TrackHHL: The 1-Bit Quantum Filter for particle trajectory reconstruction

The transition to the High-Luminosity Large Hadron Collider (HL-LHC) presents a computational challenge where particle reconstruction complexity may outpace classical computing resources. While quantum computing offers potential speedups, standard algorithms like Harrow-Hassidim-Lloyd (HHL) require prohibitive circuit depths for near-term hardware. Here, we introduce the 1-Bit Quantum Filter, a domain-specific adaptation of HHL that reformulates tracking from matrix inversion to binary ground-state filtering. By replacing high-precision phase estimation with a single-ancilla spectral threshold and exploiting the Hamiltonian's sparsity, we achieve an asymptotic gate complexity of $O(\sqrt{N} \log N)$, given Hamiltonian dimension $N$. We validate this approach by simulating LHCb Vertex Locator events with a toy model, and benchmark performance using the noise models of Quantinuum H2 trapped-ion and IBM Heron superconducting processors. This work establishes a resource-efficient track reconstruction method capable of solving realistic event topologies on noise-free simulators and smaller tracking scenarios within the current constraints of the Noisy Intermediate Scale Quantum (NISQ) era.