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arXiv:2605.09958

Quantum Nonlinear Properties from a Single Measurement Setting

Zihao Li, Datong Chen, Dayue Qin, Yuxiang Yang, You Zhou

quant-phSubmitted May 11, 2026• Announced May 12, 2026• 0 citations• Revised Sep 30, 2026 (v2)
v1 · May 11, 2026v2 · Sep 30, 2026
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Abstract

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 ${\rm tr}(Oρ^t)$ 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.

Comments
10+45 pages and 9 figures. A new sample complexity lower bound for state moment estimation is added

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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 ρρρ, such as the higher-order expectation value tr(Oρt){\rm tr}(Oρ^t)tr(Oρt) for some observable OOO, 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.
Quantum learning encounters fundamental challenges when estimating nonlinear properties, owing to the inherent linearity of quantum mechanics. Although recent advances in single-copy randomized measurement protocols have achieved optimal sample complexity for specific tasks like state purity estimation, generalizing these protocols to estimate broader classes of nonlinear properties without sacrificing optimality remains an open problem. In this work, we introduce the observable-driven randomized measurement (ORM) protocol enabling the estimation of Tr ( O ρ 2 ) for an arbitrary observable O —an essential quantity in quantum computing and many-body physics. We establish an upper bound for ORM’s sample complexity and show its optimality for observables with a large trace-norm, including Pauli and local observables, closing a gap in the literature. For these observables, ORM admits an efficient implementation with Clifford circuits. Numerical experiments validate that ORM requires substantially fewer state samples to achieve the same precision compared to classical shadows. Additionally, we introduce a braiding randomized measurement protocol for multiple low-rank nonlinear observables, reducing circuit complexities in practical applications.
Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces, tr(ρ1⋯ρK)\mathrm{tr}(ρ_1 \cdots ρ_K)tr(ρ1​⋯ρK​), and nonlinear observables such as tr(ρK)\mathrm{tr}(ρ^K)tr(ρK), for integer KKK, can be accessed through collective measurements on multiple state copies, but standard protocols based on swap tests require coherent operations on the full Hilbert space and become experimentally unfeasible for large systems. In this work, we introduce a framework for multi-copy measurements based on random projections onto lower-dimensional subspaces prior to the collective measurement, which is then performed only on the reduced Hilbert space. This procedure yields a tunable tradeoff between coherent quantum resources and statistical sampling overhead, allowing the amount of coherent processing to be matched to the capabilities of the underlying hardware. We derive explicit formulas relating the Haar-averaged projected moments to multivariate traces of the original states and analyze the sampling overhead induced by the projection procedure. Specifically, after compressing an nnn-qubit state to a reduced qqq-qubit subspace, estimating tr(ρK)\mathrm{tr}(ρ^K)tr(ρK) requires approximately O(2(n−q)(K−1))O(2^{(n-q)(K-1)})O(2(n−q)(K−1)) copies of ρρρ, with each qubit projected out increasing the sampling cost by a factor of 2K−12^{K-1}2K−1. Our results establish how coherent multi-copy operations can be traded for additional state copies, enabling multi-copy quantum protocols to be optimized for the available hardware resources.
Estimating nonlinear properties such as Rényi entropies and observable-weighted moments serves as a central strategy for spectrum spectroscopy, which is fundamental to property prediction and analysis in quantum information science, statistical mechanics, and many-body physics. However, existing approaches are susceptible to noise and require significant resources, making them challenging for near-term quantum hardware. In this work, we introduce a framework for resource-efficient simultaneous estimation of quantum state moments via qubit reuse. For an mmm-qubit quantum state ρρρ, our method achieves the simultaneous estimation of the full hierarchy of moments Tr(ρ2),…,Tr(ρk)\text{Tr}(ρ^2), \dots, \text{Tr}(ρ^k)Tr(ρ2),…,Tr(ρk), as well as arbitrary polynomial functionals and their observable-weighted counterparts. By leveraging qubit reset operations, our core circuit for simultaneous moment estimation requires only 2m+12m+12m+1 physical qubits and O(k)\mathcal{O}(k)O(k) CSWAP gates, achieving a near-optimal sample complexity of O(klog⁡k/ε2)\mathcal{O}(k \log k / \varepsilon^2)O(klogk/ε2). We demonstrate this protocol's utility by showing that the estimated moments yield tight bounds on a state's maximum eigenvalue and present applications in quantum virtual cooling to access low-energy states of the Heisenberg model. Furthermore, we show the protocol's viability on near-term quantum hardware by experimentally measuring higher-order Rényi entropy on a superconducting quantum processor. Our method provides a scalable and resource-efficient route to quantum system characterization and spectroscopy on near-term quantum hardware.
Optimal single-copy estimation of quantum state moments: why Tr⁡(ρ3)\operatorname{Tr}(ρ^3)Tr(ρ3) and Tr⁡(ρ4)\operatorname{Tr}(ρ^4)Tr(ρ4) are equally hard
Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments Tr⁡(ρt)\operatorname{Tr}(ρ^t)Tr(ρt), allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity Θ(d)Θ(\sqrt d)Θ(d​) at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer t≥2t\ge 2t≥2: for any sufficiently small constant additive error, the optimal sample complexity is Θ ⁣(d⌈log⁡2t⌉1+⌈log⁡2t⌉) Θ\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right) Θ(d1+⌈log2​t⌉⌈log2​t⌉​) with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer h≥1h\geq1h≥1, all moment orders 2h−1<t≤2h2^{h-1}<t\leq2^h2h−1<t≤2h share the same exponent. In particular, the third and fourth moments both require Θ(d2/3)Θ(d^{2/3})Θ(d2/3) copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an Ω(d1−1/t)Ω(d^{1-1/t})Ω(d1−1/t) lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every t≥4t\geq4t≥4. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond d\sqrt dd​.
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.
Given a quantum system on many qubits split into a few different parties, how many total correlations are there between these parties? Such a quantity, aimed to measure the deviation of the global quantum state from an uncorrelated state with the same local statistics, plays an important role in understanding multipartite correlations within complex networks of quantum states. Yet, the experimental access of this quantity remains challenging as it tends to be nonlinear, and hence often requires tomography which becomes quickly intractable as dimensions of relevant quantum systems scale. Here, we introduce a much more experimentally accessible quantifier of total correlations, which can be estimated using only single-qubit measurements. It requires far fewer measurements than state tomography, and obviates the need to coherently interfere multiple copies of a given state. Thus we provide a tool for proving multipartite correlations that can be applied to near-term quantum devices.