Efficient estimation of nonlinear properties is a significant yet challenging task in quantum information processing and many-body physics. Current methodologies often suffer from an exponential sample complexity or require auxiliary qubits and deep quantum circuits. To address these limitations, we propose an efficient auxiliary-free replica shadow (AFRS) framework, which leverages the power of joint entangling operations on a few input replicas while integrating the mindset of shadow estimation. We rigorously prove that AFRS can offer exponential improvements in estimation accuracy compared with the conventional shadow method and facilitate the simultaneous estimation of various nonlinear properties, unlike the destructive swap test. Additionally, we introduce an advanced local-AFRS variant tailored to estimating local observables with constant-depth quantum circuits, significantly simplifying the experimental implementation. Our Letter paves the way for efficient and practical estimation of nonlinear properties on near-term quantum devices.
Nonlinear properties of quantum states are essential to quantum information and many-body physics, but assessing them experimentally is challenging, as it typically requires multi-copy operations or a large number of measurement settings. To address this challenge, we develop a universal framework, collision-based nonlinear estimation (CBNE), for efficiently measuring nonlinear quantities of a quantum state
ρ, such as the higher-order expectation value
tr(Oρt) for some observable
O, using single-copy randomized measurements. Strikingly, our protocol requires only a single measurement setting, provided that the system dimension is sufficiently large or a few ancillary qubits are available; this contrasts with the conventional expectation that multiple measurement bases are necessary for nonlinear estimation. In addition, CBNE is observable-independent at the experimental stage, which enables simultaneous estimation of multiple nonlinear functions. It further extends to broader tasks, including the estimation of principal component properties and partial-transpose moments of quantum states. Our results provide a practical and scalable route for measuring nonlinear state properties on near-term quantum devices.
Measuring global quantum properties-such as the fidelity to complex multipartite states-is both an essential and experimentally challenging task. Classical shadow estimation offers favorable sample complexity, but typically relies on many-qubit circuits that are difficult to realize on current platforms. We propose the robust phase shadow scheme, a measurement framework based on random circuits with controlled-
Z as the unique entangling gate type, tailored to architectures such as trapped ions and neutral atoms. Leveraging tensor diagrammatic reasoning, we rigorously analyze the induced circuit ensemble and show that phase shadows match the performance of full Clifford-based ones. Importantly, our approach supports a noise-robust extension via purely classical post-processing, enabling reliable estimation under gate-dependent noise where existing techniques often fail. Additionally, by exploiting structural properties of random stabilizer states, we design an efficient post-processing algorithm that resolves a key computational bottleneck in previous shadow protocols. Our results enhance the practicality of shadow-based techniques, providing a robust and scalable route for estimating global properties in noisy quantum systems.
Extracting information efficiently from quantum systems is crucial for quantum information processing. Classical shadows enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Introducing a shallow random quantum circuit before measurements improves sample efficiency for high-weight Pauli observables and low-rank properties. However, in practice, these circuits can be noisy and bias the measurement results. Here, we propose the robust shallow shadows, which employs Bayesian inference to learn and mitigate noise in postprocessing. We analyze noise effects on sample complexity and the optimal circuit depth. We provide theoretical guarantees for the success of error mitigation under a wide class of noise processes. Experimental validation on a superconducting quantum processor confirms the advantage of our method, even in the presence of realistic noise, over single-qubit measurements for predicting diverse state properties, such as fidelity and entanglement entropy. Our protocol thus offers a scalable, robust, and sample-efficient method for quantum state characterization on near-term quantum devices. Fast and reliable characterisation of quantum systems is a key part of quantum technologies development. Here, the authors propose and demonstrate a way to embed noise characterisation in classical shadow estimation of nonlocal properties, enabling an efficient way to extract information from noisy quantum systems.
Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory. While quantum computers promise speedups for many of these tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates. Here, we propose a sample-efficient and noise-resilient protocol for learning properties of quantum states building on the shadow estimation scheme [Huang et al., Nature Physics 16, 1050-1057 (2020)]. By introducing an experimentally friendly calibration procedure, our protocol can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions. When the strength of noises can be bounded, our protocol approximately retains the same order of sample efficiency as the standard shadow estimation scheme, while also possessing a provable noise resilience. We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments, including estimations of quantum fidelity, correlation functions and energy expectations, etc., which highlight a wide spectrum of potential applications of our protocol on near-term devices.
Inferring nonlinear features of quantum states is fundamentally important across quantum information science, but remains challenging due to the intrinsic linearity of quantum mechanics. It is widely recognized that quantum memory and coherent operations help avoid exponential sample complexity, by mapping nonlinear properties onto linear observables over multiple copies of the target state. In this work, we prove that such a conversion is not only sufficient but also necessary. Specifically, we prove that the estimation of
tr(ρkO) for a broad class of observables
O is exponentially hard for any protocol restricted to
(k−1)-replica joint measurements, whereas access to one additional replica reduces the complexity to a constant. These results establish, for the first time, an exponential separation between
(k−1)- and
k-replica protocols for any integer
k>2, thereby introducing a fine-grained hierarchy of replica-based quantum advantage and resolving an open question in the literature. The technical core is a general indistinguishability principle showing that any ensemble constructed from large Haar random states via tensor products and mixtures is hard to distinguish from its average. Leveraging this principle, we further prove that
k-replica joint measurements are also necessary for distinguishing rank-
k density matrices from rank-
(k−1) ones. Overall, our work delineates sharp boundaries on the power of joint measurements, highlighting resource-complexity trade-offs in quantum learning theory and deepening the understanding of quantum mechanics' intrinsic linearity.
Accurate estimation of observables in quantum systems is a central challenge in quantum information science, yet practical implementations are fundamentally constrained by the limited number of measurement shots. In this work we explore a variation of the classical shadows protocol in which the measurements are kept local while allowing the resulting classical shadows themselves to be correlated. By constructing locally optimal shadows, we obtain unbiased estimators that are competitive with state-of-the-art methods in terms of measurement overhead, while requiring only single-qubit measurements and allowing for the estimation of any observable in pure post-processing, reducing estimation errors by orders of magnitude over standard classical shadows. We validate our approach through numerical experiments on molecular Hamiltonians with up to 40 qubits consistently observing significant reductions in estimation errors, including for estimations of multiple chemically relevant observables simultaneously.