From Hope to Heuristic: Realistic Runtime Estimates for Quantum Optimisation in NHEP

TL;DR

The paper addresses runtime estimation and scalability of QA and QAOA for track reconstruction in NHEP by casting the problem as a QUBO and simulating on classical hardware. It shows that a Fourier-based heuristic for the QAOA parameter space yields near-optimal energy with a low-dimensional representation, enabling annealing schedules derived from $T=\sum_{i=1}^{p}(|\gamma_i^*|+|\beta_i^*|)$ and corresponding $f(t_i)$. Spectral-gap analysis reveals a largely size-invariant $\Delta_{\min}$ with a minimum near $s \approx 0.85$, providing stable guidance for runtime estimates. Together, the work advocates co-design and practical heuristics to push toward quantum advantage on NISQ devices for NP-like optimisation problems in NHEP, particularly track reconstruction.

Abstract

Noisy Intermediate-Scale Quantum (NISQ) computers, despite their limitations, present opportunities for near-term quantum advantages in Nuclear and High-Energy Physics (NHEP) when paired with specially designed quantum algorithms and processing units. This study focuses on core algorithms that solve optimisation problems through the quadratic Ising or quadratic unconstrained binary optimisation model, specifically quantum annealing and the Quantum Approximate Optimisation Algorithm (QAOA). In particular, we estimate runtimes and scalability for the task of particle track reconstruction, a key computing challenge in NHEP, and investigate how the classical parameter space in QAOA, along with techniques like a Fourier-analysis based heuristic, can facilitate future quantum advantages. The findings indicate that lower frequency components in the parameter space are crucial for effective annealing schedules, suggesting that heuristics can improve resource efficiency while achieving near-optimal results. Overall, the study highlights the potential of NISQ computers in NHEP and the significance of co-design approaches and heuristic techniques in overcoming challenges in quantum algorithms.

Figures (5)

• Figure 1: Left: Energy values for ground and first excited state of the Hamiltonian given in \ref{['eq:QA']} over interpolation parameter $s$ for an example tr problem instance considered in this work. Right: Corresponding sg (i.e., the difference between energy levels of the first excited and the ground state). Lines do not provide an interpolation, but are only used to guide the eye; insets magnify the region around the minimal sg.
• Figure 2: qaoa circuit for cost Hamiltonian $H_C$ and parameter vectors $\vec{\beta}$, $\vec{\gamma}$ of size $p=2$.
• Figure 3: Average sg and the corresponding position of the minimum sg over interpolation parameter $s \coloneqq t/T$ for $98$tr problem instances with varying sizes. In each qubit range we computed sg for $20\dots30$ problem instances. Shaded areas represent the standard deviation.
• Figure 4: Energy approximation ratio obtained via qaoa optimisation for increasing values of $p$. The approximation ratio curves for $q_\text{max} \in \{5, 10, 25, 50\}$ exhibit a significant overlap.
• Figure 5: Left: Absolute parameter values $\vec{\beta}$ and $\vec{\gamma}$ obtained from the qaoa optimisation at $p = 50$ for the tr problem instance considered in this work. Right: Annealing schedules obtained from these parameters. The horizontal line denotes the position of the minimal sg.