The sample complexity of identifying an unknown pure stabilizer state has been a longstanding problem in quantum information. In this paper, we resolve the entire exact critical crossover for the uniform ensemble of pure $n$-qubit stabilizer states under arbitrary collective measurements and without side information. For $k=n+c$, with fixed integer $c$, the optimal success probability converges to an explicit positive series $P_{\infty}(c)$. In particular, $P_{\infty}(0)=0.1760837896\ldots$, while its two tails obey a quadratic-exponential law below threshold and $1-P_{\infty}(c)\sim2^{-c}$ above threshold. The pretty-good measurement is exactly optimal for every finite $n$ and $k$. The proof diagonalizes the stabilizer Gram operator through Clifford tensor-power duality and a defect filtration, including the exceptional binary sectors. The resulting integer-scale law quantifies the $n+O(1)$ copy threshold.
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Related arXiv paper: arXiv:2609.13923,"Partial Stabilizer Learning under Fixed Commuting Constraints and the Absence of a Copy-Rate Discount,"establishes the first-order copy threshold for unrestricted stabilizer-state identification, but leaves the fixed-offset regime k=n+c untreated. This submission resolves that regime by determining the limiting optimal success probability
The sample complexity of identifying an unknown pure stabilizer state has been a longstanding problem in quantum information. In this paper, we resolve the entire exact critical crossover for the uniform ensemble of pure n-qubit stabilizer states under arbitrary collective measurements and without side information. For k=n+c, with fixed integer c, the optimal success probability converges to an explicit positive series P∞(c). In particular, P∞(0)=0.1760837896…, while its two tails obey a quadratic-exponential law below threshold and 1−P∞(c)∼2−c above threshold. The pretty-good measurement is exactly optimal for every finite n and k. The proof diagonalizes the stabilizer Gram operator through Clifford tensor-power duality and a defect filtration, including the exceptional binary sectors. The resulting integer-scale law quantifies the n+O(1) copy threshold.
It is well-known that learning a pure n-qubit stabilizer state ∣ψ⟩ both requires, and can be accomplished with, access to a number of copies of ∣ψ⟩ linear in n. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that Lδ(n), the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most 0<δ<1/8, satisfies n+⌈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 n-qubit Clifford unitary from 2n+⌈log2(1/δ)⌉+4 queries, the n-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group 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 determine the optimal average global fidelity of T→T+k approximate cloning for the uniform ensemble of n-qubit stabilizer states, for every number of qubits n, every k ≥ 1 and every 1 ≤ T ≤ 3. With D = 2^n, the optimum is P_stab(n;T,k) = C(D+T−1, T) · F_{T+k−κ(k)}, where F_s = ∏_{i=0}^{s−2}(2^i+1) / (D ∏_{i=0}^{s−2}(D+2^i)) is the stabilizer frame potential and κ(k) is the largest dimension of a doubly-even binary code of length k. The result rests on an exact formula for the largest eigenvalue of the mixed stabilizer moment operators: for all a+b ≥ 1 and all n it equals F_{a+b−w}, where w = min(a,b) + κ(|a−b|) is the Witt index of an associated Z4-valued quadratic form. We then ask what is lost by measuring first. For measure-and-prepare (entanglement-breaking) cloners we show that, after the pretty-good measurement, re-preparing a better output than copies of the guess helps exactly from four output copies on, because fourth powers erase the phases ±1, ±i of stabilizer amplitudes; that for 1→4 the optimum is 3(D+48+√(D²+16D+1664)) / ((D+1)(D+2)(D+4)(D+8)) for every n, so coherent cloning wins by less than a factor 5; and that for 2→4 there are two regimes: the pretty-good-measurement strategy is optimal for n ≤ 3, while for every n ≥ 4 Bell sampling is optimal, with fidelity 3/(D(D+2)) and a coherent advantage of exactly 5D/(D+4). The upper bounds are explicit dual certificates for the PPT relaxation, covering every n. Version 2 adds Part II (measure-and-prepare cloning), Part III (open problems), a certificates appendix, and citation fixes. Part I results and theorem numbering are unchanged from version 1. Files:• Weldon_2026_Exact_optimal_cloning_of_stabilizer_states_v2.pdf — the paper (15 pages).• paper3_v2_certificates.zip — every Part II certificate with standalone checkers, run scripts, expected logs and a SHA-256 manifest (SHA-256 236f7e4c5405e8e4030d476958e51aae5fb7b2fd2349c3ec9ddcb882cddf054f).• paper3_code.zip — Part I code and certificates, unchanged from version 1 (SHA-256 de2fc42b8bc4fa710b8506de2001d475a8d8b4a011ed0df72c693110a5d417e0).• paper3_v2_source.zip — LaTeX source.
