Topological Fidelity Score: Unified Metric

• TFS is a family of fidelity-based or topology-aware constructions that quantify structural agreement between diverse objects like quantum states, band structures, and reconstructed attractors.
• It is applied in contexts ranging from diagnosing topological quantum phase transitions and edge-state effects to evaluating generative-model precision and structural fidelity in time-series forecasting.
• Its definitions vary by context, employing methods such as fidelity susceptibility peaks, zero-density metrics, manifold-distance formulations, and persistence-diagram aggregation.

Topological Fidelity Score (TFS) is not a single universally standardized quantity in the arXiv literature. Rather, the label denotes a family of fidelity-based or topology-aware constructions used to quantify structural agreement between objects as different as quantum ground states, complexified band structures, estimated data supports, and reconstructed dynamical attractors. In quantum condensed matter, TFS-like quantities are built from ground-state fidelity, fidelity susceptibility, fidelity zeros, or manifold distances to diagnose topological quantum phase transitions, edge-state physics, and disorder-induced Majorana pinning; in machine learning, the same label is used either for Topological Precision in generative-model evaluation or for an aggregate persistence-diagram score in time-series forecasting (Tian et al., 2014, Lin et al., 19 Mar 2026, Fang et al., 2024, Kim et al., 2023, Weerasekara et al., 24 Jun 2026).

1. Terminological scope and recurring constructions

Across the works considered here, TFS always combines two ingredients: a notion of fidelity or overlap, and a topological or structural representation on which that overlap is evaluated. In topological superconductors and insulators, the basic object is the overlap of many-body BdG or band-theory ground states under a control-parameter change. In complexified two-band models, the relevant object is the biorthogonal fidelity in the complex parameter plane. In support-based generative-model evaluation, TFS is explicitly identified with Topological Precision and is computed from overlap between statistically significant KDE superlevel supports. In time-series forecasting, TFS is a geometric mean of four persistence-diagram descriptor ratios derived from delay-embedded phase-space reconstructions (Tian et al., 2014, Lin et al., 19 Mar 2026, Kim et al., 2023, Weerasekara et al., 24 Jun 2026).

Context	Primary object	TFS-type construction
Disordered topological superconductors	BdG ground-state fidelity susceptibility	Peak- or area-based score around the first critical peak
Two-band topological models	Biorthogonal fidelity zeros	Minimal-distance, zero-density, or gap-reality score
Manifold-distance framework	BZ/GBZ-averaged state overlap	Normalized integral of squared overlaps
Generative models	KDE-estimated significant supports	Topological Precision (TopP)
Time-series forecasting	$H_1$ persistence diagrams	Geometric mean of four descriptor scores

A persistent source of confusion is to treat TFS as a single metric with fixed semantics. The available literature does not support that usage. The phrase instead names domain-specific fidelity scores whose mathematical form depends on whether the underlying object is a quantum state manifold, a support estimate in feature space, or a persistence diagram in reconstructed phase space. This suggests that “TFS” functions more as a research-program label than as a unique invariant.

2. Fidelity susceptibility, edge physics, and disorder in topological quantum matter

In the disordered nanowire setting, the underlying system is a spin–orbit-coupled semiconductor nanowire in a Zeeman field, proximitized by an $s$-wave superconductor and analyzed in the BdG formalism. The control parameter is the Zeeman field along $x$, $\lambda = V_x$, and the ground-state fidelity is

$$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$

with fidelity susceptibility

$$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$

For the clean effective wire, the bulk gap closes at

$V_{x,c}=\sqrt{\mu^2+\Delta^2},$

and the dominant $\chi_F$ peak coincides with this topological quantum phase transition (TQPT). In the disordered case, however, $\chi_F(V_x)$ develops multiple peaks: the first peak marks the true TQPT, while subsequent peaks correlate with abrupt relocations of zero-energy LDOS maxima from the wire ends into the interior, indicating disorder-induced pinning of Majorana bound states (Tian et al., 2014).

That distinction motivates several candidate TFS definitions in the nanowire setting. The most direct is the peak-dominance score

$$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$

where $s$0 is the height of the first $s$1 peak and the $s$2 are the heights of later peaks. A second proposal is the critical-area score,

$s$3

which measures how much of the total fidelity response is concentrated at the true TQPT rather than at post-transition pinning events. Both scores are naturally in $s$4, and both treat the first peak—not the tallest peak—as the critical reference. This is an essential methodological point in the presence of strong disorder (Tian et al., 2014).

