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arXiv:2503.03098 · Physical review. D/Physical review. D.

Quantum magic in quantum electrodynamics

Qiaofeng Liu, Ian Low, Zhewei Yin

Published Dec 2, 2025• 11 citations
arXivHTMLPDF

Abstract

In quantum computing, non-stabilizerness -- the magic -- refers to the computational advantage of certain quantum states over classical computers and is an essential ingredient for universal quantum computation. Employing the second order stabilizer R\'enyi entropy to quantify magic, we study the production of magic states in Quantum Electrodynamics (QED) via 2-to-2 scattering processes involving electrons and muons. Considering all 60 stabilizer initial states, which have zero magic, the angular dependence of magic produced in the final states is governed by only a few patterns, both in the non-relativistic and the ultra-relativistic limits. Some processes, such as the low-energy $e^-\mu^-\to e^-\mu^-$ and Bhabha scattering $e^-e^+\to e^-e^+$, do not generate magic at all. In most cases the largest magic generated is significantly less than the maximal possible value of $\log (16/7) \approx 0.827$. The only instance where QED is able to generate maximal magic is the low-energy $\mu^-\mu^+\to e^-e^+$, in the limit $m_e/m_\mu \to 0$, which is well approximated in nature. Our results suggest QED, although capable of producing maximally entangled states easily, may not be an efficient mechanism for generating quantum advantages.

