We develop a GPU-accelerated hybrid quantum Monte Carlo (QMC) algorithm to solve the fundamental yet difficult problem of $U(1)$ gauge field coupled to fermions, which gives rise to a $U(1)$ Dirac spin liquid state under the description of (2+1)d quantum electrodynamics QED$_3$. The algorithm renders a good acceptance rate and, more importantly, nearly linear space-time volume scaling in computational complexity $O(N_τ V_s)$, where $N_τ$ is the imaginary time dimension and $V_s$ is spatial volume, which is much more efficient than determinant QMC with scaling behavior of $O(N_τV_s^3)$. Such acceleration is achieved via a collection of technical improvements, including (i) the design of the efficient problem-specific preconditioner, (ii) customized CUDA kernel for matrix-vector multiplication, and (iii) CUDA Graph implementation on the GPU. These advances allow us to simulate the $U(1)$ Dirac spin liquid state with unprecedentedly large system sizes, which is up to $N_τ\times L\times L = 660\times66\times66$, and reveal its novel properties. With these technical improvements, we see the asymptotic convergence in the scaling dimensions of various fermion bilinear operators and the conserved current operator when approaching the thermodynamic limit. The scaling dimensions find good agreement with field-theoretical expectation, which provides supporting evidence for the conformal nature of the $U(1)$ Dirac spin liquid state in the QED$_3$. Our technical advancements open an avenue to study the Dirac spin liquid state and its transition towards symmetry-breaking phases at larger system sizes and with less computational burden.

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