BF-DCQO: Digitized Counterdiabatic Quantum Optimizer

• BF-DCQO is a fully quantum optimization framework that integrates bias-field feedback with digitized counterdiabatic driving to tackle higher-order binary problems.
• It employs iterative measurement-driven bias updates and first-order nested commutator corrections to suppress non-adiabatic transitions and enhance convergence.
• Empirical benchmarks show significant improvements in approximation ratio, success probability, and runtime over traditional quantum and classical optimization methods.

The Bias-Field Digitized Counterdiabatic Quantum Optimizer (BF-DCQO) is a fully quantum algorithmic framework for combinatorial and higher-order binary optimization on near-term gate-based quantum devices. BF-DCQO advances the digitized counterdiabatic quantum optimization (DCQO) paradigm by integrating feedback-driven bias fields and auxiliary counterdiabatic terms within an iteratively digitized quantum evolution, enabling robust convergence to low-energy solutions under stringent hardware and coherence constraints. The approach is natively suited for challenging higher-order unconstrained binary optimization (HUBO) tasks, operates efficiently without classical outer-loop optimization, and demonstrates significant improvements in approximation ratio, success probability, and time-to-solution scaling compared to traditional quantum and classical baselines.

1. Theoretical Foundations and Motivation

Modern combinatorial optimization problems, particularly those mapped to Ising spin-glass or HUBO Hamiltonians (incorporating 2-body and higher order, e.g., 3-body terms), present computational bottlenecks for classical heuristics and variational quantum approaches on NISQ devices. Traditional quantum adiabatic optimization requires long evolution times, which are infeasible given limited device coherence and error rates. Variational algorithms such as QAOA, while mitigating circuit depth issues, often suffer from trainability obstacles like barren plateaus.

BF-DCQO circumvents these challenges by combining:

• Digitized counterdiabatic (CD) driving—the application of nested-commutator-based approximate adiabatic gauge potentials to suppress non-adiabatic transitions.
• Measurement-informed bias fields—iteratively updated local longitudinal fields derived from computational-basis measurement statistics, guiding state preparation and evolution toward low-energy subspaces.
• Trotterized, fully quantum workflows—eliminating the need for classical parametrization or optimization loops, ensuring robustness to device noise and scalability with system size (Cadavid et al., 22 May 2024).

The principal goal is polynomial (and sometimes exponential) scaling enhancement in ground state probability and approximation ratio within coherence restrictions characteristic of current and near-term quantum processors (Hegade et al., 2022, Cadavid et al., 22 May 2024).

2. Mathematical Structure and Algorithmic Protocol

BF-DCQO encodes the optimization objective as a native $k$-local Hamiltonian: $$H_f = \sum_{i} h^z_i \sigma^z_i + \sum_{i<j} J_{ij} \sigma^z_i \sigma^z_j + \sum_{i<j<k} K_{ijk} \sigma^z_i \sigma^z_j \sigma^z_k$$ The quantum system evolves under a time-dependent digitally interpolated Hamiltonian: $H_{ad}(\lambda) = (1 - \lambda) H_i + \lambda H_f$ where

$$H_i = \sum_{i} h^x_i \sigma^x_i + \sum_{i} h^b_i \sigma^z_i$$

with initial bias field values $h^b_i = 0$.

A counterdiabatic correction is applied in the form: $$H_{cd}(\lambda) = H_{ad}(\lambda) + \dot{\lambda} A^{(1)}_\lambda$$ with adiabatic gauge potential approximate via the first nested commutator: $$A^{(1)}_\lambda = i \alpha_1(t) [H_{ad}, \partial_\lambda H_{ad}]$$ When expanded for the 2-local case (e.g., in portfolio optimization), this yields all necessary CD terms as computationally feasible single- and two-qubit operators, with explicit forms and analytic coefficients (Hegade et al., 2021).

The digitized quantum evolution is implemented by Trotterization: $$U(T,0) = \prod_{k=1}^{n_{\mathrm{trot}}} \prod_{j=1}^{n_{\mathrm{terms}}} \exp\big[ -i \gamma_j(k\Delta t) \Delta t\, H_j \big]$$ where $H_j$ index the Hamiltonian terms and $\gamma_j$ are the stepwise parameter functions. The bias field is inserted directly into $H_i$ at each iteration.

