Qubit Lattice Algorithm (QLA)

• QLA is a numerical framework that encodes classical and quantum wave equations on a spatial lattice with qubit registers, ensuring high precision in simulating dynamics.
• The method alternates local unitary collision operations with nearest-neighbor streaming, incorporating potential operators to handle inhomogeneities and nonlinearities.
• QLA achieves second-order accuracy in reproducing Maxwell’s equations, conserves energy to high precision, and is inherently parallelizable for efficient simulations.

The Qubit Lattice Algorithm (QLA) is an explicit, initial-value numerical method that encodes the evolution of classical or quantum wave equations—such as Maxwell’s equations—on a discrete spatial lattice where each site hosts a register of qubit amplitudes. The fundamental update mechanism alternates local unitary “collision” operations (entangling or mixing qubit amplitudes at a site) with nearest-neighbor “streaming” (shifting specific amplitudes between sites). For inhomogeneous or nonlinear equations, QLA incorporates local “potential” operators that encode gradients of parameters (e.g., refractive index), extending the underlying unitary sequence. Through careful construction, the QLA reproduces the target partial differential equation (PDE) to second order in the lattice discretization, conserves the appropriate physical norm (e.g., electromagnetic energy) to high precision, and facilitates automatic emergence of interface physics without explicit boundary conditions (Soe et al., 2 Dec 2025).

1. Mathematical Formulation of the QLA

The derivation of QLA implementations for Maxwell’s equations begins by recasting the curl-form equations in an energy-normalized variable set via a Dyson map. For an inhomogeneous, non-magnetic dielectric, Maxwell’s curl equations are: $$\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E}, \quad \frac{\partial \mathbf{D}}{\partial t} = \nabla \times \mathbf{H}, \quad \mathbf{D} = \epsilon_0 n^2(\mathbf{r}) \mathbf{E}, \quad \mathbf{B} = \mu_0 \mathbf{H}.$$ Introducing the Dyson-mapped vector field,

$$\mathcal{U} = W^{1/2} u, \quad u = (\mathbf{E}, \mathbf{H})^\top, \quad W = \mathrm{diag}(\epsilon_0 n^2, \mu_0),$$

yields unitary evolution in a homogeneous medium. In component form for 2D $(x,y)$: $$\begin{aligned} \partial_t q_0 &= \frac{1}{n_x} \partial_y q_5,\ \partial_t q_1 &= -\frac{1}{n_y} \partial_x q_5,\ \partial_t q_2 &= \frac{1}{n_z}(\partial_x q_4 - \partial_y q_3),\ \partial_t q_3 &= -\partial_y(q_2 / n_z),\ \partial_t q_4 &= \partial_x(q_2 / n_z),\ \partial_t q_5 &= -\partial_x(q_1 / n_y) + \partial_y(q_0 / n_x). \end{aligned}$$ Each $\mathcal{U}(x, y, t) = (q_0, \ldots, q_5)^\top$ encodes the Dyson-mapped electromagnetic field at a site (Soe et al., 2 Dec 2025).

2. QLA Operator Sequence and Lattice Update

The QLA update at each timestep comprises a minimal sequence of alternating collision and streaming operations, completed by local “potential” steps (to handle inhomogeneity):

• Collision Operators ($C_X, C_Y$): Block-diagonal unitaries acting on pairs (or higher groupings) of the six local amplitudes. In $x$-direction,

$$C_X = \begin{bmatrix} 1 & &&&& \ & \cos\theta_1 &&&& -\sin\theta_1 \ && \cos\theta_2 & & -\sin\theta_2 & \ &&& 1 && \ && \sin\theta_2 & & \cos\theta_2 & \ & \sin\theta_1 &&&& \cos\theta_1 \end{bmatrix},$$

with small angles $\theta_1 = \delta/(4 n_y)$ and $\theta_2 = \delta/(4 n_z)$.

• Streaming Operators ($S^{x\pm}_{\alpha,\beta}$): Shift a selected pair of qubit amplitudes (e.g., $(q_1, q_4)$, $(q_2, q_5)$) forward/backward along $x$ or $y$, leaving other amplitudes unchanged.
• Potential Operators ($V_X, V_Y$): Sparse, generally non-unitary matrices encoding discrete representations of refractive index gradients, with angles

$$\beta_0 = \delta^2 (\partial_x n_y)/n_y^2, \quad \beta_2 = \delta^2 (\partial_x n_z)/n_z^2,$$

for $x$ (analogous formulation for $y$).

One lattice timestep is constructed as a composition: $$\mathcal{U}(t+1) = V_Y V_X \cdot U_Y U_X \cdot \mathcal{U}(t),$$ where $U_X, U_Y$ denote composite collision-streaming sequences in $x$ and $y$ (8 steps each for second-order accuracy) (Soe et al., 2 Dec 2025, Vahala et al., 2023).

