MAGIC++: Magic in Particle Scattering

• MAGIC++ is a framework that quantifies non-stabilizerness (magic) produced in particle scattering, bridging quantum information theory and quantum field dynamics.
• It models gluon and graviton scattering as two-qubit processes, enabling quantitative analysis using metrics like the second Rényi magic (SSRE).
• The approach leverages double-copy relations, revealing a spin-suppression rule for magic generation with implications for quantum resource engineering.

MAGIC++ denotes an emerging line of inquiry at the interface of quantum information theory, quantum field theory, and high-energy particle scattering, centered on the study of non-stabilizerness—termed “magic”—as a quantifiable resource in quantum states produced via fundamental particle interactions. The term “magic” here tracks the deviation of a pure quantum state from the free resource class of stabiliser states, which are simulable via Clifford gates. In this setting, gauge boson and graviton scattering processes are recast as two-qubit quantum channels whose outputs can be quantitatively analyzed for their magic content, yielding insights into both resource theory and the algebraic relations underlying relativistic field amplitudes. Recent work presents the first exact, closed-form study of magic production in $2\to2$ gluon and graviton scattering, elucidating a universal spin-suppression phenomenon and connecting double-copy amplitude relations to “squaring rules” for magic (Gargalionis et al., 20 Aug 2025).

1. Resource-Theoretic Magic and Its Quantification

Within quantum computation, the resource-theoretic paradigm differentiates between Clifford operations, which efficiently simulate stabiliser states, and genuine quantum advantage, which requires circuit elements outside the stabiliser framework. Non-stabilizerness, or “magic,” quantifies the degree to which a given state constitutes a non-free resource. A rigorous quantifier is the Stabiliser-Rényi entropy (SRE), defined for an $n$-qubit pure state $|\psi\rangle$ by

$$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$

where $\mathcal P_n$ is the $n$-qubit Pauli group. For integer $q\ge2$, $M_q$ vanishes precisely for stabiliser states and is additive across independent systems. Studies have focused on the second Rényi magic $M_2$ (also termed “SSRE”), which is especially tractable and relevant for operational settings.

2. Gluon and Graviton Scattering as Two-Qubit Processes

At tree level, $2\to2$ scattering of massless vectored and tensor bosons (gluons and gravitons) is naturally recast as a process in the two-qubit space, as each particle’s helicity ($n$0) maps directly to the computational basis $n$1. The total outgoing state occupies

$n$2

with the physical spin entering entirely in the selection rules for allowed helicity amplitudes and their explicit kinematic dependencies. Specifically, Yang–Mills gluons (spin-1) and General Relativity gravitons (spin-2) are treated, with scattering amplitudes computed in the center-of-mass frame. Supersymmetric extensions allow interpolation between lower (spin-$n$3 gluinos) and higher (spin-$n$4 gravitinos) spin representatives, exposing the full structure of spin dependence.

3. Scattering Amplitudes and Closed-Form Magic Expressions

The tree-level amplitudes for gluon MHV (Maximally Helicity Violating) processes are: $n$5 with $n$6 the Mandelstam invariants and $n$7 color structure constants (Del Duca–Dixon–Maltoni basis). Graviton amplitudes are related via the Kawai–Lewellen–Tye (KLT) relations:

$n$8

yielding (for three physical configurations)

$n$9

Projecting onto a definite momentum and color sector, one obtains a normalized two-qubit state $|\psi\rangle$0 with amplitudes $|\psi\rangle$1, for which both concurrence

$|\psi\rangle$2

and $|\psi\rangle$3 can be computed as explicit functions of the scattering angle $|\psi\rangle$4 (with $|\psi\rangle$5, $|\psi\rangle$6, $|\psi\rangle$7).

For gluon scattering: $|\psi\rangle$8 For gravitons, one replaces $|\psi\rangle$9: $$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$0 A series expansion at small $$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$1 yields $$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$2 and $$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$3, indicating a much sharper activation threshold for magic in the graviton case.

4. Spin Dependence and Bounds on Magic

Empirical and closed-form analysis reveals a monotonic decrease of produced magic with increasing spin. Scanning over all $$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$4 two-qubit stabiliser initial states, the maximal $$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$5 achieved is

$$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$6

substantially below the conjectured absolute upper bound for two-qubit states [Liu et al. (2025)],

$$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$7

Extending to the supersymmetric sector, with spin $$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$8, the magic profile follows

$$M_q(|\psi\rangle) = -\frac{1}{1-q}\log_2\left( \frac{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2q} }{ \sum_{P\in \mathcal P_n}\langle\psi|P|\psi\rangle^{2} } \right),$$9

The profile narrows and the peak value decreases as $\mathcal P_n$0 increases. The integrated “magic power”, defined by

$\mathcal P_n$1

decreases monotonically with increasing $\mathcal P_n$2, consistent with a universal spin-suppression rule.

5. Double-Copy Physics and the Magic Squaring Rule

The central finding is that graviton amplitudes are algebraically the “square” of gluon amplitudes via the KLT/double-copy relation, so all Mandelstam variables in the gluon expressions are mapped to their squares in the graviton case. This results in all exponents in the $\mathcal P_n$3 formula being doubled for gravity relative to gauge theory, directly leading to the observed hierarchy $\mathcal P_n$4 except in a narrow region around $\mathcal P_n$5. Thus, the algebraic double-copy structure of the amplitudes is inherited by the magic content of the final quantum state, suggesting that in general, magic in a “product theory” is inherited from its factors—a phenomenon suggestive of a multiplicative or “squaring” rule for magic under the double-copy.

6. Applications and Prospects for MAGIC++

Several prospective applications and conjectures derive from these results:

• The universal spin-suppression of generated magic in relativistic scattering suggests that low-spin quanta should be preferred in quantum computing devices, simulators, or platforms aiming to maximize resource state generation.
• The mapping between double-copy amplitude relations and magic “squaring” rules points to possible new methods for engineering resource states in theories with double-copy structure, such as Yang–Mills–biadjoint–scalar correspondences.
• In condensed-matter and quantum-optical contexts, effective spin-$\mathcal P_n$6 excitations involved in scattering can be analyzed using the presented closed-form expressions, allowing prediction and optimization of resource state yields.
• The analytic formulas for $\mathcal P_n$7 in terms of kinematic invariants provide benchmarks for “quantum process tomography” in collider experiments, offering a pathway for empirical characterization and manipulation of magic in high-energy physics.

This suggests a deep, previously unexamined connection between the algebraic structure of quantum field theory amplitudes and contemporary resource theories in quantum information. A plausible implication is that further exploration of magic generation underlies future advances in both quantum algorithm design and high-energy experiment, particularly in the search for maximally resourceful quantum states produced by microscopic dynamics (Gargalionis et al., 20 Aug 2025).