Entropy-Controlled Benchmark Framework

• Entropy-Controlled Benchmark is a framework that uses controlled entropy measures to set performance boundaries across diverse domains such as quantum circuits and computational clusters.
• It employs methodologies from statistical physics and information theory to quantify system disorder, enabling fairer comparisons and precise resource trade-offs.
• Applications span from noise analysis in quantum experiments to structural optimization in optical devices, signal processing, and bioinformatics.

An entropy-controlled benchmark is a quantitative framework in which entropy—broadly, a measure of randomness, disorder, or uncertainty—is used as a fundamental parameter to design, analyze, or calibrate performance tests across diverse domains such as quantum computation, parallel computing, signal processing, and statistical learning. The defining feature is explicit control or measurement of entropy at the level of system, process, circuit, or data, then using this to bound achievable accuracy, efficiency, or robustness. Entropy-controlled benchmarks go beyond traditional performance metrics by capturing the effects of underlying disorder or noise, enabling both fairer comparisons and principled resource trade-offs across platforms and algorithms.

1. Foundational Principles and Key Definitions

Central to entropy-controlled benchmarks is the selection of an entropy measure corresponding to the system under paper. Examples include:

• The linear cross-entropy $\chi$ between outcome distributions in quantum circuits, which serves as a boundary-order parameter for measurement-induced phase transitions (Li et al., 2022).
• Entropy density $s(\rho)$, given by normalized Rényi or von Neumann entropy, as a scalar diagnostic for noise accumulation in near-term quantum devices (Demarty et al., 23 Dec 2024).
• Shannon entropy of work distributions to assess complexity and control in nonequilibrium quantum dynamics (Campbell, 2023).
• Binary or multinomial entropy encoding disorder in engineered structures such as planar optical elements (He et al., 2023).
• Graph-theoretic entropy of parallel system architectures measuring incompatibility-induced disorder (Adefemi, 12 Sep 2025).
• Entropy-rank ratios positioning individual sequences within the global spectrum of all block-wise entropy values in a combinatorial DNA context (Pastore et al., 7 Nov 2025).

In each case, the protocol either fixes system entropy as an external parameter to be controlled (e.g., tuning initial disorder, circuit depth, or process randomness) or measures the resultant entropy to benchmark or bound performance.

2. Methodologies for Quantum Systems

Measurement-Induced Phase Transitions: The entropy-controlled cross-entropy benchmark leverages the linear cross-entropy $\chi$ computed between bulk outcome probability distributions from two distinct initial states $\rho, \sigma$ propagated through otherwise identical random monitored quantum circuits. Formally,

$$\chi_C = \frac{\sum_m p_m^{(\rho)} p_m^{(\sigma)}}{\sum_m [p_m^{(\sigma)}]^2}, \qquad \chi = \mathbb{E}_C [ \chi_C ]$$

where $p_m^{(\rho)}$ is the probability of outcome $m$ given initial state $\rho$. The circuit-averaged $\chi$ functions as an order parameter: $\chi \to 1$ in the volume-law phase (minimal initial-state information in measurements) and $\chi < 1$ in the area-law phase. Experimentally, the protocol requires sampling measurement records from $\rho$, evaluating $p_m^{(\sigma)}$ for the same record via classically tractable (e.g., stabilizer) simulation, and aggregating by averaging. Sampling complexity achieves error $\epsilon$ with $O(1/\epsilon^2)$ trajectories per circuit, no postselection, and is robust under weak depolarizing noise (Li et al., 2022, Hu et al., 22 Jan 2025).

Entropy-Density Benchmarking: For near-term quantum circuits of $n$ qubits, the entropy density $s(\rho) = S^{(2)}(\rho)/n$ (with $S^{(2)}$ the second-order Rényi entropy) quantifies noise accumulation. Under global depolarizing noise of strength $P$,

$$\rho_{\text{out}} = (1-P) U \rho_0 U^\dag + P \frac{I}{2^n}$$

with $$S^{(2)}(\rho_{\text{out}}) = -\log_2 \operatorname{Tr}[\rho_{\text{out}}^2]$$ interpreted as a circuit-level signature of entropy production. Analytical models relate local gate error rates $(p_1,p_2)$ to effective $P$, providing closed-form expressions for the scaling of $s(\rho)$ with circuit depth and enabling estimation of the maximal circuit size—beyond which quantum advantage becomes unattainable—by locating the entropy threshold at which quantum output cannot outperform classical algorithms for ground-state energy or similar application metrics (Demarty et al., 23 Dec 2024, Besserve et al., 1 Oct 2025). These models are validated experimentally and link hardware-level noise directly to application-level feasibility.

3. Entropy-Controlled Benchmarks in Computational and Physical Systems

Parallel Cluster Entropy: System-level entropy for supercomputers or clusters quantifies incompatibility-induced disorder using a graph-theoretic model. Each compute node is a complete bidirectional graph of components, with edge weights reflecting vendor compatibility. The cluster entropy $E$ is given by

$E = \sum_{i=1}^n 3\,\log(1 + S_i)$

where $S_i$ is the maximal pairwise incompatibility for node $i$, normalized to account for component count and penalty scaling. Empirical studies reveal strong negative correlation between $E$ and traditional performance benchmarks (e.g., LINPACK, MLPerf, HPCC composite scores), demonstrating that hardware disorder significantly degrades computational output (Adefemi, 12 Sep 2025). Entropy-controlled benchmark suites assemble workloads that target specified entropy levels and report both raw and entropy-normalized performance, enabling reproducible, disorder-aware benchmarking.

