# Hierarchy and Composite Systems

**Hierarchy** in scqubits refers to building composite systems from individual qubits. The `HilbertSpace` class combines multiple qubit objects and adds interactions.

```python
from scqubits import Transmon, Fluxonium, HilbertSpace, InteractionTerm

# Create qubits
q1 = Transmon(EJ=15.0, EC=0.3, ncut=30, label='q1')
q2 = Transmon(EJ=15.0, EC=0.3, ncut=30, label='q2')
q3 = Fluxonium(EJ=12.0, EC=2.5, EL=0.5, ncut=30, label='q3')

# Create Hilbert space
hilbert = HilbertSpace([q1, q2, q3])

# Add interactions
g12 = 0.01  # Qubit 1-2 coupling
g23 = 0.005 # Qubit 2-3 coupling

hilbert.add_interaction(
    InteractionTerm(g=g12, op1=q1.n_operator(), op2=q2.n_operator())
)
hilbert.add_interaction(
    InteractionTerm(g=g23, op1=q2.n_operator(), op2=q3.n_operator())
)

# Get full Hamiltonian
H_total = hilbert.hamiltonian()
evals = H_total.eigenenergies()
```

## Multi-Qubit Design

With hierarchy, you design:
1. Individual qubits and their parameters
2. Coupling topology (which qubits interact)
3. Interaction strengths

This compositional approach scales to 3–5 qubits easily. For larger systems, use approximations or specialized simulators.

scqubits automates the tensor product of Hilbert spaces, making composite system analysis practical.