Exact quantum circuits for a lattice Boltzmann realization of the Dirac equation

The gist The quantum lattice Boltzmann (QLB) scheme of Succi and Dellar advances a four–component Dirac spinor on a lattice by a fixed sequence of local, exactly norm–preserving operations: a basis rotation, a collision, a streaming shift, and the inverse rotation; this unitarity is a structural property of the scheme, not an approximation <ref:2608.06570#pg2>.

The Dirac QLB Scheme

The scheme is based on replacing the field equation with the transport of a distribution on a discrete set of velocities, advanced by an alternating stream and collide update <ref:2608.06570#pg4>. The feature that distinguishes the Dirac QLB scheme from most discretizations of a partial differential equation is that each elementary operation is unitary, which means it exactly preserves the Euclidean (l2) norm of the field, thus conserving probability <ref:2608.06570#pg2>. This correspondence suggests that a QLB time step should map onto a sequence of quantum gates <ref:2608.06570#pg2>.

Quantum Circuit Construction

The paper provides an exact gate–level construction for every operation of the 3D Dirac QLB scheme, including fixed rotation gates, the collision gate, and the streaming shift as a controlled increment on a position register <ref:2608.06570#pg2>. The rotation and collision operators are explicit 4×4 matrices that are wrapped as single unitary gates and then transpiled using the Cartan (KAK) decomposition to find the shortest sequence of native gates, such as single-qubit rotations and controlled-NOT (CX) gates <ref:2608.06570#pg10>.

Qubit Encoding and Registers

The field is stored using amplitude encoding where the 4N field values are placed in the 2n amplitudes of a register of n = log2(4N) qubits, with each qubit pattern interpreted as a binary address <ref:2608.06570#pg5>. The state consists of two spinor qubits to encode the four spinor components and one position register per axis, resulting in a total of 2 + P a n(a) pos qubits <ref:2608.06570#pg5>. The combined index for the field is given by i = c + 4x + Nx y + NxNy z, which corresponds to the row-major (C-order) flattening of an array of shape (Nz, Ny, Nx, 4) <ref:2608.06570#pg9>.

Time Integration and Operations

A single QLB time step corresponds to a fixed gate sequence Ustep, which is composed of the rotation R−1a, the collision Qˆa, the streaming shift (Algorithm 1 or 2), and Ra <ref:2608.06570#pg9>. The streaming shift is realized as a controlled increment on the position register using a ripple of multi-controlled-X gates defined by Equation (15).

Potential and Boundary Conditions

The position-dependent potential V(x) enters entirely through the collision operator, where it manifests as either a phase oracle in the massless limit or a position–multiplexed collision in the massive case. Reflecting boundary conditions are realized by a single folded increment described in Algorithm 2, which doubles the line into a ring to ensure no probability is lost.

Numerical Verification and Results

The circuits reproduce the classical solver to machine precision on an emulator, with the maximum density deviation between 3.7×10−12 and 1.0 × 10−17 across one-, two-, and three-dimensional tests. The gate counts are reported for each building block, showing that streaming dominates the cost, with a free 323 step costing about 2700 CX gates.

Discussion and Outlook

The work establishes that the Succi–Dellar theory can be implemented on a (gate–model) quantum computer, and reports the associated gate counts. The construction suggests several continuations, including hardware-native compilation and simulation under noise. The result is one of exact representability, and the authors claim no more.

Reproducibility

All operators, circuits, tests and figures are in the public quantumKineticMethods library. The validation suite test port.py checks every operator and composition against its classical reference.

Author Contributions

N.S developed the quantum–circuit constructions and the open-source quantumKineticMethods library, performed the numerical verification, and wrote the manuscript. The E.Y initiated the group’s research direction in quantum computing and contributed to conceptual development through discussions, took part in verifying circuit constructions, and reviewed and edited the manuscript. K.G. and M.M contributed to the conceptual development through brainstorming discussions during the proposal stage of the project.