Magic plays a dual role in quantum computation: it promotes stabilizer dynamics from efficient classical simulability to computational universality, but also challenges fault-tolerant architectures, since non-stabilizer operations are harder to protect against noise. Magic state distillation addresses this issue, yet existing protocols remain largely tailored to specific assumptions about the input or noise model and, more fundamentally, no general information-theoretic theory of optimal magic-state conversion currently exists. Here we develop such a characterization for pure qubit states through universal magic state concentration: a fixed stabilizer protocol that converts a few copies of an unknown pure non-stabilizer qubit state into an exact target magic state. Motivated by the impossibility of exact T-state concentration, we build six- and eight-copy protocols producing an exact CCZ state, with six copies being minimal. Their success probabilities are governed by the linearized order-three stabilizer Rényi entropy M3lin. We show that this connection is structural: for up to nine input copies, M3lin fully determines the success probability of every universal Clifford-invariant stabilizer protocol. Remarkably, this characterization persists asymptotically: our constructions achieve optimal rates among universal single-output protocols and remain optimal up to logarithmic factors among arbitrary stabilizer protocols. As a corollary, we show that any unknown pure non-stabilizer state suffices for universal quantum computation via probabilistic CCZ-state injection. Together, these results identify the stabilizer Rényi entropy as a fundamental operational quantity in magic state distillation.
We prove that every pure state of stabilizer rank at most k has stabilizer extent at most 2O(klog(k+1)), and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-k operators. This implies stabilizer fidelity at least 2−O(klog(k+1)), resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an Ω(n2/logn) lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of T-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states.
As a consequence of the Clifford rank--norm inequality, we obtain an Ω(n2/logn) lower bound for exact representations of n-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank O((logn)2/loglogn), respectively, a logn improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most k, with poly(n)2O(klog(k+1)) time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.
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 n-qudit pure state ∣ψ⟩ that has fidelity τ with some stabilizer state, our algorithm outputs a stabilizer state ∣φ⟩ such that ∣⟨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), where the dimension d 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>cos2(π/8), establishing a qudit analogue of the threshold-based approach in prior qubit work.
We consider the problem of testing whether an unknown n-qubit quantum state ∣ψ⟩ is a stabilizer state, with only single-copy access. We give an algorithm solving this problem using O(n) copies, and conversely prove that Ω(n) copies are required for any algorithm. The main observation behind our algorithm is that when repeatedly measuring in a randomly chosen stabilizer basis, stabilizer states are the most likely among the set of all pure states to exhibit linear dependencies in measurement outcomes. Our algorithm is designed to probe deviations from this extremal behavior. For the lower bound, we first reduce stabilizer testing to the task of distinguishing random stabilizer states from the maximally mixed state. We then argue that, without loss of generality, it is sufficient to consider measurement strategies that a) lie in the commutant of the tensor action of the Clifford group and b) satisfy a Positive Partial Transpose (PPT) condition. By leveraging these constraints, together with novel results on the partial transposes of the generators of the Clifford commutant, we derive the lower bound on the sample complexity.