Finite-size and boundary effects further complicate fidelity-based topological diagnostics. For finite 1D Dirac/Ising-class systems, including SSH-type models, the reduced fidelity susceptibility obeys the universal scaling form

$s$5

where $s$6 is the fermionic gap and $s$7 is system length. With periodic or antiperiodic boundary conditions there are no edge states and $s$8 is dual under $s$9. With symmetric open boundaries, however, two edge states appear for $x$0, the duality is broken, and the susceptibility peaks at $x$1 with $x$2. The paper summarized in the data therefore proposes edge-sensitive scores such as

$x$3

with $x$4, to isolate the fidelity contribution of edge physics (König et al., 2016).

In two-dimensional free-fermion topological insulators and superconductors, fidelity can be extended to mixed thermal states through the Uhlmann fidelity, and the associated finite-temperature susceptibility becomes

$x$5

At $x$6, fidelity drops and $x$7 diverges at gap-closing points, while a quantity

$x$8

signals rapid eigenbasis change. At finite temperature, these signatures are smeared, $x$9 remains regular, and the cited work concludes that the studied models do not exhibit finite-temperature phase transitions (Amin et al., 2018). A plausible implication is that any TFS derived from thermal fidelity in this setting should be interpreted as a crossover detector rather than as a finite-$\lambda = V_x$0 topological order parameter.

3. Complexified parameters, fidelity zeros, and critical encodings

A distinct TFS lineage arises from the complexification of the transition-driving parameter in two-band topological models. For a Bloch Hamiltonian

$\lambda = V_x$1

with $\lambda = V_x$2, the appropriate notion is biorthogonal fidelity, built from left and right eigenstates of $\lambda = V_x$3 and $\lambda = V_x$4. The central criterion is that fidelity zeros occur precisely when, for some momentum $\lambda = V_x$5,

$\lambda = V_x$6

In finite-size systems these zeros lie on discrete vertical lines parallel to the imaginary axis; in the thermodynamic limit they accumulate into extended regions in the complex parameter plane. For the Kitaev and Haldane models, the real-part projection of the zero regions is bounded by the Hermitian topological critical points, while in the QWZ model the transitions at $\lambda = V_x$7 are identified in the same way and the additional critical point at $\lambda = V_x$8 is signaled by zeros crossing the real axis (Lin et al., 19 Mar 2026).

This framework supports several explicit TFS constructions. The first is a minimal-distance metric: $\lambda = V_x$9 For the Kitaev chain in the paper’s convention,

$$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$0

for $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$1, so the score diverges as $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$2 and vanishes for $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$3. For the QWZ model,

$$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$4

which diverges at $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$5, while $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$6 in the thermodynamic limit at special momenta $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$7 and $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$8, encoding the additional critical point at $$F(V_x,\delta V_x)=|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|,$$9 (Lin et al., 19 Mar 2026).

The same paper also proposes a zero-density metric,

$$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$0

and a gap-reality criterion,

$$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$1

with small $$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$2 for regularization. These constructions do not require a direct real-axis fidelity singularity; instead, they exploit how topological criticality is encoded by the arrangement of zero loci in complexified parameter space. This suggests a broader interpretation of TFS as a score of spectral accessibility to non-Hermitian criticality.

4. Manifold-distance formulations and normalized overlap scores

Another quantum-mechanical use of TFS is built on the “manifold distance” framework. For matched points $$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$3 and $$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$4 on two manifolds $$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$5 and $$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$6, the pointwise pure-state distances are

$$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$7

and the aggregate manifold distances are

$$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$8

On this basis, the paper summary defines a normalized Topological Fidelity Score

$$\chi_F(V_x)=2\lim_{\delta V_x\to 0}\frac{1-|\langle \Phi(V_x)\mid \Phi(V_x+\delta V_x)\rangle|^2}{(\delta V_x)^2}.$$9

for single occupied bands, and the gauge-invariant projector form

$V_{x,c}=\sqrt{\mu^2+\Delta^2},$0

for multiband or degenerate occupied subspaces. In non-Hermitian systems with open boundaries, the same construction is defined over the generalized Brillouin zone (GBZ) using same-type overlaps, such as right-right overlaps, to avoid normalization-induced artifacts (Fang et al., 2024).

