Quantum Encoding in Leptophilic U(1)' Models

• The paper introduces a modular quantum encoding technique that maps anomaly-free leptophilic U(1)' gauge models into efficient quantum circuits.
• It employs qubit-based encoding with single-qubit rotations and controlled-phase gates to enforce gauge invariance and anomaly cancellation.
• The framework enables high-fidelity simulations of scattering processes and effective operator evolution on current and next-generation quantum devices.

A quantum encoding framework for leptophilic extensions systematically maps anomaly-free $U(1)'_{\ell}$ gauge theories—where only charged lepton flavors possess nonzero charges—into resource-efficient, symmetry-preserving quantum circuits tailored for current- and next-generation quantum devices. By encoding the family-dependent leptonic charges directly in qubit registers, the gauge and anomaly constraints are enforced at the level of quantum states and circuit identities. The approach provides a modular, scalable interface between field-theoretic models of leptophilic new physics and quantum information platforms, with applications to the simulation of scattering processes, effective operator evolution, and precision measurements.

1. Field-Theoretic Structure of Leptophilic $U(1)'_{\ell}$ Gauge Extensions

Leptophilic $U(1)'_{\ell}$ models augment the Standard Model (SM) gauge group by an additional abelian symmetry acting exclusively on the three SM charged lepton flavors ($e$, $\mu$, $\tau$) through generation-dependent charges $q_{e}, q_{\mu}, q_{\tau}$. The neutral gauge boson $Z'_{\ell}$ couples to the lepton current: $$\mathcal{L} \supset g' Z'_{\mu} J^{\mu}_{\ell}, \qquad J^{\mu}_{\ell} = \sum_{\alpha=e,\mu,\tau} q_\alpha (\bar{L}_\alpha \gamma^\mu L_\alpha + \bar{\ell}_{R\alpha} \gamma^\mu \ell_{R\alpha})$$ Gauge and gravitational anomaly cancellation impose: $$\sum_{\alpha} q_\alpha^3 = 0,\qquad \sum_{\alpha} q_\alpha = 0$$ Typical solutions include $(q_e, q_\mu, q_\tau) = (1, -1, 0)$ ("$L_\mu-L_\tau$" model) and $(1, 1, -2)$ ("universal" charge). These conditions ensure a consistent quantum field theory formulation and uniquely determine the possible flavor structures of the leptophilic sector (Kara, 20 Jan 2026, Kara, 2 Dec 2025).

2. Qubit Encoding of Leptophilic Charge Structure

Each flavor mode $\alpha$ is mapped to a single qubit, with computational basis $\{\ket{0}_\alpha, \ket{1}_\alpha\}$ representing lepton occupation numbers. The number operator is

$\hat n_\alpha = \frac{I - Z_\alpha}{2}$

and the total $U(1)'_{\ell}$ charge operator becomes

$$\hat Q' = \sum_\alpha q_\alpha \hat n_\alpha = \sum_\alpha \frac{q_\alpha}{2} (I - Z_\alpha)$$

Dropping the trivial identity, the generator in the computational basis is

$$\hat Q'_{\rm qubit} = \sum_\alpha \frac{q_\alpha}{2} Z_\alpha$$

Gauge transformations act as

$$U(\varphi) = \exp(i \varphi \hat Q') = \prod_\alpha R_Z^{(\alpha)}(\varphi q_\alpha)$$

where $R_Z^{(\alpha)}$ denotes a $Z$-axis rotation on qubit $\alpha$. This encoding is diagonal and fully scalable with the number of flavors and charge assignments (Kara, 20 Jan 2026).

3. Enforcing Gauge Invariance and Anomaly Cancellation

Gauge invariance dictates that physical (computational) states satisfy $\hat Q' \ket{\psi} = 0$. The corresponding projector is

$$\hat P = \int_0^{2\pi}\frac{d\varphi}{2\pi}\, U(\varphi) = \frac1{2\pi} \int d\varphi\, e^{i \varphi \hat Q'}$$

Operationally, quantum circuits are constructed so that all logical layers commute with $U(\varphi)$, ensuring that state preparation, time evolution, and measurement remain within the invariant sector ($\hat Q' = 0$). Anomaly cancellation ($\sum_{\alpha} q_\alpha = 0$) further ensures $U(2\pi) = I$ for all qubit states, preventing global phase ambiguities and spurious representations (Kara, 20 Jan 2026).

4. Mapping Gauge-Invariant Operators to Quantum Circuits

At energies $s \ll m^2_{Z'}$, $Z'_\ell$ is integrated out and the low-energy effective Hamiltonian is

$$H_{\rm EFT} \simeq \sum_{\alpha < \beta} \frac{g'^2 q_\alpha q_\beta}{m_{Z'}^2}\, \hat n_\alpha \hat n_\beta$$

Up to overall constants and single-qubit terms, the nontrivial two-body interaction is diagonal in the $Z$ basis. Its Trotterized time evolution is implemented as a product of controlled-phase gates: $$e^{-i H_{\rm EFT} \Delta t} \approx \prod_{\alpha<\beta} \exp(i \theta_{\alpha\beta} Z_\alpha Z_\beta),\qquad \theta_{\alpha\beta} = \frac{g'^2 q_\alpha q_\beta \Delta t}{4 m_{Z'}^2}$$ Each two-qubit term $\exp(i\theta Z_\alpha Z_\beta)$ is compiled via the standard $\{\mathrm{CNOT}, R_Z\}$ circuit with depth $\mathcal{O}(1)$ per qubit pair (Kara, 20 Jan 2026).

