The Dirac-sVQA algorithm of Zuo, He, and Zhao (arXiv:2606.08053, 2026) simulates the one-dimensional nonlinear Dirac equation (NLDE) by a first-order splitting of each time step into an exactly implemented linear Dirac substep (a spinor-Fourier Dirac propagator on a joint position-spin qubit register) and a nonlinear correction performed as a variational quantum update whose cost function is built from finitely many measurements of overlap, self-channel, and cross-channel spinor observables (the D-QNPU module). The paper derives gate and shot complexity estimates and reports numerical agreement with classical solvers, but proves no error bounds; its conclusion explicitly names rigorous error analysis as future work. Establish a priori error bounds for Dirac-sVQA that couple the splitting and Fourier pseudospectral discretization error, the ansatz-expressibility and optimization error of the variational update, and the finite-shot statistical error of the nonlinear measurements, and control the accumulation of these errors over repeated state-dependent time steps. The problem is open.

Background

The nonlinear Dirac equation (NLDE) extends the 1928 Dirac equation for spin-1/2 relativistic particles with self-interaction terms. In the one-dimensional model problem on the torus T=R/(2π)\mathbb{T}=\mathbb{R}/(2\pi) with periodic boundary conditions studied by Zuo, He, and Zhao, the unknown is a two-component complex spinor Ψ(t,x)∈C2\Psi(t,x)\in\mathbb{C}^2 solving i∂tΨ=−iα∂xΨ+βΨ+λ1(Ψ∗βΨ)βΨ+λ2∣Ψ∣2Ψ,i\partial_t\Psi=-i\alpha\partial_x\Psi+\beta\Psi+\lambda_1(\Psi^*\beta\Psi)\beta\Psi+\lambda_2|\Psi|^2\Psi, where α\alpha and β\beta are Pauli matrices acting on the two spinor components, Ψ∗\Psi^* denotes the conjugate transpose, and the two nonlinearities are physically distinct: setting λ2=0\lambda_2=0 gives the Soler model of scalar self-interaction from quantum field theory, while setting λ1=0\lambda_1=0 gives a density-dependent nonlinearity used for Bose–Einstein condensates in optical lattices. Classically the NLDE is solved by operator splitting: the scheme used by Zuo, He, and Zhao is a first-order Lie-type splitting Ψ(x,t+τ)=Φτp∘Φτk(Ψ(x,t))\Psi(x,t+\tau)=\Phi_\tau^p\circ\Phi_\tau^k(\Psi(x,t)) that separates the flow Φk\Phi^k of the linear Dirac operator −iα∂x+β-i\alpha\partial_x+\beta, which is diagonal and exactly solvable in Fourier space, from the flow Φp\Phi^p of the nonlinear part, which is a pointwise ordinary differential equation in the spinor and is advanced by one explicit Euler step; the spatial discretization is Fourier pseudospectral (grid values via the discrete Fourier transform). Rigorous error estimates for such classical schemes exist in specific regimes: Bao, Cai, Jia, and Yin (2016) proved error bounds for Crank–Nicolson and splitting Fourier pseudospectral methods in the nonrelativistic limit regime (where the speed of light tends to infinity and solutions oscillate on a fast time scale), Bao, Cai, and Yue (2022) proved improved uniform long-time bounds for the weakly nonlinear Dirac equation, and Krämer, Schratz, and Zhao (2021) analyzed splitting for Thirring-type interaction (a cubic, vector-like spinor self-interaction); Xu, Shao, and Tang (2013) review and analyze NLDE methods more broadly. So the classical, deterministic part of the error is well understood. On quantum hardware the state lives in a qubit register and every gate is unitary, so a state-dependent nonlinear update cannot be a fixed circuit. Zuo, He, and Zhao therefore proposed Dirac-sVQA, a splitting variational quantum algorithm. Each time step decomposes into: (i) a spinor-Fourier Dirac propagator (SFDP), an exact circuit implementation of the linear Dirac substep on a joint position-spin register, using the quantum Fourier transform (QFT, the quantum analogue of the discrete Fourier transform) to pass to momentum space where the spin-momentum coupling and mass-induced spin rotation are diagonal; and (ii) a nonlinear correction in which the explicit Euler step for the nonlinear subflow — whose state-dependent target map Q[Ψ~(t)]=I−iτG[Ψ~(t)]Q[\tilde\Psi(t)]=I-i\tau G[\tilde\Psi(t)] is not exactly unitary — is reformulated as a variational quantum algorithm (VQA): a parameterized quantum circuit, or ansatz, of depth dd whose parameters are optimized classically (by the L-BFGS-B method, a limited-memory bounded quasi-Newton optimizer) to minimize a cost function assembled from a small set of measurable observables: overlaps and self-channel and cross-channel spinor observables. This measurement-based nonlinear module is called the Dirac quantum nonlinear processing unit (D-QNPU), and its observables are estimated from finitely many measurement shots. The paper derives gate and shot complexity estimates and gives numerical evidence in three nonlinear regimes — agreement with a normalized classical splitting solver and stable error behavior in the root-mean-square error, with error saturation beyond ansatz depth about 8 — but no rigorous error bounds. What is missing is exactly what the paper's conclusion names as open: rigorous error analysis. Three error sources of the scheme are uncontrolled in any rigorous sense: the operator-splitting and pseudospectral discretization error; the variational-update error, consisting of the ansatz-expressibility error (how well depth-dd circuits can represent the intermediate spinor state) plus the optimizer's residual; and the statistical error of the finite-shot D-QNPU observable estimates. Because the nonlinear update depends on the intermediate state, errors at one step enter the observables measured at the next, so error accumulation over many steps must be controlled; the paper's experiments show that norm control is decisive — the Dirac-sVQA solution tracks the normalized classical Fourier pseudospectral baseline (NC) closely, while the unnormalized classical baseline (C) deteriorates rapidly — and, because the explicit Euler target map is not exactly unitary while the amplitude-encoded ansatz states are normalized by construction, each variational step projects a non-unitary target onto the normalized ansatz manifold. Existing general theory does not apply: the variational quantum simulation theory of Yuan, Endo, Zhao, Li, and Benjamin (2019) and the error bounds for variational quantum time evolution by Zoufal, Sutter, and Woerner (2021) concern linear time evolution; nonlinear variational solvers (Lubasch et al. 2020; Köcher et al. 2025 for the nonlinear Schrödinger equation, the direct methodological predecessor) likewise carry no rigorous error bounds. The only quantum NLDE scheme with a proven convergence result is the non-variational quantum lattice Boltzmann scheme (a quantum analogue of the lattice Boltzmann streaming-and-collision method for kinetic equations) of Li, Zhang, and Zhang (2023) for the Gross–Neveu model (a fermionic field theory with a quartic spinor self-interaction), and earlier quantum-walk simulations of the NLDE (a quantum walk being the unitary quantum analogue of a random walk; Lee, Kurzyński, and Nha 2015) had no analysis either.