3. Continuum Limit, Accuracy, and Energy Conservation

Under “diffusion scaling” ($\Delta t \sim \delta^2$), Taylor expansion of the composed QLA operator up to $O(\delta^2)$ recovers the continuum Dyson-mapped Maxwell equations. The remaining error per time step is $O(\delta^4 / \Delta t) \sim O(\delta^2)$, confirming second-order accuracy in both time and space. This accuracy is set by the choice of the collision and potential angles, computed locally from the refractive index and its derivatives.

In the Dyson-normalized basis, energy conservation becomes an exact (formal) statement: $$\mathcal{E}(t) = \|\mathcal{U}(t)\|^2 = \iint [\epsilon_0 n^2 E^2 + \mu_0 H^2] \,dx\,dy$$ For practical QLA runs, relative drift in $\mathcal{E}(t)$ is typically less than $10^{-7}$ for $O(10^5)$ steps, limited only by non-unitary contributions from $V_X, V_Y$. For sufficiently small $\delta$ and with interleaved (half-angle) potentials, norm drift can be rendered negligible (Soe et al., 2 Dec 2025, Vahala et al., 2023).

4. Physical Boundary Handling and Emergent Interface Phenomena

A signature feature of QLA is the absence of explicit interface or jump condition enforcement. Material discontinuities, such as a step in refractive index at a dielectric interface, are encoded solely via local changes in potential angles $\beta$ (and thus in $V_X, V_Y$) at the affected sites. The time-evolving wavepacket naturally splits into reflected and transmitted components as prescribed by Maxwell’s equations.

Empirical benchmarks recover correct Fresnel reflection and transmission coefficients for normally or obliquely incident pulses, including correct amplitude and phase relations for both fields. Under total internal reflection, QLA automatically captures the Goos–Hänchen lateral shift associated with evanescent coupling, as well as the associated transient energy transfer at the interface, all without imposing ad hoc field continuity conditions. For broad spatial pulses, discrepancies with theory decrease as the pulse width increases (Soe et al., 2 Dec 2025, Soe et al., 14 Jan 2026).

5. Implementation and Parallelization

At each lattice site, the required operators are small (6 × 6) sparse unitaries (or nearly unitary), which can be explicitly constructed and, in quantum hardware, decomposed into at most a constant number of 2-qubit gates for each collision operation. Streaming is realized as basis state permutations or SWAPs. The update cycle, with all operations acting locally or on nearest neighbors, is inherently parallelizable.

The QLA update cycle summarizes as:

1. Update site amplitudes using $U_X$ (collide–stream chain in $x$).
2. Update using $U_Y$ (collide–stream in $y$).
3. Apply $V_X$ (potential in $x$).
4. Apply $V_Y$ (potential in $y$). Optionally, monitor the conserved norm and extract physical fields via the inverse Dyson map at any time (Soe et al., 2 Dec 2025).

6. QLA in Higher-Dimensional and Anisotropic Media

The formalism generalizes to higher-dimensional and anisotropic dielectrics by extending the indices in the Dyson map and the collision/potential operator blocks (e.g., for full tensor $\epsilon(x, y)$, the collision angles correspond to each principal axis). The same operator structure recovers second-order accuracy. For anisotropic refractive index profiles, additional sparsity in the potential operators accurately tracks local derivatives (Vahala et al., 2023). No boundary conditions are imposed; the scattering, interference, and field generation at complex materials are emergent consequences of the local QLA update.

7. Physical Examples and Applications

QLA has been validated in multiple regimes:

• 2D propagation and scattering of bounded electromagnetic pulses from infinite planar and localized dielectric interfaces exhibit amplitude splits in full agreement with Fresnel theory and Goos–Hänchen-type lateral displacements under total internal reflection.
• For Gaussian pulses at normal or oblique incidence, QLA accurately reproduces both the transmitted and reflected pulse shapes, phase shifts, and emergent Huygens-like secondary wavefronts in the transmitted field, dependent on the sharpness of the pulse in Fourier space (Soe et al., 2 Dec 2025, Soe et al., 14 Jan 2026).
• Autonomous, self-consistent treatment of boundary and interface phenomena confirms the suitability of QLA for simulation of transient and steady-state wave phenomena in engineered electromagnetic structures.

QLA thus provides a systematically improvable, nearly unitary algorithmic framework for electromagnetic wave simulation in classical and quantum settings, supporting both high-precision numerical studies and the development of quantum algorithms on near-term platforms. The approach is extensible to a broad range of hyperbolic PDEs beyond Maxwell’s equations, including non-Abelian gauge theories and quantum lattice Boltzmann models.