Run-length certificates in quantum learning: sample complexity and noise thresholds

TL;DR

This work reframes quantum state learning under strict copy-by-copy interaction as a stopping-time problem, where learning completes via an online run-length certificate requiring $M_H$ consecutive successes. By decomposing sample complexity into a dimension-dependent search phase and a dimension-independent certification phase, the authors derive universal bounds on stopping times and establish a sharp noise threshold: when the label-flip probability $q$ satisfies $qM_H\gtrsim 1$, halting becomes exponentially expensive. The analysis reveals a fundamental information-theoretic bottleneck induced by the one-bit feedback channel, connects the observed $O(N^{-1})$ accuracy to quantum-limit constraints, and demonstrates, through simulations, the practical implications of classification noise on adaptive quantum learning. The results offer a principled framework for designing minimal-feedback protocols and stopping rules that balance exploration and sequential certification, with implications for quantum-state learning and adaptive information acquisition in noisy, resource-constrained settings.

Abstract

Quantum learning from state samples is often benchmarked in a fixed-budget paradigm, relating error to a prescribed number of copies. We instead adopt a stopping-time viewpoint: in minimal-feedback learning, the learning completion can be defined by an online run-length certificate extracted from a one-bit-per-copy record. As an instantiation, we analyze single-shot measurement learning (SSML), introduced in [Phys. Rev. A 98, 052302 (2018)] and [Phys. Rev. Lett. 126, 170504 (2021)], which tunes a unitary and halts after $M_H$ consecutive successes. Viewing the halting as a sequential certification linking the observed counter to infidelity, we derive sample-complexity bounds that separate search (driving success probability toward unity) from certification (run statistics of consecutive successes). The resulting trade-off among $M_H$, dimension $d$, and one-bit reliability clarifies when performance is control-limited versus certificate-limited. With label-flip noise probability $q$, we find a sharp feasibility threshold: once $qM_H \gtrsim 1$, the expected halting time grows exponentially, making the learning completion impractical even under ideal control. More broadly, this shows that under severely constrained feedback, the certification can dominate sample complexity and small label noise becomes the information bottleneck. Finally, the near-optimal accuracy enabled by run-length certification aligns with the quantum-state-estimation (and equivalently, no-cloning) limits, expressed in the stopping-time terms.

Figures (3)

• Figure 1: The empirical learning probability $P(N)=\Pr[T \le N]$ for $d=2$ and several halting thresholds $M_H$ (solid), together with the fit $1-\exp(-N/N_c)$ (dashed).
• Figure 2: Noiseless SSML accuracy scaling. Each marker corresponds to a halting threshold $M_H\in\{40,80,160\}$, and the dashed line indicates a reference $1/N$ slope. Increasing $d$ shifts the curve mainly through a dimension-dependent prefactor, consistent with the factor $\mathsf{K}(d)$ in our bounds.
• Figure 3: The universal noise-induced blow-up of the run-length certificate. Markers: Monte-Carlo estimates of $q\mathbb{E}[X_{M_H}(1-q)]$ for several noise rates $q$. Dashed curve: theoretical prediction $e^{qM_H}-1$ (valid for small $q$ and accurate as a scaling law more generally). The collapse highlights $qM_H$ as the controlling dimensionless parameter and illustrates the sharp feasibility threshold at $qM_H=O(1)$.

Theorems & Definitions (30)

• Remark 1: Freezing on success and the certification viewpoint
• Remark 2: Scope
• Definition 1: Expected and high-confidence sample complexity
• Lemma 1: Certificate strength
• Remark 3: Scope of the stopping-time criteria
• Theorem 1: Run waiting time: mean and tail
• proof
• Remark 4: Linear versus exponential certification
• Theorem 2: Universal decomposition bound
• proof