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arXiv:2610.22476

Typical Superpositions are Magical

Joshuah T. Heath, Andrew Urilyon, Damien Quinn, Graham Kells

quant-phSubmitted Oct 15, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

The non-stabilizerness, or "magic", of a quantum state is a common metric for the hardness of simulation on a classical computer, and is often quantified by the stabilizer Rényi entropy (SRE). We show how random superpositions of classically simulable states saturate the SRE to the symmetry-restricted Haar-random maximum. For any large-$N$ random pure-state ensemble satisfying the conditions of Schur's lemma, the average SRE of the $m$-component superposition $\langle M_2(m)\rangle=N-\log_2[\langle W(m)\rangle]$ is governed by an average witness $\langle W(m)\rangle$ that universally decomposes into a Haar floor and an ensemble-specific excess. As the number of components $m$ increases, the excess is suppressed, resulting in a logarithmic scaling of the SRE before plateauing at the Haar-valued maximum. We derive a general closed form for $\langle W(m)\rangle$ in the thermodynamic limit, and numerically verify our result for random stabilizer states and random Gaussian states.

Comments
Main text: 7 pages, 4 figures. Supplemental Material: 25 pages, 3 figures

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Nonstabilizerness or `magic' is a key resource for quantum computing and a necessary condition for quantum advantage. Non-Clifford operations turn stabilizer states into resourceful states, where the amount of nonstabilizerness is quantified by resource measures such as stabilizer Rényi entropies (SREs). Here, we show that SREs saturate their maximum value at a critical number of non-Clifford operations. Close to the critical point SREs show universal behavior. Remarkably, the derivative of the SRE crosses at the same point independent of the number of qubits and can be rescaled onto a single curve. We find that the critical point depends non-trivially on Rényi index α . For random Clifford circuits doped with T-gates, the critical T-gate density scales independently of α . In contrast, for random Hamiltonian evolution, the critical time scales linearly with qubit number for α > 1 , while it is a constant for α < 1 . This highlights that α -SREs reveal fundamentally different aspects of nonstabilizerness depending on α : α -SREs with α < 1 relate to Clifford simulation complexity, while α > 1 probe the distance to the closest stabilizer state and approximate state certification cost via Pauli measurements. As technical contributions, we observe that the Pauli spectrum of random evolution can be approximated by two highly concentrated peaks which allows us to compute its SRE. Further, we introduce a class of random evolution that can be expressed as random Clifford circuits and rotations, where we provide its exact SRE. Our results opens up new approaches to characterize the complexity of quantum systems.
Magic is a property of a quantum state that characterizes its deviation from a stabilizer state, serving as a useful resource for achieving universal quantum computation e.g., within schemes that use Clifford operations. In this work, we study magic, as quantified by the stabilizer Renyi entropy, in a class of models known as generalized Rokhsar-Kivelson systems, i.e., Hamiltonians that allow a stochastic matrix form (SMF) decomposition. The ground state wavefunctions of these systems can be written explicitly throughout their phase diagram, and their properties can be related to associated classical statistical mechanics problems, thereby allowing powerful analytical and numerical approaches that are not usually available in conventional quantum many body settings. As a result, we are able to express the SRE in terms of wave function coefficients that can be understood as a free energy difference of related classical problems. We apply this insight to a range of quantum many body SMF Hamiltonians, which affords us to study numerically the SRE of large high-dimensional systems, and in some cases to obtain analytical results. We observe that the behaviour of the SRE is relatively featureless across quantum phase transitions in these systems, although it is indeed singular (in its first or higher order derivative, depending on the nature of the transition). On the contrary, we find that the maximum of the SRE generically occurs at a cusp away from the quantum critical point, where the derivative suddenly changes sign. Furthermore, we compare the SRE and the logarithm of overlaps with specific stabilizer states, asymptotically realised in the ground state phase diagrams of these systems. We find that they display strikingly similar behaviors, which in turn establish rigorous bounds on the min-relative entropy of magic.
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Non-stabilizerness (colloquially "magic") characterizes genuinely quantum (beyond-Clifford) operations necessary for preparation of quantum states, and can be measured by stabilizer Rényi entropy (SRE). For permutationally symmetric states, we show that the SRE depends, for sufficiently large systems, only on a constant number of expectation values of collective spin operators. This compact description is leveraged for analysis of spin-squeezing protocols, which inherently generate non-stabilizerness. Under one-axis twisting (OAT), the generation of optimal squeezing is accompanied by a logarithmic divergence of SRE with system system size. Continued time evolution under OAT produces metrologically useful "kitten" states-superpositions of rotated GHZ states-that feature many-body Bell correlations but exhibit a smaller, system-size-independent SRE that decreases with increasing Bell-correlation strength. Our results reveal connections between non-stabilizerness, multipartite correlations, and quantum metrology, and provide a practical route to quantify non-stabilizerness in experiments for precision sensing.
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