization.
• Figure 3: New Set of Fault-tolerant Gadgets. (a) New initialization gadget for preparing the logical $\ket{\overline{+}}^{\otimes k}$. (b) New syndrome measurement gadget with higher parallelism. (c) New final measurement gadget with one physical qubit less.
• Figure 4: Gadget resynthesis. (a) QAOA input problem graph. (b) A compiled partial circuit with the old initialization gadget and the first entangler layer. (c) Optimized circuit with initialization gadget resynthesis. (d) Idling errors occur in the encoded mixer layer and the old syndrome gadget. (e) Idling errors reduced with the syndrome gadget resynthesis.
• Figure 5: Optimized mixer with Z2 symmetry property. The circuit depth of a mixer layer is reduced by half by leveraging the bottom qubit.