3. Iterative Bias-Field Feedback and State Preparation

Distinctively, BF-DCQO employs a quantum feedback loop in its warm-starting protocol:

1. An initial quantum circuit is executed (using a guessed or uniform bias field).
2. Measurement outcomes in the computational basis are analyzed: the means $\langle \sigma^z_i \rangle$ across the lowest-energy (CVaR) fraction are computed.
3. The bias fields for the next iteration are set as $h^b_i = \langle \sigma^z_i \rangle$ (signed or unsigned, or with further rescaling), focusing the initialization towards high-probability solution subspaces (Romero et al., 5 Sep 2024, Cadavid et al., 22 May 2024).
4. The new initial state is prepared as the ground state of the updated $H_i$, constructed efficiently via single-qubit $R_y(\theta_i)$ rotations:

$$\theta_i = 2 \tan^{-1} \left( \frac{-h^b_i + \lambda_i^{\min}}{h^x_i} \right)$$

with $\lambda_i^{\min} = -\sqrt{(h^b_i)^2 + (h^x_i)^2}$.

This feedback is iterated multiple times, yielding convergence to tighter, lower-variance energy distributions. The process is entirely quantum; no classical optimizer is used for parameter learning (Cadavid et al., 22 May 2024), distinguishing BF-DCQO from hybrid variational approaches.

4. Counterdiabatic Driving: Construction, Digitization, and Physical Implementation

BF-DCQO digitizes the CD-corrected time evolution by first-order (typically) Trotter decomposition into low-depth quantum circuits implementable on both superconducting and trapped-ion quantum hardware. The digitized unitary for each step includes exponentials of the CD corrections. Only terms with significant rotation angles (post-thresholding) are retained to minimize gate count (Cadavid et al., 22 May 2024, Romero et al., 9 Jun 2025).

The adiabatic gauge potential, required for CD driving, is computed using closed-form expressions for local and two-body terms using the variational principle or analytical minimization, as in: $$\alpha_1(t) = -\frac{1}{16} \left[(-1 + \lambda)^2 h_0^2 + J^2 \lambda^2 \right]$$ for the nearest-neighbor Ising model (Sun et al., 2022). For higher localities (e.g., in HUBO), the expansion includes corresponding multi-qubit Pauli operator products.

These circuits are transpiled into hardware-compatible gate sets, employing parallelization, SWAP-layer scheduling, and dynamical decoupling when possible (Cadavid et al., 22 May 2024, Romero et al., 9 Jun 2025). The approach is compatible with all-to-all connectivity (e.g., trapped ions) and limited-connectivity topologies (e.g., heavy-hex superconducting qubits).

5. Performance Metrics and Empirical Benchmarking

BF-DCQO is evaluated on the basis of:

• Ground state success probability ($|\langle \psi_{gs} | \psi_f(T) \rangle|^2$),
• Approximation ratio $\mathrm{AR} = E(\alpha=1)/E_0$,
• Distance to solution $$\mathrm{DS} = 1 - \min\{E_k\}_{k=1}^{n_{\text{shots}}}/E_0$$,
• Runtime-to-approximate-solution (total QPU+CPU wall time) (Chandarana et al., 13 May 2025).

Experiments with up to 156-qubit physical systems and 433-qubit MPS classical simulations for three-local HUBO have demonstrated that BF-DCQO:

• Outperforms classical simulated annealing, Tabu search, and QAOA (even with CVaR post-processing) in both solution fidelity and runtime scaling for large/industrial HUBO problems (Romero et al., 5 Sep 2024, Chandarana et al., 13 May 2025).
• Demonstrates runtime quantum advantage, achieving comparable or better solutions $>3.5\times$ faster than top-tier classical solvers (CPLEX, SA) on the hardest instances, with further acceleration as system size increases (Chandarana et al., 13 May 2025).
• Does not require ancillary qubits for higher-order terms, avoiding the hardware and solution quality penalty of QUBO reduction.
• Remains effective under realistic noise (on both IBM superconducting and IonQ trapped-ion hardware), delivering improvement with shallow circuits (Cadavid et al., 22 May 2024, Romero et al., 9 Jun 2025).

