Variational Thermal Quantum Simulation via Thermofield Double States

We present a variational approach for quantum simulators to realize finite temperature Gibbs states by preparing thermofield double (TFD) states. Our protocol is motivated by the quantum approximate optimization algorithm and involves alternating time evolution between the Hamiltonian of interest and interactions which entangle the system and its auxiliary counterpart. As a simple example, we demonstrate that thermal states of the 1D classical Ising model at any temperature can be prepared with perfect fidelity using $L/2$ iterations, where $L$ is system size. We also show that a free fermion TFD can be prepared with nearly optimal efficiency. Given the simplicity and efficiency of the protocol, our approach enables near-term quantum platforms to access finite temperature phenomena via preparation of thermofield double states.

Figure 1

Schematic depiction of the hybrid quantum-classical mode of operation for the variational scheme for preparing a thermofield double state at temperature $T$ with respect to a Hamiltonian $H$. Given a set of parameters $(\overrightarrow{\alpha },\overrightarrow{\gamma })$, a quantum simulator prepares the wave function $|\psi (\overrightarrow{\alpha },\overrightarrow{\gamma }){⟩}_p$ defined in Eq. (2). The free energy $E_A-T{S}_A$, where $E_A$ is the energy of system $A$ and $S_A$ is entanglement entropy, is then measured and updated in a classical computer. The latter generates a new set of parameters and the process is iterated until the free energy cost function is minimized.

Figure 2

Preparation of Ising chain TFD at $\beta =2$. The error in fidelity with respect to the target TFD state is plotted as a function of the number of pairs of variational parameters $p$, for various system sizes. Perfect fidelity is achieved at $p=\lfloor L/2\rfloor$, a result which appears to hold for arbitrary temperature.

Figure 3

Preparation of Majorana fermion TFD state. (a) Semilog (base $e$) plot of the infidelity for different system sizes at $\beta =2$. (b) Semilog plot of the infidelity of $L=30$ for different temperature values. The infidelity appears to decay exponentially with a slope $-1/p_0$ which depends on temperature. (c) The dependence of the decay rate $-1/p_0$ on the temperature. The blue dots are slopes from linear fitting of the plots in (b) at each temperature. The red line is the linear fitting of the blue dots for $\beta \in [1,6]$. (d) The dependence of the infidelity at $\beta =2$ on the system size $L$ at $p=2$ and $p=3$. It is evident that the infidelity scales linearly with $L$ for large enough $L$.

Figure 4

Preparation of the $XY$ spin chain TFD state at $\beta =2$.