DOI
10.1103/l8vw-kwhz

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In quantum computing, non-stabilizerness -- the magic -- refers to the computational advantage of certain quantum states over classical computers and is an essential ingredient for universal quantum computation. Employing the second order stabilizer R'enyi entropy to quantify magic, we study the production of magic states in Quantum Electrodynamics (QED) via 2-to-2 scattering processes involving electrons and muons. Considering all 60 stabilizer initial states, which have zero magic, the angular dependence of magic produced in the final states is governed by only a few patterns, both in the non-relativistic and the ultra-relativistic limits. Some processes, such as the low-energy e−μ−→e−μ−e^-\mu^-\to e^-\mu^-e−μ−→e−μ− and Bhabha scattering e−e+→e−e+e^-e^+\to e^-e^+e−e+→e−e+, do not generate magic at all. In most cases the largest magic generated is significantly less than the maximal possible value of log⁡(16/7)≈0.827\log (16/7) \approx 0.827log(16/7)≈0.827. The only instance where QED is able to generate maximal magic is the low-energy μ−μ+→e−e+\mu^-\mu^+\to e^-e^+μ−μ+→e−e+, in the limit me/mμ→0m_e/m_\mu \to 0me​/mμ​→0, which is well approximated in nature. Our results suggest QED, although capable of producing maximally entangled states easily, may not be an efficient mechanism for generating quantum advantages.
Quantum magic, or nonstabilizerness, is a quantum resource associated with computational advantage in quantum systems. In high energy collisions, Quantum Electrodynamics (QED) is inefficient at generating magic while the weak mixing angle, a fundamental constant of nature, sits near a value that minimizes magic production in charged-lepton scattering. These observations were made in the laboratory (lab) basis, in which spin is projected along the incoming beam axis. An alternative choice is the helicity basis, in which spin is projected along the direction of motion of each particle. The transformation between these two bases is, in general, not a Clifford operation and therefore can change the amount of magic. We present a detailed study of magic production in both bases for QED and electroweak processes, and compare these results with the basis-invariant non-local magic. In the ultra-relativistic limit, magic production is generally smaller in the helicity basis due to helicity selection rules, while the lab basis generally yields less magic in the non-relativistic regime. We provide circuit realizations of the ultra-relativistic Bhabha amplitudes using linear combinations of unitaries and show that the lab basis construction contains a larger TTT-gate count at generic scattering angles. Interestingly, in both bases the physical weak mixing angle lies close to the value that minimizes magic production.
Magic, also known as non-stabilizerness, measures the usefulness of a quantum state for quantum computation. While magic is defined relative to a choice of computational basis, in some physical settings the available data determine this basis only up to local phase conventions. In this paper, we generalize the notion of magic and formulate it in a phase-independent manner, and hence define a generalized stabilizer Rényi entropy. As a case study, we consider higher-multiplicity tree-level gluon scattering, interpreting the outgoing helicities as qubits. In this setting, the helicity data naturally determine a local basis for each qubit but leave a phase ambiguity. For 3→23\to 23→2 scattering, we find that the final-state phase-independent magic is generically larger than the maximum attainable in 2→22\to 22→2 scattering. For 2→32 \to 32→3 scattering, we find a nonzero minimal value approached in the soft limit. Moreover, when the three outgoing momenta become symmetric, the magic approaches a local minimum only a few percent above the soft-limit value. In all cases considered, the color dependence cancels from the phase-independent stabilizer Rényi entropy.
Abstract Magic is a quantum resource essential for universal quantum computation and represents the deviation of quantum states from those that can be simulated efficiently using classical algorithms. Using the stabilizer Rényi entropy (SRE), we investigate two-qubit states with maximal magic, which are most distinct from classical simulability, and provide strong numerical evidence that the maximal second order SRE is ln ⁡ ( 16 / 7 ) ≈ 0.827 , establishing a tighter bound than the prior ln ⁡ ( 5 / 2 ) ≈ 0.916 . We identify 480 states saturating the new bound, which turn out to be the fiducial states for the mutually unbiased bases (MUBs) generated by the orbits of the Weyl–Heisenberg (WH) group, and conjecture that WH-MUBs are the maximal magic states for n -qubit, when n ≠ 1 and 3. We also reveal a striking interplay between magic and entanglement: the entanglement of maximal magic states is restricted to two possible values, 1 / 2 and 1 / 2 , as quantified by the concurrence; none is maximally entangled.
Stabilizer states constitute an important class of quantum states that can be generated from computational-basis states using Pauli operators and Clifford gates. Although they may exhibit substantial multipartite entanglement, quantum circuits restricted to stabilizer operations can be efficiently simulated classically and therefore cannot, by themselves, provide a quantum computational advantage. Such an advantage requires non-stabilizer resources, commonly referred to as quantum magic. In this paper, we investigate the non-stabilizerness of quantum states arising in a class of spin Hamiltonians by computing their stabilizer Rényi entropies. Using real-space renormalization-group techniques, we obtain a closed-form expression valid in the low-energy, large-distance regime. We analyze how quantum magic evolves under coarse-graining and explore its behavior across different parameter regimes and quantum phases.
The production of magic states is studied in two settings. The first is the electroweak (EW) sector of the Standard Model (SM). The second is an extension featuring a new broken U(1)U(1)U(1) gauge symmetry and a Dirac fermion charged under it. This setup resembles a dark U(1)U(1)U(1) scenario, with the additional fermion playing the role of a dark matter candidate that annihilates into SM particles through its coupling to the new gauge boson. In the EW sector, the low-energy regime reproduces earlier magic production results obtained for Quantum Electrodynamics, whereas the high-energy and ZZZ-resonance regimes generate new magic distribution functions and non-trivially reorganize the stabilizer state classes, with Bhabha scattering exhibiting the strongest sensitivity to electroweak effects. Also, a subset of fixed stabilizer states is identified, for which the magic distributions remain unchanged across the different energy regimes. In the dark sector, the main effect of the new massive mediator is the appearance of new magic distributions functions for Moller-like, Bhabha-like, and inverse pair-annihilation processes in the low-energy limit. These reach the maximal magic value at the SM-to-dark fermion mass ratios mf/mχ→0m_f/m_χ\to 0mf​/mχ​→0 and mf/mχ→1.83929m_f/m_χ\to 1.83929mf​/mχ​→1.83929.
Entanglement and non-stabilizerness (magic) quantify two distinct departures of quantum systems from classical description: the former measures non-local correlations, while the latter measures the deviation from stabilizer states that can be efficiently simulated classically. Understanding magic in physically relevant quantum field theories is essential for identifying where quantum advantage may be realized in the early fault-tolerant quantum computing era. We calculate the gauge-invariant entanglement entropy and stabilizer Rényi entropy of the ground state of the (1+1)-dimensional SU(2) lattice gauge theory formulated in a dressed-site basis that enforces Gauss's law exactly. Using tensor networks, we obtain results for system sizes up to L=100L=100L=100 (300 qubits). We find a crossover denoted by g⋆g_{\star}g⋆​ where the ground state passes from a more magic-rich regime into a regime with less magic; this is also tracked by the sharpest change of both the entanglement entropy and lattice particle density. Our large-scale study of non-stabilizerness and entanglement entropy in a non-Abelian lattice gauge theory with matter provides new insight into the interplay of magic and entanglement in gauge theories, both of which are relevant for classical and early fault-tolerant quantum simulations.