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

TL;DR

The 1-Bit Quantum Filter is introduced, a domain-specific adaptation of HHL that reformulates tracking from matrix inversion to binary ground-state filtering and achieves an asymptotic gate complexity of O(\sqrt{N} \log N)$, given Hamiltonian dimension $N.

Abstract

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.

Figures (5)

• Figure 1: Left: A simulated event in the LHCb detector. The larger grey dots represent the primary vertices of multiple simultaneous proton-proton collisions, while the small black dots indicate the detector hits generated by the produced particles. The faint grey lines depict the reconstructed tracks of the produced particles. The red dots and lines indicate the hits and particles produced in a single given collision, represented by the red circle. Right: Illustration of the definition of the Hamiltonian graph construction Nicotra_2023. Colored segments represent active variables $S_i=1$ forming valid tracks, while grey segments correspond to inactive variables $S_j=0$.
• Figure 2: Overview of the simulation tool events and algorithm performance. Left: (a) Example of an event with one collision vertex, 5 layers and 64 particles, where green lines indicate correct track segments and black lines indicate false segments. Right: (b) The performance of the segment identification algorithm as a function of the number of collision points (with 20 tracks each), showing the correct segment efficiency, fake rate, and the number of segment pairs considered/accepted.
• Figure 3: Circuit Architecture and Decomposition: (a) The complete 1-Bit Quantum Filter circuit, featuring state preparation, 1-bit phase estimation with flagging logic, and uncomputation. The additional phase gate $P(\phi)$ on the time register implements the diagonal term of the Hamiltonian. (b) The Direct Structural Synthesis (DSS) decomposition, explicitly showing that the controlled time evolution $U_{\text{DSS}}$ is implemented as a sequential application of controlled interaction gates $\prod_{k} G_k = \prod_{k} e^{i B_k t}$ corresponding to the non-zero off-diagonal terms of the Hamiltonian matrix. (c) The generalized structure of a single interaction gate $G_k = e^{i B_k t}$. For states with Hamming distance $d_H > 1$, a CNOT ladder ($S_{\text{lad}}$) disentangles the states onto a pivot ($S_{\text{piv}}$), while passive qubits ($S_{\text{pass}}$) act as controls to strictly enforce subspace selection.
• Figure 4: Left: Gate count complexity scaling for 1-Bit quantum filter implementations under (a) IBM Torino hardware compilation and (b) TKET optimization. Power-law fits with constrained logarithmic scaling ($c=1.0$) yield $b \approx 0.53$ (Qiskit) and $b \approx 0.52$ (TKET), consistent with $O(\sqrt{N} \log N)$ theoretical complexity ($R^2 > 0.999$). Right (c): Success probability $P_{\mathrm{succ}}$ versus matrix size $N$ for 3-layer and 5-layer tracking scenarios. Fitted exponents exactly match the theoretical $N^{-1/2}$ scaling up to $N=10^6$ where we see an upward deviation in the residuals.
• Figure 5: Noise resilience benchmarks.(a) Hellinger fidelity ($F_H$) relative to the noiseless baseline across five problem configurations, indicating the preservation of the probability distribution structure. (b) Signal Separation Index (SSI) quantifying the contrast between valid tracks and the dominant noise floor. The dashed line at $\text{SSI}=1$ represents the distinguishability limit where signal amplitudes drop below the background noise. Error bars denote $1\sigma$ confidence intervals derived via Monte Carlo resampling of the measurement counts.