Computing quantum magic of state vectors
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AbstractNon-stabilizerness, also known as “magic,'' quantifies how far a quantum state departs from the stabilizer set. It is a central resource behind quantum advantage and a useful probe of the complexity of quantum many-body states. Yet standard magic quantifiers, such as the stabilizer Rényi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number $N$ of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits ($d=2$) and the mana for qutrits ($d=3$) for pure states given as state vectors. Our methods compute SRE and mana at cost $O(N d^{2N})$, providing an exponential improvement over the naive $O(d^{3N})$ scaling, with substantial parallelism and straightforward GPU acceleration. We further show how to combine the fast Hadamard transform with Monte Carlo sampling to estimate the SRE of state vectors, and we extend the approach to compute the mana of mixed states. All algorithms are implemented in the open-source Julia package HadaMAG, which provides a high-performance toolbox for computing SRE and mana with built-in support for multithreading, MPI-based distributed parallelism, and GPU acceleration. The package, together with the methods developed in this work, offers a practical route to large-scale numerical studies of magic in quantum many-body systems.Featured image: HadaMAG workflow: a quantum state vector $|\psi\rangle$ with $d^N$ amplitudes is fed through $d^N$ fast Hadamard transforms, i.e., butterfly networks of additions and subtractions, to efficiently extract all $d^{2N}$ Pauli expectation values $\langle P \rangle$, from which measures of quantum magic, the stabilizer Rényi entropy $M_2(|\psi\rangle)$ for qubits ($d=2$) and the mana $\mathcal{M}(|\psi\rangle)$ for qutrits ($d=3$), are obtained.Popular summaryStabilizer states form a special class of quantum states that align with a discrete set of privileged directions in Hilbert space and can therefore be simulated efficiently on a classical computer. Magic, or non-stabilizerness, measures how far a state departs from this classically tractable set, and is a key resource behind the enhanced computational power of quantum systems. Characterizing this feature in concrete many-body states requires computing suitable measures of magic. Yet this quickly becomes difficult in practice, because standard magic measures require summing exponentially many expectation values. Here we show that this computation can be reorganized using fast Hadamard and Fourier transforms, yielding exact algorithms that are exponentially faster than straightforward approaches. This enables the computation of measures of magic: "stabilizer Rényi entropy" for qubits and "mana" for qutrits in significantly larger systems than previously accessible from state-vector data. We also develop approximate sampling methods and extend the same framework to mixed states. All methods are implemented in the open-source Julia package HadaMAG.jl, providing a practical toolbox for large-scale studies of quantum magic.► BibTeX data@article{Sierant2026computingquantum, doi = {10.22331/q-2026-04-10-2059}, url = {https://doi.org/10.22331/q-2026-04-10-2059}, title = {Computing quantum magic of state vectors}, author = {Sierant, Piotr and Vall{\`{e}}s-Muns, Jofre and Garcia-Saez, Artur}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2059}, month = apr, year = {2026} }► References [1] J. Preskill, ``Quantum computing and the entanglement frontier,'' (2012), arXiv:1203.5813. https://doi.org/10.48550/arXiv.1203.5813 arXiv:1203.5813 [2] A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature 607, 667 (2022). https://doi.org/10.1038/s41586-022-04940-6 [3] G. Vidal, Phys. Rev. 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Technol. 10, 045026 (2025). https://doi.org/10.1088/2058-9565/adfd0dCited byCould not fetch Crossref cited-by data during last attempt 2026-04-10 08:17:06: Could not fetch cited-by data for 10.22331/q-2026-04-10-2059 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-10 08:17:07: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractNon-stabilizerness, also known as “magic,'' quantifies how far a quantum state departs from the stabilizer set. It is a central resource behind quantum advantage and a useful probe of the complexity of quantum many-body states. Yet standard magic quantifiers, such as the stabilizer Rényi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number $N$ of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits ($d=2$) and the mana for qutrits ($d=3$) for pure states given as state vectors. Our methods compute SRE and mana at cost $O(N d^{2N})$, providing an exponential improvement over the naive $O(d^{3N})$ scaling, with substantial parallelism and straightforward GPU acceleration. We further show how to combine the fast Hadamard transform with Monte Carlo sampling to estimate the SRE of state vectors, and we extend the approach to compute the mana of mixed states. All algorithms are implemented in the open-source Julia package HadaMAG, which provides a high-performance toolbox for computing SRE and mana with built-in support for multithreading, MPI-based distributed parallelism, and GPU acceleration. The package, together with the methods developed in this work, offers a practical route to large-scale numerical studies of magic in quantum many-body systems.Featured image: HadaMAG workflow: a quantum state vector $|\psi\rangle$ with $d^N$ amplitudes is fed through $d^N$ fast Hadamard transforms, i.e., butterfly networks of additions and subtractions, to efficiently extract all $d^{2N}$ Pauli expectation values $\langle P \rangle$, from which measures of quantum magic, the stabilizer Rényi entropy $M_2(|\psi\rangle)$ for qubits ($d=2$) and the mana $\mathcal{M}(|\psi\rangle)$ for qutrits ($d=3$), are obtained.Popular summaryStabilizer states form a special class of quantum states that align with a discrete set of privileged directions in Hilbert space and can therefore be simulated efficiently on a classical computer. Magic, or non-stabilizerness, measures how far a state departs from this classically tractable set, and is a key resource behind the enhanced computational power of quantum systems. Characterizing this feature in concrete many-body states requires computing suitable measures of magic. Yet this quickly becomes difficult in practice, because standard magic measures require summing exponentially many expectation values. Here we show that this computation can be reorganized using fast Hadamard and Fourier transforms, yielding exact algorithms that are exponentially faster than straightforward approaches. This enables the computation of measures of magic: "stabilizer Rényi entropy" for qubits and "mana" for qutrits in significantly larger systems than previously accessible from state-vector data. We also develop approximate sampling methods and extend the same framework to mixed states. All methods are implemented in the open-source Julia package HadaMAG.jl, providing a practical toolbox for large-scale studies of quantum magic.► BibTeX data@article{Sierant2026computingquantum, doi = {10.22331/q-2026-04-10-2059}, url = {https://doi.org/10.22331/q-2026-04-10-2059}, title = {Computing quantum magic of state vectors}, author = {Sierant, Piotr and Vall{\`{e}}s-Muns, Jofre and Garcia-Saez, Artur}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2059}, month = apr, year = {2026} }► References [1] J. Preskill, ``Quantum computing and the entanglement frontier,'' (2012), arXiv:1203.5813. https://doi.org/10.48550/arXiv.1203.5813 arXiv:1203.5813 [2] A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature 607, 667 (2022). https://doi.org/10.1038/s41586-022-04940-6 [3] G. Vidal, Phys. Rev. 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Technol. 10, 045026 (2025). https://doi.org/10.1088/2058-9565/adfd0dCited byCould not fetch Crossref cited-by data during last attempt 2026-04-10 08:17:06: Could not fetch cited-by data for 10.22331/q-2026-04-10-2059 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-10 08:17:07: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.