The chief analytic claim of the manifold-distance paper is that if two manifolds have the same topology, $V_{x,c}=\sqrt{\mu^2+\Delta^2},$1 is smooth and its higher derivatives remain finite, whereas crossing a topological phase boundary produces divergent higher-order derivatives near the critical points. Representative examples include

$V_{x,c}=\sqrt{\mu^2+\Delta^2},$2

in a 1D Hermitian Kitaev-like model, and

$V_{x,c}=\sqrt{\mu^2+\Delta^2},$3

in a 2D Hermitian $V_{x,c}=\sqrt{\mu^2+\Delta^2},$4-wave/Chern-like model. The summary therefore introduces a derivative-based susceptibility

$V_{x,c}=\sqrt{\mu^2+\Delta^2},$5

which integrates the per-$V_{x,c}=\sqrt{\mu^2+\Delta^2},$6 fidelity susceptibility and inherits the same critical divergences (Fang et al., 2024).

Relative to more traditional Berry-phase or Chern-number diagnostics, these TFS/MD constructions are state-overlap-based and remain well-defined at and across gap closings. The data explicitly states that they apply to Hermitian, non-Hermitian, and topologically ordered systems, and that in non-Hermitian OBC problems the GBZ must replace the ordinary BZ. A plausible implication is that manifold-distance TFS is best viewed as a unifying differential diagnostic of phase-boundary crossing rather than as a replacement for topological invariants.

5. TFS as Topological Precision in generative-model evaluation

In the generative-model literature, Topological Fidelity Score is explicitly identified with Topological Precision, denoted TopP, within the TopP&R framework. The setting is not quantum mechanics but support estimation in feature space. Let $V_{x,c}=\sqrt{\mu^2+\Delta^2},$7 and $V_{x,c}=\sqrt{\mu^2+\Delta^2},$8 be the real and generated distributions on $V_{x,c}=\sqrt{\mu^2+\Delta^2},$9, with samples $\chi_F$0 and $\chi_F$1. Supports are estimated by KDE superlevel sets

$\chi_F$2

where the thresholds $\chi_F$3 and $\chi_F$4 come from bootstrap confidence bands and remove topological noise and low-density outliers with confidence $\chi_F$5. Persistent homology is computed on the KDE superlevel filtration

$\chi_F$6

and topological significance is quantified through persistence lifetimes, with features of lifetime $\chi_F$7 jointly significant in the $\chi_F$8 diagram-space confidence ball (Kim et al., 2023).

The TFS itself is then

$\chi_F$9

and the diversity counterpart is

$\chi_F(V_x)$0

Both lie in $\chi_F(V_x)$1. The computational pipeline consists of feature extraction by a pretrained embedder, optional random projection $\chi_F(V_x)$2, bandwidth selection by a balloon estimator or persistence-based criterion, KDE computation, bootstrap confidence-band estimation, support extraction, and then TopP/TopR evaluation (Kim et al., 2023).

The notable theoretical claim is that TopPR is statistically consistent and robust under Assumptions A–D, including adversarial noise. Under the stated conditions,

$\chi_F(V_x)$3

with an analogous bound for TopR. The paper also states that TFS remains stable under outliers and non-IID perturbations up to $\chi_F(V_x)$4 contamination, and that replacing KDE by $\chi_F(V_x)$5-NN support estimation removes the consistency guarantees (Kim et al., 2023). In this domain, therefore, “topological fidelity” refers not to a homotopy invariant but to a robust precision score over statistically significant supports.

6. TFS in topology-aware time-series forecasting

The most explicit recent use of the term appears in TopoCast, where TFS is a topology-driven score for structural fidelity in time-series forecasting. The starting point is Takens delay embedding. For a univariate series $\chi_F(V_x)$6,

$\chi_F(V_x)$7

with $\chi_F(V_x)$8. The paper fixes $\chi_F(V_x)$9 and $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$0 for all experiments, embeds ground-truth and forecast sequences into point clouds $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$1 and $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$2, builds Vietoris–Rips filtrations with Euclidean distance, computes $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$3 persistence diagrams using Ripser, and discards features with lifetime $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$4 where $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$5. TopoCast focuses on $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$6 because periodic signals form closed loops under delay embedding (Weerasekara et al., 24 Jun 2026).

