Transpilation-Induced Structural Deformation Metrics

• Transpilation-induced structural deformation metrics are quantitative measures that capture circuit restructuring, including node degree shifts, critical path expansions, and connectivity inflation.
• They analyze changes between pre- and post-transpiled DAGs using metrics Δ_deg, Δ_path, and Δ_conn to predict fidelity loss due to hardware mapping and SWAP insertions.
• Empirical studies show that higher structural deformation correlates with increased spectral complexity and measurement cost, guiding adaptive fidelity estimation in quantum circuits.

Transpilation-induced structural deformation metrics are a family of quantitative measures designed to capture how quantum circuit transpilation—specifically qubit mapping, routing, and SWAP insertion—reshapes the dependency structure of a quantum program. These metrics operate on the directed acyclic graph (DAG) representations of quantum circuits before and after transpilation, providing a principled summary of the topological and connectivity changes introduced to make logical circuits compatible with hardware constraints. They play a foundational role in recent fidelity estimation frameworks such as QuFid, where they act as both explanatory and predictive indicators for increased noise sensitivity and measurement overhead following circuit modification (Li et al., 21 Jan 2026).

1. Formal Definition and Metric Family

Given a logical circuit DAG $\mathcal{G}_0 = (V_0, E_0)$ and its transpiled (mapped-and-routed) version $\mathcal{G}_t = (V_t, E_t)$, the structural deformation is defined as the triplet:

$\Delta_\text{struct}(\mathcal{G}_0, \mathcal{G}_t) = \{\Delta_{\deg}, \Delta_{\path}, \Delta_{\conn}\}$

Each component measures a distinct aspect of DAG restructuring:

• $\Delta_{\deg}$ quantifies node degree (fan-in/out) changes, capturing local connectivity shifts.
• $\Delta_{\path}$ measures changes in the lengths of critical gate dependency chains, reflecting "long-range" structural expansions or contractions.
• $\Delta_{\conn}$ expresses the global change in edge (dependency) density, indicating overall connectivity inflation or deflation.

The metrics are formally specified as:

• Degree shift:

$$\Delta_{\deg} = \frac{1}{|V_0|} \sum_{v \in V_0} \left| \deg_{\mathcal{G}_t}(v) - \deg_{\mathcal{G}_0}(v) \right|$$

• Path expansion:

$\Delta_{\path} = \frac{1}{|\mathcal{P}_0|} \sum_{p \in \mathcal{P}_0} \left| \mathrm{length}_{\mathcal{G}_t}(p) - \mathrm{length}_{\mathcal{G}_0}(p) \right|$

where $\mathcal{P}_0$ is the collection of critical (e.g., longest topological) paths in $\mathcal{G}_0$, and length is measured as the number of edges.

• Connectivity inflation:

$\Delta_{\conn} = \frac{|E_t| - |E_0|}{|E_0|}$

Together, these metrics provide a multidimensional measure of transpilation-induced distortion to circuit structure.

2. Interpretations in Graph-Theoretic Terms

Each metric in $\{\Delta_{\deg}, \Delta_{\path}, \Delta_{\conn}\}$ encodes specific structural differences between the pre- and post-transpilation DAGs:

• $\Delta_{\deg}$ compares degree vectors over $V_0 \cap V_t$, with extra nodes (from inserted SWAPs) effectively zero-padded; it reflects local insertion/removal of dependencies and fan-out redistribution.
• $\Delta_{\path}$ assesses whether critical execution paths have been stretched by the introduction of additional gates or rerouted via SWAPs; it operates by tracking the same node sequences in both graphs and measuring the net change in length.
• $\Delta_{\conn}$ gives a normalized difference in edge density, acting as a global proxy for the introduction of new dependencies—often correlating with the number of SWAPs inserted due to hardware topology constraints.

These computations rely strictly on graph distances and connectivity statistics and do not require knowledge of underlying noise models or device parameters. This hardware-aware yet noise-model-agnostic property is central to their utility in adaptive fidelity estimation (Li et al., 21 Jan 2026).

