# Time Evolution

**Time evolution** solves the [[qutip-master-equation|master equation]] to get the system state at future times. QuTiP provides multiple solvers, each suited to different problems.

## Mesolve: Master Equation Solver

[[qutip-mesolve|Mesolve]] is the standard solver for deterministic (ensemble-averaged) evolution. It integrates the master equation using efficient ODE solvers.

```python
from qutip import *
import numpy as np

H = sigmaz()
c_ops = [0.1 * sigmam()]
times = np.linspace(0, 10, 100)
psi0 = basis(2, 0)

result = mesolve(H, psi0, times, c_ops, [sigmaz(), sigmam()])
# result.states: density matrix at each time
# result.expect: expectation values of observables
```

## Mcsolve: Monte Carlo Solver

[[qutip-mcsolve|Mcsolve]] runs stochastic trajectories, applying collapse operators randomly. Each run gives a different realization; ensemble average matches mesolve.

```python
result_mc = mcsolve(H, psi0, times, c_ops, [sigmaz()], ntraj=1000)
# Runs 1000 trajectories; averaging gives mesolve result
```

Mcsolve is useful for understanding quantum noise at the individual-event level and for benchmarking error mitigation.

## Other Solvers

- **odesolve**: Custom ODE integration
- **sesolve**: Schrödinger equation (no collapse, unitary evolution)
- **Liouvillian**: Construct the full superoperator for advanced analysis

Choose mesolve for typical problems, mcsolve for trajectory analysis or strong noise regimes.