Quantum Magic in High Energy Collision

Abstract: 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 $T$-gate count at generic scattering angles. Interestingly, in both bases the physical weak mixing angle lies close to the value that minimizes magic production.

• The paper demonstrates that scattering-frame dependence is largely kinematic: helicity selection rules reduce ultra-relativistic magic, while non-local magic separates interaction effects from basis rotations.
• Electroweak calculations show that the weak mixing angle remains near a magic-minimizing value in both lab and helicity bases, supporting a frame-independent resource interpretation of Standard Model couplings.
• Circuit analysis finds that sparser helicity amplitudes reduce simulation costs, requiring four Pauli strings and fewer non-Clifford gates than the eight-string lab-basis decomposition in generic configurations.

Motivation and scope

Magic, or nonstabilizerness, quantifies the resource beyond entanglement required for quantum computational advantage: by the Gottesman-Knill theorem, circuits composed solely of Clifford gates are classically simulable, so a non-Clifford ingredient such as the $T$-gate is necessary for universality. This paper, "Quantum Magic in High Energy Collision" (2608.19095), extends a program initiated in Refs. (Liu et al., 5 Mar 2025, Liu et al., 22 Sep 2025) that computed the second-order stabilizer Rényi entropy (2-SRE), $M_2$, generated in tree-level $2\to2$ charged-lepton scattering, and addresses a structural issue flagged there as an open question: the 2-SRE is built from Pauli-string expectation values and is therefore invariant only under Clifford operations, so the magic ascribed to a scattering final state depends on the reference frame in which the fermion spins are projected. The two natural frames are the lab basis, in which all spins are projected along the beam axis in the center-of-mass frame, and the helicity basis, in which each particle's spin is projected along its own momentum. Since the lab-to-helicity transformation is not a Clifford operation, the two bases generically assign different magic to the same physical process. The paper's central question is whether the two headline conclusions of the earlier work—QED is an inefficient magic generator, and the empirical weak mixing angle $\hat{s}_W^2$ sits near a magic-minimizing value—are robust against this frame choice or artifacts of the lab basis.

Basis dependence as a kinematic effect

The authors first quantify how much magic a change of computational basis alone can generate. Acting on the 60 two-qubit stabilizer states (which carry $M_2 = 0$), a uniform rotation $R_y^{\otimes 2}(\theta)$ produces only four distinct angular patterns, with a maximum of $\log(16/9)$—strictly below the two-qubit maximum $M_2^{\max} = \log(16/7) \approx 0.827$ (Liu et al., 24 Feb 2025). For the initial-state particles, the lab-to-helicity map is $I \otimes R_y(\pi)$, a Clifford operation that merely reshuffles the Pauli spectrum and preserves magic. For final-state particles, however, the map is $R_y(\theta)\otimes R_y(\theta+\pi)$, equivalent to a non-Clifford rotation by the scattering angle $M_2$0; part of any lab–helicity difference in final-state magic is therefore purely kinematic, set by $M_2$1 rather than by the interaction dynamics. The paper also employs the basis-invariant non-local magic $M_2$2, defined by minimizing $M_2$3 over all local basis rotations; for two-qubit pure states it is a function of entanglement alone via the concurrence $M_2$4, $M_2$5 (Gargalionis et al., 4 Mar 2026, Busoni et al., 10 Mar 2026), vanishing for both product and maximally entangled states and peaking at $M_2$6. By construction, magic measured in either physical basis can never fall below $M_2$7, and whichever basis carries less magic is expected to track it.

QED: helicity selection rules suppress magic at high energy

For the four representative QED processes (Bhabha, Møller, $M_2$8 elastic, and $M_2$9), the authors evaluate $2\to2$0 from all 60 stabilizer initial states in both bases, in ultra-relativistic and non-relativistic limits. In the ultra-relativistic limit, the helicity basis is generally less magical: after averaging over angle and over all stabilizer states, the helicity-basis $2\to2$1 is smaller in every case, for example $2\to2$2 versus $2\to2$3 for Bhabha and Møller scattering, and $2\to2$4 versus $2\to2$5 for $2\to2$6. The mechanism is explicit: helicity conservation forces many helicity amplitudes to vanish in the massless limit, so the helicity-basis scattering matrix is sparse—Bhabha scattering has only six nonzero entries versus a fully populated lab-basis matrix—restricting the final state to fewer superpositions. The lab basis, by contrast, yields less magic in the non-relativistic limit, where the lab-basis amplitudes become proportional to the identity for Bhabha, $2\to2$7, and $2\to2$8 scattering, so that scattering generates no magic at all and any helicity-basis magic is entirely an artifact of the basis transformation. In Møller scattering, the non-relativistic lab-basis matrix is instead the sparse one, and the non-local magic exactly coincides with the lab-basis magic for several stabilizer classes. Notably, the non-local magic vanishes for entangled initial states in all cases examined, extending the observation of Ref. (Robin et al., 27 Oct 2025) beyond Møller and Bhabha scattering. Two exceptional stabilizer classes (G12, G13) in ultra-relativistic Bhabha scattering show larger angle-averaged magic in the helicity basis, yet their non-local magic vanishes—indicating that the non-local quantity need not interpolate simply between the two bases.

