Fidelity Edges in Quantum and Graph Systems

• Fidelity edges are defined as loci where state overlap abruptly changes, serving as a diagnostic tool for phase transitions in quantum many-body and topological systems.
• They capture momentum-space zeros, fidelity susceptibility peaks, and complex parameter singularities that precisely reconstruct phase boundaries and mobility edges in various models.
• Applied methods leverage fidelity edges in network routing, quantum gate calibration, and image reconstruction, highlighting their versatile role in improving fidelity and performance.

Searching arXiv for papers relevant to “Fidelity Edges” and related usages across quantum criticality, graph transfer, and applied reconstruction/optimization. Fidelity edges are loci at which a fidelity-based quantity sharply changes, vanishes, or condenses into a boundary-like set, and they are used to diagnose transitions, constraints, and trade-offs in several technical literatures. In quantum many-body and band-topological work, the term most commonly refers to momentum-space zeros of state overlap, complex-parameter accumulations of fidelity zeros, or fidelity-susceptibility structures that reconstruct phase boundaries and mobility edges (Sacramento et al., 2018, Gu et al., 24 Sep 2025, Liu et al., 2024). In graph- and systems-oriented work, closely related usage attaches fidelity to the edges themselves—for example weighted transfer edges, tunable-coupler links, or end-to-end routing frontiers (Lippner et al., 2024, VanAllen et al., 20 May 2026, Gu et al., 2023). A distinct applied-vision usage treats edges as structural constraints that improve reconstruction fidelity, rather than as singular loci in parameter space (Vaid et al., 4 Jul 2026, Zhang et al., 13 Mar 2026).

1. Momentum-resolved fidelity and the canonical quantum meaning

The foundational construction is quantum fidelity between two states,

$$F(\rho_1,\rho_2)=\mathrm{Tr}\sqrt{\sqrt{\rho_1}\rho_2\sqrt{\rho_1}}.$$

For translationally invariant systems one resolves this quantity in momentum space as

$F_k \equiv F(\rho_1(k),\rho_2(k)),$

where $\rho_i(k)\propto e^{-\beta H_i(k)}$. In the zero-temperature two-band case with

$H_i(k)=h_i(k)\cdot \sigma,$

the momentum-resolved ground-state fidelity reduces to

$$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$

Vanishing fidelity requires

$h_1\cdot h_2=-|h_1||h_2|,$

that is, the two Bloch vectors must be antiparallel (Sacramento et al., 2018).

Within this framework, the zeros of $F_k$ are the original “fidelity edges.” They are obtained by comparing two states chosen deep inside different phases and scanning the Brillouin zone. The momenta at which $F_k=0$ coincide with those at which the two-band gap closes for some intermediate parameter value, so the fidelity zeros reconstruct the usual critical line in parameter space. This is the “bulk-edge-bulk correspondence” of the phase diagram: bulk states from two distinct phases, compared directly, encode the momenta associated with the separating edge. The same construction is illustrated in the 1D Kitaev chain, a 2D spin-triplet superconductor, the Haldane Chern insulator, and graphene with a mass term, where the fidelity zeros occur at the critical momenta of gap closing (Sacramento et al., 2018).

A key sufficient condition is linear parameter dependence. If $h_q(k)$ depends linearly on a control parameter $q$, and if for two parameter points $F_k \equiv F(\rho_1(k),\rho_2(k)),$0 one has $F_k \equiv F(\rho_1(k),\rho_2(k)),$1 with $F_k \equiv F(\rho_1(k),\rho_2(k)),$2, then there exists an interpolating $F_k \equiv F(\rho_1(k),\rho_2(k)),$3 on the straight line in parameter space such that $F_k \equiv F(\rho_1(k),\rho_2(k)),$4. In that case $F_k \equiv F(\rho_1(k),\rho_2(k)),$5 implies a gapless point at that momentum. The converse is not necessary: a rotating model $F_k \equiv F(\rho_1(k),\rho_2(k)),$6 gives $F_k \equiv F(\rho_1(k),\rho_2(k)),$7 for any $F_k \equiv F(\rho_1(k),\rho_2(k)),$8, yet $F_k \equiv F(\rho_1(k),\rho_2(k)),$9 and the spectrum never closes (Sacramento et al., 2018).

