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# Mcsolve (Monte Carlo Trajectories)

**Mcsolve** runs stochastic trajectories of quantum evolution, applying [[qutip-collapse-operators|collapse operators]] at random times. Each trajectory is a possible quantum history; averaging over many trajectories gives the ensemble result (matching [[qutip-mesolve|mesolve]]).

Monte Carlo simulations are useful for understanding how noise manifests at the individual-event level and for systems where many collapse operators dominate.

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

# Simple decay
H = 0.5 * sigmaz()
c_ops = [0.1 * sigmam()]
times = np.linspace(0, 10, 50)
psi0 = basis(2, 1)

# Run 1000 trajectories
result = mcsolve(H, psi0, times, c_ops, [sigmaz()], ntraj=1000)

# result.expect: ensemble average (matches mesolve)
# result.trajectories: individual runs (if stored)
```

## How It Works

Each trajectory evolves via an effective non-Hermitian Hamiltonian:

$$H_{\text{eff}} = H - \frac{i\hbar}{2} \sum_k L_k^\dagger L_k$$

At each step, with probability $p_k(dt)$, collapse operator $L_k$ is applied (normalized jump). Otherwise, evolution continues. After $N$ trajectories, expectations average to the master equation solution.

## Advantages and Trade-offs

**Advantages**:
- Natural interpretation of quantum jumps
- Efficient for many collapse operators
- Can track individual quantum events

**Trade-offs**:
- Slower than mesolve for small $N_{\text{traj}}$ (needs many runs for good statistics)
- More memory if storing trajectories
- Statistical noise in results

Use mcsolve when you need trajectory-level understanding or when many dissipation channels dominate.