Runtime Quantum Advantage with BF-DCQO (II)

22.05.2025

The Bias-Field Digitized Counterdiabatic Quantum Optimization (BF-DCQO) algorithm has recently demonstrated a compelling runtime quantum advantage in tackling higher-order unconstrained binary optimization (HUBO) problems. Notably, it can outperform established classical solvers such as CPLEX and Simulated Annealing (SA) in both speed and quality of solutions [1].

In this post, we explore how BF-DCQO stacks up against prominent quantum algorithmic approaches, such as Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA). By examining these comparisons, the aim is to understand where BF-DCQO stands in the evolving landscape of quantum optimization.

Quantum versus Quantum: Performance Benchmarks

In our previous blog, we showed how BF-DCQO, when executed on IBM’s digital quantum hardware, can outperform classical solvers such as CPLEX and SA for a class of complex problems. This runtime quantum advantage is particularly striking given the hardware disparity: the quantum processor operates at a relatively modest shot rate of ~10 kHz, whereas classical systems can evaluate up to 10⁸ samples per second on a regular computer. Despite this, BF-DCQO showed a performance boost, which would grow with system size, as the evidence suggested.

But outperforming classical solvers is only one part of the story. To fully gauge its potential, we must also compare BF-DCQO with other quantum optimization strategies. Here, we shift our focus to a quantum-versus-quantum comparison, evaluating BF-DCQO alongside prominent contenders in both digital and analog quantum processors.

Let’s begin by comparing BF-DCQO running on IBM’s digital quantum hardware with (analog) quantum annealing on D-Wave’s Advantage Prototype 2.6. This next-generation annealer features 1200+ qubits and is designed to naturally solve large-scale Quadratic Unconstrained Binary Optimization (QUBO) problems, which is the quadratic version of the more general HUBO class.

However, tackling HUBO problems on D-Wave requires a transformation pipeline. First, the HUBO instance must be reduced to a QUBO form through a mapping procedure. Then, a suitable embedding onto the quantum annealer’s hardware graph must be found. These steps introduce an overhead in the form of auxiliary qubits and added constraints, which are not intrinsic to the original problem. In contrast, BF-DCQO can address HUBO instances natively, avoiding this added complexity, offering up to a 4.3x reduction in the number of qubits for the instances here tested.

To make a runtime comparison with BF-DCQO, we executed the QA experiments using different annealing times between 0.5 µs and 2000 µs, while keeping the best performing. After recent scaling speedup claims using QAOA, in addition to QA, we also benchmarked BF-DCQO against a hardware-friendly version of QAOA, known as Linear-Ramp QAOA (LR-QAOA) [2]. Here we used up to 5 layers and kept the best performing on hardware.

BF-DCQO entirely sidesteps the bottleneck of classical parameter optimization, a feature shared by LR-QAOA, which was one of the motivating factors behind including it in our comparisons. By leveraging a linear ramp instead of variational feedback loops, LR-QAOA becomes a far more feasible candidate for execution on near-term quantum devices.

Performance Overview

In Figure 1, we present a comparative analysis of the accuracy achieved by BF-DCQO, QA, and LR-QAOA. Each bar in the plot represents the percent accuracy with respect to the optimal solution, and the corresponding runtime is displayed on top of each bar for clarity. To ensure a balanced comparison, we applied the same post-processing pipeline, originally developed for BF-DCQO, to all quantum algorithms under evaluation.

The results reveal a consistent trend: BF-DCQO delivered higher solution accuracy across all benchmark instances. More importantly, this improvement was not just in solution quality, it also came with a significantly reduced runtime compared to both QA and LR-QAOA.

These findings strongly suggest that BF-DCQO offers a more effective path to high-quality approximate solutions of HUBO problems on near-term quantum hardware. Not only did it outperform both digital (LR-QAOA) and analog (QA) quantum solvers in accuracy, but it also did so faster, even when running on digital quantum hardware with modest capabilities. This solidifies BF-DCQO as a frontrunner for achieving runtime quantum advantage in practical optimization scenarios.

Figure1. Accuracy (%) for five 156-qubit HUBO instances tackled with BF-DCQO, QA and LR-QAOA. Each bar represents the best experimentally achieved accuracy with the runtimes shown on top in seconds. Each of the instance is generated using up to 3 local terms and specific distributions. [1]

Conclusions

When we pit BF-DCQO against QA and LR-QAOA, BF-DCQO consistently outperformed both in accuracy, runtime and resources (in terms of qubit overhead for QA and circuit depth for LR-QAOA) in the tested instances. Thanks to its integration of counterdiabatic terms directly into a digitized evolution framework and the use of a bias field, BF-DCQO achieves significantly faster convergence on challenging 156-qubit instances. Despite running on hardware with a relatively low shot rate of ~10 kHz, BF-DCQO demonstrated a reduction in runtime to reach an approximate solution, compared to QA on D-Wave’s Advantage2 system and LR-QAOA on IBM Kingston.

But the story doesn't end at fixed problem sizes. One of BF-DCQO’s most powerful advantages is its scalability. While analog quantum annealing may suffer from embedding overheads and LR-QAOA may require time-consuming parameter searches, BF-DCQO maintains efficiency as the problem size grows. Its bias-field calibration and embedded counterdiabatic corrections require minimal classical overhead and scale naturally with system size. Importantly, it continues to exploit the same Hamiltonian structure, regardless of qubit count.

In practice, this positions BF-DCQO as a hardware-efficient, scalable approach to runtime quantum advantage on industrial use-cases. BF-DCQO not only demonstrated superiority on current quantum hardware but also set the stage for even greater gains as quantum processors improve in coherence, connectivity, and gate fidelity.

References

1. Chandarana P, Cadavid AG, Romero SV, Simen A, Solano E, Hegade NN. Runtime Quantum Advantage with Digital Quantum Optimization. arXiv preprint arXiv:2505.08663. 2025 May 13.
2. Montanez-Barrera JA, Michielsen K. Towards a universal QAOA protocol: Evidence of a scaling advantage in solving some combinatorial optimization problems. arXiv preprint arXiv:2405.09169. 2024 May 15.

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