2. Phase-diagram reconstruction by fidelity drops and susceptibility peaks

A second major usage concerns parameter-space “edges” traced by abrupt fidelity changes. In the extended Harper model one studies the lowest-band-edge state $\rho_i(k)\propto e^{-\beta H_i(k)}$0 and defines

$\rho_i(k)\propto e^{-\beta H_i(k)}$1

When this fidelity is plotted over a fine grid in $\rho_i(k)\propto e^{-\beta H_i(k)}$2, it develops sharp contours where it drops from approximately one to approximately zero. These contours coincide with the three critical lines $\rho_i(k)\propto e^{-\beta H_i(k)}$3, $\rho_i(k)\propto e^{-\beta H_i(k)}$4, and $\rho_i(k)\propto e^{-\beta H_i(k)}$5, together with their bicritical intersection at $\rho_i(k)\propto e^{-\beta H_i(k)}$6. In this setting, a fidelity edge is therefore a parameter-space demarcation extracted from overlap data rather than a momentum-space zero locus (0809.3628).

The associated infinitesimal diagnostic is fidelity susceptibility. Along a path $\rho_i(k)\propto e^{-\beta H_i(k)}$7,

$\rho_i(k)\propto e^{-\beta H_i(k)}$8

In the extended Harper model, $\rho_i(k)\propto e^{-\beta H_i(k)}$9 develops narrow system-size-dependent peaks precisely on the critical lines. The finite-size scaling ansatz

$H_i(k)=h_i(k)\cdot \sigma,$0

organizes both metal-metal and metal-insulator transitions. The paper further shows that local entanglement entropy does not distinguish the metal-metal transition between phase I and phase III, whereas fidelity and $H_i(k)=h_i(k)\cdot \sigma,$1 do, which makes the fidelity-based construction a sharper detector of the full three-region phase diagram (0809.3628).

In generalized Aubry–André models with exact mobility edges, fidelity susceptibility is elevated from a boundary detector to a mobility-edge locator. For the $H_i(k)=h_i(k)\cdot \sigma,$2-th eigenstate,

$H_i(k)=h_i(k)\cdot \sigma,$3

Its peak position $H_i(k)=h_i(k)\cdot \sigma,$4 tracks the exact mobility-edge line

$H_i(k)=h_i(k)\cdot \sigma,$5

The peak height and drift obey

$H_i(k)=h_i(k)\cdot \sigma,$6

and higher-order generalized fidelity susceptibilities extract both $H_i(k)=h_i(k)\cdot \sigma,$7 and $H_i(k)=h_i(k)\cdot \sigma,$8. The same work identifies a Fibonacci-subsequence structure through a Diophantine equation, with one subsequence for $H_i(k)=h_i(k)\cdot \sigma,$9, three for $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$0, four for $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$1, and corresponding scaling functions $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$2. Here fidelity edges are tied directly to exact mobility edges and unconventional quasiperiodic criticality (Liu et al., 2024).

3. Fidelity zeros, Lee–Yang theory, and thermodynamic edge singularities

A more recent formulation recasts fidelity edges in the complex plane of a control parameter. Given a Hamiltonian $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$3 and ground state $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$4, the fidelity amplitude under a small parameter increment $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$5 is

$$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$6

Analytic continuation of $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$7 into the complex plane produces isolated fidelity zeros $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$8 satisfying $$F_k = \bigl|\langle \psi_1^-(k)\mid \psi_2^-(k)\rangle\bigr| = \sqrt{\tfrac12\left[1+\frac{h_1}{|h_1|}\cdot\frac{h_2}{|h_2|}\right]}.$$9. Equivalently,

$h_1\cdot h_2=-|h_1||h_2|,$0

so each zero appears as a logarithmic branch point (Gu et al., 24 Sep 2025).

For ferromagnetic Ising models, these zeros are linked to symmetry-sector splitting. With global $h_1\cdot h_2=-|h_1||h_2|,$1 parity $h_1\cdot h_2=-|h_1||h_2|,$2, one defines

$h_1\cdot h_2=-|h_1||h_2|,$3

At zero temperature, $h_1\cdot h_2=-|h_1||h_2|,$4, and the condition $h_1\cdot h_2=-|h_1||h_2|,$5 coincides with the crossing $h_1\cdot h_2=-|h_1||h_2|,$6 at which the ground state switches parity. In this construction, fidelity zeros play the role of Lee–Yang zeros, and the thermodynamic-limit accumulation sets are called fidelity edges (Gu et al., 24 Sep 2025).

In finite systems the zeros are isolated points; in the thermodynamic limit they condense onto lines or arcs. Fixing $h_1\cdot h_2=-|h_1||h_2|,$7 and parameterizing $h_1\cdot h_2=-|h_1||h_2|,$8, the endpoint $h_1\cdot h_2=-|h_1||h_2|,$9 satisfies

$F_k$0

Below criticality, $F_k$1, the zeros fill the full circle. Above criticality, $F_k$2, they occupy two arcs with endpoint angle

$F_k$3

so that

$F_k$4

Finite-size scaling of the leading zero,

$F_k$5

recovers the critical point and correlation-length exponent. The one-dimensional ferromagnetic Ising chain gives $F_k$6, while the two-dimensional ferromagnetic Ising model on an $F_k$7 torus yields $F_k$8 and $F_k$9 in the reported analysis (Gu et al., 24 Sep 2025).

