Simulation methods for open quantum many-body systems

Coupling a quantum many-body system to an external environment dramatically changes its dynamics and offers novel possibilities not found in closed systems. Of special interest are the properties of the steady state of such open quantum many-body systems, as well as the relaxation dynamics toward the steady state. However, new computational tools are required to simulate open quantum many-body systems, as methods developed for closed systems cannot be readily applied. Several approaches to simulating open many-body systems are reviewed, and advances made in recent years toward the simulation of large system sizes are pointed out.

Figure 1

Mean-field phase diagram of the driven-dissipative Bose-Hubbard model. The numbers inside the plot represent the number of stable mean-field solutions. The yellow region exhibits two mean-field solutions, one of which is unstable. From [102].

Figure 2

(a) Writing a wave function $|\psi ⟩$ as a MPS for six sites. Each site has a physical dimension $d$. (b) A density matrix $\rho$ can be written as a MPDO, an extension of the MPS formalism. Such a construction automatically ensures positivity of the density matrix.

Figure 3

(a) Defining a MPS $|\mathrm{\Psi }⟩$ over the enlarged Hilbert space using ancillas (in red). (b) Taking the projector of the MPS with ancillas. (c) Tracing out the ancillas from the projector to obtain the MPDO $\rho$.

Figure 4

Choi isomorphism: vectorizing a density matrix written in terms of an MPO. In a TN diagram, it is simply reshaping one of the indices and gluing it to the other, thereby giving us a MPS.

Figure 5

(a) Steady-state density matrices of two systems $A$ and $B$ are first obtained using brute force. They are then expressed in their respective diagonal forms. (b) We then merge the two systems and keep only the $\chi$ most probable pair of states. The process is repeated for different $\chi$’s until we get some convergence. Larger systems can be simulated by merging more systems in step (b).

Figure 6

TN diagram for the PEPO of $\rho$ on a 2D square lattice with bond dimension $D$ and physical dimension $d$. When vectorized, it can be understood as a PEPS for $|\rho {⟩}_{♯}$ with physical dimension $d^2$.

Figure 7

Operator-entanglement entropy $S_{\mathrm{op}}$ for a block of $2\times 2$ unit cells for real time evolution of the master equation for different values of dissipation strength. Stronger dissipation implies lower entanglement growth and faster convergence to the NESS. From [96].

Figure 8

Our study based on iPEPS found bistability in the phase diagram of the dissipative Ising model for low bond dimensions $D=1,2$. The bistability is replaced by a first-order transition for higher $D$’s. From [96].

Figure 9

TN diagram for the PEPDO of $\rho$ on a 2D square lattice with bond dimension $D$ and physical dimension $d$. When vectorized, it can be understood as a PEPS for $|\rho {⟩}_{♯}$ with physical dimension $d^2$.

Figure 10

Comparison of the solutions according to the variational principle (solid line), the mean-field decoupling (dashed line), and wave-function Monte Carlo simulations for $4\times 4$ lattices for the up-spin density $n_r$ of the dissipative Ising model. The mean-field solution displays a region of bistability, while the variational solution correctly predicts a first-order transition. From [194].

Figure 11

Node structure of a restricted Boltzmann machine for open quantum systems. The vectorized density matrix is realized in terms of a physical layer ${\sigma }_i$ corresponding to a set of spin $1/2$ variables. These are coupled to the nodes of a hidden layer $h_i$, which are again coupled to the third layer ${\tau }_i$, which represents the adjoint of the physical layer.