From each $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$7 diagram $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$8, four descriptors are extracted: $$TFS_{\mathrm{peak}}=\frac{H_{\mathrm{major}}}{H_{\mathrm{major}}+\sum_i H_{\mathrm{minor},i}},$$9 with $s$00. For each positive descriptor $s$01, the component score is

$s$02

The Topological Fidelity Score is then the geometric mean

$s$03

or, in weighted form,

$s$04

By construction, $s$05, with equal weights used in the paper (Weerasekara et al., 24 Jun 2026).

TopoCast further introduces Dominant Cycle Overlap (DCO), which maps the most persistent $s$06 generator back to time indices using Ripser cocycle representatives. If $s$07 and $s$08 are the time-index sets touched by the dominant cocycles, then

$s$09

The localized score is

$s$10

This decomposition addresses a limitation of diagram-level TFS: it can register high structural agreement even under pure phase shifts. The paper’s synthetic ECG examples make this precise. A half-beat phase shift yields $s$11, $s$12, $s$13, and $s$14, whereas a smooth forecast yields $s$15, $s$16, $s$17, and $s$18. Thus TFS measures diagram-level structural preservation, while LTFS additionally penalizes temporal mislocalization (Weerasekara et al., 24 Jun 2026).

The empirical significance of this formulation is that it can invert model rankings relative to pointwise error. On Exchange with prediction horizon $s$19, PatchTST attains the best MSE $s$20 but the lowest LTFS $s$21, with severe loop injection $s$22. On ILI with $s$23, Autoformer has the highest LTFS $s$24 despite not having the best MSE. The paper explicitly attributes these discrepancies to structural failure modes such as over-smoothing, phase shifts, and spurious periodicity that are not visible to MSE or MAE (Weerasekara et al., 24 Jun 2026).

7. Comparative interpretation, misconceptions, and methodological limits

The most important comparative fact is that TFS is not an invariant of a single mathematical type. In the quantum-matter papers it is a fidelity-derived detector of TQPTs, edge-state effects, or complex-parameter criticality; in the generative-model paper it is a support-overlap precision score; in TopoCast it is a persistence-diagram aggregate. Any claim that “TFS” has one canonical formula would therefore be inaccurate given the cited literature (Tian et al., 2014, Kim et al., 2023, Weerasekara et al., 24 Jun 2026).

A second recurring misconception is that every fidelity anomaly signals a phase transition. The disordered nanowire study shows explicitly that only the first $s$25 peak tracks the true TQPT, while later peaks are signatures of Majorana pinning and relocation in the zero-energy LDOS. Energy spectra alone can become ambiguous because of disorder-induced in-gap states, so $s$26 must be interpreted jointly with gap closure and real-space LDOS (Tian et al., 2014).

A third limitation concerns the distinction between diagnosis and classification. Several of the quantum formulations are powerful detectors of gap closings or state-manifold rearrangements, but they do not themselves compute a topological invariant. The disordered-nanowire paper does not compute a Pfaffian, $s$27 index, or winding number; the two-dimensional Uhlmann-fidelity study correlates fidelity drops and $s$28 peaks with Chern-number changes but also emphasizes that fidelity-based diagnostics are not invariants by themselves; the manifold-distance paper positions overlap-based distances as alternatives to Berry-connection-based measures, especially at criticality, but not as replacements for all topological classification schemes (Tian et al., 2014, Amin et al., 2018, Fang et al., 2024).

The machine-learning variants introduce their own domain-specific caveats. In TopP&R, the feature extractor, the KDE bandwidth, and the confidence level $s$29 materially affect the estimated supports, and the paper recommends reporting $s$30 explicitly and checking sensitivity across embedders. In TopoCast, the score is focused on $s$31, requires reasonable embedding parameters $s$32, and excludes DCO on windows without a dominant ground-truth cycle; thus LTFS is intentionally specialized to oscillatory structure rather than arbitrary temporal morphology (Kim et al., 2023, Weerasekara et al., 24 Jun 2026).

Taken together, these works establish TFS as a flexible but non-unified family of fidelity-based structural scores. Its shared logic is to quantify how much salient topology or geometry survives under perturbation, comparison, or prediction. Its specific mathematical realization depends entirely on the representation deemed structurally meaningful in the target domain: BdG ground states and LDOS in disordered superconductors, zero loci in complexified topological band models, occupied-state manifolds in BZ or GBZ, statistically significant supports in learned feature spaces, or $s$33 persistence signatures in delay-embedded time series.