Entropy in Device and Data Augmentation: An alternative paradigm employs entropy as a structural or data complexity control parameter. In planar optical devices, the binary entropy of zone deviations from a Fresnel zone plate is tuned to balance subdiffraction focusing and wide-field imaging, with a theoretical equilibrium point $S_0=0.5$ maximizing combined utility (He et al., 2023). In bioinformatics, the entropy–rank ratio $R_{T,n}$ situates sequence complexity relative to the global combinatorial entropy distribution at fixed block size and $n$-mer granularity. $R_{T,n}$ guides complexity-aware data cropping for robust augmentation in convolutional neural networks, outperforming random and standard entropy benchmarks (Pastore et al., 7 Nov 2025).

4. Applications to Signal Processing, Algorithms, and Control

Audio Compression via Entropy-Controlled Dithering: The entropy of a dithered, quantized audio signal is controlled via a parameter $\alpha$ tuning between different probability density function (PDF) families (rectangular, triangular, modified) for dithering noise. Entropy $H(Y)$ per sample is jointly optimized with perceptual metrics (VISQOL, STOI) over $\alpha$, explicitly balancing fidelity against compressibility. The parameter sweep over $\alpha$ realizes a continuous benchmark of algorithmic trade-offs, and the framework generalizes to lossy codec parameterization for other signals (Murray et al., 4 Jan 2025).

Stochastic Process Control: In completion-time analysis, a benchmark exponential distribution with entropy matching that of the native process defines a constant hazard rate, $\lambda_{\text{bench}} = \exp[1-H(T)]$, which is used to contrast the process hazard rate and infer how sharp restart at time $r$ will increase or decrease system entropy. This predictive entropy benchmark provides an optimal reference for controlling the variance/randomness of stochastic timed processes (Eliazar et al., 2022).

Quantum Control Benchmarks: The Shannon entropy of the work probability distribution, $P(W)$, under two-point measurement protocols serves as a summary of non-equilibrium complexity and control overhead in quantum dynamics. $H[P]=-\sum_W P(W)\ln P(W)$ functions as a protocol complexity score, enabling direct comparison of different counterdiabatic or shortcut-to-adiabaticity strategies, with benchmarks anchored to Kibble–Zurek scaling and resource efficiency (Campbell, 2023).

5. Verification, Analytical Structure, and Benchmark Ensemble Design

Entropy-controlled benchmarks typically involve analytical modeling (as in noise accumulation or spectrally-resolved entropy), numerical scaling analysis, and experimental validation to link entropy with measurable performance outcomes. Key methodological elements are:

• Explicit estimation or control of system entropy, either as a control variable (e.g., $\alpha$ in audio, block entropy in DNA, total entropy in device design) or as a measured outcome (e.g., output entropy in quantum circuits).
• Construction of reference/benchmarking procedures that, in many cases, use known statistical baselines (e.g., random circuits, maximum entropy distributions, worst-case incompatibility graphs) to structure the comparison.
• Use of cumulative distribution or rank-based entropy spectra (e.g., $R_{T,n}$ in DNA) to overcome saturation artifacts and calibrate scores uniformly across datasets.

For complex or ensemble-based systems (e.g., quantum circuits), statistical averaging over circuit instances or disorder realizations is used to robustly extract entropy-controlled features or transitions.

6. Limitations, Interpretability, and Future Directions

Several entropy-controlled benchmarks are sensitive to the entropy measure chosen, the modeling assumptions (e.g., depolarizing noise dominance in quantum devices), and the computational tractability of entropy estimation (scaling exponentially for global quantum-state entropies, or combinatorially for block-wise metrics). While entropy-controlled approaches generalize broadly, domain-specific knowledge is required for appropriate normalization, scaling, and application-specific thresholding.

Emerging directions include integration with machine learning (e.g., generative neural models for monitored quantum circuits to reduce sample complexity (Hu et al., 22 Jan 2025)), multi-objective optimization (e.g., joint loss functions over entropy and perceptual or classification fidelity), and extension to multi-scale entropy control in both physical devices and data-centric applications.

7. Summary Table: Representative Entropy-Controlled Benchmark Prototypes

Domain	Entropy Quantity	Benchmark/Threshold Mechanism
Monitored quantum circuits	Linear cross-entropy $\chi$	Order parameter for measurement-induced transitions (Li et al., 2022)
NISQ quantum circuits	Entropy density $s(\rho)$	Circuit-depth thresholds for quantum advantage (Demarty et al., 23 Dec 2024)
Supercomputer clusters	Graph entropy $E$	Incompatibility-sensitive performance bounds (Adefemi, 12 Sep 2025)
Planar optics	Binary entropy $S$	Balance of focusing and imaging (SR vs S) (He et al., 2023)
DNA sequence analysis	Entropy-rank ratio $R$	Distribution-aware data augmentation (Pastore et al., 7 Nov 2025)
Audio compression	Sample entropy $H(Y)$	Dithering fidelity–entropy optimization (Murray et al., 4 Jan 2025)
Stochastic timing	Shannon entropy $H(T)$	Hazard-matched exponential restart (Eliazar et al., 2022)

Entropy-controlled benchmarks thus provide a technically rigorous, structure-aware, and universally applicable framework for dissecting, normalizing, and optimizing performance under randomness and disorder, grounded in well-defined information-theoretic and statistical constructs.