QCE-ACF: Adaptive Quantum Circuit Evolution

• QCE-ACF is a variational quantum algorithm that integrates genetic-inspired circuit evolution with an adaptive cost function to efficiently solve binary optimization problems.
• It employs four mutation operators to evolve circuit structures and uses a dynamic cost metric to downweight infeasible outcomes and overcome fitness plateaus.
• Experimental benchmarks show QCE-ACF outperforms QAOA, achieving optimal results with significantly reduced execution times under realistic NISQ conditions.

The QCE-ACF (Quantum Circuit Evolution with Adaptive Cost Function) framework is a variational quantum algorithmic methodology for solving binary optimization problems. It integrates a classical-optimizer-free quantum circuit evolutionary (QCE) approach with an adaptive cost function (ACF) that dynamically reshapes the fitness landscape during circuit evolution. QCE-ACF achieves efficient, noise-resilient convergence, outperforming both baseline QCE and quantum approximate optimization algorithm (QAOA) approaches in terms of execution time and solution quality under realistic (noisy-intermediate-scale quantum) conditions (Fernandez et al., 28 Mar 2025).

1. Fundamentals of Quantum Circuit Evolution (QCE)

QCE is a genetic-inspired framework that evolves both the structure and parameters of quantum circuits for constrained optimization without recourse to classical optimizers. Each “parent” in generation $g$ is a quantum circuit $U_g$ assembled from a fixed gate set $$\{R_x, R_y, R_z, CR_x, CR_y, CR_z, R_{xx}, R_{yy}, R_{zz}\}$$, with both connectivity and depth determined dynamically. Offspring are generated by applying one of four mutation operators (insert, delete, swap, modify) to the gate list, each with equal probability.

For each generation, $K$ offspring are produced, evaluated via a cost function, and the best-scoring offspring is selected as the parent for the next generation. The core challenge addressed by QCE-ACF is that local, incremental changes in circuit structure may stagnate the search in cost-plateaus where fitness does not improve, limiting convergence efficiency in the baseline QCE paradigm (Fernandez et al., 28 Mar 2025).

2. Adaptive Cost Function (ACF) Mechanism

The adaptive cost function modifies the standard measurement-based cost to accelerate convergence and steer evolution away from infeasible regions:

Let $C(x)$ be the QUBO/Hamiltonian-based cost for bitstring $x$. Denote $M_g$ the set of $N_t$ measured bitstrings sampled during evaluation of a candidate circuit in generation $g$, with frequency $f_x$ for string $x$. Partition $M_g$ into feasible ($F_g$) and violation ($V_g$) subsets according to problem constraints.

The ACF is defined piecewise:

• If $F_g \neq \emptyset$,

$$\langle C_g\rangle_F = \frac{1}{N_t - |V_g|} \sum_{x \in F_g} f_x\,C(x)$$

• If $F_g = \emptyset$, identify $x_m = \arg\max_{x \in V_g} f_x$ (most common violator), and let $V_{g,m} = \{x_m\}$,

$$\langle C_g\rangle_V = \frac{1}{N_t - |V_{g,m}|} \sum_{x \in V_g \setminus V_{g,m}} f_x\,C(x)$$

The overall adaptive cost function per generation is:

$$ACF_g(U) = \begin{cases} \langle C_g\rangle_F, & F_g\neq\emptyset\ \langle C_g\rangle_V, & F_g=\emptyset \end{cases}$$

By completely discarding infeasible measurement outcomes or, in the absence of feasible results, omitting the most frequent violation, ACF reshapes the landscape to favor circuits yielding feasible solutions, efficiently breaking cycles of stagnation (Fernandez et al., 28 Mar 2025).

3. Algorithmic Procedure and Control Parameters

QCE-ACF proceeds via the following loop for $G_{max}$ generations:

1. Mutate parent circuit $U_g$ $K$ times to generate offspring $\{U_{g,i}\}$.
2. For each $U_{g,i}$, perform $N_t$ quantum measurements to obtain bitstring frequencies.
3. Partition results into $F_g$ and $V_g$ and compute the cost via the adaptive cost function, as described above.
4. Select the offspring with minimum cost for the next generation.
5. Terminate if the minimum cost plateaus.

