Bell sampling is a simple yet powerful tool based on measuring two copies of a quantum state in the Bell basis, and has found applications in a plethora of problems related to stabiliser states and measures of magic. However, it was not known how to generalise the procedure from qubits to $d$-level systems -- qudits -- for all dimensions $d > 2$ in a useful way. Indeed, a prior work of the authors (arXiv'24) showed that the natural extension of Bell sampling to arbitrary dimensions fails to provide meaningful information about the quantum states being measured. In this paper, we overcome the difficulties encountered in previous works and develop a useful generalisation of Bell sampling to qudits of all $d\geq 2$. At the heart of our primitive is a new unitary, based on Lagrange's four-square theorem, that maps four copies of any stabiliser state $|\mathcal{S}\rangle$ to four copies of its complex conjugate $|\mathcal{S}^\ast\rangle$ (up to some Pauli operator), which may be of independent interest. We then demonstrate the utility of our new Bell sampling technique by lifting several known results from qubits to qudits for any $d\geq 2$: 1. Learning stabiliser states in $O(n^3)$ time with $O(n)$ samples; 2. Solving the Hidden Stabiliser Group Problem in $\tilde{O}(n^3/\varepsilon)$ time with $\tilde{O}(n/\varepsilon)$ samples; 3. Testing whether $|ψ\rangle$ has stabiliser size at least $d^t$ or is $\varepsilon$-far from all such states in $\tilde{O}(n^3/\varepsilon)$ time with $\tilde{O}(n/\varepsilon)$ samples; 4. Clifford circuits with at most $n/2$ single-qudit non-Clifford gates cannot prepare pseudorandom states; 5. Testing whether $|ψ\rangle$ has stabiliser fidelity at least $1-\varepsilon_1$ or at most $1-\varepsilon_2$ with $O(d^2/\varepsilon_2)$ samples if $\varepsilon_1 = 0$ or $O(d^2/\varepsilon_2^2)$ samples if $\varepsilon_1 = O(d^{-2})$.

翻译：贝尔采样是一种基于贝尔基下测量量子态两个副本的简单而强大的工具，已在与稳定子态和魔法度量相关的众多问题中得到应用。然而，此前尚不清楚如何将这一过程从量子比特推广到所有维度d>2的d能级系统——量子比特——并保持其有效性。事实上，作者先前的工作（arXiv'24）表明，贝尔采样向任意维度的自然扩展无法提供关于被测量子态的有意义信息。本文中，我们克服了先前工作中的困难，发展出一种适用于所有d≥2量子比特的有用贝尔采样推广方法。我们这一原语的核心是一个基于拉格朗日四平方定理的新酉算子，它将任意稳定子态|𝒮⟩的四个副本映射为其复共轭态|𝒮*⟩的四个副本（相差某个泡利算子），这一结果可能具有独立的研究价值。随后，我们通过将多个已知结果从量子比特推广到任意d≥2的量子比特，展示了新贝尔采样技术的实用性：1. 以O(n)样本量在O(n³)时间内学习稳定子态；2. 以Õ(n/ε)样本量在Õ(n³/ε)时间内求解隐藏稳定子群问题；3. 以Õ(n/ε)样本量在Õ(n³/ε)时间内测试|ψ⟩是否具有至少dᵗ的稳定子规模，或与所有此类态相距ε以上；4. 证明最多包含n/2个单量子比特非克利福德门的克利福德电路无法制备伪随机态；5. 当ε₁=0时以O(d²/ε₂)样本量，或当ε₁=O(d⁻²)时以O(d²/ε₂²)样本量，测试|ψ⟩的稳定子保真度是否至少为1-ε₁或至多为1-ε₂。