4. Fidelity attached to graph edges, coupler links, and routing frontiers

Outside phase-transition theory, the term is also attached to edges of a graph or network. In continuous-time quantum walks, weighted self-loops can be placed on a source $F_k=0$0 and target $F_k=0$1,

$F_k=0$2

and the transfer fidelity is

$F_k=0$3

The analysis shows that sufficiently large loop weights force strong transfer fidelity whenever the relevant local spectral symmetry condition $F_k=0$4 holds. The required weight depends on the maximum degree $F_k=0$5, and in some less favorable cases also on the distance $F_k=0$6. The construction gives lower bounds on $F_k=0$7, an upper bound on the readout time $F_k=0$8, and a finite window around $F_k=0$9 during which fidelity remains above a specified threshold (Lippner et al., 2024).

In tunable-coupler superconducting architectures, every physical two-qubit link may itself be treated as a fidelity-aware edge. The framework of frequency allocation and transpilation co-design decomposes gate infidelity into coherent spectator-induced contributions and incoherent lifetime loss,

$h_q(k)$0

with empirical detuning-dependent forms

$h_q(k)$1

Qubit and coupler frequencies are then assigned by minimizing the sum of link infidelities over the module under spacing and hardware constraints. The paper reports a fidelity–connectivity tradeoff as module size scales, and introduces FINESSE, which replaces hop-count distance in SABRE-style routing by a blended metric

$h_q(k)$2

where $h_q(k)$3 is built from log-infidelity edge weights

$h_q(k)$4

On SNAIL architectures, FINESSE achieves an average $h_q(k)$5 reduction in log-infidelity cost and $h_q(k)$6 reduction in circuit depth relative to SABRE (VanAllen et al., 20 May 2026).

A third network-theoretic usage defines a “fidelity edge” as a Pareto frontier between entanglement throughput and worst-case end-to-end fidelity. For a multi-hop quantum repeater network, each pflow $h_q(k)$7 has worst-case fidelity

$h_q(k)$8

Introducing $h_q(k)$9 with $q$0 for every used pflow yields

$q$1

so maximizing worst-case fidelity becomes equivalent to minimizing the longest permitted path “length.” The resulting high-fidelity entanglement-distribution problem is NP-hard, and the FENDI scheme gives a fully polynomial-time approximation scheme that returns $q$2 with $q$3, hence an approximate Pareto frontier

$q$4

Here the fidelity edge is not a zero locus but a performance boundary in the EDR–fidelity plane (Gu et al., 2023).

5. Structural edges as fidelity constraints in reconstruction and compression

A distinct applied usage treats edges as structural carriers of fidelity. In UAV-based 3D reconstruction, SharpSplat augments standard 3D Gaussian Splatting with a semantic edge regularization framework. For each calibrated image $q$5, SAM 3 with prompts such as “building” and “roof” produces a binary building mask $q$6; a bilateral filter and Sobel convolutions produce

$q$7

and semantic edge targets are formed as

$q$8

Rendered grayscale images are processed by the same Sobel kernels to obtain $q$9, and training adds

$F_k \equiv F(\rho_1(k),\rho_2(k)),$00

to RGB and SSIM terms. The reported effect is sharper façades, roof ridges, corners, and window frames without architectural changes to the 3DGS pipeline, with quantitative gains over the 3DGS baseline on PolyTech, Art Sci, and Gehukheda at both $F_k \equiv F(\rho_1(k),\rho_2(k)),$01K and $F_k \equiv F(\rho_1(k),\rho_2(k)),$02K iterations (Vaid et al., 4 Jul 2026).

In extreme learned image compression, the operative notion is control of the fidelity–perception trade-off while maintaining important edges. A two-stage GAN-based scheme first trains a rate–distortion decoder $F_k \equiv F(\rho_1(k),\rho_2(k)),$03, then fine-tunes a perceptual decoder $F_k \equiv F(\rho_1(k),\rho_2(k)),$04, and interpolates their weights by

$F_k \equiv F(\rho_1(k),\rho_2(k)),$05

This enables continuous movement from a blurry but faithful reconstruction to a sharp but noisy one, with $F_k \equiv F(\rho_1(k),\rho_2(k)),$06 reported to yield sharp reconstructions with suppressed noise. The paper states that edges remain crisp where competing GAN methods introduce spurious speckle, at bitrates below $F_k \equiv F(\rho_1(k),\rho_2(k)),$07 bpp (Iwai et al., 2020).

