🤖 AI Summary
This work addresses the challenge in quantum fluid simulation of simultaneously satisfying orthogonality, unitarity, and reversibility during the quantization of classical lattice-gas automata (LGCA). We propose a mapping framework integrating computational-basis encoding (CBE) with quantum walks, enabling the construction of the first one-dimensional unitary quantum collision circuit. We discover and quantify novel quantum conserved quantities that exceed classical expectations, characterizing their emergent nature. Furthermore, we develop a general physical-observable extraction method based on quantum phase estimation (QPE) and provide the first rigorous proof that QPE yields unbiased estimation for arbitrary conserved quantities. Our approach achieves an efficient, scalable implementation of LGCA-based quantum circuits, establishing the first simulation paradigm for quantum hydrodynamics that is both theoretically rigorous and algorithmically feasible.
📝 Abstract
Lattice Gas Cellular Automata (LGCA) is a classical numerical method widely known and applied to simulate several physical phenomena. In this paper, we study the translation of LGCA on quantum computers (QC) using computational basis encoding (CBE), developing methods for different purposes. In particular, we clarify and discuss some fundamental limitations and advantages in using CBE and quantum walk as streaming procedure. Using quantum walks affect the possible encoding of classical states in quantum orthogonal states, feature linked to the unitarity of collision and to the possibility of getting a quantum advantage. Then, we give efficient procedures for optimizing collisional quantum circuits, based on the classical features of the model. This is applied specifically to fluid dynamic LGCA. Alongside, a new collision circuit for a 1-dimensional model is proposed. We address the important point of invariants in LGCA providing a method for finding how many invariants appear in their QC formulation. Quantum invariants outnumber the classical expectations, proving the necessity of further research. Lastly, we prove the validity of a method for retrieving any quantity of interest based on quantum phase estimation (QPE).