3. Role in Noise Propagation Modeling

The structural deformation metrics are integrated directly into the noise-propagation model constructed over the transpiled circuit's DAG. Specifically, a weighted adjacency matrix $A \in \mathbb{R}^{n \times n}$ is built:

$A_{ij} = w_{ij} \propto \phi\left( \Delta_{\deg}, \Delta_{\path}, \Delta_{\conn} \right) \cdot \mathbb{I}[(v_i, v_j) \in E_t]$

where $\phi(\cdot)$ is a deformation-aware weighting function that accentuates edges in high-deformation regions and attenuates those in less-perturbed areas. The noise-propagation operator is then formed as the row-stochastic matrix:

$$P = D^{-1} A \qquad D = \mathrm{diag}(d_1, \dots, d_n),\; d_i = \sum_j A_{ij}$$

This construction ensures that the iterative action of $P^k$ preferentially diffuses noise along topologically stretched or structurally inflated portions of the circuit, making measurement planning sensitive to the specific transpilation-induced vulnerabilities present in the circuit instance.

4. Quantification of Circuit Complexity and Fidelity Cost

A spectral analysis of the noise-propagation operator $P$ captures the effective complexity and long-range coupling within the transpiled circuit. The leading eigenvalues $\lambda_1, \lambda_2, \dots$ are computed, and the spectral complexity is defined as:

$C(\mathcal{G}) = \sum_{i=1}^k |\lambda_i|$

for a chosen $k \ll n$ (empirically, e.g., $k=5$). Large values of $C(\mathcal{G})$ are associated with slow mixing or persistent noise propagation across elongated dependency chains, thus directly informing the adaptive allocation of measurement resources. The metric enters batch-size formulas of the form:

$$\mathbb{P} = C(\mathcal{G}) \cdot \log(\text{Depth})$$

linking structural deformation to the number of experimental shots required to achieve targeted fidelity bias (Li et al., 21 Jan 2026).

5. Empirical Case Studies and Observed Impact

Empirical investigations on IBM Quantum hardware reveal metric-dependent variations in measurement cost:

• Bernstein–Vazirani (BV) circuits, with minimal transpilation overhead ($\Delta_{\deg}$, $\Delta_{\path}$, $\Delta_{\conn}$ all small), exhibit low spectral complexity ($C \approx 1.2$–$1.5$) and require only a few hundred shots.
• QKNN circuits, heavily routed with substantial SWAP insertion, demonstrate $\Delta_{\deg} \sim 0.4$, $\Delta_{\path} \sim 2.1$, and $\Delta_{\conn} \sim 0.35$, driving $C$ to $4$–$5$ and necessitating thousands of measurement shots for equivalent fidelity bias.
• For QAOA, increasing the qubit count from $4$ to $10$ leads to path elongation (30–50%), mean degree inflation (20%), and edge count doubling, corresponding to a growth of the second eigenvalue ($\lambda_2$) from $0.4$ to $0.75$ and batch sizes rising from $\sim 600$ to $\sim 7300$ shots.

These results confirm that structural deformation metrics are predictive both of spectral circuit complexity and of practical measurement cost escalation when circuits are heavily transpiled (Li et al., 21 Jan 2026).

6. Extraction Workflow and Algorithmic Realization

Metric extraction proceeds according to a reproducible procedure. The following high-level steps are implemented in QuFid:

1. Node set alignment: $V_\text{common} \leftarrow V_0 \cap V_t$
2. Degree shift: For each $v \in V_\text{common}$, compute degree vectors and aggregate their absolute difference to form $\Delta_{\deg}$.
3. Critical path expansion: Identify critical paths $P_0$ in $\mathcal{G}_0$ (e.g., all longest chains). For each $p \in P_0$, determine its length in $\mathcal{G}_0$ and in $\mathcal{G}_t$, then average the absolute length changes to yield $\Delta_{\path}$.
4. Connectivity inflation: Compute edge count ratio to obtain $\Delta_{\conn}$.
5. Reporting: Return the triplet $\{\Delta_{\deg}, \Delta_{\path}, \Delta_{\conn}\}$, which is then fed into the random-walk-based pipeline for fidelity estimation.

This workflow is codified in Algorithm 2 of (Li et al., 21 Jan 2026), serving as the standardized approach for DAG-based structural deformation analysis in adaptive fidelity estimation pipelines.

7. Significance and Research Context

Transpilation-induced structural deformation metrics provide a principled connection between physically motivated circuit transformation processes and the observable measurement burden in noisy intermediate-scale quantum experiments. By abstracting transpiler-induced overheads into graph-theoretic descriptors, these metrics enable platform-independent, automated, and locally adaptive fidelity planning. Their agnosticism to hardware noise models and direct interpretability in terms of dependency graph transformation distinguishes them from traditional, gate-count-based or layer-depth-only metrics.

A plausible implication is that future quantum compilers and scheduling algorithms may explicitly target minimization of these structural deformation metrics to control hardware-induced noise sensitivity and measurement resource demand.