VQA with Adaptive Cost Encoding (ACE)

• Variational Quantum Algorithm with Adaptive Cost Encoding (ACE) is an approach that tailors cost functions in quantum circuits to match problem constraints and measurement noise.
• ACE techniques dynamically adjust parameters, perform selective filtering, and optimize measurement allocation to improve convergence and fidelity.
• Applications span combinatorial optimization, state preparation, image segmentation, and circuit design, offering practical benefits for NISQ devices.

Adaptive Cost Encoding (ACE) is a family of techniques for enhancing variational quantum algorithms (@@@@1@@@@) by tailoring the cost function—used as the optimization objective—to adapt to the properties of the problem, the quantum circuit, and the measurement noise. ACE encompasses parameter tuning within cost functions, problem-specific cost evaluations, dynamic shot allocation, and selective filtering of measurement outcomes, with the fundamental goal of improving convergence behavior, fidelity, resource efficiency, and problem correspondence. Research on ACE spans applications in combinatorial optimization, quantum state preparation, image segmentation, and circuit design, as well as adaptive measurement control protocols.

1. Conceptual Foundation and Motivation

ACE is predicated on adjusting the cost function driving a VQA such that it reflects the structure and constraints of the underlying problem more directly, optimizes measurement and computational resources, or adapts dynamically to circuit evolution. Traditional VQAs aim to minimize a cost function $C(\vec{\theta})$ over circuit parameters $\vec{\theta}$, but the form and implementation of $C$ critically impact both practical and physical performance.

Conventional cost functions—such as those based on fidelity error $$\mathcal{E} = 1 - |\langle\xi(\beta)|\Psi(\vec{\alpha},\vec{\gamma})\rangle|^2$$ or subsystem free energy $$F_A(T;\vec{\alpha},\vec{\gamma}) = \mathrm{Tr}[\rho_A H_A] + T \mathrm{Tr}[\rho_A \log\rho_A]$$—often require extensive measurement resources (full state tomography, entropy estimation), rendering them impractical on current noisy intermediate-scale quantum (NISQ) devices (Premaratne et al., 2020).

The ACE approach seeks to overcome these limitations by engineering more experimentally accessible and physically meaningful cost functions, adaptively tuned to the circuit and the problem. This includes:

• Expressing cost functions in terms of expectation values of easily measured operators (correlators).
• Dynamically modifying coefficients and terms within the cost function based on problem parameters or performance metrics.
• Pruning redundant measurement outcomes and focusing optimization on critical features of the quantum state or solution space.
• Integrating classical techniques such as surrogate modeling and Bayesian optimization for adaptive shot allocation in quantum measurements (Anders et al., 3 Feb 2025).
• Filtering infeasible solutions during circuit evolution and optimization (Fernandez et al., 28 Mar 2025).

ACE is thus a set of strategies unifying cost engineering, measurement adaptation, and solution filtering to maximize the efficiency and reliability of VQAs.

2. Engineering and Tuning Quantum Cost Functions

Within ACE, cost functions are systematically tuned for both experimental feasibility and problem alignment. A representative instantiation is the procedure for generating thermofield double (TFD) states in the transverse field Ising model (Premaratne et al., 2020).

• Standard Cost Functions: Conventional options include the fidelity error $\mathcal{E}$ and subsystem free energy $F_A(T)$, both of which suffer from measurement overhead.
• Experiment-Friendly Ansatz: An initial alternative is $$\mathcal{C}_0(T;\vec{\alpha},\vec{\gamma}) = \langle H_A + H_B - T H_{AB} \rangle$$, requiring only a handful of correlator measurements.
• Parametrically Tuned Cost Functions: To address poor performance at intermediate temperatures, the cost function is extended to $$\mathcal{C}_1(T;\vec{\alpha},\vec{\gamma}) = \langle c_1(\zeta,\tau)\rangle$$, with $c_1(\zeta,\tau)$ containing adjustable coefficients (e.g., $$X_A + X_B + \zeta(ZZ_A+ZZ_B) - T^\tau (ZZ_{AB} + XX_{AB})$$). Optimal values of $\zeta$ and $\tau$ are obtained by sweeping or minimizing an aggregate error metric across temperature ranges.
• Selective Pruning via Density Matrix Comparison: Further refinement yields $\mathcal{C}_2 = \sum_{i=1}^{15} a_i (r_i - s_i)^2$, where $a_i \in \{0,1\}$ prune non-essential density matrix elements, focusing optimization on the few terms critical for encoding thermodynamic properties.

These engineered cost functions preserve direct problem correspondence and can be systematically optimized using classical post-processing. The approach leverages tunable parameters and selection strategies to balance physical representability (energy, entropy) and measurement practicality.

3. Architectural Integration and Optimization Strategies

ACE methods have been integrated into various quantum algorithmic workflows:

• Circuit-Efficient Encoding for Combinatorial Optimization: In image segmentation, ACE leverages an ABE-style circuit architecture (logarithmic qubit scaling with respect to problem size) and defines its cost function as the classical min-cut metric—$$C(\vec{x}) = \sum_{1 \leq i < j \leq n} [x_{v_i} \cdot (1 - x_{v_j})] \cdot w(v_i,v_j)$$—directly computed from decoded measurement outcomes (Venkatesh et al., 23 May 2024). This eliminates the need for QUBO transformation or auxiliary penalty terms and offers a better mapping between quantum parameters and binary solutions.
• Adaptive Observation Cost Control: The SubsCoRe method (Anders et al., 3 Feb 2025) interleaves a Gaussian process (GP) surrogate model into the optimization, adaptively determining the number and distribution of quantum measurement shots so that prediction uncertainty (variance) stays below an accuracy threshold ($\kappa^2$) within confident regions (CoRe). Measurement allocation is solved analytically for equidistant sampling points, minimizing resource use while preserving solution accuracy.
• Evolutionary Circuit Tuning: The QCE-ACF framework (Fernandez et al., 28 Mar 2025) embeds an adaptive cost function within a quantum circuit evolutionary algorithm, filtering measurement outcomes by feasibility with respect to problem constraints. The cost is evaluated only on feasible measurement strings, improving convergence speed and avoiding stagnation in regions dominated by violation outcomes.

