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arXiv:2609.03944

On the geometry and typicality of quantum magic

Zhenhuan Liu, Zi-Wen Liu

quant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations• Revised Sep 8, 2026 (v2)
v1 · Sep 3, 2026v2 · Sep 8, 2026
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Abstract

We prove that, for an $n$-qubit system of dimension $d=2^n$, every state satisfying $\operatorname{Tr}(ρ^2)\le 1/(d-a_\ast)$, with $a_\ast=0.458327\cdots$, lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establish quantitative estimates for the Hilbert--Schmidt inradius and volume radius of the stabilizer polytope, and use them to characterize the typicality of magic in random induced states obtained by tracing out a $k$-dimensional subsystem from a $d\times k$-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension $k_\star$ is bounded between $Ω(d^2/\log^2d)$ and $\mathcal{O}(d^2)$. We further prove that the number of facets of the stabilizer polytope lies between $\exp[Ω(d^2/\log^2 d)]$ and $\exp[\mathcal{O}(d^2\log^2 d)]$ employing a result of Bourgain and Milman in convex geometry, substantially improving upon the previous quasipolynomial lower bound and implying that doubly-exponentially many linear inequalities in the number of qubits are required for an exact description of the magic-free region. Overall, our results reveal the near-extremal geometry of the stabilizer polytope and provide a quantitative foundation for understanding the typicality, robustness, and detectability of magic.

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Identifying the boundary between classical and quantum computation is a central challenge in quantum information. In multi-qubit systems, entanglement and magic are the key resources underlying genuinely quantum behaviour. While entanglement is well understood, magic - essential for universal quantum computation - remains relatively poorly characterised. Here we show that determining membership in the stabilizer polytope, which defines the free states of magic-state resource theory, requires super-exponential time $\class{exp} ( n^2)$ in the number of qubits nnn, even approximately. We reduce the problem to solving a 333-\class{SAT} instance on n2n^2n2 variables and, by invoking the exponential time hypothesis, the result follows. As a consequence, both quantifying and certifying magic are fundamentally intractable: any magic monotone for general states must be super-exponentially hard to compute, and deciding whether an operator is a valid magic witness is equally difficult. As a corollary, we establish the robustness of magic as computationally optimal among monotones. This barrier extends even to classically simulable regimes: deciding whether a state lies in the convex hull of states generated by a logarithmic number of non-Clifford gates is also super-exponentially hard. Together, these results reveal intrinsic computational limits on assessing classical simulability, distilling pathological magic states, and ultimately probing and exploiting magic as a quantum resource.
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Abstract Identifying the boundary between classical and quantum computation is a central challenge in quantum information. In multi-qubit systems, entanglement and magic are the key resources underlying genuinely quantum behaviour. While entanglement is well understood, magic - essential for universal quantum computation - remains relatively poorly characterised. Here we show that determining membership in the stabilizer polytope, which defines the free states of magic-state resource theory, requires super-exponential time $\class{exp} ( n^2)$ in the number of qubits nnn, even approximately. We reduce the problem to solving a 333-\class{SAT} instance on n2n^2n2 variables and, by invoking the exponential time hypothesis, the result follows. As a consequence, both quantifying and certifying magic are fundamentally intractable: any magic monotone for general states must be super-exponentially hard to compute, and deciding whether an operator is a valid magic witness is equally difficult. As a corollary, we establish the robustness of magic as computationally optimal among monotones. This barrier extends even to classically simulable regimes: deciding whether a state lies in the convex hull of states generated by a logarithmic number of non-Clifford gates is also super-exponentially hard. Together, these results reveal intrinsic computational limits on assessing classical simulability, distilling pathological magic states, and ultimately probing and exploiting magic as a quantum resource.
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