For resolved mediator regimes, the propagator phase

$$U_\varphi = \exp(i \varphi_{\rm res}(s) \hat Q') = \prod_\alpha R_Z^{(\alpha)}(\varphi_{\rm res} q_\alpha)$$

with

$$\varphi_{\rm res}(s) = \arg\left[\frac{g'^2}{s - m_{Z'}^2 + i m_{Z'} \Gamma_{Z'}}\right]$$

is also implemented as a product of single-qubit $Z$ rotations.

5. Explicit Quantum Circuits for Leptophilic Scattering

An explicit 2$\rightarrow$2 scattering quantum circuit for processes $$\ell_\alpha \ell_\beta \rightarrow \ell_\alpha \ell_\beta$$ proceeds as follows:

1. State Preparation: Apply Hadamard gates $H_\alpha \otimes H_\beta$ to both flavor qubits, generating a superposition of possible initial lepton states,

$$\ket{00}_{\alpha\beta} \xrightarrow{H_\alpha \otimes H_\beta} \frac{1}{2}\big(\ket{00}+\ket{01}+\ket{10}+\ket{11}\big)$$

1. Interaction: Apply the controlled-phase gate $\exp(i\theta Z_\alpha Z_\beta)$ encoding the effective contact interaction.
2. Measurement Basis: Reverse the Hadamard gates.
3. Readout: Measure both qubits in $\{0,1\}$.

The difference of outcome probabilities, $P(00) - P(11) \propto \sin \theta$, encodes the desired scattering amplitude,

$$\mathcal{A}(\ell_\alpha \ell_\beta \to \ell_\alpha \ell_\beta) \sim g'^2 \frac{q_\alpha q_\beta}{m_{Z'}^2}$$

The entire circuit preserves gauge invariance at each step, as all quantum gates commute with $U(\varphi)$. Trotterization errors do not break gauge symmetry, enabling high-fidelity simulations even for shallow-depth devices (Kara, 20 Jan 2026).

6. Resource Requirements and Quantum Architecture Compatibility

For $n_f$ lepton flavors, the framework requires exactly $n_f$ qubits. The maximal number of two-qubit gates equals the count of distinct, nonvanishing charge products, reaching at most $n_f(n_f-1)/2$. Circuit depth is $\mathcal{O}(1)$ for parallel gates and $\mathcal{O}(\mathrm{deg})$ on degree-limited hardware. No ancilla qubits or nonstandard measurements are necessary. In universal-charge cases, the complete protocol fits within three qubits and three controlled-phase gates, with total circuit depth less than $10$ ({\rm H, R_Z, CNOT} basis) (Kara, 20 Jan 2026).

This matches the capabilities of existing superconducting and trapped-ion quantum processors, and the methodology generalizes trivially to larger flavor sectors or additional $U(1)$ extensions by including extra qubits and phase gates.

7. Interplay with Effective Field Theory and Observables

The quantum encoding schema provides a Hamiltonian-level realization of the effective four-lepton operators arising in the low-energy limit of $U(1)'_{\ell}$ models. The EFT procedure follows:

• Write the ultraviolet Lagrangian with SM + $U(1)'_{\ell}$ symmetry and SM singlet scalars to induce $Z'_\ell$ and vectorlike-fermion masses (Kara, 2 Dec 2025).
• Ensure anomaly cancellation via suitable charge assignments and minimal vectorlike lepton completion if required.
• Integrate out $Z'_\ell$ to obtain the effective dimension-six SMEFT operators,

$$\mathcal{L}_{\rm eff}^{(6)} = -\frac{g_\ell^2}{2 M_{Z_\ell}^2}J'_\mu J'^\mu$$

with explicit charge structures for $C_{ll}, C_{le}, C_{ee}$.

• Evolve Wilson coefficients via RGEs to the electroweak scale, preserving the product-charge structure.
• Match onto LEFT at low energies to produce all relevant four-lepton and neutrino-current observables.

A single effective scale,

$$\Lambda_{\rm eff}^{(\alpha\beta)} = \frac{M_{Z_\ell}}{g_\ell \sqrt{|q_\alpha q_\beta|}}$$

governs all tree-level signatures in SMEFT and LEFT: precision $e^+e^- \to \ell^+\ell^-$, neutrino trident production, $\nu e$ scattering, parity-violating Møller scattering, and muon decay parameters. This unification enables succinct mapping from UV charge assignments directly to phenomenological predictions and experimental constraints (Kara, 2 Dec 2025).

8. Scalability, Applications, and Validity Domain

The framework is linearly scalable: increasing the number of lepton flavors or extending to multiple $U(1)$ factors only incrementally increases qubit and gate resources. Validity of the low-energy encoding requires all kinematic invariants to remain well below the $Z'_\ell$ mass and any heavy-lepton thresholds; couplings must remain perturbative ($g_\ell \lesssim 1$). All charge assignments must yield zero gauge anomalies either directly or via explicit vectorlike lepton completion (Kara, 2 Dec 2025).

Applications span theoretical quantum simulations of BSM leptophilic sectors, "plug-and-play" model building for EFT interpretations in collider and low-energy data, and practical benchmarking of quantum hardware for symmetry-protected dynamics. Modularity and gauge-commuting circuit construction provide a reusable template for mapping beyond-SM gauge symmetries to quantum simulable Hamiltonians (Kara, 20 Jan 2026, Kara, 2 Dec 2025).