This architectural diversity highlights ACE's ability to integrate with both hybrid and optimizer-free quantum algorithms. Optimization may involve classical parameter sweeps, Bayesian inference, or evolutionary feedback mechanisms, depending on the workflow.

4. Performance Characteristics and Resource Efficiency

ACE approaches frequently demonstrate improved performance metrics, such as convergence rate, fidelity, and resource efficiency, as detailed in numerical experiments.

• Convergence Behavior: Engineered cost functions with tunable parameters ($\zeta, \tau$) yield improved agreement with target states across temperature ranges, especially at intermediate and high temperatures (Premaratne et al., 2020). For image segmentation, ACE achieves faster convergence (fewer iterations) and lower relative error than alternative encodings (PGE, ABE) (Venkatesh et al., 23 May 2024).
• Resource Scaling: Qubit-efficient architectures (logarithmic scaling in $n$) enable image segmentation tasks on much larger instances than QAOA-style approaches with linear scaling (Venkatesh et al., 23 May 2024). Shot-adaptive methods like SubsCoRe can reduce measurement cost by up to an order of magnitude, as validated on molecular VQE benchmarks (Anders et al., 3 Feb 2025).
• Execution Time and Noise Robustness: The QCE-ACF method achieves solution quality identical to QAOA but with dramatically shorter execution times (up to two orders of magnitude faster for large instances). In noise-induced simulations—critical for NISQ hardware—ACE maintains robust convergence with shallow circuits (Fernandez et al., 28 Mar 2025).
• Measurement Overhead: By focusing on essential features (as in $\mathcal{C}_2$) or eliminating irrelevant or infeasible outcomes, ACE strategies systematically reduce measurement requirements without sacrificing solution quality.

A plausible implication is that ACE techniques offer meaningful improvements in practical scenarios bounded by quantum hardware and measurement limitations.

5. Limitations, Scalability, and Open Challenges

While ACE demonstrates a range of advantages, several limitations and challenges are reported:

• Parameter Tuning Dependence: The optimal selection of cost function parameters (e.g., $\zeta$, $\tau$ in $\mathcal{C}_1$ or $a_i$ in $\mathcal{C}_2$) may depend on problem-specific details, such as qubit number or Hamiltonian parameters, and may require re-optimization for new instances (Premaratne et al., 2020).
• Scalability of Selection Techniques: The explicit density matrix pruning used in engineered cost functions ($\mathcal{C}_2$) is practical for small systems but may not scale to larger qubit registers (Premaratne et al., 2020).
• Measurement Noise and Decoding Robustness: For image segmentation, the accuracy of the adaptive cost is tied to reliable measurement and classical decoding of solutions. Noise can distort binary assignments and impact cost evaluation (Venkatesh et al., 23 May 2024).
• Classical Optimization Heuristics: Discrete adaptive cost landscapes (as in ACE for image segmentation) can simplify optimization but may necessitate new robust classical optimization techniques for larger and deeper circuits.
• Generalization to Larger Problems: Many numerical experiments are performed on small system sizes (e.g., 4–16 pixels, up to 4 qubits), leaving open the evaluation of ACE scalability on larger, more complex instances.
• Adaptive Filtering Risks: In QCE-ACF, discarding violation outcomes accelerates convergence, but may also overlook potentially useful exploratory regions of the solution space in specific problem instances.

Further research is suggested to resolve these open issues and investigate the full scope of ACE's applicability on next-generation quantum hardware.

6. Broader Applications and Implications

ACE principles are finding applications across quantum algorithmic domains:

• Quantum State Preparation: ACE enables more effective generation and purification of thermal states relevant to condensed matter systems, with enhanced fidelity and tractable resource requirements (Premaratne et al., 2020).
• Quantum Chemistry and Physics Simulation: Adaptive measurement control for VQE (e.g., via SubsCoRe) improves the scalability and projective precision of molecular ground-state energy estimations (Anders et al., 3 Feb 2025).
• Quantum-enhanced Computer Vision: ACE-based segmentation paves the way for quantum algorithms in image processing, given dramatic reductions in qubit and gate requirements (Venkatesh et al., 23 May 2024).
• Binary Optimization: Classical optimizer-free circuits augmented with ACE outperform traditional QAOA variants in execution time and resource efficiency, suggesting new paradigms for constraint-handling in quantum search (Fernandez et al., 28 Mar 2025).
• Hybrid Quantum-Classical Algorithms: Shot-adaptive and cost-engineered approaches are likely to integrate with emerging hybrid methods, offering better error mitigation and resilience to NISQ limitations.

ACE thus constitutes a unifying theme in contemporary quantum algorithm research, emphasizing adaptability, resource management, and physical correspondence as central design criteria for practical quantum optimization strategies.