The first passage time (FPT) is the time taken for a stochastic process to reach a desired threshold. In this Letter we address the FPT of the stochastic measurement current in the case of continuously measured quantum systems. We find that our approach, based on a charge-resolved master equation related to full-counting statistics of charge detection, enables efficient and analytical computation of the FPT. We develop a versatile framework applicable to quantum jump unraveling and quantum diffusion scenarios, demonstrating that the FPT can be obtained by introducing absorbing boundary conditions. Our framework is demonstrated with two relevant examples: First, we examine the tightness of recently proposed kinetic uncertainty relations for quantum jumps, which place bounds on the signal-to-noise ratio of the FPT. Second, we investigate the usage of qubits as threshold detectors for Rabi pulses, showing how our method can optimize detection probability while minimizing false positives. This Letter offers insights into the applications of the FPT for continuous measurements including the signal-to-noise ratio bounds and false positive minimization strategies, advancing quantum information processing applications.

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• Escape problems
• Noise
• Quantum control
• Quantum measurements
• Random walks
• Stochastic inference
• Uncertainty relation
• Weak values & weak measurements
• Fokker–Planck equation
• Full counting statistics
• Lindblad equation
• Master equation
• Quantum Monte Carlo
• Stochastic analysis
• Stochastic differential equations
• Time series analysis