Problem	System Size	Comparison	BF-DCQO Performance
3-local HUBO	156 Qubits	QAOA, Sim. Anneal., D-Wave	$34.1\%$ AR gain vs D-Wave, $72.8\%$ DS gain vs SA, <2s
MAX 4-SAT	up to 36	Hardware/Simulation benchmarks	Optimal/near-optimal in all instances (IonQ)
Spin-glass	100, 433	ITensor (DMRG), MPS, QAOA, D-Wave	BF-DCQO circuits become shallower yet gain in fidelity

6. Limitations, Generalization, and Recent Controversies

While BF-DCQO shows robust polynomial performance enhancement and fast convergence for a wide class of non-convex binary optimization instances, several limits and ongoing debates have been noted:

• Convergence Plateaus: After several bias update iterations, the protocol can saturate to a family of correlated low-energy states, impeding further improvement without structural modifications. This has motivated the development of branch-and-bound extensions (BBB-DCQO) for exhaustive exploration (Simen et al., 21 Apr 2025).
• Quantum Contribution Dispute: Controlled benchmarking has highlighted that, for some instances and on currently available hardware, the quantum subroutine of BF-DCQO offers minimal enhancement over trivial classical "null" protocols using analogous bias-field update heuristics. In such settings, D-Wave quantum annealing devices outperformed BF-DCQO both in solution quality and runtime (Farré et al., 17 Sep 2025).
• Hardware-bound Depth: Very large or dense instances may still be bottlenecked by gate errors/decoherence in deep circuits, especially with limited connectivity or imperfect parallelization (Romero et al., 9 Jun 2025).
• Scaling Prospects: The quantum advantage appears to be most pronounced in highly non-convex, high-order, and hardware-fitted instances; less so in fully connected or classical-simulable low-order problems.

7. Impact, Extensions, and Design Guidance

BF-DCQO has established new benchmarks for digital quantum optimization, especially for intractable HUBO instances and industry-motivated high-dimensional problems where classical approaches struggle and variational quantum algorithms become untrainable. Its digitized structure accommodates arbitrary $k$-locality, high resilience to decoherence, no dependence on classical global optimizers, and straightforward extension to higher-dimensional qudits (qutrits) for further advantages in native problem encoding (Tancara et al., 14 Oct 2024).

Implementation guidance derived from empirical and theoretical results includes:

• Use shallow Trotter steps and prioritize first-order nested commutator corrections for hardware feasibility.
• Select bias field update schemes (signed, unsigned, weighted) tailored to problem data and variability; leverage CVaR post-selection to avoid noise-induced bias drift.
• Embed feedback iteratively with hardware-compatible state preparation and measurement, avoiding classical optimization loops.
• For problems displaying convergence plateau, consider integrating branch-and-bound logic or multi-stage hybrid workflows (HSQC) to expand solution reach (Simen et al., 21 Apr 2025, Chandarana et al., 7 Oct 2025).

References to Key Methods and Results

• CD protocol and nested commutator construction (Hegade et al., 2022, Sun et al., 2022)
• Feedback and bias-field digitized updates (Cadavid et al., 22 May 2024, Romero et al., 5 Sep 2024)
• Empirical hardware results: 156–433 qubits, IBM and IonQ platforms (Cadavid et al., 22 May 2024, Chandarana et al., 13 May 2025, Romero et al., 9 Jun 2025)
• Comparison to QAOA, classical heuristics, and D-Wave (Romero et al., 5 Sep 2024, Chandarana et al., 13 May 2025, Farré et al., 17 Sep 2025)
• Circuit optimization and AOQMAP codesign (Ji et al., 2023)

BF-DCQO thus represents a central method in the NISQ-era quantum algorithmics toolkit, offering both theoretically principled and experimentally validated performance for real-world combinatorial optimization at and beyond the threshold for commercial quantum advantage.