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arXiv:2510.06848

Reconquering Bell sampling on qudits: stabilizer learning and testing, quantum pseudorandomness bounds, and more

Jonathan Allcock, João Fernando Doriguello, Gábor Ivanyos, Miklós Sántha

Published Oct 8, 2025• 0 citations
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Abstract

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 $O(d^2\varepsilon_2/(\varepsilon_2-\varepsilon_1)^2)$ samples if $\varepsilon_1 = O(\varepsilon_2/d^{2})$.

DOI
10.48550/arxiv.2510.06848

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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 ddd-level systems -- qudits -- for all dimensions d>2d > 2d>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≥2d\geq 2d≥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 ∣S⟩|\mathcal{S}\rangle∣S⟩ to four copies of its complex conjugate ∣S∗⟩|\mathcal{S}^\ast\rangle∣S∗⟩ (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≥2d\geq 2d≥2: 1. Learning stabiliser states in O(n3)O(n^3)O(n3) time with O(n)O(n)O(n) samples; 2. Solving the Hidden Stabiliser Group Problem in O~(n3/ε)\tilde{O}(n^3/\varepsilon)O~(n3/ε) time with O~(n/ε)\tilde{O}(n/\varepsilon)O~(n/ε) samples; 3. Testing whether ∣ψ⟩|ψ\rangle∣ψ⟩ has stabiliser size at least dtd^tdt or is ε\varepsilonε-far from all such states in O~(n3/ε)\tilde{O}(n^3/\varepsilon)O~(n3/ε) time with O~(n/ε)\tilde{O}(n/\varepsilon)O~(n/ε) samples; 4. Clifford circuits with at most n/2n/2n/2 single-qudit non-Clifford gates cannot prepare pseudorandom states; 5. Testing whether ∣ψ⟩|ψ\rangle∣ψ⟩ has stabiliser fidelity at least 1−ε11-\varepsilon_11−ε1​ or at most 1−ε21-\varepsilon_21−ε2​ with O(d2ε2/(ε2−ε1)2)O(d^2\varepsilon_2/(\varepsilon_2-\varepsilon_1)^2)O(d2ε2​/(ε2​−ε1​)2) samples if ε1=O(ε2/d2)\varepsilon_1 = O(\varepsilon_2/d^{2})ε1​=O(ε2​/d2).
Bell sampling is a simple yet powerful measurement primitive that has recently attracted a lot of attention, and has proven to be a valuable tool in studying stabiliser states. Unfortunately, however, it is known that Bell sampling fails when used on qu\emph{d}its of dimension d>2d>2d>2. In this paper, we explore and quantify the limitations of Bell sampling on qudits, and propose new quantum algorithms to circumvent the use of Bell sampling in solving two important problems: learning stabiliser states and providing pseudorandomness lower bounds on qudits. More specifically, as our first result, we characterise the output distribution corresponding to Bell sampling on copies of a stabiliser state and show that the output can be uniformly random, and hence reveal no information. As our second result, for d=pd=pd=p prime we devise a quantum algorithm to identify an unknown stabiliser state in (Cp)⊗n(\mathbb{C}^p)^{\otimes n}(Cp)⊗n that uses O(n)O(n)O(n) copies of the input state and runs in time O(n4)O(n^4)O(n4). As our third result, we provide a quantum algorithm that efficiently distinguishes a Haar-random state from a state with non-negligible stabiliser fidelity. As a corollary, any Clifford circuit on qudits of dimension ddd using O(log⁡n/log⁡d)O(\log{n}/\log{d})O(logn/logd) auxiliary non-Clifford single-qudit gates cannot prepare computationally pseudorandom quantum states.
Learning a classical description of a quantum state is a fundamental task in quantum computation. Among the most important classes of quantum states are stabilizer states, which play a central role in quantum error correction and fault-tolerant computation. To mitigate the effects of realistic noise, agnostic learning of stabilizer states has emerged as a natural and well-motivated problem. Recently, Chen \textit{et al.} [STOC'25, p. 429-438] resolved this problem for qubit systems by using a stabilizer bootstrapping framework. However, the agnostic learning of qudit stabilizer states remains largely unexplored, since the qudit setting introduces fundamental structural differences that prevent a direct generalization of existing qubit techniques. In this paper, we successfully generalize the stabilizer bootstrapping framework to qudit systems and present the first efficient quantum algorithm for agnostic learning of qudit stabilizer states. Specifically, given copies of an unknown nnn-qudit pure state ∣ψ⟩|ψ\rangle∣ψ⟩ that has fidelity τττ with some stabilizer state, our algorithm outputs a stabilizer state ∣φ⟩|φ\rangle∣φ⟩ such that ∣⟨φ∣ψ⟩∣2≥τ−ε\left| \braket{φ|ψ} \right|^2 \geq τ- \varepsilon∣⟨φ∣ψ⟩∣2≥τ−ε with high probability. The algorithm uses only single-copy and four-copy measurements, and its sample and time complexity scale as (d/τ)O(d2log⁡(1/τ))⋅poly(n,1/ε)(d/τ)^{O(d^2 \log(1/τ))} \cdot \mathrm{poly}(n, 1/\varepsilon)(d/τ)O(d2log(1/τ))⋅poly(n,1/ε), where the dimension ddd is an odd prime. As a direct corollary, our algorithm enables efficient estimation of the magic of a quantum state, as quantified by its stabilizer fidelity. Completing the picture, we also present a streamlined algorithm for the high-fidelity regime τ>cos⁡2(π/8)τ> \cos^2(π/8)τ>cos2(π/8), establishing a qudit analogue of the threshold-based approach in prior qubit work.