Recommended parameters for successful application, as per empirical results, include $N_t=1024$ shots per evaluation, $K=4$ offspring, $G_{max}=50-100$ generations, with equal probability assignment to each mutation type ($$p_{insert} = p_{delete} = p_{swap} = p_{modify} = 0.25$$). No explicit learning rate or convergence threshold is required, as the ACF dynamically adapts search pressure (Fernandez et al., 28 Mar 2025).

4. Performance and Benchmarking Results

QCE-ACF has been systematically benchmarked on set-partitioning problems of up to 20 qubits. Experimental findings include:

Instance	QAOA Ratio	QAOA Time (s)	QCE-ACF Ratio	QCE-ACF Time (s)
14.1	1.0	19.37	1.0	1.08
14.2	1.0	18.03	1.0	0.63
20.1	1.0	1694.8	1.0	9.50
20.10	1.0	1643.6	1.0	2.34

Both QAOA and QCE-ACF achieved optimality (ratio = 1.0), but QCE-ACF did so with execution times lower by one to two orders of magnitude. Under injected noise (two-qubit and readout errors), QCE-ACF consistently retained optimal performance with shallow circuit depths ($\lesssim 30$ for 14 qubits), while QAOA runtimes increased substantially. The design of ACF, aggressively downweighting noisy infeasible outcomes, ensured robust operation in noisy intermediate-scale quantum (NISQ) environments (Fernandez et al., 28 Mar 2025).

5. Comparative Analysis and Practical Recommendations

QCE-ACF surpasses baseline QCE by preventing evolutionary stagnation on infeasible or cost-plateau regions—the ACF “reshapes” the landscape constantly by discounting measurement outcomes that do not meet problem constraints. This leads to the evolution of shallower quantum circuits, with reduced accumulation of error channels per shot.

QAOA, in comparison, runs a fixed-depth, parameterized ansatz and typically requires computationally expensive classical optimization routines. The free-structure ansatz of QCE-ACF, combined with its low-complexity selection pressure, yields equally accurate results more rapidly, especially notable as problem size and noise scale.

To apply QCE-ACF to new constrained binary optimization problems, the recommended workflow is:

1. Express the problem as a QUBO and define $C(x)$ with penalty terms.
2. Use $N_t \approx 10^3$; $K=4$ offspring per generation.
3. Implement ACF per equations (7)-(9) in the framework.
4. Monitor whether feasible strings occur; in their absence, execute the single largest violator removal protocol to assure steady progress.
5. Expect robust operation for depths scaling sub-exponentially, even under NISQ conditions (Fernandez et al., 28 Mar 2025).

6. Noise Resilience and Scalability Considerations

QCE-ACF's circuit depth control and fitness-driven discrimination contribute to its noise resilience. The exclusion of infeasible measurement outcomes in each generation means the evolutionary search is concentrated on regions less affected by quantum errors. Empirical results confirm that circuit depth stays minimal, and measurement frequency stabilization can be achieved with moderate shot counts. To maintain efficiency for larger instances, penalty weights $\rho_i$ in the QUBO must be tuned so feasible outcomes are present with sufficient frequency (Fernandez et al., 28 Mar 2025).

A plausible implication is that for problems with severely imbalanced feasible/violation measurement statistics, adaptive penalty weighting or sampling schemes may need further refinement to maintain convergence speed.

7. Significance and Directions for Further Research

QCE-ACF systematically addresses the convergence and runtime bottlenecks of evolutionary quantum algorithms for binary optimization. Its scalability, noise resilience, and parameter-free adaptation render it broadly applicable within NISQ-era constraints. The key insight of “fitness landscape reshaping” via adaptive cost metrics provides a conceptual template for future algorithmic improvements, including hybridization with more advanced evolutionary dynamics or integration into multi-level quantum-classical hierarchies.

Potential directions include tuning penalty schedules for infeasibility as problem constraints become more stringent, formal analyses of ACF-induced search dynamics, and experimental deployment on real quantum hardware to further validate the approach under device-specific noise and connectivity constraints (Fernandez et al., 28 Mar 2025).