Variational thermal quantum simulation of the lattice Schwinger model

Confinement of quarks due to the strong interaction and the deconfinement at high temperatures and high densities are a basic paradigm for understanding the nuclear matter. Their simulation, however, is very challenging for classical computers due to the sign problem of solving equilibrium states of finite-temperature quantum chromodynamical systems at finite density. In this paper, we propose a variational approach, using the lattice Schwinger model, to simulate the confinement or deconfinement by investigating the string tension. We adopt an ansatz that the string tension can be evaluated without referring to quantum protocols for measuring the entropy in the free energy. Results of numeral simulation show that the string tension decreases both along the increasing of the temperature and the chemical potential, which can be an analog of the phase diagram of QCD. Through numerical simulations on the classical computer, we demonstrate the potential of exploiting near-term quantum computers for investigating the phase diagram of finite-temperature and finite-density nuclear matters.

Figure 1

A pictorial representation of the variational quantum algorithm, which can prepare thermal states and compute the corresponding free energy. In quantum device, we design an initial product state ${\rho }_0(\theta )$ and the unitary quantum neural network $U(\varphi )$ parametrized by a serious of variable parameters. Using classical computers, the variable parameters $(\theta ,\varphi )$ are updated and optimized in order to minimize the free energy.

Figure 2

The circuit of preparing one-qubit mixed state as a subsystem. $Rx({\theta }_i)=e^{i{\theta }_i{\sigma }_i^x}$ is a one-qubit rotation.

Figure 3

A depiction of the parametrized unitary for a system of three qubits in the case $p=2$.

Figure 4

The free energy of the Schwinger model in the case $m=1$, $g=1$, $\varpi =1$. Left: the system sizes $N=4$. Right: the system sizes $N=6$. As the depth $p$ of the increases, the free energies calculated by the quantum variational algorithm will converge close to the exact values.

Figure 5

The string tension for different $ϵ$ in the case $m=1$, $g=1$, $\varpi =1$, $N=6$. (a) the string tension as a function of inverse temperature $\beta$. (b) logarithm of the string tension as a function of temperature $T$.

Figure 6

The string tension in the case $ϵ=0.5$, $m=1$, $g=1$, $\varpi =1$, $N=6$. (a) At different $\mu$, the string tension as a function of the temperature $T$; (b) The string tension as a function of the temperature $T$ and the chemical potential $\mu$.