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This paper shows that the quantum lattice Boltzmann scheme for the 3D Dirac equation can be built exactly from gate-model operations, with no approximation beyond the classical method. It validates the circuits against a classical solver on a state-vector emulator and runs a reduced two-site instance on Rigetti hardware. The authors explicitly do not claim speedup, and state preparation, asymptotic cost, and noise remain open, so readers should treat this as a representability result rather than evidence of practical advantage.
Exact quantum circuits for a lattice Boltzmann realization of the Dirac equation
arXiv:2608.06570v2 Announce Type: replace Abstract: The quantum lattice Boltzmann 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. While this correspondence has been noted qualitatively, it has remained unclear whether it can be realized exactly and unitarily as a gate-model quantum circuit. This unitarity is a structural property of the scheme, which should, in principle, enable an operator-by-operator port onto quantum hardware. 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 boundary conditions as unitary circuits. Together, these constructions provide the first complete demonstration that the scheme is exactly representable in the gate model without any approximation beyond the classical method itself. 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, thus verifying our quantum implementation. Our aim is representability and correctness rather than computational speedup; questions of state preparation, asymptotic cost, and hardware noise are identified as open avenues for future work. Appendices restore physical units, derive the scheme's Zitterbewegung in closed form and compare it with a trapped-ion simulation, and execute a reduced two-site instance on a Rigetti superconducting processor, where choosing the qubits from the live calibration lowers the state-preparation error. All operators, circuits, and figures are reproducible from the open-source quantumKineticMethods library.
Abstract:The quantum lattice Boltzmann 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. While this correspondence has been noted qualitatively, it has remained unclear whether it can be realized exactly and unitarily as a gate-model quantum circuit. This unitarity is a structural property of the scheme, which should, in principle, enable an operator-by-operator port onto quantum hardware. 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 boundary conditions as unitary circuits. Together, these constructions provide the first complete demonstration that the scheme is exactly representable in the gate model without any approximation beyond the classical method itself. 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, thus verifying our quantum implementation. Our aim is representability and correctness rather than computational speedup; questions of state preparation, asymptotic cost, and hardware noise are identified as open avenues for future work. Appendices restore physical units, derive the scheme's Zitterbewegung in closed form and compare it with a trapped-ion simulation, and execute a reduced two-site instance on a Rigetti superconducting processor, where choosing the qubits from the live calibration lowers the state-preparation error. All operators, circuits, and figures are reproducible from the open-source quantumKineticMethods library.
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph)
arXiv:2608.06570 [quant-ph]
(or arXiv:2608.06570v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2608.06570 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 368 words
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