BF-DCQO: Scalable Quantum Optimization

• BF-DCQO is a quantum optimization framework that merges counterdiabatic driving with iterative bias-field updates to steer quantum evolutions toward optimal solutions.
• It digitizes quantum evolution via Trotter-Suzuki decomposition, enabling effective implementation on gate-based hardware for complex HUBO and combinatorial challenges.
• The algorithm achieves polynomial improvements in ground state probability and runtime, outperforming classical solvers, VQAs, and quantum annealing in scalability and efficiency.

Bias-Field Digitized Counterdiabatic Quantum Optimization (BF-DCQO) is an algorithmic framework that integrates digitized counterdiabatic quantum optimization with iteratively constructed bias fields to enable efficient, scalable solutions to large-scale combinatorial and higher-order binary optimization problems on gate-based quantum computers. This protocol stands out by uniting counterdiabatic driving, a shortcut-to-adiabaticity approach that suppresses nonadiabatic transitions, with a measurement-driven adaptive bias field mechanism that iteratively “steers” the quantum evolution toward regions of solution space with high probability of ground state occupation. Unlike traditional variational quantum algorithms (VQAs), BF-DCQO is non-variational, eliminating dependence on classical parameter optimization and thus sidestepping trainability issues endemic to VQAs. Validated both numerically and on quantum hardware up to 156 qubits for dense higher-order unconstrained binary optimization (HUBO) problems, BF-DCQO achieves polynomial scaling improvement in ground state success probability, higher approximation ratios, and runtime quantum advantage over leading classical, quantum annealing, and variational quantum approaches (Cadavid et al., 22 May 2024, Romero et al., 5 Sep 2024, Chandarana et al., 13 May 2025, Romero et al., 9 Jun 2025).

1. Mathematical Framework and Algorithm Structure

BF-DCQO addresses optimization problems cast as searching for the ground state of an Ising-type Hamiltonian, which can include arbitrary k-local terms, as in HUBO models: $$H_f = \sum_i h_i^z \sigma_i^z + \sum_{i<j} J_{ij} \sigma_i^z \sigma_j^z + \sum_{i<j<k} K_{ijk} \sigma_i^z \sigma_j^z \sigma_k^z + \cdots$$ The time-dependent quantum evolution is governed by an interpolating Hamiltonian

$$H_{ad}(\lambda) = (1 - \lambda(t)) H_i + \lambda(t) H_f$$

where $H_i$ is a readily preparable initial Hamiltonian (e.g., a transverse-field mixer $H_i = -\sum_i \sigma_i^x$), and $\lambda(t)$ is a monotonic scheduling function ($\lambda(0)=0,\, \lambda(T)=1$). To accelerate the evolution and suppress diabatic excitations, the algorithm incorporates a (first-order, nested-commutator) counterdiabatic term: $$H_{cd}(\lambda) = H_{ad}(\lambda) + \dot{\lambda} A_{\lambda}^{(1)}$$ with

$$A_{\lambda}^{(1)} = -2\alpha_1\left( \sum_i h_i \sigma_i^y + \sum_{i<j} J_{ij} (\sigma_i^y \sigma_j^z + \sigma_i^z \sigma_j^y) \right)$$

and $\alpha_1$ determined analytically or variationally for optimal suppression of diabatic transitions (Hegade et al., 2022, Guan et al., 2023, Cadavid et al., 22 May 2024).

The evolution is digitized: the total propagator is decomposed (Trotter-Suzuki) into a finite sequence of short-time evolutions under each term, allowing implementation on digital quantum hardware: $$U(T,0) = \prod_{k=1}^{n_{trot}}\prod_j \exp(-i\,\gamma_j(k\,\Delta t)\,\Delta t\,H_j)$$ This digital representation enables modular circuit design and efficient adaptation to hardware constraints such as qubit connectivity.

A central aspect of BF-DCQO is the iterative introduction of a bias field into the initial Hamiltonian, updating it after each quantum run: $$\tilde{H}_i = \sum_i (h_i^x \sigma_i^x - h_i^b \sigma_i^z)$$ where the bias terms $h_i^b$ are computed as a function of the measured expectation values $\langle \sigma_i^z \rangle$ from low-energy quantum samples. This feedback mechanism creates a warm-start for the next quantum iteration, adapting the basis of the digital quantum dynamics to the evolving ground-state structure. State initialization is performed with qubit rotations,

$$\theta_i = \arctan\left( \frac{h_i^x}{h_i^b + \sqrt{(h_i^x)^2 + (h_i^b)^2}} \right),$$

ensuring the new ground state of the mixer Hamiltonian aligns with the bias field (Cadavid et al., 22 May 2024, Romero et al., 5 Sep 2024).

