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arXiv:2606.20238

Random Projections for Multi-Copy Quantum Algorithms

Xiaoyu Liu, Jordi Tura, Johannes Knörzer

quant-phSubmitted Jun 18, 2026• Announced Jun 19, 2026• 0 citations
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Abstract

Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces, $\mathrm{tr}(ρ_1 \cdots ρ_K)$, and nonlinear observables such as $\mathrm{tr}(ρ^K)$, 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 $\mathrm{tr}(ρ^K)$ requires approximately $O(2^{(n-q)(K-1)})$ copies of $ρ$, with each qubit projected out increasing the sampling cost by a factor of $2^{K-1}$. 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.

Comments
5 pages + End Matter (2 pages) + Supplemental Materials (20 pages); 4 + 8 figures

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The state frame potential is a standard diagnostic of how closely a quantum state ensemble approximates Haar randomness. In this work, we study the problem of estimating the state frame potential of order ttt to within additive error ε\varepsilonε under three progressively weaker access models: (i) query access to a multi-state-preparation oracle, (ii) general sample access, and (iii) single-copy sample access. In the query model, we establish a near-optimal query complexity of Θ~(t/ε)\widetildeΘ(\sqrt{t}/\varepsilon)Θ(t​/ε), yielding a quadratic improvement in the dependence on ttt over the previous best result of Nakata, Takeuchi, Kliesch, and Darmawan (PRX Quantum 2025). In the general sample model, we establish the optimal sample complexity Θ(t/ε2)Θ(t/\varepsilon^2)Θ(t/ε2). In the single-copy sample model, we present a store-and-estimate approach whose sample complexity depends on the Rényi entropy of the ensemble weights. As an application, we use the single-copy algorithm to assess the randomness of projected state ensembles, where the entropy term becomes the observational Rényi entropy associated with measuring one subsystem.
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