Problem Statement

For the one-dimensional nonlinear Dirac equation on the torus T=R/(2π)\mathbb{T}=\mathbb{R}/(2\pi), i∂tΨ=−iα∂xΨ+βΨ+λ1(Ψ∗βΨ)βΨ+λ2∣Ψ∣2Ψ,i\partial_t\Psi=-i\alpha\partial_x\Psi+\beta\Psi+\lambda_1(\Psi^*\beta\Psi)\beta\Psi+\lambda_2|\Psi|^2\Psi, with periodic boundary conditions, as solved by the single canonical Dirac-sVQA scheme of Zuo, He, and Zhao (arXiv:2606.08053) — a first-order Lie-type operator splitting Ψ(x,t+τ)=Φτp∘Φτk(Ψ(x,t))\Psi(x,t+\tau)=\Phi_\tau^p\circ\Phi_\tau^k(\Psi(x,t)) combining an exactly implemented spinor-Fourier linear Dirac substep (SFDP) with a nonlinear substep in which one explicit Euler update, whose target map Q[Ψ~(t)]=I−iτG[Ψ~(t)]Q[\tilde\Psi(t)]=I-i\tau G[\tilde\Psi(t)] is not exactly unitary, is carried out as a variational correction minimizing the implementable cost Cimpl(θ)=−2Eov−2τγh[(λ2+λ1)Eself+(λ2−λ1)Ecross]C_{\mathrm{impl}}(\theta)=-2E_{\mathrm{ov}}-2\tau\frac{\gamma}{h}\left[(\lambda_2+\lambda_1)E_{\mathrm{self}}+(\lambda_2-\lambda_1)E_{\mathrm{cross}}\right] (with γ\gamma the conserved mass and h=2π/2nh=2\pi/2^n the grid spacing), estimated by the D-QNPU from finitely many measurements of the overlap, self-channel, and cross-channel observables Eov,Eself,EcrossE_{\mathrm{ov}},E_{\mathrm{self}},E_{\mathrm{cross}}, with a depth-dd hardware-efficient ansatz built from RxR_x, RzR_z, and CNOT gates and optimized classically — establish rigorous a priori error bounds between the scheme's computed spinor and the exact NLDE solution at a final time TT, accounting for all three error sources arising from the scheme's three components: (i) the first-order operator-splitting time-discretization error together with the Fourier pseudospectral space discretization; (ii) the approximation error of the variational update, comprising the expressibility error of the depth-dd ansatz, the residual of the classical optimization, and the per-step projection of the non-unitary Euler target onto the normalized ansatz manifold; and (iii) the statistical estimation error of the finite-shot D-QNPU measurements. The bounds should make explicit the dependence on the time step τ\tau, the spatial resolution (number of position qubits nn), the ansatz depth dd, the shot budget, the optimization tolerance, and the final time TT, under stated assumptions on the regularity of the exact solution and the expressibility of the ansatz class, and should control the accumulation of these errors over repeated state-dependent time steps, including the effect of the per-step projection onto the normalized ansatz manifold.