2. Bias-Field Construction and Iterative Update

The bias field construction in BF-DCQO is measurement-driven and can be optimized further via conditional value-at-risk (CVaR) principles. After each digitized counterdiabatic evolution and readout, the bias for each qubit $i$ is updated using a rule such as:

• Unsigned: $h_i^b = \pm \langle \sigma_i^z \rangle$
• Signed: $$h_i^b = \pm \text{sign}(\langle \sigma_i^z \rangle)$$

The update is based on the α-percent lowest-energy samples, emphasizing low-energy regions and avoiding “trapping” in local minima (Romero et al., 5 Sep 2024). This CVaR-inspired update sharpens the sampling distribution and systematically increases ground state overlap over iterations.

In practice, the iterative scheme proceeds as:

1. Run the digitized counterdiabatic quantum circuit with current biases.
2. Measure in the computational basis; compute $\langle \sigma_i^z \rangle$ using the lowest-energy states.
3. Update $h_i^b$ for all qubits; prepare the new ground state of the mixer Hamiltonian via local rotations.
4. Repeat for a fixed number of bias-field updates (e.g., 10–11 iterations).
5. Optionally, further refine with a weighted signed bias in the final iteration to “lock in” the identified ground state solution (Romero et al., 5 Sep 2024).

Warm-starting with a classical optimizer (e.g., simulated annealing) to generate an initial bias field has been shown to further accelerate convergence and solution quality (Chandarana et al., 13 May 2025).

3. Performance, Scaling, and Comparative Benchmarks

Extensive experiments and simulations demonstrate that BF-DCQO achieves polynomial improvements in ground state success probability for all-to-all Ising and higher-order spin-glass instances, reducing the exponent in the scaling decay compared to both standard DCQO and digitized adiabatic protocols (Cadavid et al., 22 May 2024). On 156-qubit IBM heavy-hex lattice processors, BF-DCQO delivers:

• Approximation ratio improvements of ≈34–35% and distance-to-solution reductions (i.e., nearing the ground state) of ≈66–67% compared to D-Wave quantum annealing and classical solvers on HUBO benchmarks (Romero et al., 5 Sep 2024).
• Two orders of magnitude higher probability of sampling the ground state relative to standard approaches for comparable circuit-depth and parameter count (Cadavid et al., 22 May 2024).
• On benchmark suite problems (e.g., MAX 4-SAT, dense HUBO), consistent achievement of optimal or near-optimal solutions in the presence of dense, highly connected interaction graphs (Romero et al., 9 Jun 2025).

In direct runtime comparisons, BF-DCQO on gate-based processors outperforms simulated annealing (SA) and commercial branch-and-bound solvers (CPLEX) in time-to-approximate solutions for selected higher-order cases, reaching optimal/near-optimal energies in seconds—where classical solvers require 5–30× longer, especially as the problem size increases (Chandarana et al., 13 May 2025).

Table: Representative Performance Improvements (from (Romero et al., 5 Sep 2024, Chandarana et al., 13 May 2025, Romero et al., 9 Jun 2025)) | Problem Class | Qubits | BF-DCQO vs. QAOA | BF-DCQO vs. SA/CPLEX | BF-DCQO vs. D-Wave | |----------------------|--------|-------------------------|------------------------------|---------------------------| | 3-local HUBO | 156 | AR +34%, DS +67% | TT_ℛ: 2–10× faster | AR +35%, Opt hits higher | | MAX 4-SAT (dense) | 36 | Optimal found | TT_ℛ: 2–5× faster | Often higher opt. prob. | | Protein folding | 33 | Optimal found | Classical stuck, zero hits | Not directly comparable |

AR = Approximation Ratio; DS = Distance to Solution; TT_ℛ = Time-to-Approximate Solution

4. Implementation on Digital Quantum Hardware

BF-DCQO is designed to exploit the modular digital architecture of gate-based quantum processors. Key features include:

• Trotterized implementation, typically with $n_{trot} = 1–3$ steps, to capture the main counterdiabatic driving effect while minimizing circuit depth (Cadavid et al., 22 May 2024, Farré et al., 17 Sep 2025).
• Aggressive circuit compression, where all gates with rotation angles below a specified cutoff ($\theta_{cutoff}$) are omitted, ensuring that the circuit fits well within hardware coherence constraints (especially important for higher-body interactions in dense HUBOs and protein folding) (Romero et al., 9 Jun 2025).
• Hardware-optimized transpilation (e.g., swap networks only where necessary, dynamical decoupling for idling qubits), particularly relevant for architectures with all-to-all connectivity (IonQ traps) or heavy-hex/multi-row topologies (IBM) (Cadavid et al., 22 May 2024, Romero et al., 9 Jun 2025).
• Warm-starting with measurement-based feedback on the bias fields; no classical variational loop or parameter landscape optimization is required (Cadavid et al., 22 May 2024).

On IonQ's trapped-ion system (up to 36 qubits), all-to-all connectivity is harnessed efficiently for dense HUBO and protein folding problems, eliminating overhead due to nonlocal interaction mapping (Romero et al., 9 Jun 2025).

