JIMWLK on a quantum computer

TL;DR

This work establishes a concrete pathway toward quantum simulation of high-energy QCD evolution equations, with direct relevance to the physics program of the Electron-Ion Collider.

Abstract

We propose a method for solving the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) evolution equation on quantum computers. Our approach exploits the reformulation of the JIMWLK equation as a Lindblad master equation governing the rapidity evolution of the hadronic density matrix, as established in prior work. To render the problem tractable for quantum simulation, we introduce several approximations: the two-dimensional transverse plane is reduced to a one-dimensional radial lattice by assuming azimuthal symmetry of the jump operators; the gauge group is restricted to $\mathrm{SU}(2)$; and the infinite Wilson lines of the JIMWLK equation are replaced by finite Wilson links along the light-cone direction. The resulting bosonic Hilbert space is truncated using the electric field basis familiar from Hamiltonian lattice gauge theory, with states restricted to angular momenta $j\leq j_{\mathrm{max}}$. We derive the matrix elements of the JIMWLK Lindblad jump operators in this basis. As a benchmark, we demonstrate rapid convergence of the fundamental dipole expectation value with $j_{\mathrm{max}}$ for both pure and mixed Gaussian initial density matrices. For the simplest truncation, $j_{\mathrm{max}} = 1/2$, we implement the Lindblad evolution using a quantum simulation algorithm verified with the Qiskit statevector simulator by decomposing the non-unitary evolution operator into a linear combination of unitaries. This work establishes a concrete pathway toward quantum simulation of high-energy QCD evolution equations, with direct relevance to the physics program of the Electron-Ion Collider.

Figures (7)

• Figure 1: (a) Spatial lattice with Wilson links (denoted by arrows), typically used in Hamiltonian lattice gauge theory. (b) Lattice used for JIMWLK evolution in this manuscript. Note that there are no links along the $x_\perp$ direction.
• Figure 2: Pictorial representation of the radial lattice considered. The circles (dashed red lines) are averaged over.
• Figure 3: Fundamental dipole expectation value $D_{1/2}(\mu)$ as a function of $\mu$ for several values of $j_{\mathrm{max}}$. The solid thick curve shows the exact analytic result from Eq. \ref{['Def: Dipole']}.
• Figure 4: Rapidity evolution of (a) the fundamental dipole $D_{1/2}(Y)$ and (b) the purity $\mathrm{tr}(\rho^2)$ for the pure-state Gaussian density matrix of Eq. \ref{['def: pure density matrix alpha basis']}, shown for several values of $j_{\mathrm{max}}$ at $\mu = 4$.
• Figure 5: The dipole expectation value (a) and the purity of the density matrix (b) as a function of rapidity obtained from a Qiskit simulation of the Lindblad-JIMWLK equation for $j_{\rm max} = 1/2$ and the initial $\mu = 4$.