TrackHHL: A Quantum Computing Algorithm for Track Reconstruction at the LHCb

TL;DR

This paper tackles the HL-LHC data deluge for track reconstruction by recasting VELO tracking as a linear system derived from a modified Denby-Peterson Hamiltonian, enabling quantum solution via the Harrow–Hassidim–Lloyd algorithm. A 1-Bit QPE variant and Suzuki–Trotter approximations reduce circuit depth and qubit requirements, yielding depth reductions by $10^3$–$10^4$ and enabling practical readout by focusing on primary-vertex information rather than full track distributions. Post-processing uses active-segment extrapolation to the beam-line and DBSCAN clustering to identify PV candidates, significantly decreasing the number of quantum-state samples needed. Collectively, these contributions illuminate a feasible path toward quantum-enabled track and PV reconstruction at the LHC in the presence of hardware constraints, with future work exploring broader quantum-linear-transform techniques such as the Quantum Singular Value Transform.

Abstract

In the future high-luminosity LHC era, high-energy physics experiments face unprecedented computational challenges for event reconstruction. Employing the LHCb vertex locator as a case study we investigate a novel approach for charged particle track reconstruction. The algorithm hinges on minimizing an Ising-like Hamiltonian using matrix inversion. Solving this matrix inversion classically achieves reconstruction efficiencies akin to current state-of-the-art algorithms. Exploiting the Harrow-Hassidim-Lloyd (HHL) quantum algorithm for linear systems holds the promise of an exponential speedup in the number of input hits over its classical counterpart, contingent on the conditions of efficient quantum phase estimation (QPE) and effectively reading out the algorithm's output. This contribution builds on previous work by Nicotra et al and strives to fulfill these conditions and further streamlines the algorithm's circuit depth by a factor up to $10^4$. Our version of the HHL algorithm restricts the QPE precision to one bit, largely reducing circuit depth and addressing HHL's readout issue. Furthermore, this allows for the implementation of a post-processing algorithm that reconstructs event Primary Vertices (PVs). The findings presented here aim to further illuminate the potential of harnessing quantum computing for the future of particle track reconstruction in high-energy physics.

Figures (5)

• Figure 1: Illustration of an LHCb event in the VELO detector, Created by Davide Nicotra
• Figure 2: The $\mathbf{Left}$ panel represents a graph construction of the minimal event, with each doublet labeled. Pink lines indicate active segments. The $\mathbf{Middle}$ panel is a heat-map of the matrix $A$ for the same event as the left panel. The $\mathbf{Right}$ panel illustrates the output distribution vector $S$ after inverting the matrix, where threshold T is applied to distinguish active from non-active segments.
• Figure 3: On the $\bold{left}$ is an event with 5 layers and 8 particles, our largest simulated event with the corresponding distribution for 1-Bit HHL on the $\bold{right}$. Where all of the inactive segments are suppressed.
• Figure 4: The $\mathbf{Left}$ panel shows a graph of the number of samples needed to reconstruct the vector $S$ as a function of particles and 1-Bit HHL versus HHL. With dotted lines drawn at the HL-LHC requirements. The $\mathbf{Right}$ panel shows how the number of qubits needed when comparing 1-Bit HHL and HHL, with a similar dashed line drawn as the graph on the left, where the problem size $N$ relates to the number of particles as $N=N_p^2 N_{hits}$, given the number of particles $N_p$ and hits per particle $N_{hits}$.
• Figure 5: The graph on the $\bold{left}$ shows the distribution of the difference between $PV_{Tracks}$ and $PV_{Segment}$. The Graph on the $\bold{right}$ shows how the Mean Absolute Difference changes as a function of the percentage of the distribution discarded. Generated using the same $B_s \rightarrow \phi \phi$ data as Nicotra_2023