5. Applications: Portfolio Optimization, HUBO, Protein Folding, and Scheduling

The BF-DCQO framework is applicable to a variety of NP-hard combinatorial optimization problems:

• Portfolio Optimization: Mapped to Ising Hamiltonians via discretized asset allocations, leveraging counterdiabatic terms and warm-started bias fields to achieve high success probabilities (>= 1.3× improvement over QAOA for meaningful instance sizes) (Hegade et al., 2021, Cadavid et al., 22 May 2024, Tancara et al., 14 Oct 2024).
• Dense Higher-Order HUBO: Directly encodes MAX k-SAT, three- and four-local spin-glasses, and industry-motivated scheduling problems. Outperforms both classical heuristics (SA, tabu-search) and quantum annealing when the hardware connectivity is adequate (Romero et al., 5 Sep 2024, Simen et al., 21 Apr 2025, Chandarana et al., 13 May 2025, Romero et al., 9 Jun 2025).
• Protein Folding: Solves physically realistic lattice folding models, encoding up to 12 residues (33 qubits), to optimality in cases where classical solvers fail due to combinatorial explosion; benefits especially from dense, all-to-all interaction mapping (Chandarana et al., 2022, Romero et al., 9 Jun 2025).
• Logistics Scheduling: For job-shop and traveling salesperson problems, achieves order-of-magnitude improvements in success rate over QAOA for fixed-depth circuits, and circuit compression ensures NISQ feasibility (Dalal et al., 24 May 2024).

6. Limitations, Controversies, and Comparative Benchmarks

Recent work has challenged BF-DCQO’s claimed quantum advantage in direct comparison with D-Wave quantum annealing on similar problem classes (Farré et al., 17 Sep 2025):

• D-Wave’s quantum annealers yielded higher ground-state probabilities and lower errors with faster total runtime than BF-DCQO (by factors of 14–101× in some cases).
• Control experiments replacing the quantum kernel in BF-DCQO with a greedy classical sweep (“BF-Null”) produced equal or better results than the full quantum protocol.
• The data implies that the iterative bias-field mechanism may play a larger role in solution quality than the quantum counterdiabatic dynamics, and that the quantum step per se may contribute minimally in tested configurations.
• The possibility emerges that the main improvement derives from informed bias construction (or warm-starts from classical solvers), and that current gate-model implementations may not yet realize the full potential quantum advantage.

Despite this, BF-DCQO displayed significant enhancements over QAOA, classical SA, and CPLEX on selected classes of hard HUBO instances (Romero et al., 5 Sep 2024, Chandarana et al., 13 May 2025). In scenarios with all-to-all connectivity, dense higher-body couplings, or where mapping to QUBO requires prohibitive ancilla overhead, BF-DCQO demonstrates competitive scaling and practical utility—particularly as hardware and algorithmic refinements progress.

7. Extensions and Future Directions

The framework has been extended with additional algorithmic refinements:

• Branch-and-Bound Integration (BBB-DCQO): Enhances bias selection by identifying ambiguous spins for branching, recursively constraining high-uncertainty variables, and pruning via relaxation-based lower bounds, improving both convergence and resource efficiency on hard non-convex HUBO landscapes (Simen et al., 21 Apr 2025).
• High-dimensional Qudits: Moving beyond qubits to qutrits for natural encoding of multi-way partitioning and 3-cut-type problems yields up to 90× improvements in solution probability over qubit-based counterparts for certain benchmarks (Tancara et al., 14 Oct 2024).
• Circuit Optimization and Compilation: Techniques such as algorithm-oriented qubit mapping (AOQMAP), layer/symmetry exploitation, and hardware-aware transpilation further reduce gate and CNOT counts by ~30%, improving robustness to noise on NISQ machines (Ji et al., 2023).

BF-DCQO’s architecture allows straightforward adaptation to problems on hardware with improved coherence time, gate fidelity, and connectivity. As quantum processors scale and as algorithmic innovations continue, BF-DCQO and its descendants remain central candidates for large-scale, real-world quantum combinatorial optimization.

---

References:

• (Hegade et al., 2021, Hegade et al., 2022, Graß, 2022, Sun et al., 2022, Chandarana et al., 2022, Cadavid et al., 2023, Barone et al., 2023, Guan et al., 2023, Ji et al., 2023, Xu et al., 18 Jan 2024, Malla et al., 27 Jan 2024, Kumar et al., 2 May 2024, Cadavid et al., 22 May 2024, Dalal et al., 24 May 2024, Romero et al., 5 Sep 2024, Tancara et al., 14 Oct 2024, Simen et al., 21 Apr 2025, Chandarana et al., 13 May 2025, Romero et al., 9 Jun 2025, Farré et al., 17 Sep 2025)