Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems

We show how to simulate numerically the evolution of 1D quantum systems under dissipation as well as in thermal equilibrium. The method applies to both finite and inhomogeneous systems, and it is based on two ideas: (a) a representation for density operators which extends that of matrix product states to mixed states; (b) an algorithm to approximate the evolution (in real or imaginary time) of matrix product states which is variational.

Figure 1

(a) Magnetization vs transverse magnetic field of the thermal state of an $XY$ model with 60 spins, for temperatures $T=0.05$, 0.5, 5, and 50 (bottom to top). (b) Error in the density matrix of the thermal state of Eq. (8) computed using MPDO, vs temperature, for a chain with $N=8$ spins, $D=8$ (solid line) and $D=14,20,24$ (dashed line).

Figure 2

(a) Errors in simulations of a cluster state with $N=6$, 7, and 8 spins (solid line, dashed line, and dash-dotted line), under Eq. (9), for $\gamma =0.1$, $h=0$, matrices of size $L=14$ and time step $\Delta t=0.01$. (b) Decay of correlations, $c_{zz}=⟨{\sigma }_i^z{\sigma }_{i+1}^z⟩-⟨{\sigma }_i^z⟩⟨{\sigma }_{i+1}^z⟩$, for the site $i=N/2$, on GHZ states with $N=10$, 20, and $30$ spins (dots, crosses, and solid line) for $L=10$.