Figures of Merit and Constraints (FoMaCs)

Updated 28 October 2025

Figures of Merit and Constraints (FoMaCs) are quantitative measures that assess and optimize complex systems through defined performance metrics and operational constraints.
They enable rigorous comparison across disciplines like cosmology, quantum information, and materials science by computing key statistical and physical parameters.
Implementation involves defining target functions, computing metric-based evaluations under constraints, and iteratively optimizing experimental or computational design.

Figures of Merit and Constraints (FoMaCs) encompass the quantitative measures and procedural frameworks used to assess, compare, and optimize complex physical, engineering, and computational systems under both performance objectives and operational constraints. In modern research practice, FoMaCs are central to evaluating scientific instrument designs, data analysis methods, and material properties, and they operate at the interface of statistical rigor, practical engineering, and decision theory.

1. Foundational Definitions and FoMaC Principles

A Figure of Merit (FoM) is a scalar or functionally simple metric designed to quantify the quality, performance, or discriminative power of an experimental setup, data set, algorithm, or material in relation to a specific scientific or engineering goal. A Constraint, in this context, determines the feasible or allowed region in parameter space according to theoretical, physical, or practical limits.

The core properties of a FoMaC are:

Quantitativeness: FoM is computable from data, models, or system parameters.
Comparability: FoMs enable side-by-side comparison between different designs, datasets, or hypotheses.
Relevance: The selected FoM directly maps to the scientific utility, sensitivity, or discrimination task at hand.
Constraint sensitivity: The figure of merit is meaningful only within the acceptable constraint regime (e.g., parameter bounds, instrumental limits).

FoMaCs span domains including cosmology (Sello, 2011, Mortonson et al., 2010, Trotta et al., 2010, Amara et al., 2013), quantum information (Enk, 2017, Zeuthen et al., 2016, Hwang et al., 2012), condensed matter (Cahangirov et al., 2014, Szafrański et al., 2021, Zhang et al., 2019), materials science (Hung et al., 2017), communications engineering (Zhang et al., 2020), and computational science (Hopf et al., 22 Jan 2025).

2. Mathematical Formulation of Figures of Merit

Figures of merit are mathematically constructed as functions of relevant parameter covariances, physical operation metrics, or probabilistic measures. In the two-parameter cosmological context, for instance, the DETF FoM is defined as the inverse area of the confidence region in parameter space: $\mathrm{FoM} = \frac{1}{\sqrt{\det \mathrm{Cov}(f_1, f_2, ...)}}$ where $\{f_i\}$ are parameters (e.g., $(\Omega_m, w)$ ) and $\mathrm{Cov}$ their covariance matrix (Sello, 2011, Mortonson et al., 2010).

In Bayesian inference schemes, FoMs quantify the expected scientific yield or model discriminative strength: $\mathcal{E}(e) = \sum_{i} p(M_i|d) \int d\theta_i\, p(\theta_i|d, M_i) \int dD\, p(D|\theta_i,e, M_i)\, \mathcal{U}(D, e, M_i)$ where $\mathcal{U}$ is a utility function (e.g., decisiveness or evidence strength), $M_i$ a model, and $d$ current data (Trotta et al., 2010).

For device-level assessment, such as photodetectors or quantum transducers, the FoMs are functions of underlying POVM elements or transfer matrices:

Quantum efficiency: $\eta = w_i$ (Enk, 2017)
Added noise for transducers: $N(\Omega) = \langle \hat{\mathcal{F}}^\dagger(\Omega)\hat{\mathcal{F}}(\Omega)\rangle/\eta(\Omega)$ (Zeuthen et al., 2016)

For materials and structures, FoMs often emerge from physically motivated expressions combining key operational constants, e.g.:

Frictional performance: $\mathrm{FoMaC} = k_s / k_c$ , the ratio of intrinsic to critical stiffness (Cahangirov et al., 2014)

3. Application Domains and Specialized FoMaC Constructs

Cosmology and Astrophysics

In cosmological survey design, FoMs quantify the improvement in parameter constraints from observational combinations (CMB, SNe, BAO, galaxy clusters). The 68% and 95% confidence-level FoMs denote the inverse of the area enclosed in $(\Omega_m, w)$ space, providing rigorous metrics for the cumulative improvement attained by incorporating additional probes (e.g., SPT SZ clusters) (Sello, 2011).

