When T-Depth Misleads: Predicting Fault-Tolerant Quantum Execution Slowdown under Magic-State Delivery Constraints

TL;DR

This paper presents a new model for predicting execution slowdown in fault-tolerant quantum computing caused by magic-state delivery constraints, emphasizing demand-supply imbalance over traditional T-depth metrics.

Contribution

It introduces the Delta_max and slack ratio metrics to better predict execution slowdown and provides empirical evidence of their effectiveness over T-depth.

Findings

Delta_max strongly predicts execution slowdown.

Slack ratio outperforms T-depth in predicting stall risk.

Lower bounds based on Delta_max have zero violations in tested instances.

Abstract

The efficient execution of fault-tolerant quantum algorithms is fundamentally limited by the production rate of magic states required for non-Clifford operations. While circuit optimization typically targets T-depth, static T-depth does not reliably predict executable performance under bounded T-state delivery. We introduce a model that captures demand-supply imbalance using two key quantities: slack ratio, a structural indicator of scheduling flexibility, and Delta_max, a measure of cumulative demand surplus. We show that Delta_max is a strong schedule-level indicator of execution slowdown and yields a provable lower bound on executable makespan for a fixed schedule. Empirical evaluation on constructed directed acyclic graph (DAG) families, with arithmetic circuits and exact quantum Fourier transform (QFT) traces providing additional grounding, shows that slack ratio is a stronger structural predictor than T-depth for stall and inversion risk, while Delta_max is the strongest predictor of slowdown. Across 4,904 instances, the lower bound shows zero violations, with 88.9% of cases within one cycle. These results highlight the importance of explicitly modeling delivery constraints in fault-tolerant quantum compilation.