References

  1. Qian Zuo, Ying He, Xiaofei Zhao, Numerical solution of the nonlinear Dirac equation by a splitting variational quantum algorithm, arXiv:2606.08053 (2026), DOI 10.48550/arXiv.2606.08053, https://arxiv.org/abs/2606.08053
  2. Weizhu Bao, Yongyong Cai, Xiaowei Jia, Jia Yin, Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime, arXiv:1511.01192 (2015), published in Science China Mathematics 59 (2016) 1461–1494, DOI 10.1007/s11425-016-0272-y, https://arxiv.org/abs/1511.01192
  3. Weizhu Bao, Yongyong Cai, Feng Yue, Improved uniform error bounds on time-splitting methods for the long-time dynamics of the weakly nonlinear Dirac equation, arXiv:2203.05886 (2022), https://arxiv.org/abs/2203.05886
  4. Patrick Krämer, Katharina Schratz, Xiaofei Zhao, Splitting methods for nonlinear Dirac equations with Thirring type interaction in the nonrelativistic limit regime, Journal of Computational and Applied Mathematics 387, 112494 (2021), DOI 10.1016/j.cam.2019.112494, https://doi.org/10.1016/j.cam.2019.112494
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  6. Ningning Li, Jing Zhang, Yongqian Zhang, Convergence of a quantum lattice Boltzmann scheme to the nonlinear Dirac equation for Gross–Neveu model in 1+1 dimensions, arXiv:2302.00245 (2023), https://arxiv.org/abs/2302.00245
  7. Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, Simon Benjamin, Theory of variational quantum simulation, arXiv:1812.08767 (2018), published in Quantum 3, 191 (2019), DOI 10.22331/q-2019-10-07-191, https://arxiv.org/abs/1812.08767
  8. Christa Zoufal, David Sutter, Stefan Woerner, Error bounds for variational quantum time evolution, arXiv:2108.00022 (2021), https://arxiv.org/abs/2108.00022
  9. Michael Lubasch, Jaewoo Joo, Pierre Moinier, Martin Kiffner, Dieter Jaksch, Variational quantum algorithms for nonlinear problems, arXiv:1907.09032 (2019), published in Physical Review A 101, 010301 (2020), DOI 10.1103/PhysRevA.101.010301, https://arxiv.org/abs/1907.09032
  10. Nikolas Köcher, Hendrik Rose, Sachin S. Bharadwaj, Jörg Schumacher, Stefan Schumacher, Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm, arXiv:2407.02989 (2024), published in Scientific Reports 15, 23478 (2025), DOI 10.1038/s41598-025-05660-3, https://arxiv.org/abs/2407.02989
  11. Chang-Woo Lee, Paweł Kurzyński, Hyunchul Nha, Quantum walk as a simulator of nonlinear dynamics: Nonlinear Dirac equation and solitons, arXiv:1512.08358 (2015), https://arxiv.org/abs/1512.08358