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arXiv:2602.18939

Predicting Magic from Very Few Measurements

J. M. Varela, L. L. Keller, A. de Oliveira Junior, D. A. Moreira, R. Chaves, R. A. Macêdo

quant-phcond-mat.str-elSubmitted Feb 21, 2026• Announced Feb 24, 2026• 0 citations
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Abstract

The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using any set of $m$ $n$-qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements $m$ and polynomial in the number of qubits $n$. By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless $\mathrm{P} = \mathrm{NP}$, no algorithm polynomial in $m$ can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.

Comments
20 pages, 3 figures

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The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using any set of mmm nnn-qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements mmm and polynomial in the number of qubits nnn. By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless P=NP\mathrm{P} = \mathrm{NP}P=NP, no algorithm polynomial in mmm can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.
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.
The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using set of m n -qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements m and polynomial in the number of qubits n . By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless P = NP , no algorithm polynomial in m can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.
Nonstabilizerness, commonly referred to as magic, is essential for universal quantum computation. Yet, for pure nnn-qubit states this property is directly linked to the hardness of direct fidelity estimation. We propose a scheme to bound the fidelity on non-stabilizer states by choosing operators with low measurement overhead and using parallel local Pauli measurements to estimate compatible observables. We selectively measure operators that share a permutational symmetry without imposing any assumption on the physical system. This selection reduces the optimization to a linear program with only O(n)O(n)O(n) variables for bounding fidelity. Our methods can be extended to bound nonlinear quantities like the purity via convex quadratic programs. For hypergraph states that can exhibit extensive nonstabilizerness, we demonstrate for up to n=500n=500n=500 qubits that our framework substantially reduces the measurement overhead under both local and global depolarizing noise. These results show that symmetry induced by measurement selection reduces both measurement and computational costs, enabling scalable characterization of magic resources in quantum computing and quantum networks.
Nonstabilizerness, also known as magic, quantifies the number of non-Clifford operations needed to prepare a quantum state. As typical measures either involve minimization procedures or a computational cost exponential in the number of qubits NNN, it is notoriously hard to characterize for many-body states. In this paper, we show that nonstabilizerness, as quantified by the recently introduced stabilizer R\'enyi entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension $\ensuremath{\chi}$ and integer R\'enyi index n>1n>1n>1, we show that the SRE can be expressed in terms of the norm of an MPS with bond dimension ${\ensuremath{\chi}}^{2n}$. For translation-invariant states, this allows us to extract it from a single tensor, the transfer matrix, while for generic MPSs this construction yields a computational cost linear in NNN and polynomial in $\ensuremath{\chi}$. We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes. We analyze the SRE near criticality and investigate its dependence on the local computational basis, showing that it is, in general, not maximal at the critical point.
Detecting nonstabilizerness requires full tomography and an optimization over exponentially many stabilizer states. A limited Pauli measurement set promises resource-efficient magic certification, yet the resulting reduced stabilizer polytope is generally difficult to characterize. We trace this difficulty into two coupled obstructions: the simultaneous measurability of measurements captured by their frustration graph structure, and the consistency of sign dependencies from stabilizer formalism. We show that the sign dependencies can be discarded exactly whenever active dependencies are absent, and that perfect frustration graphs then make this reduced polytope efficiently solvable. This solvable regime derives a closed form bounded by the clique number of the frustration graph, revealing a tradeoff between witness capacity and simultaneous measurability. Clifford covariance allows rotated measurement sets to enlarge the detectable state space without raising the capacity. Graph structure therefore emerges as both a certificate of tractability and a design principle for scalable magic resource detection.
We introduce a monotone to quantify the amount of nonstabilizerness in mixed quantum states. The monotone gives a necessary and sufficient criterion for detecting the presence of nonstabilizerness for both pure and mixed states. The monotone is based on determining the boundaries of the stabilizer polytope in the space of Pauli string expectation values. The boundaries can be described by a set of hyperplane inequations, where violation of any one of these gives a necessary and sufficient condition for nonstabilizerness. The monotone is constructed by finding the hyperplane with the maximum violation and is a type of Minkowski functional. We also introduce a faithful witness based on similar methods. The approach is more computationally efficient than existing faithful mixed state monotones such as robustness of magic due to the smaller number and discrete nature of the parameters to be optimized.