Electroweak sector: the weak mixing angle result is basis-independent

For charged-lepton scattering mediated by photon and $2\to2$9 exchange at $\hat{s}_W^2$0, the paper tests whether the striking result of Ref. (Liu et al., 22 Sep 2025)—that $\hat{s}_W^2$1 agrees at the sub-percent level with the magic-minimizing $\hat{s}_W^2$2—survives the change of frame. The answer is affirmative: in both bases, most individual stabilizer-state magic distributions $\hat{s}_W^2$3 exhibit a minimum near $\hat{s}_W^2$4, and the doubly averaged $\hat{s}_W^2$5 shows a local minimum near the Standard Model value in either basis. The magic-minimizing $\hat{s}_W^2$6 deviates substantially from the value $\hat{s}_W^2$7 toward $\hat{s}_W^2$8 in the range $\hat{s}_W^2$9, with agreement improving at higher energies. The lab basis reproduces $M_2 = 0$0 more accurately at $M_2 = 0$1 specifically, so the sub-percent agreement of the earlier work is a lab-basis feature, but the qualitative conclusion—that the electroweak coupling structure minimizes magic production—is frame-independent. As in the QED case, the helicity basis carries systematically less magic at these massless-fermion energies, consistent with helicity selection rules.

Circuit cost: sparser amplitudes are cheaper to simulate

The final section connects magic content to fault-tolerant resource counts via a linear-combination-of-unitaries (LCU) construction (Childs et al., 2012) of the ultra-relativistic Bhabha scattering matrix. The helicity-basis matrix diagonalizes under the Clifford unitary $M_2 = 0$2 into four Pauli strings, requiring two ancilla qubits; the SELECT operator factorizes entirely into Clifford gates, so the $M_2 = 0$3-count stems solely from approximating the three non-Clifford rotations in state preparation, giving $M_2 = 0$4. The lab-basis matrix decomposes into eight Pauli strings, but since the two bases are related by a unitary, a two-ancilla implementation exists with two additional generic $M_2 = 0$5 rotations, adding $M_2 = 0$6 $M_2 = 0$7 gates at generic scattering angles. The basis with less magic is thus also cheaper to simulate, and at $M_2 = 0$8 or $M_2 = 0$9 the extra overhead vanishes—at $R_y^{\otimes 2}(\theta)$0 the full helicity circuit requires zero $R_y^{\otimes 2}(\theta)$1 gates—mirroring the angular structure of the magic distributions.

Limitations and open questions

The analysis is confined to tree level, to $R_y^{\otimes 2}(\theta)$2 charged-lepton scattering, and to frames fixed event-by-event so that amplitudes carry no azimuthal dependence; a frame with fixed transverse axes would introduce $R_y^{\otimes 2}(\theta)$3-dependence and requires separate treatment. The claim that the non-local magic tracks the less-magical basis is argued from the observed cases rather than proven in general, and the two exceptional Bhabha classes where the ordering inverts, yet $R_y^{\otimes 2}(\theta)$4, are not fully explained. Whether the near-coincidence of $R_y^{\otimes 2}(\theta)$5 with the magic-minimizing value reflects a deeper principle of the Standard Model, or extends beyond tree level and to non-abelian sectors, remains unresolved. The phase-convention dependence of the single-qubit computational basis, discussed in (Chu et al., 14 Jul 2026), is also set by convention rather than analyzed.

Conclusion

This paper establishes that the frame dependence of magic in scattering is controlled and largely kinematic: a basis rotation can generate at most $R_y^{\otimes 2}(\theta)$6, helicity selection rules make the helicity basis less magical in the ultra-relativistic regime, and the ordering reverses non-relativistically where lab-basis amplitudes become trivial. The two principal physical conclusions of prior work survive the change of frame—QED remains an inefficient magic generator, and the weak mixing angle sits near a magic-minimizing value in both bases—while the sparser helicity amplitudes translate directly into a lower $R_y^{\otimes 2}(\theta)$7-gate cost in LCU circuit realizations. The result places the "Standard Model minimizes quantum resources" hypothesis on a frame-independent footing and ties the magic content of a collision process to the concrete cost of simulating it.