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arXiv:2509.24000

Exponential Advantage from One More Replica in Estimating Nonlinear Properties of Quantum States

Qi Ye, Zhenhuan Liu, Dong-Ling Deng

Published Sep 28, 2025• 0 citations
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Abstract

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 $\mathrm{tr}(ρ^{k} O)$ 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.

DOI
10.48550/arxiv.2509.24000

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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)\mathrm{tr}(ρ^{k} O)tr(ρkO) for a broad class of observables OOO is exponentially hard for any protocol restricted to (k−1)(k-1)(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)(k-1)(k−1)- and kkk-replica protocols for any integer k>2k>2k>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 kkk-replica joint measurements are also necessary for distinguishing rank-kkk density matrices from rank-(k−1)(k-1)(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.
Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as tr⁡(ρt)\operatorname{tr}(ρ^t)tr(ρt), but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order t≥3t\ge 3t≥3, existing protocols show that ⌈t/2⌉\lceil t/2\rceil⌈t/2⌉ replicas already suffice for polynomial-sample estimation of tr⁡(ρt)\operatorname{tr}(ρ^t)tr(ρt), yet it has remained open whether one fewer replica must necessarily incur a sample-complexity barrier growing with the dimension. We prove that this is indeed the case in the sample/copy-access model with replica-limited joint measurements: any protocol restricted to ⌈t/2⌉−1\lceil t/2\rceil-1⌈t/2⌉−1 replicas requires dimension-growing sample complexity, while ⌈t/2⌉\lceil t/2\rceil⌈t/2⌉ replicas suffice by prior work. Thus the exact replica threshold for fixed-order pure moments is ⌈t/2⌉\lceil t/2\rceil⌈t/2⌉. Equivalently, for fixed-order pure moments, one additional coherent replica is not merely useful but marks the exact threshold between polynomial-sample estimation and a dimension-growing regime in the replica-limited model. We further show that the same threshold law extends to a broad family of observable-weighted moments tr⁡(Oρt)\operatorname{tr}(Oρ^t)tr(Oρt), including Pauli observables and other observables with bounded operator norm and macroscopic trace norm. Coherent replica number therefore acts as a genuinely discrete resource for nonlinear quantum-state estimation.
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​.
Can measurements chosen adaptively, in response to earlier outcomes, be exponentially more informative than measurements fixed in advance? In many quantum settings, adaptivity helps at most polynomially; known exponential advantages require coherent measurements across copies or restricted measurements. We show an exponential advantage measuring one copy at a time, with no quantum memory kept between state copies, even when the nonadaptive learner may perform arbitrary global measurements on each copy. We construct explicit families of multi-qubit states in which a scalar parameter is encoded in one of exponentially many candidate observables, with an auxiliary marker indicating which one. An adaptive protocol identifies the relevant observable and then measures it using only single-qubit measurements and one round of adaptation. At fixed accuracy, it needs logarithmically many copies as the qubit count grows, whereas every nonadaptive protocol needs exponentially many. The same mechanism yields an exponential gap in local asymptotic minimax mean squared error constants, tight up to the exponential base. The states are separable and classical in label-dependent product bases; no entanglement or other nonclassical correlations are required. The gap comes from measurement incompatibility: the candidate observables cannot be measured jointly, so a fixed measurement must divide its information budget among them, while adaptivity selects the informative one. Accordingly, the separation disappears when the candidate observables commute, and cannot occur with fully observed classical samples. Measurement incompatibility thus emerges as a resource for quantum learning, distinct from entanglement and quantum memory, that classical feedback alone can exploit.
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.
Learning properties of quantum states from measurement data is a fundamental challenge in quantum information. The sample complexity of such tasks depends crucially on the measurement primitive. While shadow tomography achieves sample-efficient learning by allowing entangling measurements across many copies, it requires prohibitively deep circuits. At the other extreme, two-copy measurements already yield exponential advantages over single-copy strategies in tasks such as Pauli tomography. In this work we show that such sharp separations extend far beyond the two-copy regime: for every prime c we construct explicit learning tasks of degree c, which are exponentially hard with (c - 1)-copy measurements but efficiently solvable with c-copy measurements. Our protocols are not only sample-efficient but also realizable with shallow circuits. Extending further, we show that such finite-degree tasks exist for all square-free integers c, pointing toward a general principle underlying their existence. Together, our results reveal an infinite hierarchy of multi-copy learning problems, uncovering new phase transitions in sample complexity and underscoring the role of reliable quantum memory as a key resource for exponential quantum advantage.
While entanglement is known to enable exponential improvements in the sample complexity of quantum learning, it remains unclear which properties of entangled resources are responsible for such improvements. We address this question through the reduction criterion, a condition obeyed by all bound-entangled states. In nnn-qubit Pauli-channel learning, we show that restricting either the input states or the measurement effects to satisfy this criterion rules out an exponential advantage for incoherent adaptive protocols. An exponential lower bound persists for the one-sided coherent adaptive protocols considered here, even when the unrestricted side retains quantum correlations across channel uses. Using conditional min-entropy, we further quantify how the sample-complexity lower bounds weaken as larger violations of the reduction criterion are allowed. Finally, we show that the same obstruction appears in conjugate-state learning: restricted joint measurements cannot reproduce the logarithmic-sample advantage of unrestricted joint measurements on ρ⊗ρ∗ρ\otimesρ^*ρ⊗ρ∗. These results identify violation of the reduction criterion as a necessary condition for an exponential advantage in the learning tasks considered here.