Accelerated quantum circuit Monte Carlo simulation for heavy quark thermalization

Thermalization of heavy quarks in the quark-gluon plasma is one of the most promising phenomena for understanding the strong interaction. The energy loss and momentum broadening at low momentum can be well described by a stochastic process with drag and diffusion terms. Recent advances in quantum computing, in particular, quantum amplitude estimation (QAE), promise to provide a quadratic speedup in simulating stochastic processes. We introduce and formalize an accelerated quantum circuit Monte Carlo (aQCMC) framework to simulate heavy quark thermalization. With simplified drag and diffusion coefficients connected by Einstein’s relation, we simulate the thermalization of a heavy quark in isotropic and anisotropic mediums using an ideal quantum simulator and compare that to thermal expectations. With Grover-like QAE, we calculate physical observables with quadratically fewer resources, which is a boost over the classical MC simulation that usually requires a large sampling number at the same estimation accuracy.

Figure 1

The schematic quantum circuits of the QCMC and the aQCMC involving $N_t$ time steps in direction $i$ constructed using $\mathcal{S}$ and $\mathcal{W}$ registers (a) with reset gates in the depth-oriented QCMC and (b) with QAE in the breadth-oriented aQCMC. Quantum gates $U_L$, $U_W$, and $U_A$ are used to represent the initial distribution loading, stochastic diffusion, and quantum evolution, respectively.

Figure 2

Quantum simulation using the QCMC method in a one-dimensional system. Panels (a) and (b) show the expectations of $⟨q⟩$ and $⟨|q|⟩$ for differential initial momenta (marked in colored lines).

Figure 3

Probability density distribution for quantum simulation with an anisotropic medium at late time $\tilde{t}=3$, compared with analytical thermal equilibriums distribution $f_{\mathrm{eq}}(q)$ in Eq. (5).

Figure 4

Quantum simulation using the aQCMC with Iterative QAE [36] (in shaded area) compared to QCMC with direct measurement to evaluate expectations (a) $⟨q⟩$ and (b) $⟨|q|⟩$ for the earlier time steps.

Figure 5

Quantum advantage using the aQCMC approach with Iterative QAE [36] to estimate physical observable with quadratically less resources.

Figure 6

Momentum distribution with (a) isotropic and (b) anisotropic mediums for two-dimensional heavy quark thermalization. The vertical axis represents $q_y$; the horizontal axis represents $q_x$. Heat maps at times $\tilde{t}=0$, 0.5, 1.0, 1.5, 3.0 are presented.

Figure 7

Elliptic flow over time steps using quantum simulation, compared with analytical thermal equilibrium limit in Eq. (12).

Figure 8

Schematics of quantum arithmetic gate for (a, b) additions and (c) multiplications.

Figure 9

Quantum amplitude estimation using (a) QPE circuit with inverse quantum Fourier transform and (b) QPE-free circuit.