Hybrid quantum-classical simulations of semiclassical gravity

Abstract

We propose a hybrid quantum-classical algorithm for the simulation of real-time dynamics in interacting quantum field theories coupled to classical fields, focusing on the self-consistent estimation of semiclassical backreaction. By discretizing space and time, we construct an iterative protocol that simulates the Trotterized dynamics of the quantum fields subject to the dynamical classical fields. By estimating certain quantum expectation values through a set of projective measurements, we source the equations of motion of the classical fields, and solve them numerically to feed them forward to the quantum simulation in an iterative self-consistent loop. Semiclassical backreaction is relevant in various fields of physics, particularly in cosmology, where quantum matter fluctuations affect the gravitational field dynamics, and a controlled renormalization must be carefully considered to get a sensible continuum limit. We benchmark our algorithm in this context, focusing on scalar-tensor theories of modified gravity exhibiting a chameleon mechanism, such that a light classical scalar field driving cosmic acceleration becomes massive in high-density regions, effectively screening any possible yet unobserved fifth force. By focusing on numerically tractable regimes, we explicitly show the convergence and robustness of our algorithm when considering the continuum limit and the effect of quantum shot noise. Our work paves the way for future experiments exploring other non-tractable regimes, including non-perturbative interactions of the quantum fields and how these can change backreaction and the gravitational dynamics.

Figures (3)

• Figure 1: Pictorial representation of the hybrid classical-quantum algorithm.
• Figure 2: (a) Representation of the effective potential for the classical scalar field in the absence and presence of backreaction. (b) Solutions of the backreaction equations \ref{['eq:ST_classical_equation']}-\ref{['eq:ST_quantum_equations']} for the chameleon model \ref{['eq:model_functions']}, in the absence and presence of backreaction. In the latter case, the exact (continuum) and the algorithm-generated solutions are represented. Parameters are fixed, with respect to a reference energy scale $a_0$, to $ma_0=0.1$, $M_{\rm Pl}=0.5$ (backreaction) or $M_{\rm Pl}=\infty$ (no backreaction), $\lambda a_0^2=0.5$, $\xi=1/2$, $T/a_0=20$, with initial conditions $\phi_0=0.5$ and $\dot\phi_0/a_0=0$. The dashed line shows the real-time evolution of the scalar field obtained by the hybrid algorithm. The algorithm parameters are fixed as $L/a_0=4$, $a/a_0=0.5$, $N_S=8$ and $\tilde{\delta t}=0.1$.
• Figure 3: (a)-(b) Comparison between the continuum solution and the algorithm solutions for the time evolution of $\phi$ and $\braket{\hat{\overline\Psi}\hat{\Psi}}^{\rm ren}$, respectively, for different values of the lattice spacing $a$. The parameters, in $a_0$ units, are fixed as $ma_0=0.1$, $M_{\rm Pl}=0.5$, $\lambda a_0^2=0.5$, $\xi=1/2$, $T/a_0=5$, with initial conditions for the scalar field $\phi_0=0.1$ and $\dot\phi_0/a_0=0$. The algorithm parameters are $L/a_0=4$, $\tilde{\delta t}=0.1$. (c)$\mathrm{L}^2$ distance between the algorithm-generated time evolution of the classical scalar field and the continuum solution, with matching IR cutoff. The parameters are the same as in (a)-(b). (d)$\mathrm{L}^2$ distance between the algorithm-generated time evolution of the classical scalar field in presence and absence of shot noise as a function of the number of shots $N_{\rm shots}$. The plot points are given by the mean of the $\mathrm{L}^2$ distance for $N_{\rm rep}=20$ repetitions, and the error bars by their standard deviation. The parameters are fixed, with respect to a reference energy scale $a_0$, to $ma_0=0.1$, $M_{\rm Pl}=0.5$, $\lambda a_0^2=0.5$, $\xi=1/2$, $T/a_0=10$, with initial conditions for the scalar field $\phi_0=0.5$ and $\dot\phi_0/a_0=0$, and algorithm parameters $L/a_0=32$, $a/a_0=0.5$, $\tilde{\delta t}=0.05$.