Acknowledgements

This work builds on the quantum lattice Boltzmann formulation of S. Succi and the three-dimensional isotropy analysis of P. Dellar and co-workers. The authors thank Wesley Jones for initiating and coordinating the quantum computing activities at the center, as well as Obadiah Reid and Ross Larsen for participating in discussions during the proposal stage of the project. Computations used GPU resources at the NLR, supported by internal funds Scalable Parallel Accelerated Computing (SPAC) and HPC User & Application Support (HPCAPPS).

A Physical Units

The scheme is formulated in lattice units, so physical values follow once a single dimensional scale, the lattice spacing ∆x (a length, in meters, m), is fixed; the time step ∆t (in seconds, s) is then set by the light–cone rule c ∆t = ∆x. The dimensionless quantities of Section 7 convert as follows, with mass and potential inverting the couplings of Eq. (9). The tests thus describe relativistic Dirac fermions at Compton–scale resolution, with sub-MeV to few-MeV masses, momenta and potentials, on picometer boxes evolved over tens of zeptoseconds. The group velocity row of Table 4 rewards a closer look because the one-dimensional spread is set by mass, whereas the differences among the massless two- and three-dimensional movers are a purely lattice (dimensional–splitting) effect.

A.3 Group velocities of the test packets

In two and three dimensions, an oblique mover must divide that single unit of speed among the axes, leaving no component at c. A direct fingerprint is the two-dimensional mover, which reuses the axis-aligned one-dimensional spinor u = Rx (0, 0, 1, 1)⊤/√2 but is launched with k = (0.6, 0.5): although kx ≈ ky, its measured drift v ≈ (0.62, 0.33) is strongly biased toward x. The reflecting-box packet reaches 0.97 c for two reinforcing reasons, one being its carrier momentum k ≈ (1.00, 0.68, 0.15) is strongly anisotropic and dominated by a single axis.

Table 3

Table 3 lists the measured two-qubit (CX) gate count and circuit depth of each building block after transpilation to these gates, as a function of the per-axis register size npos = log2 N qubits. Streaming grows with npos and dominates the cost. The spinor operators are constant-size; streaming grows with npos, and it dominates a time step (Eq. (22)). The potential phase oracle, for npos = 4, has a depth of 9. The periodic streaming for npos = 3, 4, 5, and 6 has depths of 106, 331, and others.

1. Improve quantum simulation of Dirac field dynamics by implementing exact gate-level constructions for lattice Boltzmann operations. The improved AI system can perform exact simulations of three-dimensional Dirac equations on a gate model quantum computer, as it reproduces the classical QLB solver to machine precision.

2. Enhance the accuracy of lattice kinetic solvers by allowing the system to use unitarity as a structural property rather than an approximation. This allows for time steps that map onto a sequence of quantum gates, ensuring exact preservation of probability at every step.

3. Develop quantum algorithms capable of simulating physical phenomena like Klein tunneling and Zitterbewegung with high fidelity. The system can accurately model the reflection fraction (R ≈ 0.19) when mass breaks Klein protection in massive potentials, achieving agreement up to 3.7 × 10-12 compared to classical solvers.

4. Create a reusable quantum circuit compiler that translates any unitary operator into an optimized gate sequence using the Cartan decomposition or Quantum Shannon decomposition methods. This system can verify that the compiled circuit reproduces the target matrix on the whole 2n-dimensional space up to an irrelevant phase, providing a guarantee of exact match.

5. Implement a multiplexed collision mechanism for massive potentials, allowing the system to apply different two-qubit spinor gates based on position register values. This enables simulating a uniformly controlled gate: the position register selects, like the address of a lookup table, which spinor gate the two spinor qubits receive.

6. Create efficient streaming operations by using a single loop parameterized by register size rather than generating separate circuits for each lattice dimension. The system can construct streaming circuits by hand and show that the algorithm is fixed once and for all, not a separate circuit for each N.

7. Design a phase oracle capable of imprinting site-dependent potentials on the position register with high efficiency. The system can synthesize this as a diagonal unitary on the npos position qubits, achieving it with O(N) elementary gates or even cheaper piecewise constant barriers for structured potentials.