Exact quantum circuits for lattice Boltzmann realization of the Dirac equation

• URL: http://arxiv.org/abs/2608.06570v1
• Date: Thu, 06 Aug 2026 20:26:27 GMT
• Title: Exact quantum circuits for lattice Boltzmann realization of the Dirac equation
• Abstract summary: We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme.<n>We then compose them into single-axis, two- and three-dimensional time steps.<n> circuits reproduce the classical QLB solver to machine precision.
• Score: 0.11999555634662634
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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, which suggests that a QLB time step should map onto a sequence of quantum gates. Here we make that mapping explicit. We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme: the fixed rotation gates, the collision gate, the streaming shift as a controlled increment on a position register, the position-dependent potential as a phase oracle, and periodic and reflecting (bounce-back) boundary conditions as unitary circuits. We then compose them into single-axis, two- and three-dimensional time steps. On a state-vector emulator the resulting circuits reproduce the classical QLB solver to machine precision (maximum density deviation between $3.7\times10^{-12}$ and $1.0\times10^{-17}$ across the one-, two-, and three-dimensional tests), so the circuits are the scheme rather than an approximation of it. The scope is narrow: we establish that the Succi-Dellar theory can be implemented on a (gate-model) quantum computer, and report the associated gate counts. We make no claim of computational advantage; state preparation, measurement, and asymptotic cost are discussed as open questions. All operators, circuits, tests, and figures are reproducible from the open-source quantumKineticMethods library.

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