Estimating the trace of quantum state powers, Tr ( ρ k ) , for k identical quantum states is a fundamental task with numerous applications in quantum information processing, including nonlinear function estimation of quantum states and entanglement detection. On near-term quantum devices, reducing the required quantum circuit depth, the number of multi-qubit quantum operations, and the copies of the quantum state needed for such computations is crucial. In this work, inspired by the Newton-Girard method, we significantly improve upon existing results by introducing an algorithm that requires only O ( r ~ ) qubits and O ( r ~ ) multi-qubit gates, where r ~ = min { rank ( ρ ) , ⌈ ln ( 2 k / ϵ ) ⌉ } . This approach is efficient, as it employs the r ~ -entangled copy measurement instead of the conventional k -entangled copy measurement, while asymptotically preserving the known sample complexity upper bound. Furthermore, we prove that estimating { Tr ( ρ i ) } i = 1 r ~ is sufficient to approximate Tr ( ρ k ) even for large integers k > r ~ . This leads to a rank-dependent complexity for solving the problem, providing an efficient algorithm for low-rank quantum states while also improving existing methods when the rank is unknown or when the state is not low-rank. Building upon these advantages, we extend our algorithm to the estimation of Tr ( M ρ k ) for arbitrary observables and Tr ( ρ k σ l ) for multiple quantum states.
Estimating the trace of quantum state powers, Tr ( ρ k ) , for k identical quantum states is a fundamental task with numerous applications in quantum information processing, including nonlinear function estimation of quantum states and entanglement detection. On near-term quantum devices, reducing the required quantum circuit depth, the number of multi-qubit quantum operations, and the copies of the quantum state needed for such computations is crucial. In this work, inspired by the Newton-Girard method, we significantly improve upon existing results by introducing an algorithm that requires only O ( r ~ ) qubits and O ( r ~ ) multi-qubit gates, where r ~ = min { rank ( ρ ) , ⌈ ln ( 2 k / ϵ ) ⌉ } . This approach is efficient, as it employs the r ~ -entangled copy measurement instead of the conventional k -entangled copy measurement, while asymptotically preserving the known sample complexity upper bound. Furthermore, we prove that estimating { Tr ( ρ i ) } i = 1 r ~ is sufficient to approximate Tr ( ρ k ) even for large integers k > r ~ . This leads to a rank-dependent complexity for solving the problem, providing an efficient algorithm for low-rank quantum states while also improving existing methods when the rank is unknown or when the state is not low-rank. Building upon these advantages, we extend our algorithm to the estimation of Tr ( M ρ k ) for arbitrary observables and Tr ( ρ k σ l ) for multiple quantum states.
We investigate the computational complexity of estimating the trace of quantum state powers tr(ρq ) for an n-qubit mixed quantum state ρ, given its state-preparation circuit of size poly(n ). This quantity is closely related to and often interchangeable with the Tsallis entropywhere q = 1 corresponds to the von Neumann entropy. For any non-integer q ≥ 1 + Ω(1), we provide a quantum estimator for Sq (ρ ) with time complexity poly(n), exponentially improving the prior best results of exp(n ) due to Acharya, Issa, Shende, and Wagner (ISIT 2019), Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (ESA 2024). Our speedup is achieved by introducing efficiently computable uniform approximations of positive power functions into quantum singular value transformation.
Optimal single-copy estimation of quantum state moments: why Tr(ρ3) and 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), allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity 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≥2: for any sufficiently small constant additive error, the optimal sample complexity is Θ(d1+⌈log2t⌉⌈log2t⌉) with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer h≥1, all moment orders 2h−1<t≤2h share the same exponent. In particular, the third and fourth moments both require 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) lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every t≥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.
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 m-qubit quantum state ρ, our method achieves the simultaneous estimation of the full hierarchy of moments 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+1 physical qubits and O(k) CSWAP gates, achieving a near-optimal sample complexity of 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.
Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces, tr(ρ1⋯ρK), and nonlinear observables such as tr(ρK), for integer K, 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 n-qubit state to a reduced q-qubit subspace, estimating tr(ρK) requires approximately O(2(n−q)(K−1)) copies of ρ, with each qubit projected out increasing the sampling cost by a factor of 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.
The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of estimating the von Neumann entropy, S(ρ), and Rényi entropy, Sα(ρ) of an unknown mixed quantum state ρ in d dimensions, given access to independent copies of ρ. We provide algorithms with copy complexity O(d2/α) for estimating Sα(ρ) for α2) for estimating S(ρ), and Sα(ρ) for non-integral α > 1. These bounds are at least quadratic in d, which is the order dependence on the number of copies required for estimating the entire state ρ. For integral α > 1, on the other hand, we provide an algorithm for estimating Sα(ρ) with a sub-quadratic copy complexity of O(d2-2/α), and we show the optimality of the algorithms by providing a matching lower bound.
We investigate the computational complexity of estimating the trace of quantum state powers tr(ρq) for annqubit mixed quantum state ρ, given its state-preparation circuit of size poly(n). This quantity is closely related to and often interchangeable with the Tsallis entropy Sq(ρ) = 1—tr(ρq)/q—1, whereq= 1 corresponds to the von Neumann entropy. For any non-integerq≥ 1 + Ω(1), we provide a quantum estimator for Sq(ρ) with time complexity poly(n), exponentially improving the prior best results of exp(n)due to Acharya, Issa, Shende, and Wagner (ISIT 2019), Wang, Guan, Liu, Zhang, and Ying (TIT 2024)| Wang, Zhang, and Li (TIT 2024), and Wang and Zhang (TIT 2025). Our speedup is achieved by introducing efficiently computableuniform approximationsof positive power functions into quantum singular value transformation. Our quantum algorithm reveals a sharp phase transition between the case ofq= 1 and constantq> 1 in the computational complexity of the Quantumq-Tsallis Entropy Difference Problem (TsallisQEDq), particularly deciding whether the difference Sq(ρ0)−Sq(ρ1) is at least 0.001 or at most −0.001: • For any 1 + Ω (1) ≤q≤ 2, TsallisQEDqis BQP-complete, which implies that Purity Estimation is also BQP-complete. • For any 1≤q≤1+1/n— 1, TsallisQED q is QSZK-hard, leading to hardness of approximating the von Neumann entropy because Sq(ρ) ≤ S (ρ), as long as BQPQSZK. The hardness results are derived from reductions based on new inequalities for the quantumq-Jensen–(Shannon–)Tsallis divergence with 1 ≤q≤ 2, which are of independent interest.