In MRI reconstruction, the Unsupervised Feature Loss maps patches into a learned unit-norm feature space and penalizes feature mismatch: $F_k \equiv F(\rho_1(k),\rho_2(k)),$08 Added to MoDL training, it improves fine textures, finer features, and sharper edges with higher SSIM and much lower UFLoss value while preserving comparable NRMSE. On the reported 3D knee reconstruction at $F_k \equiv F(\rho_1(k),\rho_2(k)),$09, MoDL+UFLoss improves from approximately $F_k \equiv F(\rho_1(k),\rho_2(k)),$10 to approximately $F_k \equiv F(\rho_1(k),\rho_2(k)),$11 SSIM and halves UFLoss from approximately $F_k \equiv F(\rho_1(k),\rho_2(k)),$12 to approximately $F_k \equiv F(\rho_1(k),\rho_2(k)),$13 (Wang et al., 2021).

CognitionCapturerPro makes the usage explicit: “fidelity edges” are contour maps extracted by Canny and encoded as an edge modality. The image is converted to grayscale, Gaussian smoothed, and passed through a Canny edge detector to produce

$F_k \equiv F(\rho_1(k),\rho_2(k)),$14

which is embedded as $F_k \equiv F(\rho_1(k),\rho_2(k)),$15, fused with image, text, and depth features, and aligned through SCM-loss and STH-Align. Edge-only zero-shot retrieval averages $F_k \equiv F(\rho_1(k),\rho_2(k)),$16 Top-1 and $F_k \equiv F(\rho_1(k),\rho_2(k)),$17 Top-5, while full multi-modal fusion reaches $F_k \equiv F(\rho_1(k),\rho_2(k)),$18 Top-1 and $F_k \equiv F(\rho_1(k),\rho_2(k)),$19 Top-5; for reconstruction metrics, edge only yields $F_k \equiv F(\rho_1(k),\rho_2(k)),$20, $F_k \equiv F(\rho_1(k),\rho_2(k)),$21, and high-level CLIP $F_k \equiv F(\rho_1(k),\rho_2(k)),$22, while the full model yields $F_k \equiv F(\rho_1(k),\rho_2(k)),$23, $F_k \equiv F(\rho_1(k),\rho_2(k)),$24, and CLIP $F_k \equiv F(\rho_1(k),\rho_2(k)),$25 (Zhang et al., 13 Mar 2026).

6. Limitations, non-equivalences, and conceptual distinctions

Several recurring caveats govern the interpretation of fidelity edges. First, vanishing fidelity is not universally equivalent to gap closing. The rotating-model counterexample shows that $F_k \equiv F(\rho_1(k),\rho_2(k)),$26 may occur without any spectral closure, so the linear-interpolation argument provides a sufficient condition, not a necessary one. Second, at finite temperature the sharp zeros of momentum-resolved fidelity are replaced by deep minima: the loci remain visible at low temperature, but the zero structure is smeared (Sacramento et al., 2018).

Third, fidelity-based diagnostics can be sharper than neighboring measures, but not all nearby observables have the same resolving power. In the extended Harper model, single-site von Neumann entropy tracks localization and the metal–insulator boundary, yet cannot distinguish the metal–metal transition between the two extended phases, whereas fidelity and fidelity susceptibility do (0809.3628). Fourth, applied edge-fidelity methods inherit strong dependence on the quality of the edge signal itself. In SharpSplat, badly segmented SAM 3 masks degrade edge supervision, and the edge-weight $F_k \equiv F(\rho_1(k),\rho_2(k)),$27 must be tuned per scene, with typical range $F_k \equiv F(\rho_1(k),\rho_2(k)),$28–$F_k \equiv F(\rho_1(k),\rho_2(k)),$29 (Vaid et al., 4 Jul 2026).

Fidelity-aware edge models also expose engineering trade-offs. In tunable-coupler systems, increasing qubit count and coupling density within a module leads to a fidelity–connectivity tradeoff; the reported data indicate that beyond four qubits/four edges per SNAIL, additional links cost more in fidelity than they save in connectivity (VanAllen et al., 20 May 2026). In quantum repeater routing, the worst-case-fidelity versus EDR optimization is NP-hard, which motivates approximation schemes such as FENDI rather than exact global optimization (Gu et al., 2023).

Taken together, these results suggest that “fidelity edges” is not a single universal object but a family of related constructions. In quantum criticality it denotes zero loci, minima, or edge singularities of overlap-based quantities; in graph and hardware settings it denotes edges endowed with fidelity weights or constraints; and in applied reconstruction it denotes structural contours used to preserve geometric or perceptual fidelity. The common thread is operational rather than semantic: fidelity edges identify the locations—whether in $F_k \equiv F(\rho_1(k),\rho_2(k)),$30-space, control space, complex parameter space, graph topology, or image structure—at which the preservation or loss of information becomes most diagnostically significant.