It is well-known that learning a pure nnn-qubit stabilizer state ∣ψ⟩|ψ\rangle∣ψ⟩ both requires, and can be accomplished with, access to a number of copies of ∣ψ⟩|ψ\rangle∣ψ⟩ linear in nnn. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that Lδ(n)L_δ(n)Lδ​(n), the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most 0<δ<1/80<δ<1/80<δ<1/8, satisfies n+⌈log⁡2(1/δ)⌉−3≤Lδ(n)≤n+⌈log⁡2(1/δ)⌉+4n+\lceil\log_2(1/δ)\rceil-3\leq L_δ(n)\leq n+\left\lceil\log_2(1/δ)\right\rceil+4n+⌈log2​(1/δ)⌉−3≤Lδ​(n)≤n+⌈log2​(1/δ)⌉+4. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown nnn-qubit Clifford unitary from 2n+⌈log⁡2(1/δ)⌉+42n+\left\lceil\log_2(1/δ)\right\rceil+42n+⌈log2​(1/δ)⌉+4 queries, the nnn-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group Z4n×F2n(n−1)/2\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}Z4n​×F2n(n−1)/2​, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.
We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: We prove that Ω(n) T-gates are necessary for any Clifford+T circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. Given an n-qubit pure quantum state |ψ⟩ that has fidelity at least τ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least τ − ε. The algorithm uses O(n/(ε2τ4)) samples and exp(O(n/τ4)) / ε2 time. In the regime of τ constant, this algorithm estimates stabilizer fidelity substantially faster than the naive exp(O(n2))-time brute-force algorithm over all stabilizer states. In the special case of τ > cos2(π/8), we show that a modification of the above algorithm runs in polynomial time. We exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory.
Stabilizer states are central to quantum computing, underlying quantum error correction, benchmarking, and efficient classical simulation. Yet their learnability exhibits a striking gap: an nnn-qubit stabilizer state can be learned from Θ(n)Θ(n)Θ(n) copies using two-copy Bell measurements, whereas non-adaptive single-copy measurements require Ω(n2)Ω(n^2)Ω(n2) copies. Here we show that adaptivity completely closes this gap. We give a polynomial-time adaptive algorithm that learns an arbitrary nnn-qubit stabilizer state from Θ(n)Θ(n)Θ(n) single-copy Clifford measurements, matching the optimal sample complexity of Bell sampling without any multi-copy measurements. The same ideas yield a sample-optimal single-copy tolerant tester and, with kkk qubits of quantum memory, the optimal testing tradeoff Θ(n−k+1/ε)Θ(n-k+1/\varepsilon)Θ(n−k+1/ε) at infidelity ε\varepsilonε. Finally, we show that this adaptive mechanism extends beyond exact stabilizer states: states of stabilizer nullity at most rrr, including states prepared by Clifford circuits with a bounded number of TTT gates, can be learned using O(n2r)O(n2^{r})O(n2r) single-copy measurements.
Clifford circuits acting on product magic states provide a compact ansatz exhibiting extensive magic, volume-law entanglement, and even classically hard sampling under standard complexity assumptions. Here, we efficiently recover its hidden product subsystems from two-copy Bell sampling. Quadratic relations between Bell samples determine the irreducible blocks after removing Pauli stabilizers, and binary linear algebra constructs a Clifford disentangler. For logarithmic-size blocks, approximate learning of the full state is efficient whenever the state remains inverse-polynomially separated from acquiring additional Pauli stabilizers. We efficiently learn nnn-qubit states prepared by random TTT-doped Clifford circuits with TTT-gate density below one via O(n2)O(n^2)O(n2) Bell samples, and disentangle hidden product blocks in Clifford-augmented matrix product states. For unitary learning, we exactly learn TTT-depth-one circuits using O(n2)O(n^2)O(n2) queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.