An infinite hierarchy of multi-copy quantum learning tasks

TL;DR

This work studies how the sample complexity of learning properties of quantum states depends on the allowed measurement primitive. It introduces an infinite hierarchy of learning tasks: for every prime $p$, there exists a degree-$p$ task that is $(p-1)$-copy hard but solvable with $p$ copies using shallow circuits, with sample complexity scaling as $O(n p \\log p \\ poly(\\varepsilon^{-2p}))$; a sharp transition occurs at $p$ copies, analogous to known two-copy vs shadow tomography separations. The authors extend these results to all square-free degrees, demonstrating a general principle behind finite-degree quantum learning tasks and underscoring the value of reliable quantum memory for exponential quantum advantage. They also discuss how to extend efficient multi-copy amplitude estimation to full Weyl–Heisenberg tomography via a mimicking-state construction, at the cost of additional classical and quantum resources. Overall, the paper reveals an intricate, scalable hierarchy of learning problems and motivates practical multi-copy strategies for quantum data analysis.

Abstract

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.

Figures (2)

• Figure 1: Learning challenge with degree $c=3$, i.e. an efficient 3-copy strategy exists, but it's hard for any 1- and 2-copy strategy. The underlying learning task is formulated for $n$-qutrit systems ($d=3$) and asks for estimating all generalized Pauli observables, also known as Weyl-Heisenberg matrices, in modulus. a) a qutrit-based 3-copy learning protocol that allows for jointly predicting all third powers of generalized Pauli observables (think: a qutrit extension of the destructive SWAP test for qubit systems). b) a transpilation of this qutrit-based circuit into qubits: $\vert0\rangle_3 \to \vert00\rangle$, $\vert1\rangle_3 \to \vert01\rangle$, $\vert2\rangle_3 \to \vert10\rangle$ (the same can be also done for the underlying learning challenge). c) Numerical simulations that compare the sample complexity of one particular single-copy strategy (orange) -- local classical shadows for qutrit systems zhu2025quditwilkens2025qudit -- and the advertised 3-copy strategy (blue) required to complete the learning task with (at least) $70\%$ success probability. The overall plot shape resembles Figure 2 in Ref. huang2022quantum which addressed a degree-2 variant of this learning challenge on the Google sycamore chip.
• Figure 2: Visualization of the efficient 6-copy protocol for the combined learning task to obtain estimates of $\mathcal{W}^{(2)}\otimes\mathcal{W}^{(3)}$ as elaborated in \ref{['sec:square-free']}. The boxes indicate the parallel execution of the generalized-Bell-basis measurements on the respective sites, to predict all the corresponding values of the Weyl-operators.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Remark 5
• Theorem 6: Theorem 1 in chen2024optimal
• Lemma 7: Reduction to separation instance
• proof
• Proposition 8
• Theorem 9
• Theorem 10