Universal Sequential Changepoint Detection of Quantum Observables via Classical Shadows

TL;DR

The paper tackles sequential changepoint detection in quantum systems where changes are defined by constraints on a finite set of observables, under a universal measurement framework. It introduces eSCD, which combines a universal random measurement policy via classical shadows with e-detectors to achieve nonparametric, ARL-controlled detection, and analyzes its theoretical guarantees and practical performance. Key contributions include finite-sample bounds on detection delay, sublinear strongly adaptive regret betting schemes (CBCE) for parameter adaptation, and extensive experiments comparing universal shadows against observable-specific baselines across single and multiple observables and measurement ensembles. The results demonstrate that eSCD achieves competitive performance with observable-specific strategies while maintaining universality, with joint Clifford measurements offering additional gains for larger systems. This framework enables robust, observer-agnostic quantum changepoint detection with practical relevance for variational algorithms, quantum sensing, and device-independent settings.

Abstract

We study sequential quantum changepoint detection in settings where the pre- and post-change regimes are specified through constraints on the expectation values of a finite set of observables. We consider an architecture with separate measurement and detection modules, and assume that the observables relevant to the detector are unknown to the measurement device. For this scenario, we introduce shadow-based sequential changepoint e-detection (eSCD), a novel protocol that combines a universal measurement strategy based on classical shadows with a nonparametric sequential test built on e-detectors. Classical shadows provide universality with respect to the detector's choice of observables, while the e-detector framework enables explicit control of the average run length (ARL) to false alarm. Under an ARL constraint, we establish finite-sample guarantees on the worst-case expected detection delay of eSCD. Numerical experiments validate the theory and demonstrate that eSCD achieves performance competitive with observable-specific measurement strategies, while retaining full measurement universality.

Figures (5)

• Figure 1: A sequence of quantum states $\{\rho^t\}_{t\geq 1}$ is processed by a measurement device that selects a measurement setting $\mathcal{M}^t$ and obtains a measurement output $X^t$ at each time step $t$. The pair $(\mathcal{M}^t,X^t)$ is then processed by a detection algorithm that aims to determine whether the expected value $\langle O_i\rangle_{\rho^t}$ of at least one observable $\{O_i\}^n_{i=1}$ has become positive. The measurement device must be designed to support different detection algorithms tailored to different sets of observables $\{O_i\}^n_{i=1}$.
• Figure 2: Illustration of the betting scheme for observable $O_i$ based on the Coin Betting for Changing Environments (CBCE) algorithm jun2017improved. The time horizon is partitioned into sub-intervals of doubling length, and a universal portfolio algorithm is instantiated within each interval. At each time $t$, the active experts are those whose intervals contain $t$. The CBCE prediction is formed as a convex combination of their outputs, with weights determined by their past losses through the CBCE meta-algorithm jun2017improved.
• Figure 3: Violin plot showing the distribution of the ARL (left) and the detection delay (right) for eSCD and eMCD as a function of the pre-change parameter $\theta_0$ (left) and as a function of the post-change parameter $\theta_1$ for $\nu=200$ and $\theta_0=-0.5$ (right). The ARL requirement is set to $1/\alpha = 1000$ (red, dashed line, left), means are shown as straight segments within each violin plot, and quantities are averaged over 100 runs.
• Figure 4: Violin plots for the distribution of the detection delay for eSCD, eMCD, and adaptive eMCD as a function of the number of observable $n$. Quantities are averaged over 100 runs, and all schemes target the ARL constraint $1/\alpha=1000$.
• Figure 5: Violin plots for distribution of the detection delay as a function of the post-change parameter $\theta_1$ and of the number of qubits $d$ for eSCD with local Clifford ensemble and joint Clifford ensemble. Quantities are averaged over 100 runs, and all schemes target the ARL constraint $1/\alpha=1000$.

Theorems & Definitions (6)

• Proposition 1
• proof
• Theorem 1: ARL Control
• Theorem 2: Efficiency Guarantee
• Corollary 1: Efficiency Guarantee under CBCE betting
• proof : Proof of Proposition \ref{['prop:bound_o_hat_general']}