A quantum algorithm for track reconstruction in the LHCb vertex detector

TL;DR

The paper tackles real-time track reconstruction for HL-LHC by formulating the problem as the ground state of an Ising-like Hamiltonian and solving a relaxed linear system with the HHL quantum algorithm. It maps hits into binary doublets, expresses the objective as $\mathcal{H}(\mathbf{S})$, and simplifies to a predominantly angular coupling suitable for VELO’s straight tracks. Classical sparse-matrix inversion achieves competitive tracking performance (~97% efficiency, ~4.3% fake rate) on LHCb-simulated data, while a quantum implementation via HHL is demonstrated only on toy models due to current Hamiltonian-simulation limitations, highlighting substantial barriers to hardware-ready quantum speedups. The work delineates a clear pathway to quantum advantage contingent on efficient Hamiltonian simulation and readout strategies, and it provides a concrete codebase for further development.

Abstract

High-energy physics is facing increasingly computational challenges in real-time event reconstruction for the near-future high-luminosity era. Using the LHCb vertex detector as a use-case, we explore a new algorithm for particle track reconstruction based on the minimisation of an Ising-like Hamiltonian with a linear algebra approach. The use of a classical matrix inversion technique results in tracking performance similar to the current state-of-the-art but with worse scaling complexity in time. To solve this problem, we also present an implementation as quantum algorithm, using the Harrow-Hassadim-Lloyd (HHL) algorithm: this approach can potentially provide an exponential speedup as a function of the number of input hits over its classical counterpart, in spite of limitations due to the well-known HHL Hamiltonian simulation and readout problems. The findings presented in this paper shed light on the potential of leveraging quantum computing for real-time particle track reconstruction in high-energy physics.

Figures (10)

• Figure 1: Schematic representation of the VELO. The 52 detection modules are arranged on the so-called A-side (left side) and C-side (right side). The interaction region is located in the area with the highest density of modules and depicted in red.
• Figure 2: (a) Event display with all the possible doublets represented: the ones that are part of a real trajectory are highlighted while the others are left in grey. (b) Diagram showing an example of a track bifurcation. (c) Diagram explicitly representing the angle $\theta_{abc}$ between two adjacent doublets $S_{ab}$ and $S_{bc}$.
• Figure 3: Plot of $\cos^\lambda(\theta)$ for several values of $\lambda$, compared with the step function $f(\theta, \varepsilon)$
• Figure 4: Quantum circuit that implements the algorithm. In red, the $\ket{\Phi}_b$, which encodes the $\mathbf{b}$ input vector and the $\mathbf{S}$ output vector. In green, the $n_q$-qubit ancilla register used for QPE. In blue, the additional ancilla qubit used for the matrix inversion.
• Figure 5: Event display of the hits (in red) coming from a collision event in half of the VELO together with the tracks reconstructed using the classical linear Ising solver presented in this work (in black).