Alternative cosmological FoMaCs include "model-breaking" measures based on the Kullback-Leibler divergence between constrained and unconstrained forecast data distributions: $\Phi = \frac{1}{2} \ln|\mathbf{C}_1\mathbf{C}_0^{-1}| + \frac{1}{2}\mathrm{tr}[\mathbf{C}_1^{-1}(\mathbf{C}_0-\mathbf{C}_1)]$ where $\mathbf{C}_1$ and $\mathbf{C}_0$ are model-imposed and empirical covariances, respectively (Amara et al., 2013).

Bayesian decisiveness and expected evidence metrics further generalize FoMacs for survey optimization, embedding full model and parameter uncertainty (Trotta et al., 2010).

Quantum Systems and Information

FoMaCs in quantum information are formalized through figures derived from POVMs, including bandwidth, efficiency, number resolution, timing/spectral entropy, and dark count rates, all as explicit functionals of detector operators on the input Hilbert space (Enk, 2017). In quantum transduction, performance is primarily characterized by transfer efficiency $\eta$ and input-referenced added noise $N$ (Zeuthen et al., 2016).

Quantum state discrimination tasks with symmetric figures of merit reveal a remarkable universality: the optimal measurement and associated conditional probabilities adopt the same analytic form for any monotonous, symmetric FoMaC (e.g., $P(j|i) = \alpha \cos^2\frac{|i-j|\pi}{2M} + \beta$ ), fixed by the no-signaling principle (Hwang et al., 2012).

Materials, Sensors, and Devices

For thermoelectric materials, the optimum figure of merit $ZT_{\mathrm{opt}}$ is theoretically unified across dimensionalities by a universal curve: $ZT_{\mathrm{opt}} = \frac{W_0^2(\alpha)}{2} + W_0(\alpha)$ with $\alpha = (\mathrm{PF}_{\mathrm{opt}}/\kappa_l)T$ and $W_0$ the principal branch of the Lambert W function (Hung et al., 2017). This enables direct, dimension-independent optimization and cross-material comparison.

In frictional nanostructure assessment, the ratio $k_s/k_c$ discriminates between stick-slip (energy dissipative) and superlubric (adiabatic) regimes (Cahangirov et al., 2014). For piezoelectric sensors and hydrophones, hydrostatic figures of merit such as $d_h g_h$ or $d_h g_h/\tan\delta$ integrate fundamental electromechanical parameters and dielectric loss (Zhang et al., 2019).

Engineering Systems and Simulation Optimization

In the evaluation of building layouts for wireless performance, the interference gain ( $g_I$ ) and power gain ( $g_P$ ) are defined as explicit closed-form functions of physical geometry and indoor propagation, offering fast, reproducible measures for architects to optimize designs for wireless-friendliness (Zhang et al., 2020).

In radio interferometry arrays, the spatial dynamic range (SDR) and Fourier domain "gap" parameter ( $\Delta u / u$ ) function as prime FoMaCs linking array configuration to imaging fidelity, with the latter tightly connected to instrumental limitations on PSF shape and dynamic range (Lal et al., 2010).

4. Methodological Workflows for FoMaC Evaluation

Effective use of FoMaCs involves:

Defining the target function (parameter, model, property) relevant for the scientific or engineering objective.
Constructing (analytically or through simulation) the covariance, error, or performance matrix underpinning the figure of merit.
Computing the FoM with and without inclusion of new probes, data sets, design elements, or theoretical priors, enabling direct quantification of improvement or trade-off.
Applying constraints (physical, operational, or theoretical) to restrict the feasible parameter space and ensure meaningfulness of the FoM.
Using FoMaCs for ranking, prioritization, and resource allocation, such as justifying new survey components, instrument upgrades, or material processing routes.

Systematic comparison using FoMaCs is essential for robust, reproducible, and transparent decision-making in experimental and computational design.

5. Advantages, Limitations, and Trade-off Analysis

The primary advantage of FoMaCs is their capacity to distill complex, multidimensional performance into a manageable set of scalar measures directly interpretable for design, benchmarking, and optimization. Their use enables standardized evaluation across experiments, data sets, instruments, or computational pipelines.

However, FoMaCs are limited by their dependence on the metric chosen, the adequacy of the model or likelihood function, and the coverage of constraint parameter space. In high-dimensional problems, improvements in aggregate FoM may not directly translate to improved discrimination of physically relevant models, especially if parameterized directions are prior-dominated (Mortonson et al., 2010). Additionally, practical trade-offs—such as between spectral stability and fabrication tolerance in integrated photonics (Kaulfuss et al., 2022), or between efficiency and added noise in quantum transduction (Zeuthen et al., 2016)—require multi-objective FoMaC analysis.

Parametrization dependence is a significant caveat: constraints and FoMs for the same physical quantity may differ by up to an order of magnitude depending on the functional or binning scheme adopted (Dossett et al., 2011).

6. Practical Impact and Survey/Experiment Optimization

FoMaCs are integral to experiment and survey planning, justifying resource allocation toward the configurations, data sets, or probes offering maximal scientific return under feasible constraints. For instance, the introduction of SPT SZ cluster data resulted in a 56% improvement in the dark energy constraint FoM at 68% confidence, compared to only an 18% improvement from a major supernova compilation (Sello, 2011).

In quantum information, the recognition that classical and quantum FoMs can, in principle, be simultaneously optimized (as they originate from independent physical properties) clarifies device design directions and sets fundamental benchmarks for photodetector and transducer development (Enk, 2017, Zeuthen et al., 2016).

Optimization under Bayesian FoMs that include the full model and parameter uncertainty leads to different experiment ranking compared to Fisher matrix-based approaches, impacting strategic cosmology planning (Trotta et al., 2010). In building design, closed-form wireless performance FoMs provide actionable, low-complexity metrics that integrate early into architectural workflows (Zhang et al., 2020).

7. Future Directions and Generalization

The proliferation of complex experimental and computational methods continues to demand FoMaCs that are statistically principled, physically motivated, and practically computable. Bayesian approaches and machine learning-based FoMs now provide even greater robustness, as in quantum circuit compilation where supervised models achieve a 49% improvement in predictive correlation for execution quality over previous metrics (Hopf et al., 22 Jan 2025).

Extension of FoMaC philosophy into interdisciplinary spaces—such as smart infrastructure, nanoscale characterization, and quantum device engineering—will require constant refinement in the definition and evaluation of both figures of merit and constraint schemes, always tethered to physically meaningful benchmarks and maximum reproducibility.

Summary Table: Representative Figures of Merit in Recent Research

Domain	FoMaC Expression/Type	Application Context
Cosmology	$(\det\,\mathrm{Cov})^{-1/2}$	Dark energy, modified gravity, model-breaking sensitivity
Quantum devices	$\eta$ , $N$ , entropy measures	Photodetectors, quantum transducers, state discrimination
Materials science	$ZT_\mathrm{opt}$ , $k_s/k_c$	Thermoelectrics, frictional superlubricity, piezoelectrics
Engineering	$g_I$ , $g_P$ , $\Delta u/u$	Wireless architecture, array configuration, imaging arrays

FoMaCs unify the quantitative assessment of performance and constraints across scientific disciplines, shaping the design, deployment, and interpretation of experimental and computational advances. Their consistent application fosters objectivity, cross-comparability, and informed optimization in scientific and engineering practice.